Sang Tao Va Giai Phuong Trinh He Phuong Trinh Bat Phuong Trinh P4

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(1)B a i t o a n 2 3 . G^a^ he phuang trinh Bai. t o a n 24. Gio. he ph^Cng. IP^f^lU'i^g^X^^ ZlUl. [. trinh. {. ch&ng han 8x^ + x'^ + bx = \/3 - x + x'^. Trong vi phdi,. thay X bdi y ta dUdc Sx^ + de phicang trinh. + 5x = \/3 - y + y'^. "Tron " x vd y vdo hai vi. dU0c "sinh. dong". Sx^ +. (4). • (!. Giai. C a c h 1. T a c6 (1) ^. ^. \^4y. >| 111. H a m so y ( x ) = l o g i x la h a m ngUdc. y y = logi X .. hdn, chang han. ciia h a m / ( x ) =. + 5x = \/3 - y + x ^. (1). /1\. y =. V i d u 12. Xet mot phu&ng trinh bac ba nao do : 8x^ + 6x = VS. Ta bien ddi thdnh phuang trinh,. trinh. = I + 8.. + ^Z. + g. (a) B a i t o a n 2 6 . Gidi he phuang. j. ". , do do do t h i cua hai h a m nay doi x i i n g nhau qua. difdiig phan giac ciia goc plian t u t h i i nhat y = x. B d i vay ( x , y) l a nghiem Thay x bdi y vd thay y bdi x, ta dUOe 8y^ + x^ + 5y = \/3 - x + y^. Ta c6 hdi toan sau. t o a n 2 5 . G^d^ he phMng. Bai. trinh. |. + ^2 + 5^ = ^. ciia (1) k h i va c h i k h i x = y, nghia la (1). X X. H a m so f{t). = 8t^ - 2t^ + At c6 fit). =. (1). - 4t + 4 > 0, V i G R nen dong. bien t r e n R, d o do tvr (1) c6 / ( x ) = / ( y ) , hay y = x. T h a y vao he d a cho, ta dUdc =. _ X +. X^. 8X^ + 6 X =. \/3 <^ 4x^ + 3x =. (2). Theo bai toan 1 d t r a n g 118, suy r a (2) c6 nghiem d u y nhat. "=2. 1. /. t>\).. (c). ^ t. =. —. 2. ). 1 2. ^. ;^. -. =. ^( = 1. ydo. =. nghich bien. n la nghiem d u y nhat ciia he. V2'2J /I. V i d u 1 4 . Chon phuang. trinh (x + y)2 + (x - 1)2 + (y _ i ) 2 ^. /. \. 0 < a < ^ (do. (1) diioc. Tirdo (x;y) =. ' v ^ W 7 ^ 3/v/3-v/7. QV. con ham g nghich bien, do do (c) c6 khong qua m o t nghiem. T h a y x = - vao. —. + 5.T;. = a^, vdi a =. V4;. Tren (0; + 0 0 ) , x e t hai ham so / ( x ) = x, ry(x) = ậ De thay h a m / dong bien. 1. ^.T;3 + J ; 2. =. - y + x^ - y^. <^8x^ - 2x2 + 4x = 8y^ - 2y2 + 4y.. x = y = ^.. C a c h 2. De (x; y) la nghiem ciia he t h i x > 0 va y > 0. K h o n g m a t t i n h tong quat, gia sii" y > x. D a t y = tx, k h i do i > 1. T h a y vao (a) dudc. Z ^ + ^2. G i a i . Dieu kien x G R v a y G R. Lay hai phildng t r i n h tr\l nhau, t a dUcJc 8x^ + y^ + 5x - 8y^ - x^ - 5y =. | ^^. 2(x'2 + y^ + x y -. X. - y + 1) = 0 4 ^ y +. X -. 1 = y2 + x y + x^.. (1). Vay he da cho c6 nghiem d u y nhat biit rdng khi gidi he doi xiing loai hai, niu lay hai phuang se xudt hien. / (x;y) = \. 16. 16. 16. 16. (y-x)/(x,y) = 0 ^ [ ^ ( - - ) = i o /. V i d u 1 3 . Ham so y = logi x la ham so ngU0c cua ham so y =^ [ -. tren. ^dy de tao ra m.dt he doi xiing loai hai, ta se nhdn hai vi cua (1) vdi (y - x ) , d&n tdi ' (y. (0; + 0 0 ) . Do do ta co hdi toan sau.. -. .T)(y +. X -. 1) = (y - x)(y2 + xy + .x^). <!=^y2 ~ x"^ + X - y = y^ 244. trinh trie nhau,. <^ y^ - y'^ - X + C = x^ - x"^ - y + C..

(2) Tii day ta se tao ra rat nhieu he doi xiing loai hai, chang han x^=x^. + y+ C. P-S-S' S = 1 5 =. .. r. 2. ^ .. .. B a i toan 29 ( D H - 2 0 1 2 A ) .. ^{y. • K h i ?y +. r 53. x'^ + xy. +. -. X -. ?y -t-. f. 1= 0. ^ (x + y)^ + ( x - l ) V ( y - i y ^ = 0 4 ^ ( X i r 2 ^ [ 2/ = 1. °. ( v o ngWem.) V /. doi xiJng loai 1 .. G i a i . T a c6 (1) SI. l^^^tyf^l. «{?(3=is)l2 « 246. '. t ^Il^ai t o a n. ^^^^=2. D a t { 1^1^. ;- 0. ;. [. {S^ > 4 P ) . K l u <16. P = 3 - 5 S''^ - 35 + 2 = 0. i. „. 3. ; (^" + ^2/+ ^ = 5 I X + y - x^y - y^'x = - 3 .. B a i t o a n 31 ( D H S P H N - 2 0 0 0 ) . Gidi he phuang. C a c b a i t o a n ve h e doi xiJng loai 1 v a c a c b a i t o a n difdc dufa ve he. |. 22. - 3 P 5 + 3(52 - 2P) - 9 5 =. B a i t o a n 3 0 . Gidi he phuang trinh. C a c bai toan r e n luyen v a nang cao.. B a i t o a n 2 8 . Giai he phuang trinh. -. 25^ + 6 5 2 + 4 5 5 + 82 = 0. Vay nghiem cua he l a. 1 + \/5 1 + \/5 , 1 - \/5 1 - \/5 Vay cac nghiem ciia he l a (0,0), ( — r — , — - — ) , ( — — , — ^ — ) •. 4.1.4. 9 ( i + y) = 22. - 3 P 5 + 3 ( 5 2 - 2P) - 9 5 = 22. 52 - 2 P + 5 =. r 53. .. + xy + x^, t a c6. 1=. X -. (^-'2/^. Dat 5 = y + i , P = yt. Ta dudc he. I ^^^^1. •r'X. y 3 ^ 3y2 _ 9y. + y 3 + 3«2 + 3y2 -. --x){y + x-l) = {y - x){xf + xy + x^) [ y= X y + X - I = y"^ + xy + x^.. - X = 0 ^ x(x''^ - x - l ) = 0 ^ x G jo,. '{). trinh. Hu'dng d S n . D a t i = - x . He t r d thanh. . •. • K h i y = X. thay vao he t a diToc. -. - 2?/ + 1 = 0 <^ u = i. + y= l. x^ + y^-x. a; - y + y^ - x^ = y^ - x^. .. Gidi he phuang. + 22 =. - 3x2 _. Giai. 3. he phuang trinh | ^ + ^2 Z ^Js G i a i . Lay hai pluWiig t r i n h cua he trii: iihau t a dUdc. 5 = 2 P=l. Vay ( 5 ; P) = (2; 1), do do x va y la n g h i f m cua nghiem ciia he da cho la (x; y) = (1; 1).. Tic he (2) nay, neu y = x thi .r^ - .r^ - .r - C = 0, do do ta nen chon^ C sao cho he phuang trinh c6 nghiem "dep" vd bai toan khong qua kho. Chang han chon C = 0 thi thdy ngay rang he phuang trinh (2) nhan (0; 0) Idm nghieni vd ta duac hdi toan sau. B a i t o a n 2 7 ( C z e c h A n d Slovak M a t h e m a t i c a l O l y m p i a d 2 0 0 8 ) .. o. 2. trinh. f x2 + y2 + xu = 7 :. :,U. [. x'« +. 3 2 . Gidi phuang trinh. y4. + x V = 21.. Vx + 1 + v'3 - x - v^(x + 1)(3 - x). G i a i . Diau kion x G [ - 1 ; 3]. D a t ( " = v / p ^ ^ ^ (. V = \/s — X >[j.. K h i do '. / u2 + ?;2 = 4 ^ < - > / ( « + v)^ -2uv = A \u-{-v-uv = 2 \u + v-uv =2 247. '. *. i.

(3) Dat I. P > ^ °. ^^^^. -. '^^^^ '^^^. [ g=0_2 5 = 2 P = 0.. P = 5 - 2 5-2 , Vay ( 5 ; P ) = (2; 0) do do u va D la nghiem ciia. (loai). p a i t o a n 3 4 ( D e t h i c h i n h t h i J c O l y m p i c 3 0 / 0 4 / 2 0 1 0 , I d p 1 1 ) . Gicii fie phUdng trinh ( x + y- v/Sy=_3. Vay. G i a i . Dieu kien x > - l , y > - l , x y > 0 . Phitdng t r i n h thu: hai cua he tirong diWug v 6 i. '^^'^^. -^2t = ^ 0 ^. I Z ^. v/r+^ = 0. u = 0 w= 2. X + 7/ + 2 + 2 V ' ( x + l ) ( y + 1) = 16. \/3^ = 2. u = 2 V = 0. • \n = 4.. X =. —. Vay neu dat 5 = x + y, P = x y , dieu kien 5^ > 4 P t h i t a t h u diroe. x = 3.. x/r+^ = 2. X + y + 2 \ / x y + x + y + 1 = 14.. v/3^ = 0 B a i t o a n 33 ( D e t h i c h i n h thiJc O l y m p i c 3 0 / 0 4 / 2 0 1 0 , Idp 1 0 ) . Gidi. / 5-v/p= 3 \ = 2v/5 + P + l = 14. f P = ( 5 - 3)2 \ + 2^/5 + ( 5 - 3 ) 2 + 1 = 14. (3). Taco he, phuang. trlnh. 1. +-L]. f4= +. i l ( ^. + 11 = 18.. •. '. (3) <^ 2 ^ 5 2 - 5 5 + 1 0 = 14 - 5 ^ {. (2). • •';„;.. - 5 5 + 10) = 5 2 - 285 + 196 5 = 6. • G i a i . Dieu kien x / 0 va y 7^ 0. D a t u =. u=. T h a y vao (1), l a. dvtdc ^ + l ) ( i ; + l) = 18. \ + v) {u. , {u + vf - 3uu ( u + i;) = 9 I lu + v)(uv + u + v^-l)-=l%.. (2). (3). r x + y = 6 ^ r y = 6 - x r y = 6 - x ^ / x = 3 lxy = 9 ^ l x ( 6 - x ) = 9 ^ ( x2 - 6x + 9 = 0 ^ ( 2/= 3.. 3. y. He CO nghiem d u y n h a t (x; y) = (3; 3 ) . C a c b a i t o a n v e h e d o i xiJng loai 2 v a c a c b a i t o a n ditdc du'a v e h e doi xiJng loai 2. „, J,. 5^ - 9 = 54 - 352 - 3 5 ^ 5 3 + 352 + 3 5 - 63 = 0. B a i t o a n 3 5 . Giai H phuang. ( 5 - 3) (52 + 6 5 + 21) = 0 ^ 5 = 3.. trinh. rf... ,. Thay vao (b) ditrtc P = 2, thoa man dieu kien 5^ > 4 P . Vay. { a;y + x2 = 1 + y \y + y2 = 1 + x. (1; 1)' (-^;. -0. ' (-1 -. Do do. V) (v<5i y G M , t u y y ) . ' X -. (x; y) = 1 \. ,,,,, ,. D a p so. Nghiem ciia he la. u;v} = (2; 1). ! u + Vi = 3 \ =. 1. <. 26 1225 • K h i 5 = — — , thay vao (2) t a dUdc P = —r--, khong thoa m a n dieu kien.. (a). T i t ( « ) va {b), til CO. ^. ^ | 5 G | - y , 6 |. • K h i 5 = 6, thay vao (2) t a dUdc P = 9, vay. Dat u + v = 5 , uv = P, dieu kien 5^ > 4 P . T i i (2), t a t h u diWc r5^'-35P = 9 ^ f 3 5 P = 5 3 - 9 \ 5 ( P + 5 + 1 ) = 18 ^ \= 1 8 - 5 2 - 5 . (6). ^{35^+^85-156 = 0. -: 1. ('••87. = (i;2). 248. Jai t o a n 36 ( D H Q G H N - 9 7 ) . Gidi he. 3y = 4 -. •. *. y - 3x = 4 - . y. B a i t o a n 3 7 ( D H Q G H N - 1 9 9 8 ) . Gidi he phitdng. trinh. |. ylzly^gi..

(4) B a i t o a n 38 ( D e n g h i O l y m p i c 3 0 / 0 4 / 2 0 0 6 ) . Gidi phuang. trinh. gn (0; 1], xet h a m so 5(u) = log2 ( 1 + 3 u ) - loggw. K h i do 3. (1). ( l 6 c o s ' ' x + 3)'* = 2 0 4 8 c o s x - 768. G i a i . Tap xac d i n h R. D a t u = 2cosx. T h a y vac (1), t a duoc (u^ + 3 ) ' = 4^ ( 4 « - 3 ) .. ,. if):. (2). fit. ^. ". g'{u). = 0. (2). Dat V = ^/^{T^. X = 1 sin y = 1. 1. -.. jg]. L^Y (4) trir (5). 0. —. 1. D o do. cos X =. -. r. o. (3). > 0. T\i (3) t a c6 he | "4 ij^ 3 ^. *. = UQ = ^ ^ ^ J ^ ^ ^ ^ ^ ^ S (0; 1].. 3u l n 3 = (1 + 3u) l n 2. cos a; = 1. + 3= 4^/4^.. v. (1 + 3u)u. I n 2 . I n 3 ". b a n g bien t h i e n v a (4) suy r a. cos a; =. -. uln3 ~. cos. 3 De u la nghiem ciia (2) t h i dieu kien la u > - . K h i do '""'^'^'••''(^^ .... 3 M 1 U 3 - ( l + 3u)ln2. 1. ( 1 + 3u)hi2. ,. I. ^. K 0. -. +. 1. gi"). smy=Vay c a c n g h i e m c i i a h e d a cho l a. thco vc, t a diWc ,. a,,. ,. -. = 4(w - u). {u - v)[{u+v){u^. + w^) + 4] = 0.. (6). k2n. y = 2 + ^27r. 3 V i ?; > 0 va u > - nen t i t (6) t a c6 u = v. Thay vao (4) t a diWc .'. X = arccos ^ + n2iT. X = - arccos ^ + n27r 1 ^ y = arcsm - + m2n y = arcsin - + m27r o o 1 1 X = arccos - + n27r X = - arccos - + n27r. X =. - 4?/, + 3 = 0. {u - l)2(w,2 + 2 K + 3) = 0. Vay (1) <^ 2 c o s x = 1 <f=^ cosx = i. 1. . 1 y = IT - arcsm - + o. • 1 y = TT — arcsin - + m2n o. 7i = 1.. m2n.. x = ± ^ + fc27r, A; G Z.. B a i t o a n 3 9 ( H S G T p H o C h i M i n h , n a m h o c 2 0 0 3 - 2 0 0 4 ) . Gidi he. ^. log2 (1 + 3 c o s x ) + log3 ( c o s i ) = log2 (1 + 3siri2/) + logg ( s m y ) . Tren khoang (0; + 0 0 ) , xet ham so f{t). ^ ' ^ ^ " ^ ^\. V i ^+. \2 + 9 i =. ^^^y { W = 2. G i a i . Dieu kien | y ^ 2 G i a i . Dieu kion cosx > 0, s i i i y > 0. Lay (1) t n r (2) then v6, t a diMo. i. B a i t o a n 4 0 . Gidi he phuang trinh. x^.. khong la nghiem ciia he phiTdng. t r i n h da cho. Lay (1) trir (2), t a duoc (3). \/x2 + 91 - Vy^ToI = ^ y - 2 -. = l o g a l l + 3f.) + loggf. K h i do. ^ (x - y) Vay ham so / dong bien t r e n khoang (0; + 0 0 ) . T t f (3) t a c6 •^x =. / (cos x) - f (sin y) <^ cos x = sin y. T h a y vao (1) t a ditdc. y. ^. + 2/^ -. y-x. x^-y^. v / y ^ + V x - 2. s/x^ + 91 + ^y^ + 91 x +• y. 1. v/x2 + 91 + x/y2 + 91 ^ do. v^r=^ +. x + y v^x2 + 91 +. + (y - x) (y + x). ^Jy^. Vx - 2. + x + y. 1 + 91 ^ v ^ ^. +. v/^^. (4) v/x2 + 91 = v ^ x - 2 + x^. ^50. 251. = 0. + x + y > 0. Thay x = y vao mot t r o n g hai phitong t r i n h cua h?, t a ditdc. l o g 2 ( l + 3 c o s x ) - l o g 3 ( c o s x ) = 2.. (2).

(5) <(^\/x2 + 91 -. ^. 10 =. - 1 +. - 9 ^ v/x-^ + 91 + 10. -. ~ ^ + \ / ^ ^ + l. 9. 4.2 4.2.1. 1. x + 3 = X + 3 + V ^ ^ + 1' . v/x2 + 91 + 10 1 > 5, t r o n g k h i do V6i .T > 2 t h i x + 3 + \/x^+ 1 + 91 > X Vay (3). ^ x 2 + 91 + 10 > X + 3. H e CO chiia mot nhat) bac hai.. = 0. (3). I. r,«.. '2M I. - 9 - X - 3j. X = 3. H e C O yeu to dang cap. phu-dng t r i n h d S n g c a p. trong he co chi'la m o t phudng t r i n h dang cap bac hai (khong chiia cac g6 hang bac nhat va t u do) t h i ta xet cac trirdng h^Jp , TrircJng hdp 1 : x = 0. T l i a y vao ho do t i m cac nghiem dang (0; y) neu c6. tlniSn nhat de t i m t. Sau do t i m x va y.. ,. ^. <1.. Bai t o a n 42 ( H V N H - 2 0 0 1 ) .. B a i t o a n 41 ( D e t h i c h i n h thiJc O l y m p i c 3 0 / 0 4 / 2 0 0 6 ) . Giai he phuang trinh / x 3 ( l + 3y) = 8 \(y3 _ 1) = 6.. / x2 - 2xy + 3y2 = 9 \13xy + 15y^ = 0.. Gidi he. If.:. r 15 ( ' y ) ^ , 1 3 . 2 / + 2 = Vx/ x. 0 ^. y. 1. x = 5y. X. y. 2^-. . X. Khi X = 5y, thay vao (1) t a diTdc y^ = i. (*)^. (x; y) =. Lay (1) trfr (2) thoo vo, t a dildc y){e. + ty+. y'^ + 2>) =Q ^. \. t = y.. •^"i - =. = 1.. D a t i = 2u, t h a y vao (3) t a dildc 8u^ - 6u = 1. 4u^ - 3'u = cos - .. (4). Theo bai toan 3 d t r a n g 119, suy ra cac nghiem cua (4) la cos—, cos-g-i c o s - ^ . Vay t a t ca cac nghiem cua (3) la 2 c o s ^ , 2 c o s - ^ , 2 c o s - - ^ . Nghiem y y y y cua (*) la '. X =. 1. — TT. cosy = 2cos-. X. =. cos. 57r. 2/ = 2 cos 252. X =. cos. 7n. y = 2 cos. ' 2. , {x; y) =. \. 2 4 9 1 - 2.- + 3.- = ^ 3 9 x-^. (3) 7r. 2. %. 2. chia ca hai ve cua (1) cho x^ ta dudc. T h a y vao (1), t a diTdc ^-2,1. y = ± — . Vay. 2. X. t^-y^-3{y-t)=0^{t-. (1) (2). Giai. Ngu X = 0 t h i t h a y vao (2) dudc y = 0, thay vao (1) thay khong thoa man. Xet X ^ 0. C h i a ca hai ve ciia (2) cho x^ t a dUdc. G i a i . Dg thay x = 0 khong thoa man ( * ) . T i e p theo xet x 7^ 0. K h i do. ~ }. thu4n. , Trirdng hdp 2 : x 7^ 0. D a t y = tx (hay t = ^ ) , thay vao phiTdng t r i n h. + 91 + 10 X = 3. He phiTdng t r i n h c6 nghiem d u y nhat x = y = 3.. Dat t = ^ , t a dUdc | ^3. (tiJc l a. 9 x^. = 1. X =. ±3.. nem {x; y) = (3; 2), (x; y) = ( - 3 ; - 2 ) . He da cho c6 b6n nghie:. K. A\/2. (. s/2\ .(3; 2), ( - 3 ; - 2 ) .. 5\/2. Qai t o a n 4 3 . Gidi he phUdng tnnh inh {xl-2xy-2y^. = Q \2 + y2 ^ 2x + 3y = 19. ^•. J.. ^ap s6. /. I. 14. X =: 3 J/= 1. . '. 57. f. y =. - 10. (. l-3v/l7 -l+3\/l7. 253. X. =. ; <. '. 2/ =. l + 3vT7 -1 - 3 \ / l 7.

(6) 4.2.2. H e C O h a i phrfdng t r i n h b a n d a n g cap bac h a i .. a^x"^. +. b-ixy. +. C2y^ =. , Khi = 16, he (A) c6 nghiem, chfing han (x; y) = (0; s/U). ^, , I<hi nL i=- 16. He [A) c6 nghiem k h i va chi k h i (3) c6 nghiem ( v i neu c6 i j-jii CO X cho b6i (2) va c6 y tilt y = i x ) , nghia la. d-^.. Dua ve dang phttCng t r i n h da xet 6 muc 4.2.1. T i l hai phudng t r i n h ban dan^ cap bac hai nay, t a tao ra mot phitdng t r i n h dang cap bac hai n h u sau dx {aax^ + h'lxy +. = d-^ (aix^ + bixy + ciy^) .. C2y'^). B a i t o a n 44 ( D H Q G T p H o C h i M i n h - 1 9 9 8 ) . Cho he phMng. trinh. CO. (A). y^y tic hai todn 44 nay ta thu diMc kit qua sau: Neu edc s6 thuc x, y thod fiian dieu kien 3x^ + 2xy + y^ = 11 thi bieu thiic x^ + 2xy + 3y^ cd gid tri nho nhat /d 22 - l l \ / 3 m gid tri Idn nhat /d 22 + l l y ^ . Bai t o a n 45. Cho cdc so thUc x,y thod man dieu kien x ^ - x y + y^ = 3. gid tri Idn nhat vd gid tri nho nhat cua bieu thiic G = x^ + x y - 2y^.. ^ = 17 + m .. He (A) c6 nghiem k h i va chi k h i. 11.3 = 17 + m. Hu'd'ng d a n . Goi T la tap gia t r i ciia G. Ta com CO nghiem:. m = 16.. I. = 11. x2 + 2 t x 2 + 3i2x2 = 17 +. m.. ^. x2 (3 + 2t + t2) ^ II { x 2 ( l + 2t + 3f2) = 17 + m .. (1) (2). l + 2t + 3^2. 11. 3 + 2 i + /2. ^. max G = - 1 + 2\/7, m i n G = - 1 -. ( m - 16)f2 + 2(m + 6)/- + 3 m + 40 = 0.. (3). a) K h i m = 0, tir (3) c6. -laf. + 12i + 40 = 0 <^ 4^2 - 3t - 10 = 0. • K h i /, = 2, thay vao (1) t a dUdc l l x ^ = 11 ^ (x; y) = (-1; - 2 ) la nghiem ciia he. ' =. -4-. x = ± 1 . Vay ( x ; y ) = (1;2).. (A).. • K h i i = - 7 , thay vao (1) dUdc ^ x ^ = l l < ^ x 2 = ^ < ^ x = 4 / \ ^4v/3 5v/3\ , / 4v/3 5v/3\ la nghiem ciia he. , (x; y) = Vay (x; y) = ^ , ^ K h i / f t = 0 he (/I) CO bon nghiem (l;2),(-l;-2),. ^>. Tien hanh t i M n g t u nhir bai toan 44 t a suy ra dudc ket qua : Ho (*) c6 nghiem k h i va chi k h i - 1 - 2 ^ 7 < m < - 1 + 2%/7. Vay. Lay (2) chia (1) thco ve t a ditdc 17 + in. 4v/3. 5N/3'. ^. 4\/3 ' ~ '. 254. Tim. k h i va chi k h i he sau. f x 2 - x y + y2 = 3 \2 + xy - 2 y 2 = m .. T i e p theo xet x 7^ 0. Dat y = tx. T h a y vao he {A) dvtdc f 3x^ + 2tx^ + t'x'. < m < 5 + 11 v ^ .. 5 - l l \ / 3 < m < 5 + l l v ^ <^ 22 - l l v ^ < 17 + m < 22 + I 1 V 3 .. a) Gidi he khi m = 0. b) Tim m di he c6 nghiem. 3y. + 10m + 338 > 0 <^ 5 - 11. j ^ l t luan : He {A) c6 nghiem k h i va chi k h i 5 - l l \ / 3 < m < 5 + l l \ / 3 . C h u y 1- 7a. 3x'^ + 2xy + y^ = 11 + 2xy + 3y2 = 17 + m .. l i a i . Neu X = 0 t h i |. A ' = -rr?. 2\/7.. L i f t i y . K h i t i m gia t r i Idn nhat, gia t r i nho nhat bang phUdng phap tap gia tri, ta khong can chi ro gia t r j ciia bioii so do biou thiic dat gia t r i Idn nhat, gia t r i nho nhat. , Bai t o a n 46.. (. X?. -\- xy. I. X. y. Gidi he phMng trinh <. iJap s6. ( x ; y ) - ( 2 ; l ) ,. - y^ = 5. 2. (x; y) = ( - 2 ; - 1 ) .. ^^•2.3 H e d a n g c a p b a c h a i . aix"^. +. bixy. +. ay^. +. di. ^. a2X^. +. b^xy. +. C2y'^. +. d2. ^. 5^3^ 3 255. 0. 0.. xy'.

(7) G i a i . Vai y = 0 t h i he t r d thanh | ca hai ve ciia (1) cho. ( v o nghiem^ . X e t y 7^ 0. Chi;i. ^^-f^-^. va dat - = i , t a dUdc. ,^. y^{2e-U • Neu. - 1) (2t - 1) < 0. i. ^lf,±l = ^. ( {x + y)^ + y^<Q. ^/^= 0. j { | t luan : He da cho c6 nghiem duy nhat k h i va chi k h i a = 0.. |;. ^ + l) = l ^ y \ t ~ \ ) { 2 t - l ). (x; y) = {s/a; 0) la nghiem ciia h^. Suy ra ngu a > 0 t h i h § c6 i t nhat hai pghiem. X e t a = 0, t a c6 he j,. (3). t ^ - t < t + l ^ t ^ - 2 t - l < Q ^ \ - y / 2 < t < \ y/2. T h a y x = ty vao (2) t a diWc. I. = l.. (4). .. < t < 1 t h i (4) vo nghiem. Suy ra he vo. [. gai t o a n 50. Tim a de he sau c6 nghiem. 5x2 ^. , ^. + 2j/2 > 3 1 O 2 < 2a - 1. : < -.^2 _. G i a i . He da cho viet lai. nghiem. 1\. • X e t ( i - 1) (2« - 1) > 0 <^ < €. f -^.^'-^xy-ly^<-l. U (1; + 0 0 ) . K e t hdp vdi (3) ta c6. 2; dieu kien ciia. 1 - \/2; 0. i\&t^. 7x2 - 4 x y +. 2 7 / < ( 2 ) 2a + 5. V. U ( 1 ; 1 + ^ ] . K h i do (4) cho t a nghiem :. (1). Cong (1) va (2) theo ve t a dUdc y =. ±^. 16 2 (i-l)(2t-l). —X. (^-l)(2i-l)'. 3. 16 4 2 -6 - —xy 3 " + -y 3 " -< 2 a + 5 1.. 't +• \. vdi i G. <^16x^ - 16xy + 4y^ <. 1\ 1-v^;U ( l ; l + V2. 2a+ 5. ^. (4x - 2yy '. <. 2o + 5. Vay he da cho tUdng dUdng vdi X? -. 2xy < a x'^+ 2xy-'2y'^ <2a+l.. B a i t o a n 4 8 . C/io he. a) Chiing minh rhng vdi moi a. he da cho ludn c6 b) Tim a de he c6 nghiem. (. ^. '. r 5 x 2 + 4 x y + 2y2 > 3. '. (1). nghiem.. duy nhat. Tit (3) suy ra 2a + 5 < 0. a <. T a xet he. Giai.. r. a) K h i X = 0, h f t r d t h a n h |. ~J2y^<'2 a + 1 |2a + l |. {x;y). <^ ^. 2/2 > - a 2 > 2a + 1. 1. Suy ra 5x2 _^ + 2j/2 = 3 ^ (4x - 2yf = 0. ( y = 2x I 21x2 ^ 3. b) T i r kgt cjua cau a) suy ra : K h o n g ton t a i a de he (*) c6 nghiem d u y nhat.. Tigp theo t a chiing m i n h , vdi a <. B a i t o a n 49. Xnc dinh cnc gin tri ciln a de he snu c.6 nghiem. (a^o; ijo) la nghiem ciia he ( H ) , k h i do. r x'^ + 2xy + 2f < a \ Axy -y^ <a. (2). duy nhat. + y"^ < a. Vay neu a < 0 t h i h?. da cho vo nghiem. T i e p theo xet a > 0. N h a n xet rang k h i do (x; y) = (0; 0). t h i he ( I ) c6 nghiem. T h a t vay, goi. | ( 4 . . - 2 „ ) ^ = o < ^ ,. ^ % ra (xo; yo) la nghiem ciia ( I ) , suy ra he ( I ) c6 nghiem. Vay he da cho c6 'Nghiem k h i va chi k h i a < - - . 4b. 256. (11). r 5xg + 4xoyo + 2y^ = 3. (1) H. G i a i . B a t phUdng t r i n h (1) v i l t lai (x + yf. ^ ^ ' ^ < (^•; y). la nghiem ciia (*) vdi m o i a.. = (0; 0, vdi ( > max. 2. {x; y). 257.

(8) f a can xac d m h cac gia t r i ciia — do (*) c6 nghiem, ti'rc la phirong t r i n h. L t f u y. D i g m mau chot cua Idi giai la tim ra dufdc bat phvtdng trinh (3). Vj^y (3) dUUc tim l a nliU the nao ? Vdi m < 0, ta c6. {. (!!l^l)t^+(-. 5mx^ + Amxy + 2rm/^ < 3 m. o. ( 5 m + 7)x^ + ( 4 m -4)xy. <^7. 77i<. o. + (2m + 2). + —-5. 2a — 1. < 3m +. (—Y \ '. Vay m < khi. ^ ^.. = 0c6. /m. 2. Cong hai bat phudng trinh cua he, ta dUdc. T a can chon m sao oho. + At. 36 f — ) + 4 < 0 <^ 61 < — < 62, ^^1,2 Va / a. <. y 2 ( 1 - 62) x^ + x y + 2y2 = 1. ^. <. y=. ±-. 62 + 4. (nhan).. l2. [2(1-62)]. 2). m = - -. 2^74. suy ra gia t r i Idn nhat ciia m la 62, dat dUdc k h i va chi. 2v/5m + 7.V2m + 2 = ± (4m - 4). ( 5 m + 7) ( m + 1) = 2 ( m - 1)^. 18 +. 62+4. X. bang binh phUdng cua mot bieu thiic nao do. M u o n vay thi. (loai). =. 62+4. {V5m + 7x)^ + ( 4 m - 4) x y + (\/2m + 2y)^. m = -5. - 1 ^ 0 n^en. \m. 0, 5 m + 7 > 0, 2m + 2 > 0 va. = » \ / 5 m , + 7. V 2 m + 2 = ± (2m -. nghiem. V i -. 4.2.4. 62 + 4 2(1-62). + 2. H e dang cap bo phan.. Vay t a se nhan bat phiWng t r i n h t h i i nhat cua he vdi - ^ , sau do cong v6i. K h i gap he dang cap bo phan, t a thudng sii dung phep the de tao ra he dang cap. Nhieu k h i da biet chfic dua dUdc ve he dang cap t h i t a lai khong lam dieu nay m a giai luon bKng each dat x = ty hoftc y = tx.. (2) va thu ditdc (3).. B a i t o a n 53. Giai he phuang trinh. o. r 3J:2 + 10x?y - 5?;^ < - 2 Tim a dd he sau c6 nghicm : < 2,0 7 2 ^ 1 - a +zxy ^2/^1^^-. B a i t o a n 51. D a p so. a <. G i a i . Ho phUdng trinh da cho vict lai + y3 = 1 x2y + 2xy2 + y3 = 2. -1. • Khi y. B a i t o a n 52. Xdc dirih gid tri l&n nhdt cila m dc hC aaii c6 iighiern. 7^. X. 0.. 2) X e t x^ + x y + 2y^ > 0. Dat x^ + x y + 2y2 = a, 0 < a < 1. K h i do m _ x^ - 4xy + 5y^. a. x"^ + xy + 2y2. 771. b) N6u y ^ 0 t h i m _ 7. ~. Dat - = t, thay vao (4) diroc 2f y Ngu /, = 1 t a. CO. he { f ^ f. N6u t = -\a C O he { f^fy Neui = i t a c 6 h e {. a) Neu y = 0 t h i — = 1 4^ m = o. - 4t + 5 t2_,.i + 2 ' 258. ^ _. X. ~ y'. Vay he C O nghiem la. i. (2). (1). (3). f - ] " - 2 f - ) +1-0 \yJ \yJ. ' - V \vJ. Giai. t h i x2 + xy + 2y^=0,. \ + y^ - x^y - 2xy2 = 0. .7:3+y3 = l. /. ^. x^ + y-^. ^. 0. T a co. 1 x 2 + xy + 2y2 < 1 \ Axy + 5y2 = m .. 1) Ngu { J = 8. ^2 + y^ = 1 t^y + 2xy2 + y3 = 2.. |. = 1 " ^. - f - 2t + I = 0 <^ t = ±1, O. .T. ^. = y =. { x i. f^£'='^ /. 1. (4). -y 3 '. 2^^. 1. \. 3 '. 259. 3. s/2-. nghiem).. 1 -. 2.

(9) B a ii t o a n 54. Gidi he phuang trinh. { "^^'^^. =. + Vs/v. •dix = y t a duoc { ^2. G i a i . Dion kion x > 0, y > 0. • X e t 2/ = 0, khong thoa m a i l he phvtdng t r i n h . • X e t y T^O, dat = t^, dieu kien i > 0 va t 7^ 1. He t r d thanh. t^y-8 = t + y ^1 - 1) = 5. tifo «•/ i T&co. ^ ^. \ ^ i X = 2y duoc: { - r , ^. (do. (1;1),. 3 5 dUdc t = - . Vay y = -g 4. 3= 4 =4-. =. 3. = 3. (-1;-1).. 2A/-. B a i t o a n 56 ( D e n g h i O L Y M P I C 3 0 / 0 4 / 2 0 0 9 ) .. '^. trinh. Giai. • Xet y = 0. T h a y vao he t a dUdc. kien t a duoc nghiem ciia he phiTdng t r i n h la (x; y) = (9; 4). „ . ™^. Gidi he^phudng. ^ + 8y^ - 4xy2 = 1 2x4 ^ 8^^4 _ 22; - y = 0.. x = 9. K e t hop vdi dieu. B a i t o a n 55 ( D H - 2 0 1 1 A ) . Gidi he phuang trinh r 5 x 2 y - 4 x / + 3 y 3 - 2 ( a : + y) = 0 (1) .. " =. 1). /2 ^. 1 3, - 8i^ + t + 3 = Q ^ t e { 1 , - - , - } . D o i chieu dieu kien ta. {*). ^. Cac nghiem cua he la. ^23T~^ = ^ ' +«2 _T 1 ^ i 2 - l. ^2. ^ (x; y) = ( ± 1 ; ± 1 ) .. ^ { y2. ^2. vay. (.T; y) = (1; 0) la nghiem ciia he da cho. • Xet y ^ 0. D a t x = ty. K h i do. «. y3(t^ - 4( + 8) = 1. G i a i . Ta c6. i/(2t'' + 8 ) = 2 i + 1 (do y 7^ 0). 2*^/ + 8y4 - 2fy - y = 0 (2) ^ xyix"^ + y2) + 2 =. + 2/^ + 2xy. - 4i + .. 1. 2/4 + 8. + y'^){xy - 1) - 2(xj/ - 1) = 0 xy = 1 <=^(xy-l)(x2 + y 2 _ 2 ) = 0<^ x^ + y- 2 - 2 . ^. •^i^ -. 1 Tru'dng h d p 1 :?; = - , thay vao (1) t a diWc X 4 3 2 6 3 5 x - -X + — - 2 x - - = 0 < S ^ 3 x - - + ^ = 0 X <^3x4 - 6x2 + 3 = 0 <^ 3 (x^ - 1)^ = 0 x^ = 1 <j=!. X = ± 1 .. 2/4-1. ^ 2/4 + / - 8/2 ^- 4/ + 10/ + 8 = 2/4 + 8. + 12/ = 0 <^ / e { 0 , 2 , 6}. hi / = 0 t a. CO. (x; y) = ( 0;. 1. K h i / = 6 t a c6 (x; y) =. ^. (x; y) = ( 1 ; 0), (x; y) =. . K h i / = 2 t a c6 (x; y) = ^ 1 ; ^ ) . \. . He CO bon nghiem. (o; ^) , (x; y) =. (l;. , (x; y) -. ^. ). .. Vay (x; y) = (1; 1), (x; y) = (-1; - 1 ) la nghiem ciia he. = 2. T h a y vao (1) t a dudc. Tru-dng h d p 2 : x^ +. B a i t o a n 57. Gtwi, he phuang trinh. 5x2y - 4xy2 + 3y^ - (x^ + y^){x + y) = 0 <i=>5x2y - 4xy2 + 3y^ - x^ - y^ - x^y - xy^ = 0 <;^4x2y - 5xy2 + 2y^ - x^ = 0.. G i a i . Xet y = 0. 5X. y. „. X^. +2--^=0<^ yi 260. x = 2y X = y. L y. '^y^^^^Jzl^. x = 0. Xet y 7^ 0. Dat / = - <^ x. ty. He da c l. (3). Neu y = 0 t h i tiJt (3) suy ra x = 0, m a u thuan vdi x^ + y2 ^ 2, vay tiep theo t a chi xet y 7^ 0. K h i do chia ca hai ve ciia (3) cho y^ t a diWc 4.T;2 ( 3 < » ^ yi. |. Chia (1) cho (2) vc theo ve t a duoc. ^^_:±±l = i z i ^ 3 / 3 _ 7 / 2 _ 3 / /2 + / -. 3. / - 2. / = -1 + 7 = 0.=. '=3-. 261.

(10) H e d a c l i o c6 b o n n g h i e m ( 0 ; 0 ) , (1; 1), ( - 1 ;. 1),. B a i t o a n 60. ( H S G Q u o c gia n a m hoc 2006-2007).. trinh 3^3_. Gidi he phMng. B a i t o a n 58.. (1). 3 ^. 1. X + iJ. trinh x'+y'. -. 12 y +. Gidi. he. phudng. v/5 = 2 '. 3x.. (2). = l. I \ G i a i . N h i n v a c he t a de d a n g n g h i d e n v i e c t h e so 1 t i t p h u d n g t r i n h. (2). vao i ) h \ m n g t r i n h (1) dfi r o phirclng t r i n h t h u a n n h a t ( d a n g c a p ) . D i o n kioii CO (2) 4=> (x2 + y2)2. x + yj^O.T^. G i a i . D i ^ i k i e n x > 0. y > 0, y + 3x ^ 0. T h a y x = 0, y = 0 vao he t h a y k h o n g t h o a m a n . V a y lie p h i f d n g t r i n h t i t d n g d i t d n g. = 1. T h 6 vao (1) t a d u d c : 1. + y)={x'. i3x'-y')ix ^ S x - i -y'. + 3x^y. y'f. - xy^ = x* + y'+. + 'Sx^y - 2x^y^. ^2x*. +. -. 1 +. 2x^y\. - xy^ - 2^"* = 0 X =. 2. y +. 'Sx. 1. 6 \. sjx]. •(1). /y. -^1... -2y. . 2x2 ^. ^ Q. 1. 9. X. y. -12 y +. (y - 9.x) (y + 3 x ) + 12.xy = 0. 3.r. y =. • K h i 2x2. y =. 4- y2 ^ 0 t a c6. 2 x ' + x y + y2 = 0 <^ x^ + ( x J 1^. = 0 <f=> x == yy = 0,. + ^. k h o n g t h o a x^ + \f = I . T n t 5 n g h d p n a y l o a i . • Vdi. X. -. 1. 1. y t a c6 2x^ = \ = ^;2 vay x = y =. 1. . V a i x = - 2 y. tac6y^ = -.Vayy. 1 2 = - ^ , x = - ^ , y. - 1 =. 1 2 ^ : , ,X . =- ^ .. K e t I r a n : H e c6 n g h i e m ( x ; y ) l a f-2. 1. .7!' 7!;'. B a i t o a n 59.. 2. (2). y + 3x. yfy. N h a n (1) v a (2) t h e o ^•e t a d u d c. px = y <^ ( x - y) ( x + 2y) {2x^ + xy + y^) = 0 <^. 12 y + 3x 12. - 1. Gidi cdc he phMng. ' innh. ( P + 4yy - ?y^ - 16x = 0 | y 2 = 5x2 ++ 4. 1. 7i;'. Vv/2'. ^j. 4= +\/3x -i= = 1 - ^ 1 V-i'. + V3. = v^^x. B a i t o a n 61. ( D e t h i c l n ' n h thu-c O L Y M P I C. phiidnq trinh. jj;, ,. r x^ - 8 x =. D a p so. a) H e CO n g h i e m ( x ; y ) l a (0; 2 ) ; ( 0 ; - 2 ) ; ( 1 ; - 3 ) ; ( - 1 ; 3).. + 2y. ^. ^- -. 1 + 1 -. 1. he. 2.. 2. x + y X. +. y/. |. =. 4v^.. B a i t o a n 6 3 ( D e n g h i O L Y M P I C 3 0 / 0 4 / 2 0 0 9 ) . Gidi he. phiMng. V4. Gidi.. ' ' y-M2x,. b ) H e CO n g h i e m ( x ; y ) l a x + y ; 2 ^ + ^ j x + y 262. 2^3.. 30/04/2000).. P a i t o a n 6 2 ( H S G q u o c g i a - 1 9 9 6 ) . Gidi he phtidng Lrinfi. Ix2-3y2=6.. (3;l);(-3;-l);. = 4 +. K e t h d p d i e u k i o n . ,suy r a he c6 n g h i e m d u y n h a t | ^ I I ^ ( 4 + ^ x / 3 ). 3 + \/. sau :. ,,. D g ( x ; y) l a n g h i e m c i i a he t h i x > 0, y > 0, do d o y = 3 x . T h a y vao (1) d i t d t. ^3-. 1. 3x -9x.. 263. j. = 1.. Innh.

(11) Giai. Dieu kien x > 0, y > 0, x + y =>;^ 0. De t h i y x = 0, y = 0 khong thoa man he, vay chi can xet x > 0 va j / > 0. Dat u = > 0, v = ^ > 0. Khi do 2u + v. 1. 2ii + i '. + v. 1 4. 2u + i ; u'^ +. 2u. 2 ^. 1. .^18 + 9(1),3 = - 1 0/ ? ; \\ 30 g ) - 1 0 .. 1. (1). Vdi y = tx thi phirong trinh thii nhat ciia he trd thanh '. (2). 2.x3 + ^3.^3. ^ JO ^. ^. 2u + t; 2iL + v 4v - u u V v?' + z;-^ uv + i;^ <^4t;u^ + 4i;^ - uv^ = 2u^v + uw^ - 2uu2 = 0. Vay. + -L. (2uu^ - u^) + (4^^ - 2uv^) = 0 u = 2t).. = 2(v/2 + l ) ^ . = 4 ( 3 + 2^/2).. = A^^. r u = 8(3 + 2v/2). \?; = 4(3 + 2v^). 10. • V 2+. •. 2 +. V i du 2. V ^ i { J ^ 1 to. CO. I. /2±3N/2\. ^4 ~ ^ 4 ^ ^. ~_~3^. ^. Q. ^^'^^. phudng phdp titdng til nhu giai he co yeu to ddng cap, ta can xdy dung phudng trinh th:ti hai cua he sao cho no khong chiia so hang tU do. Ta c6 bdi todn sau.. f X = 64(17+12N/5). y = 16(17+12^).. +. ^. 10. (x;y) = ( l ; 2 ) , (x; y) =. Bai toan 65. Giai he phUdng trinh He CO nghiem duy nhdt { ^ I. X =. /. = 2?' vao (1) ta (hruc -|=. ^. Vay ling vc3i ba nghiem t tim difdc d tren thi he da cho c6 ba nghiem la. ^u^{2v - u) + 2u2(2u - u) <t4- (u^ + 2v^){2v - u) = 0 Thay. 10. 2 + t^. 1. ^2vu^ + 4w^ -. t= 2. 9t^ - 30t^ -f lot + 28 = 0 <^ ( i - 2)(9t^ - 12< - 14) = 0. y/v. Nhan (1) va (2) theo ve ta diWc 4. j'^^'V. Dat t = - , khi do. 2 _ J _ _ 4u + 2u. +. ^^^ISx'^ + 9y^ = -lOx^'y + SOxy'' - lOx"*. | ^ 4 ^ ^4 + J_~3y ^ Q. v|) Hirdtng dan. Khi y = 0, he da cho trd thanh | ^ 4 ^ ~1 Q <^ X = - 1 . Vay. 4.2.5. nghiem co dang (x;0) ciia he la (-1;0). Tiep theo xet y ^ 0. Dat x = ty, thay vao ho ta dudc. Phufdng phap sang tac bai toan mdi.. V i du 1. V6i { J = 1 to. CO. I + ^. giai he, ta se di(a ve mot phMng. _9.. tnnh hac ba theo i —. hon niia phuong X. 2x'^todn + 2/3sau. ^ 10 trinh nay c6 "nghiem, dep" t = 2. Ta c6 bdi. x^y - 3xy2 + x^ = - 9 .. Bai toan 64. Giai he phuang trinh Giai. Khi x = 0 , he da cho tr6 thanh | g _r_^g .3 = 6. Vay he khong c6 nghiem. dang ( 0 ; y). Tiep theo xet x 7^ 0 . TiJt he da cho, ta co. 9 {2x^. + y3). = _ 10 (x^y - Sxy^ + x^) 264. ty^. 2/ = 2x, nen khi. tV. - 2iy.y2 = - 1. + y* + ty-Sy. =0. <io^o J. ((^ _ 2t) = - 1 y3 ( ( 4 +. 0. Suy ra -1 - 2t ^3t^ +. 1) = 3 _ t.. 3-t /4. +1. <=> -^t'' - l = 3t^ -t* - 6t + 2t. - 6« + 1 = 0 ^ (i - 1) (3*2 + 5 i - l ) = 0 ^ i € | 1, - ^ ^ ^. Lifu y . Ta biet chac ch^n rang se co ( = 1 la do vdi x = y = 1 thi < = - = 1y. 265.

(12) C h u y 2. Viec sang tdc cdc he phUdng trinh duac gidi bang each dit.a ve cdc. phicang trinh dang cap bdc hai nhic cdc hai todn 60, 6 1 , 62 se duac trinh bay a muc 4.6.2 d trang 311. Sau day ta .se tiinh bay mot cdcli sung tdc klidc cuug khd nhanh chong cho cdc he phiMng trinh dang nay {xem vi du 3 ) .. at x^ = i ta ditdc he. D. V i d u 3. Xct , = (2 y = 4.T. + v^y. Khi do (1 + —KA V y + oxj. y. + 5x = ^. ^. = 9x. y + bx. 77 +. 9 5x7. X. + 4= = 2 + vAr. 9. 1 -. = - =^ 1 +. 9. _ ^. 1. y + 5x. =. l + a^ I--2a = (1 + a'){2 2\ + 2a - 1 + 2a) = (1 + a'^){Aa + 1). l + a^ 2 +2a 1 + fv2 2 r-'l; , 1+^2 11 = (l + a 2 ) . 9 . 2 11. X'. + 2 - \/3 = 4. M d i khdc. 1 - 2a 2 +2a. 4 + 4a - 11 + 22a = 26a - 7.. * ' !? -aJ. VI DX 7^ 0, Va nen neu 4a 4- 1 = 0 t h i D = 0, he v6 nghiem. T i e p theo t a xet a. \. K h i do. y + 5xJ. 2. C (1 + f v ^ ) z + (1 - 2fv)x = 2 (1 + a'-^)^ + 2(1 + a ) x = 11 Ta tfnh cac dinh th^ic y = ax, x^ = z.. 26a - 7. Or. D. Ta CO bai todn sau.. 4a+r '. D. (l + a 2 ) ( 4 a + l ) '. Dieu kien x!^ = z cho ta phUdng t r i n h B a i t o a n 66. Gidi he phiMng trinh. 81. 26a - 7. (4a+1)2. (l + a 2 ) ( 4 a + l ). <^ 81(1 + a^) = (26a - 7)(4a + 1). ... ..v.... a = 2. 44 a = -23'. <^81a2 + 81 = 104a2 - 2a - 7 4* 23a2 - 2a - 88 = 0. 4.3. He bac hai tong quat.. 4.3.1. e, V6i a = 2, t a diMc { J = 2.. N h a n dang v a phifdng p h a p giai.. X. He bac hai vcli hai an x va y la | •. •. \. ^^^^J + b2xy. 1 '''A t w'^" t. +. C2y'^+. d2X. +. C2y. ^. (*). h-. Mot so trirSng hop dac biet (doi xi'mg loai 1, loai 2, dS,ng cap...) da difdc xet (1 cac phan tntdc. K h i cac t m l i chat dac biet khong con t h i he (*) d\trtc giai theo mot so do chnng se dudc t r i n h bai trong cac bai toan 67 d trang 2GG, bai toan 68 6 trang 267. T i i y nhien phirong phap nay khong i)hai la t o i uu. N h i n chung c:ac dang thitcJng gap dcu dita t r c n nigt vai dac t h u cua dang bfic hai. Nen biet khai thac cac tfnh chat dac biet do ta se t i m ditrJc Idi giai ngiln ggn. B a i t o a n 67. Gidi he phitcing trinh. 2 + y 2 + a; - 2y = 2 • ^ + y ^ + 2(x + y) = 11.. {5. Giai.. =. 44 Vdi a = - — , t a dildc 23'. / y =. 9 153 ~. 9 44. -. 44\. 2 3 \4. 23. 17. 23 17. 17'. Vay he da cho c6 hai nghieui (x; y) = (1; 2 ) , (x; y) =. J^!. J^)". L i f u y. Bai toan 67 se dUdc giai nhanh hdn neu ta n h i n thay dUdc : L a j ' hai phitdng t r i n h cua he trir nhau, t a se t h u dUdc mot phitdng t r i n h bac nhat theo hai an x va y, tit day r i i t y theo x, giai bang phifdng phap the.. 4.3.2. S a n g tac cac he bac hai tong quat.. ''. = 0. Khi do { ""1 + y' " + ^y = - 3 [ x ' ' - x y + y ' ' + X - 2y = 1^Vay ta thu ditOc mot he bac hai tong quat, he do ch&c ch&n c6 mot nghiem "dep" la {x;y) = {3;0). Ta CO bai todn sau day. . ,.. V i d u 1. Xet hai s6x = 3vdy. K h i X = 0 t h i he viet lai |. _,_ 2y - 11. "SliiC-ni.. K h i X 7^ 0. Dat y = a x , thay vao he da cho t a ditOc 4.(i_2rk)x = 2 ( l + a ^ ) x ' ^ + 2 ( l + a ) x = 11. (l+a2):;;2. x 2 + ft'-^x^ + X - 2ax = 2 I x^ + d^x^ + 2x + 2ax = 11 2GC. B a i t o a n 68. Gidi h$phmng. trinh. { ""l^~ H'[ X — xy + y +. X —. ,o Zy — i^-.

(13) ^ / x2 + 22/2 + a;y + x - 1 0 ? / = - 1 2 ^ \2 - 2/2 - x y + 15x + 4 y = - 8 .. Giai. • K h i .T = 0 t h i he viet lai | ^2 ^ 2y = 12^. "^^. nghiem.. fa thu dUdc bdi todn. • K h i X 7^ 0. D a t y = ax, thay vao he da cho t a dilOc •+ .1. (1 + a 2 ) x 2 + 2 ( Q - 2 ) i = - 3 ( l - a + a2)x2+(l-2a)x=12. Dat. .?•. sau.. B a i t o a n 6 9 . Giai he phudng trinh ^ • ^. [ 2y2 + x y + x - lOy = - 1 2 \2 - y2 - x y + + 4y = _ 8 .. Htfdng d a n . D a t x = w, - 2 va y = ?; + 3, t a dUdc he dang cap bac hai. = z t a dvTdc he (1 + Q 2 ) 2 + 2 ( Q - 2 ) X = - 3 {l-a + a^)z+{l-2a)x=12 y = ax, x^ — z.. I 3u2 -uv-v^. 1 + ^2. 1- a +. 2a - 4 1 _ 2Q. 2(Q-2) 1 - 2a. 1 + a2 l - a + a^. = - 4 a ^ + 7a2 - 8a + 5.. + 2n.u + a2 + 2t;2 ^ 4^,; + 2^2 + uv + bu + av + ab +. = - 1 8 a + 45. -3 12. = \.. 'Tf^. >'. LuM y- Phep d f t t x = u - 2 v a y = i; + 3 dUdc t i m r a n h u sau : Ta dat j ; = u + a va y = t; + 6, vdi a, 6 t i m s a u . K h i do, t h a y vao phitdng t r i n h thit nhat c i i a ho, t a d u d c. T a t i n h cac d i n h t h i i c. -3 12. , ^. M. + a - 10?; - 106 =. -12.. De t h u dutdc phUdng t r i n h dang cap bac hai t h i dieu k i e n la. = ISa^ - 3a + 15.. ^1. / 2a + 6 + 1 = 0 \6 + a - 1 0 = 0. K h i D = 0, tufc la a = 1 t h i he v6 nghiem. T i e p theo chi xet a ^ 1. D i o u kicu x^ = z cho t a p h i M n g t r i n h de xac d i n h a. r. ^ a = 6 = 3.. -2. Vay ta dat X = u - 2 va y = ?; + 3. Ngoai ra t a c6 the lam nhanh hdn nhiT sau: L a n lUdt dao h a m hai v c phUdiig t r i n h t h i i n h a t thco b i c n x ( x e n i y la. X. z ^. =. hkng so), theo bien y (xem x l a h a n g so) c i i a he t a dUdc. t. D. (I5a2 - 3a + 15)^ = ( - 4 a ^ + 7a - 8a + 5) (45 -. 18a). <^153a'* + 216a^ + 360a = 0 <^a ( I 5 3 a ^ + 216a2 + 360) = 0 <^. a = 0 a = -2.. K h i a = 0 t h i D = 5, Z)^ = 15, s u y r a x = ^. = 3 ^ y. K h i a = - 2 t h i D = 8 1 , D^, = 81, suy T& x = 1 He d a cho c6 h a i n g h i e m | ^ = Q. va | ^ ^. y =. •. Bai t o a n 70. Giai he phUdng. ,, (1) (2). Hwdng d i n . Lan htdt dao ham hai vc phitdng t r i n h thi't n h a t theo bicn x. I2. + ^"o ~ f. ta se thu. ^dt u = x + 2,. 3. Khi do x^ + 4 x + 4 + x y - 3 x + 22/ - 6 + 2y2 - 12?/ + 18 = 4 3x2 + 12x + 12 - x y + 3 x - 22/ + 6 - y2 + 62/ - 9 = 1 268. 1 i. -2.. V i d u 2. Tic mot he. dang cap bac hai, hang each tinh tien nghiem,. V = y -. trinh. 1 x 2 + 3y2 + 4xy - 18x - 22y + 31 = 0 \ 2 x 2 + 4 y 2 + 2 x y + 6 x - 4 6 y + 1 7 5 = 0.. (xeni y l a hang so), theo bien y (xem x la hang so) ciia h e t a dUdc. phudng p h a p d a t r i n h b a y 6 t r e n t a luon giai dUdc c a c he b a c h a i tong quat.. ". -2. Tit do C O phep dat x = u - 2 va y = ?; + 3.. 0.. Lufu y . V i c a c p h u d n g t r i n h d a thi'rc b a c khong q u a 4 luon giai diWc n c n vdi. dUdc mdt he bac hai tSng qudt. Xet he I. r2x + y + l = 0 ^ r x = \y + X - 10 = 0 ^ \ = 3.. „. r. 2x + 4y - 18 = 0 ^ / x = \x + 6 y - 2 2 = 0 ^ \ = 7.. -5. -3. • v a y , thitc hien phep doi bien x = - 5 + iz, y = 7 + t;, t a ditdc. Jl. ju"^+ 3v^+ 4uv = 1 2u2 + 4?;2 + 2?iu = 1.. (3) (4). '. .f. '. MAO. '' ' (11).

(14) He nay la he dang cap, c6 the giai theo each thong thucJng, nhung I m i y la tr(t hai phUdng t r i n h (3) va (4) v^ theo ve t a c6 ngay u'^+ v'^-2uv = 0 ^ u = y \=. V =. 2\/2 -1. Tacohe ( 7 / ) o { 8 w ! = l. p a p s 6 . He CO nghiem l ^ - 2^) =. + 1- ^. ^ , , ^ ^ , , f x2 - 2xy + 2y + 15 = 0 B a i t o a n 74. Giai he phuang tnnh | _ ^^^^ + ^2 ^ 5 ^ 0. Vay ( / ) CO nghiem D a p so. He CO n g h i § m (x; y) la {2^2 + 1; 3 ^ 2 + l ) ; ( - 2 \ / 2 + i ; - 3 ^ 2 + 1).. 2v/2' 5;. —i + 7. 4,4. B a i t o a n 71 ( D l n g h i O l y m p i c 3 0 / 0 4 / 2 0 1 1 ) . Giai he phuang trinh. Phifdng phap dung t i n h ddn dieu c u a h a m so. T i n h ddn dion c.iia ham so la mot cong cu hfru hic\ dfi sang tac va giai phUdng t r i n h , van de nay da diTdc t r i n h bay ci bai 1.3 : PhUdng phap dua phUdng t r i n h ve phirdng t r i n h ham (5 trang 15). Trong bai nay t a se khai thac t i n h (Idu dicu ham so dc giai ho phUdng t r i n h . M o t so van dc ve pluidng phap giai da CO a t r a n g 15 nen khong neu r a d day m a t a se di vao nhiing vi du, bai toan cv the.. a.2 + 4 y 2 _ 4 ^ + 1 2 y + l l = 0 x2 + 4y2 - 2a;y - X + 4y - 12 = 0. H i f d n g d a n . N c u tinh tao nhhi nhan thi thay ugay rang day la bai toan dS ' 3.7; — 23 ' lay hai phitdng trinh trijf nliau t a ditdc y = —, the vac phitdng trinh. t o a n 75 Gidi he / W I T ^ + ^1+7) = 1 (1) B a i t o a n 7 5 . Giai h^ ^ ^^^^ - 2xy + 1 = 4xy + 6x + 1. (2). thii nhat ciia he dUdc phUdng trinh bac 4 va may man la phiTdng trinh nay CO t 6 i hai nghiem "dep" x = 1 va x = 4. B a i t o a n 7 2 . Giai he phudng trinh ( H\+ f 2^," ' ^ \x^ + y^ + x + y-4. G i a i . Dieu kien 6x - 2xy + 1 > 0. T a c6 + 2/+ 2 = 0 (1) = 0. (2). phudng trinh b^c h a i vdi an y, con x la thara so, c6 5 = x + 1 va. -1. P = - 2 x 2 + 5a; _ 2 ^ - (x - 2) (2x - 1) = {-x + 2) (2x - 1 ) .. ( ^ 2 x 2 + 6x + 1. 13^. > 0, do do fit) dong bien tren R. nen. - I) = x\. x/2x2 + 6 x 4 - 1 = 3x. 25x2. (3). \/2x2 + 6x 4-1 = - 2 x . (4). - 7 ; - "T •. (x + y - 2) (2x - y - 0) = 0. [ ^ c 6 ( 3 ) ^ { 2 ^ i 6 x + l = 9x2. =. [Tac6(4)..{2<t6x.l=4x2. ^ ^=. thu dUdc y = 2 - X , y = 2x - 1 bang each tach, them bdt,... ro rang kho khS" hdn nhieu so vdi phan tich - 2 x 2 ^ 53. _ 2 thanh (2 - x ) (2x - 1).. | | y he ( / ) CO hai nghiem (x; y) - (1; - 1 ) ; (x; y) = (. B a i t o a n 7 3 . Giai h$ phuang trinh. ^ • x ' -re n - ' i ' B am toan 76. Giai he. -. - 2 ^ + 2y = - 3. U r - 2 x y + 2x = - 4 . 270. ^/W^.. < ^ x \ / 2 x 2 + 6x + 1 = 2x2 + 6x 4-1 - g^2. -i=-0.. _ 5^ + y + 2 = 0 ^\^^^. /. vdi f{t) = t +. x V 6 x + 2x2 + 1 = - 4 x 2 + 6x + 1. ( y = 2-x { y= 2x-l l x 2 + 2 / 2 + x + y - 4 = 0 ' \ + i/ +x + y. L i f u y. De bien doi 2x2 + x y -. - 2 / + \/y2 + 1 .. y + \/y2 + 1. (1) ^ X = - y . T h e vao (2), t a dudc. CO hai nghiem yi = 2 - x, y2 — 2x - 1. D a n den. He CO hai nghiem (x; ?/) = ( 1 ; 1 ) , (x; ?/) =. /n. Ma (1) Ition CO nghia vdi m o i x G R, y G R va (1) <^ fix) = fi-y). Ta chon P = {-x + 2) (2x - 1) de ( - x + 2) + (2x - 1) - x + 1 - S. Vay (1). 4. 1. (1) ^ X + v/x2 + l = - y + V'y2 + 1 ^ fix) = fi-y),. H i f d n g d a n . T a c6 (1) o y'^ - ( x + l)y - 2x2 + 5^,- - 2 = 0. T a coi day la. (. 1' ^1 + l ) •. + l)^(. I. = zl^^. ^. ;. 1 7 3 7 ^ 4 - 4 ( 2 x 4 - 1 ) = y y ^ + 3y + y) (2x - y) + 4 = - 6 x - 3y. 271. .. )• (1) (2). (i).

(15) G i a i . D i c u kieii .x >. o. > 1. T a c6:. ^ x ( x ^ + x^ - 3x + 3) = 0 ^ [ /. (2) <^ (x + y){2x - ?;) + 4 + 4(x + jj) + (2x - ?y) = 0 ^{x + y + l){2x-y + 4) = 0. x'* + x2 - 3x + 3 = .T^ + (^x - - J l i e m d u y nhat {x;y) = (0,1).. 7 . ^\/ = 2x + 4 (do t i r dieu kien suy r a x + y + l > - > 0 ) . Thay vao (1), t a dUdc :. '. B a i t o a n 78. G.dz/.e. V. \ / 3 ^ ^ + 2x - 8 = v / 2 ^ + 3. 3\. ^. <:»2 (3x - 1) + v / 3 x - 1 = 2 (2x + 3) + V2x + 3.. (3). _ 3^ ^ 3. 3 + ^ > 0 nen (4) vo nghiem. Vay ( / ) c6. f (41/2 + 1) + 2 (x2 + 1) =6 | ,2^ (2 + 2 y v T l ) = x +. 22/(1. (3) 4^ / ( V s T ^ ) - / (y2irr3).. / (v^3x^) = /. ^ VSx^. = V2F+3. ^ X = 4 =^ y = 12.. (2). (^). G i a i . Dieu kien : x > 0. Neu x = 0 t h i t i t (1), t a c6 0 = 6 (sai'). Vay gia sii x> 0, chia ca hai ve ciia (2) cho x^, t a dUdc. Xet ham so / (i) = 2f + t vdi ( > 0. K h i do. M a /'(O = 4t + 1 > 0 Vf > 0 nen ham so / dong bicn t r c n [0; +oo), do do. :. + ^/VTT) = i. ('i +. yjTi^. (3). Xet ham so f (t) = t (l + V T + F ) , vdi t G M, phitdng t r i n h (3) viet lai thanh / ( 2 y ) = / ( i ) . T a c6 f'{t) = 1 + ^ T T ? + ^ i = = > 0, V i G 1 , do do t i l. Vay he phirong t r i n h ( / ) c6 nghiem d u y nhat (x, y) = (4,12).. / ( 2 y ) = / ( - ) t a CO 2y = - . Thay vao (1) t a dUdc B a i t o a n 7 7 . Gidi he phMng. trinh. j^s + ^s^^.^'l^o ^. ^. X. (2). .x^ + Y tifdng. R a t ti.r nhien t a n h i n vao tiifng phiWng t r i n h dc danh gia vdi muc dich t i m m o i quan he gifra hai bien. Txi (1) t a thay rang 2 ve l a 2 da thiic doc lap ciia 2 bien x, y va ciing bac. N h u vay viec ap dung phitdng phap sut dung t i n h ddn dicu c6 cd h o i thanh cong rat cao. V a day cQng l a liic chung t a dung t d i k y t h u a t he so bat d i n h . D a u t i c n , t a chon m o t d a thi'tc bat k i lam chuan d (1). De thay nen chon da thiic ben ve t r a i v i n h i n no ddn gian hdn. Vdi y tulcing do t a dUdc ham so dac trUng f{t) = t^ + t-2, nhuT vay viec ciia chiing t a can lam do l a phan tich : y^ + 3y^+Ay. = g\y) +. (3). Xet ham so f{t) = + i - 2, i G R. T a c6 f'{t) = 3^^ + 1 > 0, V( G R. Suy ra / dong bien t r e n E . Vay (3) / ( x ) = f{y + 1) <^ x = y + 1. T h e vao (2) : x^ + x^ - 3x2 + 3x = 0 272. + 2 (x2 + 1). - 6 = 0 4=> x^ +. X. - 6 = - 2 (x^ + l ) V ^ .. (4). cd g'{x) = 3x2 + 1 > 0 va /^/(^•^ = _ 2 f 2 x v ^ + < 0, Vx > 0. V 2v/x J /ay g{x),h{x) ddn dicu ngUdc chieu t r c n (0;+00) va g{l) - h{\) nen (4) cd nghiem d u y nhat x = 1, suy r a y = ^. D o do ( / ) cd nghiem (x, y) = ^ 1 , ^ B a i t o a n 7 9 . Giai he phudng. r (\/^2^-3x2y. R6 rang g{y) c6 dang g{y) = y + h t i t day t a khai trien va dong nhat he so dUdc 6 = 1 . N h u vay t a c6 phitdng t r i n h + x - 2 = (y + 1)^ + (y + 1) - 2T6i day t h i y titcfng giai b a i toan da ditdc hoan thien. G i a i . T i t (1) t a c6. x^ + (x - 1)^ + 1 = 0. X. p t cac ham so 5 (x) = x^ + x - 6, / i (x) = - 2 (x^ + l ) ^ x , vdi x G (0; +00).. g{y)-2.. x3 + x - 2 = ( y + l ) ' + (y + l ) - 2 .. X. . x2y. - X +. trinh + 2) ( v / V T T + l ) = 8 x 2 , /. (1). (2). 2 = 0.. G i a i . Vdi x = 0 hoSc y = 0 t h i thay vao he (I) dan t d i v6 l i . G i a sii x 2/ 7^ 0. Phitdng t r i n h (1) tUdng dildng vdi. ^^^^tizl^'^ + ^4y2 = 8x2y3 ^ ^\/x2. ^^^-^=^'y. + l - 4x2y + X = 2x2y v/4y2 + 1 - 2x2y. <!=>\/x2 + 1 + X = 2x2y (V4y2 + 1 + 1) 273. + ^ ^ 2x2y. (I). 0 va.

(16) 1 X. -o + 1 + 1. + 1+ 1. J{2yf. = 2y. ). Xet ham so f{t) = t (^fi?T\ l ) c6 /'(O = 1 +. rng nhat he so t a dUdc (3). >0. neri. / dong bien tren R. T i r (3) t a c6 / ( - I = / (2y) <^ - = 2y <^ 2x1^ = 1. Thf> X. vao (2), t a c6 : 2x'^y - 2 x + 4 = 0 < ^ x - 2 x + 4 = 0 < ^ x = 4=>?/ = ^. 8 Ket luan : He c6 nghiem d u y nhat (x; y) = 4; - . V 8/ X^ -. B a i t o a n 80. Giai he. 3X2. + 2=. Ta (/). Y t i f d n g . Chung t a lai bat dau t i m t o i t i i cai ddn gian t d i phiic tap. Tvi (]) de y rang t a da c6 dang g{x) - h{y) nhit mong muon, n h u vay y tu6ng dting t i n h ddn dieu de xet ham dac trUng da xuat hien. Se t o t hdn neu g{x), h{y) la ham da thiic. Vay t a thijf b i n h phudng de loai bo can thirc :. ^. - 3x2 + 2 =. + 3. N h u vay t a se dat a = \/y~+3 => y = - 3, y-^y + 3 = {a^ - 3)o = Ham dac t r u n g se la f{t) = - "it. Do do can phan tich x^-3x2 + 2 = 5^(x)-3g(x). De thay r^ng g{x). = x + 6. T i r do. ^x^. _ 3(^ ^ ^). - 3x2 + 2 = x^ + 36x2 ^ ^3^2 _ 3^^. ^ ^3 _ 3^. - 3 i la ham dong bien tren [1; + 0 0 ) . N h u vay y. x>2. ( x>2. y2 + 8 y > 0. [ yG(-oo;-8]U[0;+cx)). f ^ o. (1) ^ x^ - 3x2 + 2 =. ^. (3. _ 1)3 _ 3(^ _ 1). {^/yT^f. -. 3\fyT^-. Ta CO s/y + ^ > ^3 > 1, x - 1 > 1. Xet ham dac t n m g f{t) = - 3t, Vi > 1 CO / ' ( < ) = 3*2 - 3 > 0, V < > 1. Suy ra ham so / dong bien tren [1; + 0 0 ) , do do tit /(x - 1) = /(VyT^) t a CO X - 1 = sfyT^ ^y = x^ -2x-2. T h e vao (2) ta dudc. - 3«. x = 3. <^(x - 3)(x3 - x2 + 5x - 2) = 0 <^. x^ - x2 + 5x - 2 = 0.. Xet Q(x) = x'^ - x2 + 5x - 2 c6 Q'(x) = 3x2 - 2x + 5 > 0, Vx € g^y jay la mot ham dong bicn tren R. Lai c6 x > 2 Q(x) > Q(2) = 13 > 0, suy ra phUdng t r i n h Q{x) = 0 vo nghiem. Vay he phildng t r i n h (/) c6 nghiem duy nhat ( x ; y ) = ( 3 ; l ) . N h a n x e t 1. NhUng bai toan tren da cho thay c6 nhieu each de dUa hai vi cua mot phuang trinh ve ham dac triing. Tuy nhien mot so bai loan kho hdn sc. dai hoi phai bicn doi cdc phuang trinh cua he de' tim ra ham dac trUng. B a i t o a n 81. Giai he phudng. trinh. |. + 3y) - 1. I x(y* - 2) = 3. j G i a i . Vdi x = 0, the vao (I) thay v6 H. G i a sii x 7^ 0, tit (I) ta c6. 2 =. —. , vdi fit). -. X. 275 274. ^ ^. CO. 2 + 3y =. x^ - 3x2 + 2 = (x +. - 3\/^T3. ^x"^ - 4x3 + 8x2 - 17x + 6 = 0. Cong viec tiep theo la t i m ham dac tritng. De thay h{y) = + 3y2 la lira chon t o t v i day la ham so ddn gian va dong bicn tren [0; + 0 0 ) . Ta se c6 g d n g phan tich {x^ - 3x2 ^ 2f = q^{x) + 372(x). Dong nhat he so se t i m diTcic q{x) = x2 - 2x - 2. Suy ra x2 - 2x - 2 = y (chu y dieu kien c6 nghiem l a x^ - 3x2 + 2 = (x - i ) ( ^ 2 _ 2x - 2) > 0 x2 - 2x - 2 > 0, do X > 2). Nhmig cau hoi dat ra la, vice k h a i tridn va dong nhat he so v6i (.r^ - 3x2 _|_ 2)2 j^jj/^ phiic tap. Lai chii y rang ham so dac trUng khong phai la d u y nhat. Lieu co mot ham so nao ddn gian hdn ? Vay dieu tU nhien la t a se d i t i m each d a t an p h u : mot ham chiia can nao do dg khong phai luy thita. De y rang ^. Vt.f. 9(x - 2) - y2 ^. 8y <^ 9(x - 2) = (x2 - 2x - 2f + 8(x2 - 2x - 2). (x^ - 3 x 2 + 2 f = y3 +32/2.. (1) <^ x^ - 3x2 + 2 =. ham dac trUng /(() = tiffing da ro rang. G i a i . Dieu kien. I. ^ (2). (1) ^. 6 = - 1 . Do do. (x - 1)^ - 3(x - 1) = i^/y + zf. + 1+ ,. \ /. - 3 = 36 0 = 36^ - 3 2 = 63 - 36. = t^ + St..

(17) Co fit). f 2x2_^x'^ + 4x - 1 = 2x^(2 - y)^^ \VFT2= -^14 - X y / S ^ ^ + 1 ( ^x + y + l + ^/¥Ty = 5 \ + xy + 4 + >/?y2 + xy + 4 = 1 2. = 3*2 + 3 > 0,Vt e K, suy ra / la ham dong bien tren M, do (j^. y = —. T h a y vao phUdng trinh thu: nhat cua he, t a dUdc X. 3 x ^ ( 2 + - ) = 1 « - 2 x ^ + 3x2 - 1 = 0 < ^ x X. TM. 1 ' -, x = ^. -l.. a). -y*. -2. =. - 2y. Zx-Zy. + yr=^ - Z^2y. - y2 + 2 = 0.. lai ta thay he ( I ) c6 hai nghiem (x; y) = ( - 1 ; - 1 ) , (x; y) = ( ^ ; 2).. L i f u y. Bang each dat t = -,t?i. 4.5. dua vc ho doi xilfng loai hai theo t va y.. 4.5.1. B a i t o a n 82. Gidi he phUcJng trinh x2. +l. (2). G i a i . D i l u kien ( | \ " * o l ^ n ° ^ ^ t ham s6 : f{t) = e\t +l),te \^x + y + Z > yj.. [0, + 00). '0. V i /'(/,) = e^{t + 1) + e* > 0, V^, > 0 nen / la ham dong bien tren [0; +oo). do (1) <^ ế(x2 + 1) = ey\y^ + \ ) ^ / ( x ^ ) = f{y^) ^x^ = y^^x=±y. log2 [ ( x + 2y + 6)^] = logs [2(x + y + 2f. (2) ^. (x + 2y + 6)^ = 2(x + y +. (3). 2f.. • Neu x — y t h i thay vao (3), t a ditdc. (4). (3x + 6 f = 2(2x + 2)2. Theo dieu kien t a c6 x > - 1 . L a i c6 (3x + 6)^ - 2(2x + 4)^ = (x + 2)^(27x + 46) > 0. (3x + 6)^ > 2(2x + 4 f .. D o do (3x + 6)^ > 2(2x + 4)^ > 2(2x + 2 ) ^ suy ra (4) v6 nghiem. • Neu X = - y , thay vao (3), t a dUdc : ( - X + 6 f = 2(2)2 < - > ( 6 - x ) ^ = 8 ^ 6 - x = 2 < ^ x = 4=J>j/ = - 4 . Vay he (*) da cho c6 nghiem duy nhat la (x, y) = (4, - 4 ) . B a i t o a n 83. Gidi cdc h$ phUdng trinh sau : ?(8x-3)\/2^^-y-4y3 = o. I 4x2 _ 8x + 2y^ + y2 _ 2y + 3 = 0 ..3/ rx-*(3-; + 55:) = 64 I xy{y'^ + 3y + 3) = 12 + 51X 276. N h a n d a n g v a phu'dng p h a p g i a i .. 1. N h a n d a n g .. (1). 31og2(x + 22/ + 6) = 21og2(x + y + 2) + l .. H e lap ba an (hoan vi vong quanh).. He lap ba an la he phudng t r i n h c6 dang. I. = /(y) y — f(z) X. z =. (trong do / la ham so).. f(x). {too:;. (*). 2. Phu'dng p h a p giai. Xet he lap ba an (*), vdi / la ham so CO tap xac dinh la D, tap gia trj la T va T C D , ham so / dong bien tron T. C a c h 1: Doan nghiem roi chiing minh he c6 nghiem d u y nhat. T h u d n g de cluing minh he c6 nghiem duy nhat ta cong ba phudng trinh ciia he ve theo vc, sau do suy ra x = y = z. C a c h 2: Tir T C D suy ra / ( x ) , / ( / ( x ) ) va / ( / ( / ( x ) ) ) thuoc D. D i (x; y; z) la nghiem ciia he thi x e T. Neu x > / ( x ) thi do / tang tren T nen ta c6 / ( x ) > / ( / ( x ) ) . Vay / ( / ( x ) ) > / ( / ( / ( x ) ) ) . Do do .. ^ > / ( x ) > / ( / ( x ) ) > / ( / ( / ( x ) ) ) = x.. Dieu mau thuan nay chiing to khong the co x > / ( x ) . T u d n g tir cung khong thg CO X < / ( x ) . Do do / ( x ) = x . Viec giai he (*) dUdc quy ve giai phUdng trinh / ( x ) = x . H d n nfra ta c6 : X =. fiy). y = z=. Hz. X =. fiy M. y =. (. fix. z =. X =. f(y). I z = /(/(/(z))). fifiy)). 'J. I. z = I{z.. z = f{z). I. I Jhii y 3. He lap ba an con ducic gidi hhng each dm he vi dang cd ban. {flz).C. = f(x). if. f 277.

(18) vdi A, B,C >\ f{y), f{z) > 0 {xem bdi todn 101 d trang 286) hoQ.c diia ve he ( fHx) = {y-x)>'.A f\y) = {z-yr.B. I. ,. \=. tr y > 2 va tir (2) ta c6 ^3. ; ^;. 2>. _ 8 = %y{y - 2) > 0. i :. 2.. *. Vay 0 = (x - 2f + (y - 2)^ + (2 - 2)^ > 0. Day la dieu vo Ii. • Neu 0 < X < 2 (ta CO ngay x > 0 vi theo (3) thi x^ = 6(2 - 1)^ + 2 > 0) thi tif (3) suy ra 62(2 - 2 ) = X ^ - 8 < 0 = ^ 0 < 2 < 2 .. {x-zr.C,. trong do A > 0, B > 0, C > 0 vd k, m, n la cdc so nguyen duang li {xem bdi todn 101 d trang 286). C h u y 4. Khi ham f khong thod cdc dieu kien da noi 3 phan phuang phdp gidi thi ta phdi c6 nhUng each xii li khdc, chang han xem bdi todn 90 d trang 280, bdi todn 109 d trang 292, bdi todn 99 d trang 284, bdi todn 107 d trang 289.. Ket hop vdi (2) suy ra 0 < y < 2. Vay 0 = (x - 2)^ + (y - 2)^ + ( 2 - 2)^ < 0. Day la dieu vo li. Vay x = 2, tir (1) ta c6 y = 2, thay y = 2 vao (2) ta c6 ^ = 2. Vay (2; 2; 2) la nghiem duy nhat ciia he.. 4.5.2. C h u y 5. Doi vdi he lap ba an thi c6 mot sai lam rat tinh vi, kho phdt hien do la sai lam: Do x, y, 2 cd vai trb nhu nhau nen khong mat tinh tSng qudt gid sti X > y > z ". Thuc ra x, y, 2 hodn vi vbng quanh nen phdi xet hai thii tU khdc nhau x > y > z vd y > x > z.. C a c bai toan.. C Bai toan 84. Gidi he phMng trinh I. I. _ + 12x - 8 = 0 (1) - 6y^ + I2y - 8 = 0 (2) - 62^ + 12z - 8 = 0 (3). y3. Bai toan 85. Gidi cdc he phudng trinh sau. Giai.. X = f{z] C a c h 1. He viet lai ^ j / = / ( x ) vdi / ( x ) = v^6x2 _. a) + 8. Khi do ham so. • U = /(?/), / xac dinh va hen tuc tren R. Tiep theo ta t i m tap gia t r i T cua / . Ta c6 ^. fix). / b). d). x^y^+yl+y - 2. \ = zl + zl+z - 2 I z = x-^ +x^ + x- 2 { 12.T2-48X + 64 = ,/. { 12y2-48y + 64 = 2^ I 1 2 2 2 - 4 8 2 + 64 = x3. ( x - sin y = 0. _ n ^ ^ - i. ,. ^ ^. •. 2x+l = y^ + y^+y 2y + l = z^ + z'^ + z 22 + 1 = x^ + x^ + X x^ - 9y2 + 27y - 27 = 0 y3 - 92^ + 272 - 27 = 0 2^ - 9 x 2 + 27x - 27 = 0. ^ ( 6 x 2 - i 2 x + 8)2. y - sin 2 = 0 1 2 - sin X = 0. 1. Vay tap gia t r i cua ham / ( x ) la T -. Hufdng dan. Xet ham so / ( x ) = sinx c6 tap xac dinh la R va tap gia t r i la [ - 1 ; 1], / dong bien tren [ - 1 ; 1]. He da cho viet lai. Ta CO (4) ^ x^ - 6x2 + 12x - 8 = 0 <^ (x - 2)^ = 0 ^ X = 2. Vay he da clio. Ta chiing minh ditdc x = / ( x ) va | y =. [ v ^ ; +oo . Ta CO /• dong bien tren -oc 1 [ l ; + o c . nen / dong bien tren [ ^ ; + o o ) . 0 f(x) Theo phan phitdng phap giai, he da cho f(x) dUdc Viet lai { ^. _ 1^6x2 - 12x + 8 (4).. viet hii I ^ f ^. ^. I y= 2. g'{x) = 1 - cos X. C a c h 2: Cong ba phiTdng trinh ciia he ve theo ve ta dUtJc (4). Ta CO (2; 2; 2) la mpt nghiem ciia he. Ta se chiing minh (2; 2; 2) la nghiem duy nhat ciia he. • Neu X > 2 thi tir (1) ta c6 2/^ - 8 = 6x(x - 2) > 0 278. y > 2.. | ^ Z ^f(^y. Xet phvtdng. trinh x = sinx tren [ - 1 ; 1]. Xet ham so ^(x) = x - sinx tren [ - 1 ; 1]. Ta c6:. He c6 nghiem duy nhat la (2; 2; 2).. (x - 2)^ + (y - if + (z - 2)^ = 0. ^fi^. >. 0, Vx e [ - 1 ; 1].. Ham g dong bien tren [ - 1 ; 1], 5(0) = 0. Do do x = 0 la nghiem duy nhat ciia phudng trinh x = sinx tren [ - 1 ; 1]. Vay (0;0; 0) la nghiem duy nhat ciia he. nrf. Bai toan 87. Giai he phUdng trinh. x^ - 3.T2 + 6 x - 6 + ln(x2 - 3 x + 3) = y. y3 - 3y2 + 6y - 6 + \n{y'^ -2,y + i) = z 2^ - 32^ + 62 - 6 + ln(22 - 32 + 3) = X. 279.

(19) G i a i . X e t ham so /(.x) =. - Sx^ + 6x - 6 + hi(x2 - 3x + 3). H a m so nay c6. tap xac dinh la M va. Vay ham s6 / ( x ) dong bien tren cac khoang. ,. 1 \I 1 \ 1 -cx); - v/3;'VV3'y3;'Vv/3 Hdn niJa t a c6 l i m fix) = ± o o va va lim. Vay f{x). dong bien t r e n E . He da cho viet lai: < y = / ( x ). x = / W. = / ( x ) - X =^ li{x) = / ' ( x ) - 1 > 0,Vx G R.. Vay li{x) dong bion tren R. Hdn nifa /i(2) = 0. Do do /i(x) = 0 do (*). x = 2. Do. I ^ ^ ^ ^ ^ He da cho c6 mot nghiem d u y n h i t l a (2; 2; 2).. B a i t o a n 88. Gidi he phiiOng. + 32 - 3 + ln(z2 - z + 1) = X .. 2004 =. trmh. 30y. 7 r \. 2 '2y. 2. 30x. 1002 ± 2v/550971. r2. +00,. hm. =. -oo.. va a 7^ ± - . K h i d o ! 6 - 3x _ t a n ^ g - 3 tan a 3x2-1. = tan3Q.. X = t a n a = f{z) = /(tanOa) = tan 27a.. sau. + Az. a = ^^^^^. vk cac hoan v i ciia no.. + 4x. 3tan2a-l. ~. Vay t a c6. 'lie la a ^ 27a - kn. 9 + 4y. 2004 =. 26. (A; e Z ) . Vay nghiem ciia h$ la (0;0;0),. hoan v i , ( t a n ( ^ ) ; t a n ( ^ ) ; t a n ( ^ ) ' 26 26 /. B a i t o a n 9 1 ( H S G q u o c g i a n a m h o c 2 0 0 5 - 2 0 0 6 , b a n g A ) . Gidi h$ .. .wP,v:. B a i t o a n 90 ( D e n g h i t h i O l y m p i c 3 0 - 0 4 ) . Gidi he phuang ^. — 71. _ I y = tan3a lo do <^ z = t a n 9a ' x = : t a n 27a.. B a i t o a n 8 9 ( d e nghj t h i O l y m p i c 3 0 - 0 4 ) . Gidi he phudng. D a p so: x = y = z =. a e. y. 2004 =. lim. X = ±—r^ khong thoa m a n phudng t r i n h nay nen de x l a nghiem ciia phUdng v3 1 x^ - 3x trinh nay t h i x khac ±—7=, k h i do y = - — 5 — - . Do do t a dat x = t a n a , vdi v3 ,„ . 3x^-1. trinh. ( x^ + 3x - 3 + ln(x^ - X + 1) = y <^ y 3 + 3 y - 3 + l n ( y 2 - y + l ) = 2 I. = -oo,. Vay tap gia t r i ciia / ( x ) l a R. Tkp xac d i n h ciia h a m so / l a con, thitc su cua tap gia t r i ciia ham so / nen t a khong the ap d u n g each giai n h u d a t r i n h bay trong phan phiTdng phap giai. X e t phudng t r i n h x^^ - 3x = y(3x^ - 1). V i. T i e p theo t a giai phifcing t r i n h f{x) = x <^ / ( x ) - x = 0. D a t h{x). lim. T u o n g t i t nhit. cac v i du trutdc t a dUdc:. 1. = +0O,. r x^ - 3x = y(3x2 - 1) <^ y ^ - 3y = z(3y-^ - 1). G i a i . De thay he da cho tiTdng dudng vdi. r. y -. trinh. f. \/x2-2x + 61og,(6-y)^x. - 2 y + 6Iog.,(6-z) = y. 1. v / z 2 - 2 2 + 6 1 o g 3 ( 6 - x ) = 2.. /. »iai. Dg (x; y; z) l a nghiem ciia he da cho t h i dieu kien l a x, y, z nho hdn 6. p da cho tUdng dirdng vcli •" • /(^O fiz). t r o n g do. logs (6 -- y ) log3(6 l o g 3 ( 6 --x). .. Vx^ - 2x + 6 y. \/y'. - 2y + 6. ' Vz^ - 2 2 + 6 280. 281. (1) (2). (3)..

(20) hay. Ta. . ,, , m (2) vdi fix) (3). ( \ogM -y) = fix) log3(6 ^ z] = f{y} \olti6 - x) = f(z) CO. fix). =. ^ Vx^ -2x. B a i t o a n 95.. la han,. B a i t o a n 96.. quat gia siif x = max(.r, y, z) t h i c6 2 trirdng hop:. B a i t o a n 97.. N' M Do. z. Do fix). x>y>. tang nen fix). > f{y). > f{z),. log3(6 - z) > log3(6 - x).. log3(6 -y)>. suy ra .^ij,. giani nen suy ra. r x2(x+l) = 2(y3-x) + l trinh I y'^{y + 1) = 2 ( 2 ^ - y) + 1. Giai he phuang. I z'^iz + l) = 2{x^-z). /(^). f fix) = gi^y) (1) y3 + y2 ^ 2y = 2z^ + 1 hay <^ / y) - g z) (2 vdi ? + 22 + 2 z - 2 x 3 + 1 I f\z)=9{x) (3) + ^2 + 2(. Do dang cln'mg m i n h d i w c / va g la nhOng G i a sii rang (x; y; 2) la nghiem ciia h$ va khong giam > y. K l i i do t i t (1) va ( 2 ) suy r a. ^ 9{y) > 9{z). > f{y). V$.y X > y > z > X. y >. >. 2''°. =^ z > x.. (. x = y = 2 = 2 cos u (n 377 57r1 UG | y , y , y | .. B a i t o a n 93 ( H S G q u o c g i a n a m h o c 2005-2006, b a n g B ) . Gidi he r x3 + 3x2 + 2x - 5 -. y < yJ + 3y2 + 2 y - 5 = 2 [ 2^ + 32^ + 22 - 5 = x. B a i t o a n 94.. Gidi he phuang. Hu'dng d a n . D i o i i kion. (4x2 ^. Gidi he phuang. trinh. r log5X=:log3(4+y^) < logg y = log3(4 + y/z) I log5^ = log3(4 + \ / i ) .. .. ;. 282. _. ^. COS x = log2 (8 COS 2 - cos 2 x - 5 ) cosy= log2f8cosx-cos2y-5") cos2= l o g 2 ( 8 c o s y - c o s 2 2 - 5 ) .. trinh. C h u n g ) . Gidi he phuang. i (4x2 + l ) x + ( y - 3 ) ^ / 5 ^ \2 + y2 + 2 v t ^ = 7. trinh. = 0 (1) ^ ^ j.^ (2) ^^-y^'^^. Phudng t r i n h (1) viet lai. ^ (3 ^ y ) ^ 5 - 2y. Xet h a m so dac t r i m g : f{t). ^. 2x _. (2\xf_. v/(5-2y)'. ^ / 5 ^. =. 3". 3*2 2 ^° ^'^^'^ = —. + - > 0,Vt G R. Suy ra. / la h a m dong bien t r e n R, t ^ do 2x = s/b - 2y. T l i a y vao (2) t a ditOc 5 - 2y + y 2 ^. 2^/^^^. = 7 <^ (y - 1)^ + 2 v / 3 ^ ^ = 3.. Dat D - y - 1. K h i do | f + 2v^_:_4x = 3 L^^J \ =^ \J6 ~ 2v. " •. X = y = 2. Do do he da cho tuong duong v d i. '. una kob. + l.. r x3 + x2 + 2x = 2y3 + 1. H i f d n g d a n . T a c6 I I <7(f) = 2^3 + 1 va f{t) = ham dong bien t r e n M. t6ng quat, c6 the coi x. Gidi he phuong. B a i t o a n 98 ( D H - 2 0 1 0 A - P h a n. = y = 2.. Tru'dng h d p 2: x > z > y. T i l d n g t i t nhir t r e n suy r a x = y = z. Phuong t r i n h g{x) = f{x) c6 nghiem duy nhat x = 3. Vay h f c6 nghiem duy nhat la (x;y;z)-(3:3;3) B a i t o a n 92.. trinh. e^ y — y ev - e!/-^ = z z — _ C „z—xX. <. 6-y<6-2<6-a;<s^a;<2<y=i>a;. 2cosx(cos^y + 1) = (1 + c o s y ) 2 2 c o s y ( l + cos^z) = (1 + 0 0 8 2 ) ^ 2 cos 2(1 +cos^x) = (1 + c o s x ) ^ T. - 2.r + 6) Vx2 - 2.T 4- 6 tang, con g(x) = log3(6 - x) la ham giiim vdi x < 6. Neu ix;y;z) la mot nghiem ciia he p h i M n g t r i n h t a chiing m i n h x = y = z. K h o n g mat t i n h t6ng (.T2. TrvfSng hdpl.. trinh. + 6. > 0,Vx < 6, suy ra fix). =. Gidi he phuang. u = 2x. K h i do. i>2 + 2 V 3 - 2u = 3 1/. = v/3 - 2v. , { v'^ + 2w = Z Dat w = v / 3 ^ ^ . K h i do { u;2 ^ 2u = 3 [ u2 + 2i; = 3.. Neu u < 1 thi. w'^=3-2u>l=^w>l=^v'^ =^ti <. 1. = 3 - 2u > 1. = =^ u >. 3-2w<l. 1,. ;Den day t a gap m a u t h u a n , vay khong the c6 u < 1. TitOng t U , triTdng h0p u > 1 cung khong t h i xay ra. Vay u = 1. Suy ra u = y = 2, thoa man d i e u kien. 283. = i f = 1. Do do x = 2 '.

(21) Lixtu y. Viec t i m ra 2x = \/5 — 2y bang t i n h ddn dieu cua ham so ditdc tien hanh bang phUdng phap he so bat d i n h n h u sau : Ta se b a t ' d a u phan tich t i t (1). PhUdng t r i n h nay c6 x, y tach bict ucn k h a nang d i i n g ddn dieu la cao. Vay t a bien doi phUdng t r i n h ve dang g{x). = 5 - a 2. (3-. 1+ ^. > 0. Vay ham s6 g d6ng bi^n. sin. = 0 nen he phitong t r i n h c6 nghi^m duy nhat x = y = z = n g h i cho k y t h i c h o n h o c s i n h gioi c a c. C h u y e n k h u vu-c D u y e n H a i v a D 6 n g B ^ n g B i c B o ) .. ". trinh. va. (3-y)v/5-2y =. =. B a i t o a n 100 ( D e. h{y)<=>{Ax^+l)x={3-y)./5^.. Dat s/5 - 2y = a => y =. K h i do g'{x). ax^ + 1=. 2. +. R6 rang p(x) c6. dang mx + n, d u n g he so bat d i n h t a t h u diWc m = 2; n = 0 =^ p{x). = 2x.. n g h i cho k y t h i c h o n h o c s i n h gioi c a c Gidi. x = cos. trvfdng. —y ,3^3. a?y +. -. =. 2. (2). az +. -. =. X.. (3). •. X. X >. TT. z = cos. G i a i . X e t h a m so / (x) = cos. / (x) =. .3v/3 ;= S i n. 3v/3. X. .373^ . K h i do. atif2. Ta. <1.. fix). I —:=x V3v/3. z}. x. =. y =. g (x) =. X. z.. Tit. — cos. 284. do. c6 •K. / (x). a ^ i t j - 1 > 0.. =. x.. Xet. ham. f. f{^)^9{y). I /(^. - x| < |x - y\> |x - y\ \y - z| = |z - x|.. =i-. > 1. CO. !• |2. >. t; > 2 ^ va t, ^ t^, t a c6. Vay / (t) la ham dong bien tren [2^; +oo). H a m <? ( 0 = < la ham dong bien tren [2v^; +oo). Hg da cho t r d t h a n h. \f'{0\\{y-z)\<\y-z\.. Titdng tir t a c6 - y\ \y - ^1 <. 2v/^ ; 2 > 2 v ^ .. Xet / (i) = at + j , i e [2v/a; +oo). V6i t, > 2^,. T i t do, theo d i n h l i Lagrange t a c6 \x-y\ \fiy)-f{z)\. la. 'hoo bat dfing thftc Cauchy, t a c6 ax + - > 2 v ^ =^ y > 2^/^. T i t d n g t i t. y ' cos. {x,y,. xz.. h$ phiCOng. trinh. max. C)' .10.1 n.feo,} 11:. (1). ax-\--=y. C h u y e n k h u vulc D u y e n H a i v a D o n g B a n g B a c B o ) .. Slit X =. phuang. G i a i . T i t he suy ra xyz ^ 0 va x,y,z cung dau. Ta thay ngu ix;y;z) nghi$m ciia he t h i ( - x ; -y; -z) cung la nghieni cua he. • Xet trifdng hdp x, y, z cung dudng. He da cho titdng ditdng vdi he sau. Nhir vay ham so dac t n t n g chinh la /(«) = y + 2 '. Gia. trtfdng. = x + ^ chinh la ham dac t r i t n g ma t a can t i m . Vay. can phai phan t i c h {4x^ + l ) . x = 4x^ + x =. B a i t o a n 99 ( D e. xy. + 1 = yz. ay2. az"^ + 1= Ta hy vong r^ng f{t). he. ~ .. sau vdi a e (^\\. 5 -. 2. Gidi. ma. so:. 5 ('»lK,f;|). Clia sit X = max {x; y; z}. K h i do x>2/=>/(x)>/(2/)=^g(2/)>y(.)=>y>2.. 3\/3 285. ; !. ..li.''.

(22) y > z =^ f (y) > f {z) ^ 9 (z) > 9 ix) x>y>z>x^x=^y. 'han cac ve cua b a phUdng t r i n h vdi nhau t a dudc. X.. (vi X > 0,y > 0,z>. = z =. Vav ho da cho c6 nghiom \ k x. z>. _1. = y = z=. \x, _ = y^ = z- =. Lifu. (x - 2) (y - 2) (2 - 2) [(x + 1)2 [y + 1)2 + 1 ) 2 + 1. 0).. + 2x3 - 4x - 2 = y2 y 4 + 2 y 3 - 4 y - 2 = z2 ,4 ^ _ 42 _ 2 = x2.. Htrdng dan. He d a cho viet l a i f (.r^ - 4) + 2r' - 4.T = ?/2 _ 2 { ( y 4 _ 4 ) + 2 y 3 - 4 y = z 2 _ 2 <=> [ {z^ - 4) + 2z3 - 4z = x2 - 2 f ( x 2 - 2 ) ( X + 1)2 + 1 ^ 0 / ^ - 2 ) (y+iy^ + l I {z^-2) iz + lf + 1 Do. (x + 1)2 + 1 > 1, (y + 1)2 + 1 > 1,. 1)2 + 1 > 1 nen t\l (1). suy ra d l. (x; y; z) la nghieni ciia he t h i. y'-2 22-2. (y + 1)' + 1^ ( z + l ) ' + l'. = =. ~ 2 ( x + 1)2 = 0 y ' - 2 (y+ 1)2^0 2 2 - 2 (2 + 1)2 = 0. x2. x2-2. Nhan cac ve cua ba phUdng t r i n h vdi nhau t a dUdc. Lan htdt thay x = 1, y = 1, 2 = 1 vao ho (*) t a dou t h u dudc kot qua (x; y; 2) = (1; 1; 1). Vay he c6 nghiem d u y nhat (x; y; 2) = (1; 1; 1).. (. Bai toan 102. Gidi he phuang trinh sau Giai.. Ho da cho tirdng ditdng vdi. r x 3 - 3 x - 2 =y -2 \-3y-2 = 2 - 2 ^ I 2^ - 3 2 - 2 = 2 - X. [. {. 286. 422 - 92 + 9 = 3x.. f (2x-3)2 = 2 ; - X I (2y - 3)^ = 2 (2 - y) i (22-3)2 = 3 ( x - 2 ) .. D o vg t r a i ciia. x>z>y>x=i-x. = y = z.. 4x2 _ I2x + 9 = 0 47/2 - 12y + 9 = 0 422 - 122 + 9 = 0. ^^^y^^^. Thay vao ho t a dUdc. X.. (x-2)(x+l)2 = y - 2 (y-2)(y+l); =z - 2 ( 2 - 2 ) ( 2 + l)2 = - ( x - 2 ) .. He da cho tUdng dudng vdi. 4x2-llx + 9= y 4y2 - lOy + 9 = 22. ba phudng t r i n h nay deu khong am nen. x^ -. 3x = y y j _ 3y = 2 2^ - 32 = 4 -. x= 1. ( x - l ) ( y - l ) ( 2 - l ) [ ( x 2 + l) ( y 2 + l ) (22 + l ) + l. Giai.. 22-2 x€ {-l,-v^,\/2} y e {-l,-v^,v/2} 2 G {-1,-/2,^2}.. 3 ^. /3 3 3 \ .He CO nghieni d u y nhat ( x ; y ; 2) = ( 2' 2' 2 j ". r). ^^^^.^^^. (x-1) (2/2+n=-(2-l) _ 1) ^ ^ _ _ J) (2 _ 1) (^2 ^ 1^ ^ _ _ 1). ( x722 y 2 _- ^2 y 2 + yx_- 1l^=Jl_- ^2 ^ ) 2x2 - x2 + 2 - 1 = 1 - y. Tiit day, ket hdp v d i (1), t a dvtdc. (x+1)^ + 1. 2x2 + 2 + y = x2 + 2. ,. >^.xM-i\. He da cho tUdng dudng vdi. | x 2 - 2 | > | 2 2 - 2 | > | y 2 - 2 | > | x 2 - 2 K | x 2 - 2 | = |22-2| = |y2-2|. x2-2. xy'' + x + 2 = y'' + 2 y22 + y + X = 22 + 2. !. Giai.. (.7;2-2)(.T2 + 2.T + 2) = ? / 2 - 2 ( j / 2 - 2 ) ( 2 / 2 + 22/ + 2) = z 2 _ 2 ( 2 2 _ 2 ) ( 2 2 + 22 + 2 ) = a . - 2 - 2 x2-2|[(x+l)2+l] = |y2_2| |y'2-2| (y+1)2 + 1 = | 2 ^ - 2 | | 2 2 - 2 | (2 + 1)2 + 1 = 1 x 2 - 2 !(1) (2 +. X= 2 y= 2 2 = 2.. L^n hrdt thay x = 2, y = : 2 , 2 = 2 vao he (*) t a deu t h u dUdc ket qua (x; y; 2) = (2; 2; 2). Vay he c6 nghiem d u y nhat (x; y; 2) = (2; 2; 2).. -1 r-. y . Co t h g sijf dung dao h a m de chiing m i n h h a m so / dong bien nhauh gon hdn.. Bai toan 101. Giai he phuong trinh. = 0^. |ii toan 105. Gidi he phuang trinh 287. C x2 + 2y2 = X (1) < y2 + 22x = y (2) I 22 + 2xy = 2. (3). j ,.

(23) G i a i . L a y ( 1 ) trtf ( 2 ) , t a ditdc x'^ - y'^ + 2z{y-x) Lay. Hoan v i vong quanh (thay (x; y; z) hdi (y; z; x ) ) t a dvfdc. = x-y<^{x-y){x. + y-2z-l). = 0.. (4). = y-z^{y-z){y. + z - 2x - 1) = 0.. (5). (2) t i i t (3), t a ditdc y 2 -z'i. • T r u d n g hdp 1 : x = y = z. T h a y vao he ta dildc 3x^ = x •i^ x e |o,. ( x(x + 2z-l) \ + z - 2 x - l. tir. (6). suy. ra. (0;0;0), (1;0;0), (0;1;0), (0;0;1), 1. 3'3'~3. /2 ^3'. fx. I. + y^ + z^ = 3 y + z^ + x^ = 3 z + x^ + y^ = 3.. (1),. (2),. (3). 1,. dan. din. ,s j , , , , ,. vr. . 1. ^. -. ^ ". '. i. X =. 4 y - y^ y = 42 - 2^. (1) (2). 2 = 4x-x2.. (3). »s. ,. Cpng ba p h i T d n g t r i n h. ta d U d c. x^ + y^ + 2^ = 3 (x + y + 2) <^ p2 - 2g = 3p.. f,x = 4 y - y 2 y = Az-z^ I 2 = 4 x - x2. f (y-Ufy-S) = 3-x 2 - 1 2 - 3) = 3 - y I (x - l ) ( x - 3) = 3 - 2.. I. _ i _ n 3. ' ". (5) 6. c.. (7). ,. -..i-:. ^ ^. .. (x - 3)(y - 3 ) ( 2 - 3)[(x - l ) ( y - \){z - 1) + 1] = 0.. trinh. (8). T r i t 6 n g h d p 1: x = 3. T h e vao (5), (G), (7) ditdc (x; y; z) = (3; 3; 3). Trifdng h d p 2: y = 3. The vao (5), (6), (7) d u d c (x; y; 2) = (3; 3; 3).. | T r i t d n g h d p 3: 2 = 3. The vao (5), (G), (7) d U d c {x\y\z). I Tritdng. h d p 4:. '; ". " '. = (3; 3; 3).. x = 0 (y = 0, 2 = 0 la t U d n g t u ) . The (7) d U d c 2 = 0, the. = 0 vao (G) diWc y = 0. Vay (x; y; 2) = (0; 0; 0) la nghiem ciia he da cho.. = 0. (4). Trirdng h d p 5: x ^ { 0 , 3 } va y ^ { 0 , 3 } va 2 ^ { 0 , 3 } . T i f (8) t a c6 (x - l ) ( y - 1 ) ( 2 - 1) + 1 - 0 ^ (x - 1) ( y 2 - y - 2 + 1) + 1 = 0. (5). •^xyz -xy-xz. t a dudc. X (1 - x 2 ) + 2 / ( 2 / - 1) = 2 y (1 - y2) + ( x. (4). Nhau (5), (6), (7) vdi nhau dudc. y {1 - y"") + z{z - 1) + x' {x ~ 1) = 0. Thitc hien (4) - z{5),. >. 2. Do t a n h a m d U d c x = y = 2 = 3 l a nghiem nen se bien doi he da cho t h a n h. (1) (2) (3). + z^ {z-l). c6. G i a i . D a t p = x + y + z, q = xy + yz + zx, r = xyz.. G i a i . L a n Ivrdt lay (1) t r i t (2) va lay (2) tru: (3), t a dudc x{l-x^)+y{y-l). (7). {. Tim cdc so x > 0, y > 0, z > Q thod man h$ phudng. {. txl. f'. Q;^;^) , 3'. do. 1 t i i (6) suy ra y < 1, do do tuf (7) c6 2 < 1, dan den. X <. Do he k h o n g d6i k h i thay (x; y; z) hdi mot hoan vj ciia no nen theo tren ta. B a i t o a n 106.. (7). Vay (x; ?/; 2) = ( 1 ; 1; 1) la nghiem duy nhat ciia he.. dUdc t a t ca cac nghiem cua he la. 2. do. x + y'^ + 2^ < 3 ( v o H ) .. = z (mhu thuan^ .. 3x-3z-0^x. 1. 1,. X. khac nhau t t o g doi, t a suy r a. 1 2. >. {x + 2 2 - 1) = 0 z = x+l. = 0 ^ = 0 ^. x + 2z=l X - z = - I. 3'~3'3. y. x + y' + z^>3iv6li). Neu. {x; z) = (0; 1). f X + y - 2z - 1 = 0 { y + 'z-2x-l=0. nen. ^1.. • T n r d n g hdp 2 : x = y 7^ z. TiT (1) va (5), t a c6. 1. '. Xet t r i r d n g h d p ( x ; y ; 2) 7^ ( 1 ; 1; 1). Neu x > 1 t h i do gia thiet y > 0 , 2 > 0. + 2x{z-y). TrUdng hdp 3 : x,y,z. y ( l - y ) ( l + y + yx) = 2 ( 1 - 2 ) ( l + x + x 2 ) .. + x-yz. + y + z = 0'^r-q+p. = 0.. (9). E u n g do t a n h d m diTdc x = y = 2 = 0 1a nghiem nen se bien doi he da cho - 1). <=>x (1 - x ) (1 + X + xz) = y (1 - y) (1 + 2 + z y ) .. ihanh (6). r x = yf4-y} { y = zh-z) { z = x(4 - x).. 288 289. •.

(24) Ct cat: dang thiic. Tiep tuc nhan ba phitdng t r i n h vcii nhau va lutu y den trucJng hdp dang xet (Tritdng hdp 5), t a dudc xyz < ^ (.T. <^xyz. = xyz (4 - x) (4 - y) (4 - z) <=> {x - 4). - 4) [yz - Ay - 4z + 16) =. - 4) (z - 4) =. 4cos : r - 3 cos — = cos a o . . a + 27r. -1. -1. - Axy ~ Axz + 16x - Ayz + IQy + 16z - 64 = - 1. <»r-4(7+16p-64=-l<4>167J-4(7. >. >. 4cos'^ —. ;. + r = 63.. (10). Theo (4), (9), (10) t a C O he. i-' \. f 6 p ~\q + r = 63 r-q + p = Q. <^ ( \bp -Iq I r = q-p. = ?3. (11) '. r. ;\. •. ^. 2. 2 (15p - 63). - -^—h^. o. n = cos. : cos (a - 27r).. o. o. . " =. rv + 27r. ;!|*>i" liin. ', '' < i. ;>\ :. n = cos. JiWW/. !. /•ifioii--' , j i. '. Mil,!,. a - 27r. Nghiem cua he da cho la (0; 0; 0 ) , (3; 3; 3),. ,. ^ 42 ^ 0 <^. : c o s ( a + 27r). 3 cos — - —. TT 57r ^ 77r\ , 2 + 2 c o s - ; 2 + 2 c o s — ; 2 + 2cos — \ 9 9 /. - 3p = 0 ^ 3ju2 _ 30p + 126 - 9p = 0. <»3p2 - 39jy + 126 = 0 <^ p2 _. a + 27r. — " ™ o - 3 27r ^ 3 n: - 3 27r 4cos — r 3 cos yuy ra phUdng trinh (*) c6 cac nghiem. TCr (11) thitc hien phep the ta diTdc. V. o „. va cac hoan v i ,. p = D.. 7. • Trirdng hop 5a: (p; f/; r ) = (6; 9; 3). Theo d i n h H V i - e t dao t a c6 (x; y; z) la nghiem ciia phirong t r i n h - 6?)^ + 9?; - 3 = 0.. 2^7. 3+. «. 7 , 2v/7. 3 -^"'3'3"". 3. cos. a + 27r 7 , 2\/7 _ a cos 3 "'3"^ 3. 27r^. va cac hoan v i . L i A i y. Bai toan nay con c6 m o t each giai khac ngtln gon hon nhieu, Idi giai tren t u y dai dong n h u n g chiia difng m o t so k y t h u a t rat hay.. Dat w = 6 + 2, t a dUdc 6^ - 36 = 1. Dat b = 2m, t a diTdc Am^ - 3m. = ^ . Theo. bai toan 3 6 t r a n g 119, t a duoc. B a i t o a n 108. (Middle European. Mathematical Olympiad. Giai he. phUcJng trinh. (1). ( 2x^ + 1^3zx. \3 + 1 = 3x2/ (2) I 22^ + 1 = 3yz.. j.' = 2 + 2 c o s ^ , t; = 2 + 2 c o s y , t; = 2 + 2 c o s y . • Trirdng h(?p 5b: (p; q; r) = (7; 14; 7). Theo d j n h h' V i - e t dao t a c6 (x; y; z) la nghiem ciia phitdng t r i n h ^ 7 Dat i = a + - , ta ditoc. - 7/2 + 14i - 7 = 0.. 4n^-3n = - ^ .. Dat. = cosa, vdi a G. 2t/3 + l = 2/3 + y3. 290. = 3x2. = 32/2. =. 3z\. _. (3) (4) (5). )au dang thi'rc d (3) xay ra <^ x = 1, dau dang th>i:c d (4) xay ra <^ j/ - 1, \u dang thulc d (5) xay ra. 2 = 1. T i t (3), (4), (5),ta c6. (2x3 + l)(2y^ + l ) ( 2 2 ^ +. TTJ. K h i do. 4?i^ - 3n = cos a = cos ( a + 2-K). + i> 3 ^. 2z^ + l = z^ + z^ + l>3</?. (*). (*). Giai. • TntcJng hdp 1 : x > 0, y > 0 va z > 0. Theo bat dang t h i i c Co-si, t a co 2x3 + l = x^ + x^ + l > 3 v ^. 7 7 2 v/7 - - a = - — . D a t a = - y - n , t a dUdc. .. (3). 2012).. l)>27xV^^. ^dau dang thitc xay ra k h i va chi k h i x = y = z = l. M a t khac t i t he (*) ta c6 =. cos ( o -. 27r).. (2x3 ^ i)(22/3 + 1)(223 + 1) = 27x^y^z'^, do do x = y = z = 1. 291.

(25) • Trildng hop 2 : Trong ba so x, y, z c6 diing mot am, hai so con lai khoiig am. Khong mat tinh t5ng quat, gia sii x < 0, y > 0, 2 > 0. Khi do ta c6 0 > 'ixy'^°=^ 27/3 + 1 > ij r • Trirdng hdp 3 : Trong ba so x, y, z c6 dung hai am, so con lai khong am. Khong mat tinh tSng quat, gia siif x < 0, j/ < 0, ^ > 0. Khi do ta c6 2z^ + 1 > 0 > iyz, dieu nay mau thuan vdi (3). • TrUdng hdp 4 : Ca ba so x, y, z deu am. Khong mat tinh tong quat, gia sit X — max {.T, y, z). Ta c6 , ^ 1 + 2x3 > 1 + Tit. X. = y > 2, ta. =^ 3xz > 3i/2 <!=i> X < y. X = y.. CO. 2y3 +. 1>. + 1 =^ 3xy. > 3y2. X <. 2. X =. Nhif vay x = y = z. Thay vao he (*) ta (hruc. 2.. ,. ]\/iat khac, ta c6 / ' ( t ) = Qi^ _ 4f + 1 > 0, V< e R. Suy ra / (t) la ham so dong bi^n tren R. Khi do y > 2 =^ / (y) > / (2) 2+ y > x + 2 y > x. Vay 2; = y. Suy ra / (x) = / (y), hay y + x = y + 2, hay x = 2. Thay x = y = z vao he phitdng trinh, ta c6 3x3 - 2x2 - x = 0. ph^((jjjg tj-^nh ^ay c6 3 nghiem 2; = 0, x = l , x = - - .. 0. Vay he i)hitdng trinh da cho c6 3 nghiem (x; y; 2) la (0 ; 0 ; 0 ) , ( l ; l ; l ) , ( - l ; 4 ; 4 ) . -. LuM y- Neu ta klioiig cpng them x, y, 2 vao tifng phitdng trinh de thu ditdc ham so dong bien / (t) = 3t^ -2i^ + t, teR nhit da trinh bay trong IcJi giai 0 tren ma xet trite tiep ham so g{t) = Zt^ - 2t^ thi se gap rat nhieu kho khan, do ham g{t) = 3^3 - 2t^ khong phai la ham dong bien. Thiic te d ky t h i do chi CO mot t h i sinh giai difdc tron ven bai nay bang each dita ve he. r X = 1 (loai) 2x3 + 1 = 3x2 ^ (x - l)2(2x + 1) = 0 <!:^ X -. Vay nghiem ciia he la ix;y;z) 4.5.3. = (1; 1; 1), {x;y;z). 2 = 323 -|. z y 4- x = 3x3 - 2x2 + X 2 + y = 3y3 - 2y2 + y. X +. 2-. = (-^;-^; -|).. 2' 2' 2^. Phu'dng p h a p sang tac cac bai toan ve he lap b a § n .. V i d u 1. Xet mot ham so thoa man hai diiu kien : f{t) khong phdi la ham dong bien tren R va f{t) + t la ham dong bien tren R. Chang han ham so (x = f(z) - 2*2, t G R. Xet he \ = f (x) Ta ducfc bai toan sau. [ ^ = fly)B a i t o a n 109 ( H S G G i a L a i , b a n g A , n a m h o c 2 0 1 0 - 2 0 1 1 ) . Gidi he phiCdng trinh. V i d u 2. Xuat phdt tic (x + l)(y + 1) = z + 1, co xy = 2 - x - y. Tii day Ian hidt thay bo (x;y) bdi {y;z), {z;x) ta dUcJc bai toan 110. Luu y rang do hay va kho cua de bai phu thuoc rat nhieu vao phuong trinh xuat phdt. B a i t o a n 110 ( C a n a d a N a t i o n a l O l y m p i a d 2004). Gidi hephuang xy xz yz. f (t) =. X = 323 - 2^2 y = 3x3-2x2 (*) l2-3y3-2y2. G i a i . Gia siif x = max {x, y, 2}, the thi x > y > 2 hoS-c x> z>y. Xet trudng hdp x> y > z (trUdng hdp x > z > y titdng tit va cac nghiem triing vdi cac nghiem cua tnrdng hdp da xet). He phitdng trinh da cho titdng ditdng vcti X + 2 = 323 - 22^ + 2 y + X = 3x3 - 2x2 + X 2 + v = 3y3 - 2y2 + y. (. {. Xet ham so / {t) = 3*3 -2t^ + t,te. R. Khi do, he (*) c6 dang. r. x + z = f(z) < y + x = J (x) [ z + y = f(y)292. •. = = =. trinh. z- X - y y - X - 'z X - yz.. l a i . Cong hai ve ciia phitdng trinh thi't nhat vdi (x + y + 1) ta difdc xy + x + y + l = 2 + l - ^ ( x + l)(y + 1) = 2 + 1. Tudng tit ta CO. (7)^. f ( x + l ) ( y + 1 ) = 2 +1 { x + l)(2 + l ) = y + l [ (y + l)(2 + l ) = x + l .. (1). (2) (3). Nhan (1), (2), (3) theo ve ta dudc [(x + l)(y + 1)(2 + 1)]2 = (x + l)(y + 1)(2 + 1). Neu. X. ^7^-1.. = - 1 thi tit (1), (2), (3) suy ra y = 2 = - 1 . Gia siS x Khi. do. ( 4 ) ^ ( x + l ) ( y + l)(2 + l ) = l .. 293. 7^. (4) -1, y. 7^. - 1 va.

(26) T i f (5) va (3) suy ra (x 4-1)^ = 1<=> a;^ + 2x = 0 • Thay x = 0 vao he ( / ) t a ditdc. x e {0, - 2 } .. '/ipn n = 3,. rfo 3p2 - 30p + 27 = 0. y = z. ^. y;z). =. i-2;-2). hQnp=. • Thay x = - 2 vao he (/) t a dUdc -2y. =. ^+. p2 - lOp + 9 = 0. x'* + 3x = ^^.^ ^ ^ . T-(/: /mm .so ytay. I , ta dUcJc ham so f{x). ^. y^l] =. X. mx . Tiirdc het (le y rang he so cua nx^ + p nhimg van khong lam mat t i n h long quat. Ta c6 =. x"^ +. , , x^ + mx f{x) - X = K nx'^ + p. Lai. B a i t o a n 111. Giai he phihlmj. X a y dvfng m o t so Idp h a m de s a n g t a c b a i t o a n m d i . M o t m o i l i e n h e giiJa h e phvfdng t r i n h v a d a y s6. 1. L d p h a m f{x). x =. x^ + mx-nx'^-px. {I - n) x^ + {m - p) x. nx'^ +p. nx^ + p. ^^^2. / \^) —. jj-j. ^. (x^. _. +. x i = a;. Xn+i. =. .. .. >, <•. ^ ^ / _ ^ ^ , V x € R. K h i do de dang thay rang. fix). -. X. =. - x ( 2 x 2 - 2) 3x2 + 1. ; /(x)-x = 0^xG. {0,-1,1}.. (nx2 + pf x = ±1.. +pf. Dicni kien de nx^ + (3p - nin)x'^ + inp c6 dang [Ax^ +. (luc nay t a c6. A. 0. /. > 0. t A = (3p - muY - ^mnp = 0 ^ \ - mmnp + m^n^ = 0.. I +. / ' ( x ) > 0, suy r a ham so / dong bieu) la. 294. (). +. :>. (. 0. + _^-\«-. X e ( - 0 0 ; - 1 ) t i l l / ( x ) thuoc (-00;-1). (i. -. va. fix). >. X.. Do. do v6i m o i n = 1 , 2 , . . . t h i G (-00;-1),. hay. 14 ^han tren. T a co X2 = / ( x i ) = / ( a ) > a = xi, ma fix) 295. -1.. T i f bang ben t a c6: Vdi moi. x„. • {^'n }t=i. 1: a <. IVu-dng h d p. 1. r. +. ^ / " > 0 ^ \^ - lOnp + 3n^ = 0.. 0. /(.V). V i d u 3. C/ipn m — 3. A"/// (/u n > 0 \2 - 30np + 9n2 = 0. = 0. ^"^^'^ ^ (3x2 + 1)2'. nx'^ + {3p - mn) x^ + mp [nx^. r. ,Vn = l , 2 , . . .. =/(x„),VnGN*. Taco. ] r ~ ^2 (nx^ + v). ". r •/» >. 34 + 1. Giai.. 3nx'* + 'ipx^ + mnx^ + mp - 2nx'' - 2mnx'^. ^. trinh. Chung minh rang day so tren cd gidi han hHu han vd tim gidi han do.. Xn+i. 77ix). ^. B a i t o a n 112 ( D e t h i H S G quoc g i a , n a m h o c 1 9 9 7 - 1 9 9 8 , b a n g B ) . Cho so thuc a. Xet day so { a ; „ } ^ ^ nhu sau:. C a c h 1. X e t ham so / ( x ) = ^. (3y2 + 1) = y3 ^ 3,^. y (3^2 + l\= + 'Sz z hz^ + 1) = .T^ + 3x. '. bang 1. CO (3x2. f{z). Wo).. Nghierii ciia he la ( 0 ; 0 ; 0 ) , ( - 1 ; - 1 ; - 1 ) , ( - 2 ; - 2 ; 0 ) va cac hoan vi.,) 4.5.4. tinel ki duac. fiai hdi toan sau, do Id hdi toan 112 vd bdi loan 111. De thn duac bai loan vi he doi xttng ba an, ta chi can xet he. 2-y. y = yz = 0. p = 9. ;V;:';;,tiJ ;. day. so. dong bi6n.

(27) 'it gia thiet ta c6 tren ( - 0 0 ; - 1 ) nen quy nap theo n suy ra x„+i > x„, Vn = 1,2.... Vay khj. 1 ^ ^=. o < - 1 thi day { x „ } ^ ^ tang va bi chan tren nen hoi tu. Dat hm x„ = khi do 61 thoa man dieu kifn 61 < - 1 va 61 G {0, - 1 , 1 } . Do do 61 = - 1 hay Hin. .T„. = -1.. '. n—•+00. Xn + di. • •. 4-1 +3xn-i n. Q 2. .. ^~. ^ n - l + 3x„_i. 4 - 1 - 3 x"•^ ^ _,1i +- r3 x „ _ i ~- ^1 _ \(x„_i - 1)3 5~"2—r~; = 3-<-i + 1 ~ 3xl_, + 1— •. _ X ^ _ i + 3xf,_j + 3x. 34-1 + 1. 34-1 + 1 •. 34-1 + 1. vay. Tru'5ng h d p 2 : a = - 1 . Khi do x„ = - l , V n G N*, suy ra hm x„ = - 1 .. x„ -. 1. 1. /X„_i -. Trirdng h d p 3: - 1 < a < 0. Tit bang tren ta c6: V6i moi x G (-1;0) thi i/(x,r"-'"'. /(x) G (-1;0) va /(x) < x. Do do v6i moi n = 1,2,... thi x„ G (-1;0), vay day so. {xn}n=i. chan ditdi. Ta c6. X2 = f{xi). = f{a). < a = xi,. dong bien tren (-1;0) nen quy nap theo n suy ra x„+i < Xn,yn. ma. XI. f{x). = 1,2,.... at b =. •ri - 1 xi + 1. Vay khi - 1 < a < 0 thi day {x„},"^^ giam va bi chan dudi nen hoi tu. Dat hm x„ = 62, khi do h2 thoa man dicu kicn 62 = n—>+oo. 3,-1. . Khi do. x„-l Xn +. hm x„ < x i = a < 0 va n—»+oo. 6+1 = 6 x„ - 1 = 6x„ + 6 <^ x„ = l ' • ' "~" " 1-f. ay neu a = - 1 thi x„ = - l . V / t G N*, neu a 7^ - 1 thi. 62 G { 0 , - 1 , 1 } . Do do 62 = ~ 1 , vay hm x„ = ~ 1 . n—>+oo. 1 +. Trxidns hdp 4: a = 0. Khi do .T„ = 0. Vn = 1,2,... Suy ra hm .T„ = 0. n—»+oo. Tru-dng h d p 5: 0 < a < 1. Tit bang tren ta c6: Vdi moi x G (0; 1) thi /(x) G (0; 1) va f{x) > x. Do do vdi moi n = 1,2,... thi x„ G (0; 1), suy ra day so {x„}^^^ bi chan tren. Ta co X 2 = /(xi) = /(a) > a = x i , ma /(x) dong bien tren (0; 1) nen quy nap theo n suy ra Xn+l. > x„,Vn = 1,2,.... Vay khi 0 < a < 1 thi day { x „ } ^ ^ tang va bi chan tren nen hoi t u . Dat hm x„ = 63, khi do ^3 thoa man dieu kien bs = hm x„ > x i = a > 0 va ri—'+00. 63 G {0, - 1 , 1 } . Do do 63 = 1 hay hm x„ = 1.. n—»+oo. Xri. =. 'a-. a+1. 1. .. O M - I. ^ , V n = l,2,.... n—>+oo. T r i f d n g h d p 7 : 1 < a. Vdi moi x G (1; +00) thi /(x) G (1; + 0 0 ) va /(x) < x. Do do vdi moi n = 1,2,... thi x„ G (l;+oo), vay day so { x „ } ^ ^ bi chSn difdi. Ta c6 X 2 = f{x\) = f{a) < a = xi, ma /(x) dong bien tren ( 1 ; + 0 0 ) nen quy nap theo n suy ra x„+i < x„,Vn = 1,2,... Vay khi I < a thi day {xn}n^i giam va bi chan dirdi nen hoi tu. Dat hm x„ = 64, khi do 64 thoa n—•+00. 0 7 ^ - 1 , ta CO cac bien doi titdng ditcJiig sau a-. 1. = l < = > a - l = a + l < ^ 0 a = 2 ^v6 nghiem j .. a- 1 = - l 4 = > a - l = - a - l < ! ^ a = 0. « +1 a - 1 < 1 <^ {a -ly 2 < {a+ ily\ -2a < 2a a> a+1 a 1 > 1'^ {a - if > (n + 1)^ -2a >2n<^a<0. a +1. Cach 2 . De thay rang vdi moi n, luon ton tai x„. Xet ham so /(x) =. X X +. 1 r1. ,. 0.. ' ' ^. ay ta xet cac tritdng hdp sau : irdng h d p 1 : a = 0. Khi do x„ = 0, Vn = 1,2,... Suy ra hm x„ = 0. n-»+oo. xidng h d p 2 : a = - 1 . Khi do x„ = - l , V n G N*. Suy ra hm x„ = - 1 . n-»+oo. ifdng h d p 3: a > 0. Khi do. mfm dieu kien 64 > 1 va 64 G 10. - 1 , 1 ) . Do do 64 = 1 hay hm x„ = 1. n—»+oo. (*). 1 -. ^. Tru-dng h d p 6: a = 1. Khi do .T„ = 1, Vn = 1, 2 , . . . Do do hm x„ =^ 1.. hm »i—»+oo. o - i y. a+iy. = 0^'. hm x„ =. n—'+oo. 297 296. + 1 /. 1..

(28) T r i r d n g hdp 4: - 1 7^ a < 0. K h i do > 1. a+1 V i d u 4.. ' Ta. Hm. fa-. ^ — 1\. {*) = 0=^ Hm. n—>+oo. n—>+oo. Xn. —. ( x = fiy). !•. i z = /(x)-.T. duac. bai toan. B a i t o a n 113.. -y. (. X. (3y2 + l) = 2 {y -. \ 3x2 + 1. ,. —1.. T a cdn c6 thi lam kho hon bai toan 111 nhu sau. Xet. ;. he. ,/). I. z =. f\x).. ai t o a n 114. ( D e n g h i t h i O l y m p i c 3 0 - 0 4 - 2 0 0 9 ) . Giai va bien ( 2x(2/2 + a2) = y(j/2 + 9a2) iheo tham s6 a) he phudng trinh < 2^(22 + a2) = z{z'^ + 9ffl2) l 22(x2 + a2) = x ( x 2 + 9a2).. X (3y'^ +l)=2{yy^) y (3z^ + 1 = 2 - z^) z (3x2 + = 2 (x - x^) .. Gidi he phUcJng trinh. f. ( x + y = f{y) y + z = f{z) \ + X= f{x),. if. X > y > z ho&c x > z > y. X e t trirdng hop x > y > z (trifdng hdp x > z > y titOng t i t va cac nghiem t r i i n g vdi cac nghiem ciia tritdng h d p da x e t ) . K h i do x>V=>f {^) > f {y) =^ 3; z > X + y z > y. V a y y = z. S u y ra / (y) = f {z),hay vao he phiTdng t r i n h , t a c6. x + y ^ y +z,. ludn (1). r 2xy2 = 2^3 iai. K h i a = 0, he (1) trd t h a n h < 2yz'^ = z^. Sx^y^z^ = x^y^z^. S u y r a. I 22x2 ^ ^.3. xyz = 0. V a y {!) <^ x = y = z = 0. T i t day v c sau t a x e t a 7^ 0. X e t h a m so ,ii'>i-j. x'' + 3x ^ ^ la ham dong bien. G i a s i i x = m a x { x , y , 2 } , the thi. nffui. ,.';;o!i. Chi do f,. _ x(x2 + 9a2) _. .. - x ^ + 7a2x 2{x'. 2(x2 + a2). X -. X —. . / W. \ = 0 = ^ ^ [ x = ±av/7. \/3a. hay x = z. T h a y x = y = z fhy ham so / dong bien trcn R. L a p luan hoan toan tUdng i\i nhit each 2 d. (3x2 + 1) = 2 (x - x^) ^ 5x^. -. X =. 0. ^. X. G. jo,. luc phitdng phap giai, t a dildc. .. aoit. u = 2 - x = 0 x = ±V^ft. X =. (1). ^. {. X. = /(x)'. ^. V a y he phvtdng t r i n h da cho c6 3 nghiem (x; y; z) la (0;0;0),. x ( x 2 4-9«2). =2. sau.. = fiy) -y y = f(z)-z z = j{x)-x. X. m = Qa^, to fto/c /icVm so. Tif /la"'- so nay ta thiet he duoc hai bai toan sau, do la bai toan 114 va bai odn 115. De' thu duac bai toan vi he doi xiing ba an, ta chi can xet he. Hu-dng d a n . H e viet lai. vdi / ( x ) =. Chon p =. m, =. < . . •. 2/ = 2 = 0 = y = z = as/l [ X = ij = 2 = - a \ / 7 . X = X. I^et luan: K h i a = 0 he (1) c6 nghiem d u y nhat l a (0; 0; 0). K h i a 7^ 0, h ^ (1) l o b a nghiem l a ( 0 ; 0 ; 0 ) , ( a v / 7 ; a v / 7 ; a v / 7 ) , ( - a v / 7 ; - a - / ? ; - a v / 7 ) . ; 5 ' 5 '. V i d u 5. Chon n = 2. Khi. Jai t o a n 115.. Cho so thiCc a > 0. Xet day so. do. nhu sau:. x„(x2+9a2) Xi. 9p2 _ 20mp + 4?u2 = 0 => 9 (—Y \rn/. {xn}n=i. - 20 f — ) + 4 = 0 \m/ 298. fChiing I m. 9". =. b;. Xn+i. —. ,1.. • ,. ''•. •. 2(x2+a2). minh rhng vdi moi so thuc b, day so tren c6 gidi hg,n hHu han vd tim. tgidi han do.. ! 299.

(29) 2(x^ + a^) '^^ ^ ^ ' ^^'^^ ^ °. G i a i . X c t h a m so /(x) = Xn+i. dang thay rang. = / ( a - „ ) , V n G N * . T a c6. 2(x2 + o2). 2(x2 + a2) f r2 - Srj2^2. = 2(x2 + a2)2 ^ 0; /'(^) = 0 ^. X. =. X. =. X. =. X. =. 0. ±a\/7. \/3a -\/3a. (jY-u'dng h d p 5: 0 < 6 < a\/7. T i r bang tren t a c6: Vdi m o i x € (0; a\/7) t h i ({x) G (0; a\/7) va /(x) > x . Do do v d i m o i n = 1 , 2 , . . . t h i x „ G (0;a>/7), jjay day so {xn}n=\i chan tren. T a cd X 2 = / ( x i ) = /(a) > a = x i , m a /(x) jQjig bien tren (0; a\/l) nen quy nap theo n suy r a a;n+i > X n , V n = 1,2,.... ^. ^ay k h i 0 < 6 < a\/7 t h i day { x „ } ^ ^ tang va b i chan tren nen hoi t u . D a t = i-s, k h i do L 3 thoa man cheu kien. lim. =. j,._,+0C'. +. Tru*dng h d p 1 : 6 <. -a\/7.. T i t bang ben t a c6: Vdi nioi X e (-oo; -aV7). t h i /(x) thufx-. ( - o o ; - a / 7 ) va /(x) > x . Do do vdi m o i n =. 1,2,... thi. b i chan tren. Ta c6. dong bicn tren (-oo; -as/l). X2 =. f{xi). =. f{a). > a = x i , m a /(a). - a v ^ , « V 7 } . Do do L 3 = as/l hay. fru'cing h d p 6: 6 = a\/7. K h i do x„ = av/7. Do do. Trifdng h d p 7: 6 > a\/7. T i r bang tren t a co: Vdi m o i x G (aV^; + 0 0 ) t h i /(x) G ( a v ^ ; + 0 0 ) va /(x) < x. Do do v d i n = 1 , 2 , . . . t h i x„ G (a>/7; + 0 0 ) . Ta CO X 2 = / ( x i ) = /(a) < a = x i , m a /(x) dong bicn tren (av/7; +cx)) nen {xn}n=i. S'^"' "^^ b i chan dudi nen hoi t u . D a t. lim. Xn —. a\/l.. n-*+oo. ,. Vf d u 6. Chon n = 3. Khi do V. n—>+oo. - 30mj!; + 9m' = 0 =^. = ^ 3 ( ^ y - 1 0 m + 3 = 0=.. L i G {O,-a\/7, a\/7} . Chpn — = 1 771 3. n—+00. m — 3p. CTion m = 3 a. T r i f d n g h d p 2: 6 = -a\/7. K h i do x „ = -a\/7, Vn = 1 , 2 , . . . suy r a lim. Trtfdng h d p 3 : -nV7. x„ =. 3. p = a, to duac ham so. x'^ + 3 a x ,,. ^. ,. -ay/l.. < x. Do do x,. G (-a\/7;0), vay day so. {xn]l^i. t o a n 1 1 6 . Gidi vd bi$n luan {theo tham so a khong am) h$ phuang (3(/2 + a)x = y{y^ + 3a) (32^ + a)y = ^(^2 + 3a) (3x2 + a)z = x(x2 + 3a).. < a = x i , m a /(x) dong bicn. nen q u y nap theo 7i suy r a x ^ + i < x„,Vn = 1 , 2 , . . . Vay. khi - a \ / l < a < 0 t h i day { x „ } ^ ^ giam va b i chan ditdi nen hoi t u . Dat hm. m. ^il ham so nay ta thiet ke duoc bdi todn 1 1 6 vd bdi todn 1 1 7 .. bi chan dudi. T a c6 X 2 = /(a^i) = /(a) tren {-a^/l;Q). ,. < 6 < 0. TiT bang tren t a c6: Vdi m o i x G (-flV7;0). t h i /(x) G (-av/7;0) va f{x). - lOmp + 3m^ = 0 = 3 A, m L. h m x „ = -ov/7.. n—>+oo. h m x „ = L 4 , k h i do L 4 thoa n—^+cx>. x „ = L i , k h i do Li thoa man dieu kien L\ -a\/7 va. Do do L i = -a\/7 hay. h m x„ = a\/7. Tl—» + 0 0. Vay k h i 6 < - A V ? t h i day {.T„}^^ taug va b i chau t r e n nen hoi t u . Dat hni. h m .x„ = a\/l.. man dieu kien L 4 > a\/7 va L 4 G {O, -a\/7, av/7}. D o do L 4 = aV7, hay. nen q u y nap theo n suy r a Xn+i > x „ , V n = 1,2,.... V ;. S. h m x „ > x\ a. > Q va n—>+oo. quy nap theo n suy r a x„+i < x„,Vn = 1,2,... Vay k h i b > a\/7 t h i day. x „ G ( - o o ; - a \ / 7 ) , hay day so {xn}n=i. .•ifv: inlr, " ' f. x „ = L 2 , k h i do L 2 thoa man dieu kien L 2 =. n—>+oo. h m x „ < x i = 6 < 0 va n—•+00. L2 G {0. - a v / 7 , a\/7}. D o do L 2 = - a \ / 7 , vay. h m x„ =. Trtrdng h d p 4 : /; ^ 0. K h i do .r„ = 0, Vn = 1 , 2 , . . . Suy r a. -a\fl. h m .T„ = 0.. t o a n 1 1 7 . Cho a > 0 c6 dinh. Xet day so { x ^ } nhit sau: (. XQ>0 ""^^=. jxl + 3a) 3x2+a ' V U G. n—•+c». 300. 301. '. trinh.

(30) Tim tat cd c/ir so dudng X Q sao cho day hap do hay tinh gidi han cua day.. co yidi han vd trony cdc trudu,^. B 1 a i t o a n 1 2 0 . Cho trade hai .so a, b thoa man dieu kien b > a > Q. Gidi he. phUdng. trinh. bai toan 117 nhit sau : X e t ham so g{x) =. .. ( {a + b)x =^ y'^ + ab \ + b)y = zl^ab ,„ {a + b)z = x^ + ab.. •. L u ^ y. T a ( o thS t i m du'rtc so hang tfing quat cua day so t r n y hoi clio tioi|„. I. T i r gia thiot (-a co. /. . >. £)ap so. He phirdng t r i n h co hai nghiem (o; a; a), (6; 6; h). \fa =. Xn-. x f , _ i + 3ax„-i 3xf,_i + a. - \fa. nx^. xl^ 1 - 3x!^„ 1 v/^ + 3ax„,^i - ay/o _ ( x „ - i - >/») 3x2_, + a 3x2_, + a Titdng ti.r t a c6 x „ -j-. Xn. =. ( x n - i + \/uy. 3xti +a. — . Tntdc het de y rang he so ciia + p. nhifng van khong lam mat t i n h tong quat. T a co f'(. \ (^^''. . Bdi vfxv. '. bc^ng 1 °. ". '^") ("^^ +P) ~ 4nx^ (x^ + m x ). 5nx^ + 5;)x^ + mnx^ + rnp — 4nx^ — 4mnx^ (nx" + pf. 3. /x.. - va. \. 2. L d p h a m f { x ) =. _ nx^ + {bp - 3mn) x^ + mp. !\:. •»>l ,,,1. {nx'^+pf. Dat T Xn. - ^/o y/o.. r-. -. x „ ++ 'v/ft Xn. ( a + 1) \/a. r-. .. Dieu kien dg cho nx* + {bp - 3mn) x^ + mp co dang {Ax"^ + Bf / ' ( x ) > 0, suy r a ham so / dong bien) la. (luc nay. r ">« ^ ^/">0 I A = (5p - 3mn)^ - 4mnp = 0 ^ \^ - 34mnp + Qm^n^ = 0.. ° 3". ''i d u 7. Chon n = 2, khi do Vay. Xn. =. 1. X(). -. V". 3". 25p2 - 68;/tp +. 36/At'^. - 68 (£•) + 36 = 0. = Q<^2b. • 111". B a i t o a n 1 1 8 ( D e n g h i O L Y M P I C 3 0 / 0 4 / 2 0 0 6 ) . Cho day { x „ } , vdi ''C/ipn — =z ^. xo = \/2. trong do a>2.. 1. [xj + 3a). - _p = 2 m 18 P_. =^ p =. C/jpn / n = 25a'*, /:/?,?: (fo p = 18a'*, to diMc ham. so. 3x^ + a. '"^^. Ch{(ng minh rang day { x „ } co gi/li han. Tinh giM han do.. x^ + 25a4x 2x4 + i8a4 •. ti ham, so nay ta thie.t ke ducic hoi loan 121 vd bai toan 122.^' Hu'cfiig dan. Bai toan nay la mot t n t d n g hdp rieng ci'ia bai toan 117, vdi. Xo =. \/2.. B a i t o a n 1 1 9 . Gidi he phuong. trmh I y =. I. z=. - 6 ^/W^.. D a p so. He co ba nghiem la ( 1 ; 1; 1). (2; 2; 2), (-3; -3; - 3 ) .. 302. 25. •. B a i t o a n 1 2 1 . Cho hai so thuc b vd a> 0. Xet day so {xn}^^^ '"'^''^ , "^ = '=. x„(x^ + 25a'') ^ ^„ ' ' \ 2x^ + 18a4 , V n = l , 2 , . . . u o . , . ^ . : :. Chiing minh rang vdi moi so thUc b, day so tren co gidi han hHu han vd tim gidi. han do. 303.

(31) B a i t o a n 122.. Gidi vd bien ludn (theo tham so a) he phuang 2x {i/ 2y{z' 2z{x*. + 9a^) = y (7a^ - y^) + 9a^) = z(7a^-z^) + 9a'')=x{7a'' -x').. trinh. U'dng h d p 2. a > c > 6. K h i do f{a) (1, '. '^. < f{c). =^3c<36=>c<6=^c=6=^a = 6=c.. ; ,,. "ay trong moi tmtfing hrip t a don c6 a = 6 = c = 1, suy ra 2 = 1 y/lO - 3x = 1 v/4 - 3 A / 1 0 - 3x = 1. X -. 4.5.5. Svt d u n g h e l a p b a an d i s a n g t a c m o t s 5 p h u ' d n g t r i n h he phu'dng t r i n h chiJa c a n .. V i d u 8. Xudt phdt tic mot lie lap dcin gidn, ch&ng han. Cach. ,,,fjv. 2. Dieu kien 10 - 3x > 0. {. h-^ = A-. <^ X = 3 (tlioa dieu kien).. \x < 4. 3a. ^. r 10 - 3x > 0. ^. 1 90 - 27x-< 16. ^. 27 -. ^ -. 74 • '. <. X. 10. <. ^ ". T'. Ta thay x = 3 la mot nghiem ciia (1). Vdi diiu kien a, b, c khong dm thi he tren gidi a = b — c = 1. Vdi x = 3, ta c6. diCdc. di dang vd kit qud Id. • Neu — < X < 3 t h i 3 ^ 4 - 3v/10 - 3x < 3, trong k h i do (x - 1) (x - 3) < 0 => 4x - x^ > 3.. a =. X. - 2 = 1, 6 = v/10 - 3x = 1,. \JA~ 3^10 - 3x.. "J.. JJ1 ;fb * / ij\;A. 74 Vay — < X < 3 khong thoa (1).. Mat khdc, theo tren ta c6. I. . Neu 3 < X < — t h i 3 ^ 4 - 3 / 1 0 ^ 1 ^ > 3, trong k h i do 4x - x^ = 4 Ta thu diiac bai toan B a i t o a n 123.. = 3c = 3\/4 - 36 = 3^/4 -. 3^/10^^.. -. ' ' i ^ ''^' n'". ". (x - 1) (x - 3) > 0 =^ 4x - x^ < 3.. sau.. Gidi phuang. trinh. 4x - x^ = 3\/4 - S/TO - 3x.. (1). Vay 3 < X < — khong thoa (1). Do do x = 3 la nghiem duy nhat ciia (1). 3. Giai.. V i d u 9. Xet. C a c h 1. Diou kion. I. f 10-3x^0 I 3v/10 - 3x < 4. ^ r i 0 - 3 x > 0 ^ \0 - 27x"< 16. a = x - 2 , b = t v^lO - 3x > 0, c = \/A-3^/W^^ ra a > 0. T i r (1) t a c6 he ( a^=4-3c \2 = 4 - 3 a c 2 = 4 " 36. . .. ^. ^74 ^ ^ ^ ^. mot he lap ba an dan gidn : < = z + Q {thong thudng ta { z^ = x + Q nen chon cdc he. sao cho vice gidi no Id khd dan gidn, vd chi c6 nghiem thod man dieu kien x = y = z). Tic he nay ta c6. 10 J -. > 0. T i r dieu kien suv Ta dicac bai todn. r 3a = f{b) , 36 = / ( c ) ( v d i / ( « ) = 4 - i ^ , ^ > O) . i 3c = / ( a ). H a m / nghjch bien tren [0; + 0 0 ) , v i /'(<) = -2t tong quat, gia sir a = max {a, 6, c}. Trifcfng hdp 1. a > 6 > c. K h i do. B a i t o a n 124.. < 0, Vf > 0. K h o n g mat t i n h. < / ( c ) ^ 3 a < 3 6 = ^ a < 6 = ^ a = 6=!>a = 6 = c.. 304. •. Giai.Datj. sau. Gidi phicang trinh. v ^ = - ^. ta CO he j^hirdng t r i n h J. x^ - A / G + s / a T + l = 6.. ^ { <-^^zl x^ = y + G y3 = . + 6 2'^ = x + 6. 305. (1,. T i r (1), t a c6 x ^ - 2 / - 6 = 0. Vay. (2) (3) (4). (*).

(32) N h a n t h a y , lie (*). k h o n g t l i a y d 6 i k h i h o a n v i v o n g q u a u h d o i v d i x,y,z. k h o n g m a t t i n h t o n g q u a t , t a c6 t h g g i a t h i e t x = m a x {a;, y, z) n h a t t r o n g b a so x, y, z, h a y x > y, x > z). ra ,. y-^Q. o. •. = x^ > y^ = z + Q^. y>. ,. z=^y^. 2 > _ i. ne^. ( x l a so lot). N e u x > y t h i t i t (2) v a (3). n e n y ^ - 1 , 2 =^ - 1 v a .r 7^ - 1 . N h a n b a p h i t o n g t r i n h ( 2 ) ,. (4) t a dUdc x y z = 1. M a t k h a c , c o n g (2), (3), (4) t h e o ve t a d u d c. suv 3 = x^ + y2 + 2^ > 3 ^xh/z'^. thco (3),(4). >. z + Q>. x + Q^. z>. (3),. J:. x^y^z"^ < 1.. po d i n g thiic xay ra nen. ;. a:^ = y^ = z^ = I ^. D e n (lay t a g a p d i e u m a u t h u a n v(5i x > z. V a y x — y. K e t h d p v 6 i (2) va (3). X. = y = z = I (do X. ;. , .,. - 1 , y ^ -I,. z ^ -I). .. t a d u o c y — z. V a y x = y = z. P h i W n g t r i n h (1) t r d t h a n h p h i t d n g t r i n h co n g h i e m d u y n h a t x = 1. - X - 6 = 0 4=> (.X - 2) (x^ + 2 x + 3) = 0 <i=> x = 2.. ". Bai t o a n 126 ( D e. T h i l : l a i , n g h i o m n a y t h o a m a n b a i t o a n . V a y p h i t r i n g t r i n h t r o n g dau b a i co n g h i e m d u y n h a t x = 2.. n g h i c h o k i t h i h o c s i n h g i o i c a c tru-ftng. Chuyen. k h u vvTc D u y e n H a i v a D o n g B a n g B a c B o n a m 2 0 1 0 ) . Gidi trinh. X = y/S^.Vi-x. +. + V^-x.V5-x. phudng. v/S-xVS-x.. f x2 + X - I = y G i a i . D i e u k i e n x < 3. D a t s/S - x = a ^ v ' 4 Vi. du. Xet ham la (]{t) = Kit. 1 0 . Xri. mM. so f{t). 2. he. lap. ha an ddn (jidn,. clianq. y2 _|_. = t'^ + t - 1. Tren khodng - - ; +oo , ham . Nghia la, tu z^ + z-1 = x, ta co z =. hdp vdi he ta. _ ^ _ i.. X =. ngiCdc cua. 2. f{t) .. ^i±^±l^.. Do d o. 3-a^ i - P b -. n g h i O l y m p i c 3 0 / 0 4 / 2 0 1 1 ) . Gidi. phuong. G i a i . Dieu kien x >. a + b =. trinh. (1). •. b + c =. , d i e u k i e n z>. ^. ^.6. 1 ^ , — . l a co. 2z + 1 = \/5 + 4x => 4^2 + 42 + 1 = 5 + 4x =^. = ab+bc + ca = ab + bc-\-ca =^ ab + bc + ca. ^. =. a.6 + 6.C + c.a. ^'. ( (a + b) ( c + ^ 3 < (b + c){a + b)=i [ {c. + a) {b + c) = 5.. e.. \. [. 2(2. r x(x+l)=:y+l y(?y+l) = 2 + l 3. + 2 - 1 = .r.. +. 1. 30G. =X. +. 1.. (2). r. 2/15. 3. a + b + c=. v/T5. -—5. \/l5. + ——4. +. yi5 3 .. G71 X. =. 240. . Thi'r lai thay t h o i i m a n .. Suf dung can bac n cua s6 phiJc de sang tac giai he phifdng trinh.. C h o so p h i r r 2 = r (cos^? + i sinv?), ?• > 0. K h i d o cac c a n b a c n c u a z l a. T a CO he p h i f d n g t r i n h. f x2 + x ~ l = y \+ y - l ^ z ^ {. -. c + a =. D a t y = x^ + x - 1. T h a y vao (1), t a d i t d c. Suy(• r a. - 1 + v/5 + 4x. = 4 - 6^ = 5 -. N h a n b a i ) h U d n g t r i n h t a d i t d c (o + 6) (6 + c) {c + a) = 2VT5. S u y r a. 2 ( x 2 + x - l ) ^ + 2x^ + 2 x = 3 + y 5 T 4 ^ .. Dat z =. 3 -. sau.. B a i toan 125 (De. - x = b; \/5 - x = c, d i e u k i e n. a, b, c l a so t h u c k h o n g a m . T a co. dUdc. (.x^ + .X - 1)^ + (.x^ + X - 1) - 1 Ta CO bai toan. <. han. 2. =. 0-. cos. , A; = 0 , 1 , 2 , . . ? i - 1-. + ism. 4. 307. va.

(33) • Cac can bac hai ciia so phiic z = i- (cos ip + i sin 0 l a. Tir day, ngitdc lai t a da t i m ditOc nghiem ciia he | ^^2y ^^^3 Z }. zo=^ \/r (cos ^ + i sin ^ ) , zi = - v 9 (|cos ^ + i sin. . ,. • Cac can bac ba ciia so phiic z = r (cos ip + i sin. . T = ^ c o s ^. , r > 0 la. f x - ^ c o s ^. y = ^ s i n ^ 20 = v / ? - ( c o s | + i s i n | j ,. n^Yi.rf3. ^ ^. feos ^. +. ,. isin^i^). i[„h,, .. .. M o t phitdng t r i n h nghiem phiJc f{z) = 0, vcfi z = x + iy, t a bien doi thanh h(x,y). + ifj{x,y). =. 0<^. 'I. y = ^ s i n ^. 'l. ITTT. y=v/2sin—. 12 1. \/3. S a n g t a c c a c h e p h u ' d n g t r i n h b a n g e a c h l u y t h t e m o t so phiJc cho tru'dc.. . Gia sit x + yi. 2. + -T'-. Ta (/T/tfc 6di i o d n sau.. ^x'-Zxy'+{ixS-y')i. = \. ^-^i. , >. 5 5 v / 3 . '"^"^ ^^•x-' + 3x^yz + 3 x y ^ r + y^i"* = - + — i u. | f^^^ ^xy^ = 1. <^(x + y i ) ^ = 5. Giai he nay t a t i m diMc x va y, t i t do c6 z. T u y nhien, c6 t h g t i m z bang each k h a i eSn bac ba ciia 1 + i nhit sau : T a c6 1 + i = \/2 ^cos ^ + i sin. (2) '. 2x^ - 6xy2 + (6x2y - 2y^)i = 5 + bVSi. + i<:>x^ + 2,x^yi + 3xy^z^ + y^i^ = 1 + i. ^x^ - 3x2/2 + (3x2y - y^)i = 1 + 1. | ^ ^ 2 ^ _ ^ 2 y 3 I 5^3. G i a i . N h a n h a i ve ciia (2) vdi i r o i cong vdi (1) t a dudc. Do t i m can bac ba cua so phiic 1 + i, t a t i m so phiic z = x + iy, x- G E , y € 1 sao cho. .. v/3\. (l. 2 + T^. Vay x + y i la can bac ba ciia so phiic 2 = 5. \. 1. v/3. 2. + V/. = 5 (cos -. 7r\. + .sin-J. M a z CO cac can bac ba la. Vay cdc cfin bac ba ciia 1 + i la. • X. zo = zi. =. V V2 VV2. (cos -. + ^ sm -. cos. J=. v^2 (cos -. h I sin - 2 — — f+47r. 22 = Y V 2 (cos ^ - y -. h I sin. f + 47r\. + z sin -. = V2. Ztt = S/S ( c o s ^ + i s i n ^ ) ,. j,. cos — + t sin —. ITTT , . .. ^ — j = V 2 (cos. 308. .• o'l). v/3 '. B a i t o a n 1 2 7 . Giai he phuang trinh. (x + iyf ^l. ^cos-. i h(x,y) = 0 \{x,y) = 0.. Nghia la m o t phirong t r i n h nghiem phiic, b i n g each tach phan thuc va phan ao luon CO t h e dita ve he phvtdng t r i n h . 4.6.1. '. V i d u 1. Xet so phiic z = 5 (cos ^ + i sin ^ 1 = 5 V 3 3/ la so phiic thod man dieu kien (x + yif = 5. =. X. =. — ^ +I s m j. zi = Vr [cos. la. .. 177r\. —. + i sin. S/5. ,. ( V /. —. Z2. ^ + 27r cos. h i sin 3. ? + 47r. cos^^. + i sin. f+27r\ — 3 y f + 47r\ /. 309. 77r\. cos — + i sin — V 9 9 ; / 137r . . 137r^ N / 5 cos — — h 7 sm — \ 9 ,.

(34) Vay cac nghieni ciia h§ phiMng t r i n h la X. TT. =. 3/F. cos — ^. 3/F. ^TT. .T = v^Scos. .T = v o c o s —. '. y=\/h cos -. g da cho CO 4 nghiem :. y=. 137r. y = v^cos -. x = x/2 cos. dung. t i n h (tang cap cua he, r o i dua ve phuong t r i n h da thufc bae ba. iyY =. +. x^ + 4x^yi + Gx^fi^ + Axx/i^ + \fi^ = v^S + i K.:. 4.6.2. x" - 6x2y2 + y^ = ^3 4(x'-*y - y^x) = 1.. = V^ ( 1 ). x'-6xV+f. B a i t o a n 128. Gidi he pfMOng trmh. .r^y - y^x = - .. (2). z"* = (x + y i ) ' = x"* + 4x''yi + (Jx^i/P. 6xV + 2/'') + 4 (x^y -. + 4xi/i^. D/. (cos -. + t sm -. ] , v/2 (^cos —. + . sin 37^. 257r 4/- / 257r V 2 cos - + i sill 24. . Vay z = x +. ' 24. X. yi ^ ^ ( c o s i L. + yi. x + yi= X +. cos v/2. ( cos. ^^^^ ' ''"'^80. —. + i sin. j. 24 257r. —. . .. + i sin. 1377. 24. 5. 7V5i. z. z. . ( zi + Z2 = 7 I 2122 = 7 ^ i + 5.. ^. •^x • ' •+ yi + <^x +. 377r\ 24. X.. + y^. 7y/5iz. zz. zz. 7y/Ei. = 0. H. x2 + y2. x2 + y2. x2 + y2 7\/5x - 5y x2 + y2. rr~l. x^+y^ y+. -7+. 5x + 7 v ^ y. + y^. = 7. 7\/5x - 5y. 5x + 7\/5y. 5x + 7y5y. X +. X,. = 7. 7y5x - 5y x2 + y2. ,.. /=0. 7 =0. : 0.. a CO bdi todn sau.. . . 257r + I sin —. 4/- / 377r . 37?: yi = V 2 I cos -r-r + I 24 24;. 5z. fc:#+ l 4 ^. x + y. +. + '""24). 137r. 310. 24 •. id sli z = X + yi, vdi x, y G R. Khi do phUdng trinh tren viet lai ,>. Ha\ X +. 3^. Vdy Z] vd Z2 Id nghiem cua phUdng trinh. y^x) i .. phitc 2 i; cos ^ + i sin ^ ) la b. 24. zi = 7- VEi Z2 - s/hi. {. ,4,; 4. ^ + ^'^"'i)' ^^ ' "^. la mot Ccln bac boii cua so plu'rc 2. = ^ ssin i. S a n g t a c c a c h e p h u ' d n g t r i n h tu" h a i s o p h i J c c h o t r i f d c .. + y^i. = ^3 + ( = 2 ^cos ^ + i sin. nen tir he da cho t a c6. cos. + z'^ ~7z + 5 + 7V5i = 0 ^ 2 - 7 + -. G i a i . X e t so phi'tc z = x + yi, vtJi x, y c M. V i. (x''-. X = 1^2. 24 377r. V i d u 3. Xet hai so phiCc z\ z^ nhu sau. Ta CO hai Loan sau.. -. 257r. i;>'W;,. <^ (a;'' - Gx^y^ + y'') + 4 (x^y - y^x) i = \/2, + i. V. y = v/2sin. 2^1 v^sin. y =. ..)/v^..f. ta cd. = ^cos-. y = ^ s i n ^. cos —. Lvfu y . Cach gi^i nay rat doc dao va nhanh hdn nhieu so vdi each su V i d u 2. Til{x-V. 137r. x = ^ c o s ^. X. , a i t o a n 129. Gidi he phuang trinh <. H. bx + 7y/5y X. 7^ =. x2 + y2 7v/5x - 5y. x^ + y^ 311. 7.

(35) Hifdng dan. Tit each sang tac he phuOng t r i n h ta thay he c6 hai nghieni 1^ (. 7. X =. y/5.. y =. x + B a i t o a n 130. Gidi he phUdng. trinh. 3x. -. x'. ,21 y. „ .. <2 iz == ixix - - y,. + \^ — \zf =. J?. nghiem. u +. z.z.. y. + i. ^ = 3 x^ + ir. -. ^. X'^. ,. +. ^. =. ^•. \ .. VI+. 6it. J. (iv. —. \. + v^ J. 2 = 3\/2 + z u. 3x-y __9. ,. x2 + y2. x2. ,. .9^. + y^. 3(aH-yi). = 3. X -. yZ +. y - xi X ^—+ y'^ST = 3. —3-2 7 ^ -)- y 27i. •. 3^2. = 1.. V 2 + 3 ^ + ^ == 3 < ^ !2= > + z +31 i_ + 1 = 3 ^ ^2 z.z z z. 3 1 + 3 + i = 0.. ^I. a2 - 62 ^. -3. f. -3. ^' +. y. 6 /1 + ^ V x + y 6 '. r ^ i a i t o a n 131. Gidi he phuong. trinh. V. ^/y. Ta CO A = 9 - 4(3 + i ) = - 3 - 4 i . Xet so phiic a + bi thoa m an dieu kien. ^ {. \ CO bai todn sau. = 1.. ^ ^Q 1 -. ' u = s/x, V = sjy, ta dU0c •. lay. - 3 - 4i = (a +. =. —. 6. x + 3y. 9 +. 1+. 6. Lay (1) t r i t (2) theo ve t a dUdc ^ - y- +. +. '*. = 1. -. i = 3. 3\/2 + z. =. V. trinh viet lai. + v^ V. (2). J. •ifi + v'^ +. 4=>u +. (1). + 2i trinh. cua phitany. (3\/2 + i ) z + 6 = 0 < ^ z + - = 3\/2 + i <^ ^ + -. He phitdng t i ' i n h da cho viet lai x. Z. (Jja sTi z — u + vi, khi do phuong. ^ 0. Xet so philc z = x + i y . K h i do _. va(I 22 ^(i -. +y. x^ + G i a i . Dieu kien x?. Vi du 4. Xe'^ hai so phiic |. =0. .T. x +. = V2. y ). ^ «. B a i t o a n 132 ( H S G Q u o c g i a n a m h o c 2 0 0 6 - 2 0 0 7 ) . Gidi he trinh. phuang. K h i do Vay (a;. -. = - 3 <^ a"* + 30^ - 4 = 0 ^. (1; - 2 ) ;. = 1. a = ±1.. (a; /;) = ( - 1 ; 2 ) . Do do A = - 3 - 4i c6 hai can bac hai. la ± ( 1 - 22). Suy ra. y. + 3x. iai. Dieu kien x > 0, y > 0, y + 3x 7^ 0. Dat u = \/3x > 0, u = hay vao he, t a diTdc 1 1 +. 12 n2 + i;2 ; 12. 73 V. = 2. u—. = Q. V. +. 12u + v' I2v. Nghia la X -. yi =. X -. yi. =. 2 1+. i. ^. i. x;y| = f 2 ; l ) x;y) = ( l ; - l ) .. Vay he phudng t r i n h da cho c6 hai nghiein la | ^ Z ^ 312. > 0.. •. + v^. = 2v/3. (1). = 6.. (2). ban phitdng t r i n h (2) v6i i , sau do cong vdi phurtng t r i n h (1) t a ditdc va. r x = 1 y = -l.. u + vi. \. = 2v/3 + 6 i . 313. (3)'.

(36) /. Xet, so phi'rc z — u + vi, v6i u > 0, v > 0. K l i i do (3) viet lai 2 -. i + S Ta. V. ^. z.z £. .. =. B a i t o a n 134. Giai he phUdng. 12 z - ^ = 2(v/3 + 3 0. 2\/3 + 6i. =. ^. 2 ^ - 2 ( v / 3 + 3 0 2 - 12 = 0.. (4). CO. A' = {V3 + 3if + 12= - 6 + 6v/3/ + 12 = 6 + oVSi = 12 Vay A ' '. CO. 2+. 2. T i t (4) t a. +. M. ». y. t. =. 1 +. xy. trinh. G i a i . Dieu kien x - 1 , y 7^ - 1 . He treu t u y la doi x i i n g loai I nhirug bac kha cao, vice dita vc 5 = x + y va P = x y c6 the gap nhieu kho khan. Nhirng neu t a biet bien d o i. t h i bai toan t r d nen ddn gian. Tritdc het t a chiing m i n h (1). T a c6 - ^. + ^. y + 1 \/3 1 . = ± (3 + v ^ i ) . ^ + 2^. = ±/l2. <»x^ +. X +. X. = l ^ x { x + l ). + y{y+l). = {x +. •. l){y+l). 1. ^. + y^ + y = x y +. X. + y+ 1. ' >. x''^ - x y + y^ = 1.. Vay (1) dung. D a t — ^ = a, -^^r^ ' ^. phitdng t r i n h m6\. CO. 2 = v/3 + 3/: + 3 + v ^ i = v/3 + 3 + (3 + \/3)« 2 = \/3 + 32 - 3 - v/3i = v/3 - 3 + (3 - v/3) 2. Do ti > 0 va. V >. 0 nen. u= v/3 + 3 v = N/3 + 3. [a^. ^l2a2-2a = 0 ^. + b'' = l. 2/= (v/3 + 3). 1. 2. = 4 + 273 y = 3 (4 + 2v/3) .. B a i t o a n 1 3 3 ( H S G q u o c g i a - 1 9 9 6 ) . Giai he phmng ^3x1 +. 1 .T +. ( 1- ^. \ y. X. trinh. ^. + y + xy = 3. 4. 5y + 9. (1). ,. x+ 6. . „ , = ^ .. l + (x + l ) ( y + 2). ;. (2). 2. Y tifdng. T a c6 (1) <i=^ (x + l ) ( y + 1) = 4. C) (2) cung c6 x + 1 va y + 1 nen t a nghi t d i dat a = x + 1; 6 = y + 1. T u y nhien m a u chot ciia bai toan chinh trinh. = 2. ) = 4x/2.. la dat c = ^ de tvr (1) c6 a6c = 1. K h i do t a lai c6 menh de sau : 1 "-^""-^^. l+a. 1 + ab^ l + b + bc^. Phu'dng p h a p b i e n d o i d a n g thu'c.. Nhieu bai he phitdng t r i n h t u y nhin phi'rc tap nhimg c6 the giai bang nhGng dang thiic ddn gian. M a u chot giai nhitng bai toan dang nay l a t a phai nhin l a quan he giifa cac an so, tvr do lap nen nhirng hang dang thirc t h i c h hdp. M o t so dang tln'rc r o t h g khong quen thuoc, nen phUdng phap nay doi hoi k i n h nghicni va sir t i n h y.. 1 _ ^ l+c+ca. T i t do t a d i den Idi giai. G i a i . Dieu kien x 7^ - G , y 7^. 314. [ ( a ; = ( 1 ; 0)-. Tir do t i m r a nghiem ciia he l a (x; y) = (0; 1); (x; y) = ( 1 ; 0). B a i t o a n 135. Giai he phmng. He da cho c6 nghiem duy nliat | ^ ~ 3 ( 4 + ^ 7 3 ). 4.7. I. +. = 12(cos- + . s m - j .. hai can bac hai la. ± \ / l 2 ( c o s ^ +ism'^^. 9. X. D a t x + 1 = a, y + 1 = 6, - = c. T i t (1) t a. CO a?>c = 1 va 1 + a + a6 = xy + 2x + y + 3 = 1 + (x + l ) ( y + 2), y + 1 5y + 9 , l x + 1 ^+6 l + 6 + 6c = l + y + l + ^ ^ = l + c + ca = 1 + - + - J - = Ta CO he phUdng t r i n h. {. abc = 1 l + a + ab ;. 1+h + br. 315. 1+c + ca. 2.

(37) VSi abc = 1, ta c6. ^.8. 1 1 1 + + 1 + a + ab 1 + b+hc l + c+ca 1 + a + a6 l a ab = 1. 1 + a + ab'^ a + ab + abc^ ab+ abc + a?bc l + a + ab x+ 1 <^x = l , d o d 6 t / = l . Vay he c6 Thay vao phitdng trinh sau dUdc 1 =. +. he phmng. xy = 3. + 15x2/(x + y) = 32.. 2 +. Y tifdng. Ta can tim moi quan he giifa cac hang t i i trong he. Muon vay ta chii y tai hang dang thiic (x + yf ^ + y^ + bxy[x + y){x^ + xj/ + y^). Giai. He da cho duoc viet lai nhu sau: + yl + xy = 2> \x'5 + 2/^ + 5.3xy(x+y) = 32. ' {xl. :. ^ / x 2. + y2^xy =. ^. + iy5 + 5.Ty(.x2 + . r y +. r x^ + y = z. + y =. z. + y =. ^ ( x. \ + y) = xy. ^. \. z ^ C x. 1 (x +. = xy. B a i t o a n 137. Giai he phUdng. + y) =. y)2 =. 1. 1. <=>^. liJ (1). x''. Bai t o a n 139. Gidt he phuang trinh |. /. + 2/^ + + y^ + yz -. (x + y ) 2 - 2 ( x + y ) + 2 2 - 3 y ) 2 - zix + 1 = 0.. Dat H = X + y , I' = X - y =j> X =. (.). x y z. trlnh. 1. y"'. 2 xz. z^. xy 1\ 1. z^j. (2). z^. 2/2. =. 0^. 2. /. feW. 'Jf^. =^x•^y = ^ =. 0.. + 2xy - zx - 2y = 3 2xy = - 1 .. zx -. X. 1. ^. 316. ^'. z^. -(^ X. T. I. v.. ;. ^'^^. (2) CO nghiem khi va chi khi. -''i. i P ". ''. he da cho CO nghiem la (1;0;2), ( - 1 ; 0 ; - 2 ) .. . f \xy - 18| = 12 •^1 t o a n 140. Giai he phuang trinh I _9_|_1 2 * 3 -2.. '. + 1 = 0. • Vdi 2 = - 2 thay vao (2) ta co w = t; = - 1 . Suy ra { yZQ^ %. = y =:. '. He (1) trd thanh. y=. _. = 0. •^61 2 = 2 thay vao (2) ta c6 « = v = 1. Suy ra | y = Q. 2. 1 z z. The vao he ta tim dudc nghiem (x; y; 2) =. .Srt6t/.t f. \ IV 2 1 - + - + = \x y z) xy 12 —2 + —2 = 2 yz zx xy z'^ 2 1\ = 0. Vy. 1\ / I 1\ zj +1 - + -. -+-. Vx. = 4=». + - ^ + - ^ + —2 +. [x^ /I. 2/. xy M. i «. Den day ta gap mau thuan vdi dieu kien. Vay he vo nghiem.. 32. Giai. Dieu kien x ^ Q,y ^ 0, z ^ 0. Ta, c6 : 1\. z ''^'^. Giai. He da cho turtng dirdng. xy. 1. + y =. 3. ' i + i + i = 2. - + - + Vx y. ;x •. ^|(,''+|)V^=o. =^{x2';y+xy = 0. \. Vay he c6 nghiem duy nhat (x; y) = (1; 1).. /I. 1. . *x + y i z-. \x -. (1). ^--'rh. Cfiai. Dieu kien xyz 7^ 0. Khi do he da cho vi6t lai ( x. irinh. so he khong m a u m L f c .. 0 a i t o a n 138. Giai he phuang trinh. nghiem duy nhat la (x; y) = (1; 1). B a i toan 136. Giai. Mot. x2. Qiai. Ta co |.Ty. - 18| = 12 - .x^ =^ 12 -. xy^-d+^y"^^. \x\ 2^3 |y| 317. .T^. > 0 ^ |x| < 2 v ^ |x| > 2v/3. '•'. •.

(38) Hifo'ng d i n . Ta CO (1) o 4x (2x2y - 2x2) _. Suy ra | x | = 2s/3 => xy = 18. K h i x = -2\/Z t a c6 y = -ZVZ. K h i x = 2y/^ ta CO y = 3 \ ^ . Thilf lai, thoa m a n he da cho. Vay nghiein cua he la. ^ 8.r3 (y - 1) - (4x + 1) (y - 1) = \ / f e + T ( y - 1) y. (x; y) e { ( - 2 v ^ ; - 3 v / 3 ) , ( 2 ^ 3 ; 3 ^ ) } .. + sJ2x + y + 2xy + 1 = 1. S/S^in: = 8x3 - 2y - 1. X >. =. 1. ! i',;,-. (1). B a i t o a n 1 4 3 ( D e n g h i cho k y t h i h o c s i n h gioi c a c truTcJng C h u y e n k h u vu'c D u y e n H a i v a D o n g B a n g B a c B o n a m 2 0 1 0 ) . Gidi he-phudng trinh i-. (2). 0.. (x - y) (x2 + x y + y2 - 2) = 6 In. y+_v^2^. (1). x''y — 3xy — 1 = 0 .. G i a i . T a c6 (1) ^ (2x + 1) - 2 (y + 1) + v/(2x + 1) (y + 1) = 0.. I. (1) ^ (x - y) (x2 + x y + y2 - 2) = 6 in. =0. „3. X"^. ^. (2). G i a i . T a c6. Dieu kien (2x + l ) ( y + 1) > 0. M a x > 0 nen y + 1 > 0. B d i vay. (1) <^ (\/27TT - v / y + l ) ( \ / 2 x + 1 + 2 vVn). . ,.;. 8x3 _ 4^. _ 1 = ^ 6 x + 1.. B a i t o a n 1 4 1 ( D e t h i h o c s i n h gioi c a c tru-dng C h u y e n k h u vuc D u y e n H a i v a D o n g B a n g B a c B o n a m 2 0 1 0 ) . Gidi he 2x-2y. _^ J) ^ s/6^TT (y - 1). ^ J) ^. \/2x + 1 - -v/y + 1 = 0 <^ y = 2x.. J/+. ^2/2+9. \^x + y ; ; ^ 2 : r 9 /. T^M^). - 2x + 61n ( x + v/x2 + 9 ) = y3 - 2y + 61n ( y +. .. (*). :et ham so / {t) = -2t + G In + Vt^Td^, « € M. Ta c6. T h a y vao (2) t a diWc ^ 6 x + 1 = 8x3 - 4a: - 1. (g^ j^Y)^. ^ 6 x + 1 = (2x)^ + 2x.. /' (/) =. G. -2 +. = 3/2 +. + 9. v / ^. 3;-. '. H a m so f{t) =^t^ ^-t dong bien tren M. Do do ,c6. (3) <^ x/feTT = 2x. 4x3 _ 3x = i .. (4). De thay rftng x > 1 khong la nghiem ciia phirdng t r i n h . X e t 0 < x < 1. Dat X. = cos a, vdi 0. < Q. < - . T i t (4) t a co. 2 V72T9 <2 + 9 27. 2 3. 2 ^ v / ^ ^. 1. 29 3. 26. 29. / ^ ^ 27 29 , 26 29 29 >1+ = 3 - 3 3 3. 29 „ =0. 3. fSuy ra / ' (0 > 0, V< e M. Vay ham s6 / dong bir-n va lien t n r tron L M a. W y he c6 nghiem (x; y) = ^cos ^ ; 2 cos. B a i toan 142 ( D e nghi O l y m p i c 3 0 / 0 4 / 2 0 1 1 ) . 4x (2x2y - 2x2 + 1) _ - 3x = 4.. ^. 1. (*) ^. Do 0 < a < ^ nen " =. „. + v/t2T9 + v. , 26 , 2 «N > 1 + — ( r + 9) - 2 7 ^ ' ' cos 3a — ^. 2. Gidi he phMng. (43. ^_ i ) + j ^ ^ / 6 ^ q n : ( y - 1) (1) (2). 318. . trinh. / (•^•) = fiy)^^. = y-. fj. I T h a y vao phiWng t r h i h (2) ci'ia he ta ('6 x*^ - 3x2 _ 1 = 0 .. (3). ' Dat x2 = u, dieu kien u > 0. K h i do u3 - 3u = 1. 319. (4).

(39) Dat u = 211 (dieu kien v > 0), thay vao (4) t a dUdc 4ir-3v. =. Vdi. -.. (5). 9. 9. v/3x + 4 + (x +. 0;. (x;y) =. , V = COS • „. 9. Do dieu kien v > 0 nen chi nhan i ; — cos ^. Tit do x = ±./2cos - . He c6 hai nghiem la. x^ - y^ + if_. G i a i . Digu kien { Gidi he phmng. (2012 - 3x) y / I ^ + (6y - 2009) y / S ^ ^ = 0. (1). 2^/7.7; - 8?y + 3V/14.T - 18?/ = .r^ + 6.T + 13.. (2). dUdc. \&\ =. 1\. , (x;?y) = ( - l ; - l ) .. 2;. - 3x. -2. .. = 0. (1). x2 + y r ^ ^ - 3 v / 2 2 r ^ + m = 0. (2). -W2cos^;-y^co^^. B a i toan 144 ( D e nghi Olympic 30/04/2012).. — + 1 > 0, vay (5) cho t a v/Sx + 9 + (x + 3). B a i t o a n 1 4 5 . Tnn cac gid Iri thuc ciia m de he. phuong trlnh sau c6 nghiem. , 2cos|; j 2 c o s ^. 2) +. cac nghiem x = 0, x = - 1 . Vdi x = 0 t a duoc y =. J. 7n. = COS. 4 thi. y = - 1 . T h i i lai t a dUdc cac nghiem ciia he phUdng t r i n h da cho la. Theo ket qua bai toan 3 d trang 119, suy ra cac nghiem ciia (5) la V = COS—,V. X <. tnnh. ijf/^^. ^ { 0<|. V.'-. , ,^. < 1 . ' ^ ^°. (1) ^ x^ - 3 x = ( 2 / - 1 ) ^ - 3 ( 2 / - 1 ) .. (3). x2+ \ / r ^ - 3 \ / r ^ + m = 0 < ^ m = 2 \ / r ^ - x 2 .. Xet ham so f{t). (3) Xet h a m so g{t). = (3i + 2000) ^Tt, V( > 0. K h i do. suy ra h a m so / dong bien tren [0; + 0 0 ) . D o do /(4 - x ) = /(3 - 22/) ^ 4. -. X =. 3 - 22/ <^ y =. (4). Thay (4) vao (2) t a dUdc 2sjlx. - t, Vi e [0; 1]. V i g'{t). =. -1. - 2 < x^ + v / l -. - 3y/2y -. 2v/3x + 4 - 2 (x + 2)] + [3v/5x + 9 - 3 (x + 3)] =. V 7 x + y + V2x + y = 5 V'2x + y + x - y = 2.. + x. G i a i . Dieu kien m i n {7x, 2 x } > -y. D a t s/lx + y = a va v/2x + y = h. TiT he. -2x(x+l). -3x(x + l) H — 7 7 = x ( x + 1) v / 3 x T 4 + (x + 2) \ / 5— ^ ^ ^+ (x + 3) v/3x + 4 + (x + 2). +. 320. 1 v'Sx + 9 + (x + 3). Gidi he phuang. trinh. <=i>2\/3x + 4 + 3v/5x + 9 = x^ + 6x + 13. 1. - 1 < 0 nen. < 1.. B a i t o a n 146 ( H S G Qu6c g i a n a m h o c 2 0 0 0 - 2 0 0 1 ) .. - 4(x - 1) + 3v/l4x - 9(x - 1) = x^ + 6x + 13. <^x(x + 1). = 2vT^. (4). ham g lien tuc va nghich bien tren [0; 1], suy ra tap gia t r i cua h a m g la [t/(l); g{0)] = [ - 1 ; 2]. Vay he da cho c6 nghiem 4^ phUdng t r i n h (4) c6 nghiem tren doan [ - 1 ; 1], ti'rc la m thuoc tap gia t r i cua h a m g, hay - 1 < m < 2. L\iu y. Tfr bai toan tron t a con t h u ditdc ket qua : Neu hai so thitc x G [ - 1 ; 1] va y G [0; 2] thoa m a n dieu kien - y^ + 3y^ - 3x - 2 = 0 t h i ,. / ( „ = 3v/;+5i±|lL»>o,v,>o. (3). ,. l a m so f{t) = t'^ - 3t c6 f'{t) = 3*^ - 3 < 0, G [ - 1 ; 1] nen / lien tuc va " g h i c h bien tren doan [ - 1 ; 1]. V i vay, t i f - 1 < x < 1, - 1 < 2/ - 1 < 1 va tir (3) t a C O / ( x ) = f{y -l)<^x = y- l-i^y = x + l. Thay vao (2) dUdc. G i a i . Dieu kien x < 4, 2/ < 2' 7x - 8?y > 0, 14x- - ISy > 0. T a c6 (1) ^ [3(4 - x ) + 2000] \ / 4 ^ = [3(3 - 2?/) + 2000] sJZ - 2y.. ' •. + 1 = 0.. phUdng t r m h da cho t a C O he (5). a + 6= 5 (1 6 + x - y = 2 (2. 321.

(40) Nhan thay. - 6^ = 5x. Do do. 4=> cos'^. {a + h){a-b). K i t hdp (3) vdi (1) suy ra 5. -. = 5x'^"=i'\-b. h = X. ^,. = x.. (3). the vao (2) ta ditdc. A C 3 B ^ + cos'' ~ + cos^ —- = ^ <^ cos A + cos B + cos C = ^ 3. <^cosyl + cos/3 +cos(7r - (/I + S)) =. + x - 2/ = 2 < » x = 2 2 / - l .. (4). The (4) vao phiTctng trinh thii hai cua he ta dUdc. XI + X2. a;20i3 + 2:2014 = a;f 14 X20U. =. 9 74. ^. "'2"' V 7. .. ". ^. I'. Y T =. C 7' ^ ' ^ i 0<A,B,C<7r.. Thay. 2X > X2"14 ^ 2 > X^"!'^ (vi X >0). j. Af B C\ C A 1 - tan —tan — ^ tan - (^tan - + tan ^ = ^ " ^^n - tan ^ ^ tan ^ = ^-2. C. -1. tan — + tan — 2 2 + _ = _ _ _ + ^ . ^ ^ ^ + 2? + C = 7r + ^2^.. Do dieu kien 0 < / I , Z^, C < TT ta c6 ^ + Z? + C = TT. PhifOng trinh (2) trd thanh. •+. l + tan2^ ' l +. 1. tan^l. +. 1. ' 1 + tan^ | 322. ^. (2)-. Tft (1) va (2) suy ra X^oi^ = ^2013 = 2. Ngliia la ta c6 X I = X2 =. B. (1). Lap luan mot each titdng tit ta cung di den 2 < y2013. , B B C C A tan - tan - + tan - tan - + tan - tan - = 1. 1. 0. Lai do ton tai k e {1,2, ...,2014} sao cho xk = X nen suy ra. vao (3) ta dudc. <^tan- = cot(^- + - J ^ -. jt;. 2 X > xfi4_VA;= 1,2,..., 2014.. (2). (3) f. ;: ;. • V,. Titdng tit doi vdi cac phudng trinh ciia he ta c6 :. (1). *„. „.. • V T ". , L. X,=xf'^. 2X > X I + X 2 = x f i 4 .. Hifdng dan. Do x, y, z difcJng nen. Hut. +. Giai. Gia .si't ( x i ; x 2 ; . . .;x2oi4) la nghiem ciia he. Goi X la gia t r i Idn nhat ciia cac so X j , i = 1,..., 2014 va Y la gia t r i be nhat cua chung. The thi tit phUUng trinh dan ta co : ,, ,. B a i toan 147. Tim cdc so thuc dudng x, y, z thoa man he phuong trinh. + yz ' y + zx ' z + xy. = x f 14 x f 1". X2 + X3 =. Bai toan 148. Giai he phuong trinh :. \. K X. ,,ti:i;=. 4=> cos >1 + COS B - cos(yl + i?) = ^ - ^ y l = B = C = ^ (do bai toan 161 d trang 111). Vay x = y = 2 = -1. j. . ,. v / 5 y - 2 + y - l = 2 ^ v/52/ - 2 = 3 - y ^ / 2 / < 3 . 5y - 2 = 9 - 62/ + y2 J y <3 ^ 11-^77 I -11?/ + 1 1 = 0 = —2—• S: / The vao (4) suy ra nghiem cua he la (x; y) = I 10 - v/77; n-V77\. •'. -. • ''"^ '. Thit lai, ta ket luan : (xj; duy nhat ciia he.. •••. = X20H. X2;X2014). =. = ( ""v/2;. Bai toan 149. Giai he phuang trinh { ^ (v^+y. v2.. '"'s/2;...; '"'s/2) la W^^). 1^. v/x + y + V x = G i a i . Dieu kien x > 0, x + y > 0. (1) titdng ditdng vdi :v {s/TTYi. + s/TT^. X +. - V. nghiem. 3 (1). 3.. = ( T + y) - (x + 3) ^ x = ^/T+Ti - Vx + 3.. ay (2) trft (3) dildc Vx + 3 = 3 ^ 2 X H / 3 + 2\/X2 + 3 X = 9 323. (2) (3).

(41) PhUdng t r i n h t h i l nhat ciia he viet lai. 2x-l)f2y-iUl.. Vay (x; y) = ( 1 ; 8), thoa man dieu kien. B a i t o a n 1 5 0 . Gidi h$ phudng trlnh. + l=4xy. \. = 1. (1). \. ( l ) < ^ x + \ / l + x2 = V ' H - 2 / 2 - y < ^ a ; + 7 j + \ / l + <^ x + y + —. =. fix). = 0. >. /(2) =. Bai toan trinh. X + y = 0.. Khi. X. > 0, t a. CO. /3 ^ 2 + 1 \ x2. W- + 2+ = -4 + - + V^ x'^ x. ^JtVi = t-\. K h i X < 0, ta CO - J-. 152. (China. \/u. + 2 =. u. \. 1 \ + - + 4T. X. X. -. ^/ V. yJ. > - 2. Girls M a t h OIympiad-2005). 1\. X +. -. 2. Gidi he phudng. + -. (1) (2). ( I ) ^ 5 ( . x 2 + l ) ^ 1 2 ( y 2 + 1 ) ^ 1 3 ( 2 2 + 1). -4 x^. x2. (2). dan den T i r (2) suy X , y,. X. X'^. - 4.. B a i t o a n 1 5 1 . G i d i he phicang trlnh [. V. 2y -. G i a i . Dieu kien x 7^ 0, y 7^ 0, z 7^ 0. T a c6. X^. Dat i = - +. 2x - -. / 1\ 1\ = 12, y + =13 \ xy + yz + 2x = 1.\J /. + 2 + ^ = - 4 + - + -^. Dat u = - +. V dan don -. = x2 f-4. / ( y ) > / ( I ) = 1 =^. Vay (3) <^ (x; y) = (2; 1) va do la nghiem duy nhat ciia he.. T l i a y 2/ = - X vao (2) diMc x\/3x + 2 x 2 + 1 = - 4 x 2 + 3^ + 1 ^ ^. X ,. 2. x2->/l+y2=Q. 0. (.T + y) ( V l + x2 + V T T y ^ + X -. (3). tang. Do do v6i x > 2 va y > 1 t h i. H u - d n g d i n . Dieu kien 3x - 2xy + 1 > 0. Vi t + Vl + t^ > 0 nen. x'^ - 7/2 = —. VliV. X yJ 2 = 2 i - - , vdi t > 1. De dang chiing m i n h ditoc / la ham. Xet h a m so f{t). (2). + 3x+l.. ;. A. ,),. |, svYr^ir^. + v / T T ^ ) (y + y r r ^ ). f. i. ciing dau. Ngoai ra, neu (x; y; z) la nghiem ciia he t h i. ( - x ; - y ; - 2 ) cung la nghiem ciia he, nhir vay t a chi can t i m cac. nghiem. ditong. T i r (1), suy ra t o n t a i a, /?, 7 € (0; TT) sao cho. (^^^ " l ) (2y^ - l ) = ^^^y. a +. /3 + 7 =. 7r,. a /3 x = t a n - , y = tan 2^ ^. 7 = tan-.. [ x2 + y2 + x y - 7x - 6y + 14 = 0. T h a y vao (2) t a ditdc. G i a i . Phitdng t r i n h thiJ hai ciia h f viet lai x2 + ( y - 7 ) x + y 2 - 6 y + 14 = 0.. 5 ( l + tan2^) 5(1. (1). Dieu kien de (1) c6 nghiem la. tan-. (y - 7)'^ - 4 (y2 - 6y + 14) > 0. 3y2 - lOy + 7 < 0 ^. 1 < y < -. 3. sin a. y2 + (x - 6)y + x2 - 7x + 14 - 0.. (2). sin. ft. 324. X <. — .. 7 tan-. sin. (3). 7. Theo d i n h l i ham so sin, t h i tir (3) suy ra a, (i, 7 la ba goc ciia mot t a m giik" CO do dai cac canh tUdng iltng la 5, 12, 13. T a m giac nay la t a n i giac vuong 7 =-,. 3x2 - 16x + 20 < 0 <^ 2 <. 13 ^1 + tan2 ^ ). 13. CO. Di^n kien d^ (2) c6 nghiem la. 1 + tan2 0 tan-. 12. Phirong t r i n h t h i i hai cua hO viet lai. (x - 6)2 - 4 (.x2 - 7x + 14) > 0. 12. 5 . ^ 12 sniQ = — , sm/3 = — . 13 13' 325. ,.

(42) • K l i i x^ = y"' <=> a- = t/, thay vac ( 1 ) , t a dvWc sin Q. Ta CO. =. 2t l + 19. 1+. x^ - 3 X = 0 « • X = ^ fdo dang xet x 7^ 0. 1O 13. •H-'""'. Vay ( x : y ) = (^g^ g j. a. TT. -K. ,^. rt. ,. ^. 0 < a < - = > 0 < - < - = i > ( ) < tan ^ < 1 2 2 4 i. Dat. (V. 1 A O. tan 77 = F •. ' ^ i " l O t n g h i e r n c i i a he.. ^. Khi x ' +. = - . ( I l i a p h u o n g t i i n h (1) c h o x . c h i a phUcJng t r i n h (2) c h o y ,. (hrri<. ^:^^ + ^ - y ^ - ^ = 0. 2v. 12. 2v. '{[. 9. o'J. 0 = t a n ^ . K h i do .. ; •'. 5. (5). 9. 2 3 12i;'^-2G« + 12 = 0 = ^ u G 1 3 , 2 ' png (5) va (6) ta t h u diwc. Taco y. 0 < / 3 < ^ = > 0 < ^ < 7 = ^ 0 < tan ^ < 1 => tan ^ = ^. 2 2 4 z 1 6. X. 4. y. (7). xy. 9. V a y cac nghieni ciia he la. 1. , ,. 1. l a x ^ + y ' = - l i e n t h a y v a c ( 7 ) , t a d u d c xy ^ - -i^ x ^ y ^ = - . N h i f v a y x^. /1 2. \. \o. J. 6. /. 1. 2. 8. \ \. B a i t o a n 153 ( L i t h u a n i a n M a t h e m a t i c a l phuang. X. va y ' la nghieni ciia phitOng t r i n h J. /. Olympiad. 2006).. r. Giai he. .,. 9. 1. + - = 0 <^ 8/,- ^ 9< + 1 = 0 o. Irinh -xy'. -\x. 1,. (i). = {). § y4 + ^.2 „.. _. (*) =. 0. 1. - ^x^ = 0. Z + xV-yV-fy'^o.. t a dm.K:. ". 8'. 0'-:y)-(l;^) U-;.v) =. ( x ; 0 ) , v 6 i X- ^ 0 va ( 0 ; y), v(n y 7^ 0. T i e p t h e o , gia s u r 7^ 0 va y 7^ 0. N h a n. pliUdng t i i i i l i (1) vdi^x^ v a nhan phUdng t i i n h (2) v 6 i. - x'y'. 1 8. (2). G i a i . D 6 t h i y {x\y) - ( 0 ; 0 ) la nghieni ci'ia he va he khong c 6 nghieni dang. x« +. Do do. / = 1 _ 1. (^;i).. j i j C a c u g l i i e i i i e i i a he l a. ( x ; y ) = ( 0 ; 0 ) , ( x ; y ) = ( g ; ^ ) , ( x ; y ) = (1;. ,1 ( x ; y ) = ( i ; 1).. (3) (4). L a y p l n t d n g t i i n l i (3) tnx p h i l d n g t i i n h (4) ve t l i c o ve, t a t h u d u d c <',X, (f:),i. y' = l { x ' - y ' ) ^ { x ' - y ' ) { ? ^ y ' - l ) = () • x ^ - y-^ = 0 x^ + 2y^ =. 326. 9. 327.

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