De thi HSG Toan 9 nam 2014

<span class='text_page_counter'>(1)</span>so Grio nuct vA DAo r,{0 I}EN'TITE. Tt{I HOC SINII (;I0I L()I'9'I'RUN(; IIOC CO S() t,{AN,{ IIOC 2(}t3 - 20t4. DE. TIri'i gian:. MOn: f},iX phrit (kh6ng k0 lrhf t etO). 150. Ciu I (5 di0rn) lu) I{Lrt r:t.rn biCu thric: A =. l-t-. {. \. h) Cho. l.r = 5 + viiJ . Ti'h 5-. .. ll:x---1. Vx-JIs-Jx*Jtu t/r'r/x -18 rtvi x)Jts :l. tri cira bieLr thirc: +) 2? -Y -r+r. g iri. 1\. Chu 2 (4 di6m). -. 2.r -l- 2023. -0 Giai he phuong trinh. :. Ciu 3 (3 di6rn) Tirn nghi0rn nguy6n duong cua phuo-ng trinh: r, + 1.., - I 3(x - l/) = 0 Cf,u 4 (3 di0rn) CtLo hinh thang ABID (AB // CD), giao (i€rn hai dubng tlreo la O. Dubr-rg rh3 g qua O song song vdi AB cit aO va BC IAn luqr rai M vd N. a) Chung minh:. Y*Y AB CD =r.. b) BiCt Soou = m2i Scoo = n2 . Tinh Sorc, theo m vi n (v6i Sn,,u, Sc,rr, Sno.,, lAn luqt la dign tich tam gi6c AOB, di0n tich tam gidc COD, dion tich t['gi6c ABCD). CAu 5 (5 di6m) Cho tam gi6c nhon ABC n6i ti6p trong dudng tron tdrn O vit IdB = 450. K6 c6c AA' vd BB'. Goi H Id truc tdm tarn giiic ABC, M r,,ti N tuong u,ng la truns diOrn ag vi CH. a) Chring rninh ring A'MB'N ld hinh vu6ng. b) Chring rninh ring A'B', MN, OH d6ng quy. du"crng cao. HET.

<span class='text_page_counter'>(2)</span> SO GI,qO DIIC VA DAO TAO TiEN'I]I1E. HTJoNG DAN crrAnn. xV. rut cHoN rroc. srNn (;ror. Lop e rnuNc Hec co sO ruAm Hec 2013 lll0n: TO,.ilV. -_. Ciu la. NEiiruiig-. Di6rn. bi€r-r thu'c:. t. (2,5d) r-\. I. /. --. A:{ Vx- JtS -Vx+JtS 1rlt, ./x' -lti \ /' /--.. rtr.i r. ). vTs. Taco: A: :(Jx-vTs -J*. *JT). (- -. Ji,r). o' : (* -vffi *x + J8 -z,F - rs;(.. a' : (2" -2J.' -ls)(...6t- 8). 0.5. rl=;. 0.5 0.25. 1. A2:2(x'-x'+ts). 0.5. flt2 -Jo -tr. Nhung do theo gi6 thitit ra rh6y. o. tT. : (J.-Jr8 - J,..Jtr)J* - J-. Cho. ru. 0.25. .9. 0.5. 2x:5 +..13 . Tinh gi6 tri c0a bi6u thf;:. B:x5 2x =. -. -\. +A:-6 1b. 5* JiJ. *. 5x4. 4x3. - 2; - nx *. :)2x- 5 - .,fiJ (2* -5)' = 13 =. (2.sd). 2023 0.5. 1y2-5x+3=0 B. :x5. -. 5x4. +. 0.5. 4; -2; -. t2x +2023. B_. (r'-. ). 5xa. - ;hi i;iii-. :hri chri. Rrit gon t. -zatl. +3x')+(r,. B; (; -5x+:). +. ulal. trinh. B:2014. h. -5-r2 +3x )+. (r*, -15x+9)+. 2014. *.(? -5x+r) + 3 (; -5x +3) +2014 :. 0.5. 0.5 0.5. (4d). I l i. I.

<span class='text_page_counter'>(3)</span> - :.r')' - I T("r'' -. 9.1,'. - +(t. ). + -i.r.)'. ). ,r+J.l'_r__-) ' .r. I. 5(.r. *3r,. - :r,)' - I 7(x' * 9,),' ) -. u( t. * i.,,)' =0. (/). -r+3r,* 3 *=5 .r-iv ki6nxiicdinh:r+3_i,.. Di6u. x=--r\.thi. Ndu. (/)o{. f. --s4r;.r'' =. I |. I. I. o. I. <=. .)-.. -\ -J. /)',. *I. :hePT(. 0.5. l0. I ) vo nghi6m. FN6u x *+3y Chia2 vd phuong triph (l ) cho (;r. +:y)(, - 3).). ra c6. :. -3y)'. {,'t, lrc* -3Y. I. x+3v. -l I x+3y+ ^l/. 0.5. t. Edtt-n '. ?,' - - '/ th\. x+3y. 4-1 (*)ots/-tt-!=o \./ =[,-i)[,.+)=o<+/= f-_. 35. t. +Voi Thay. A. /=-J n. thi. x+3v 4 x-3y 3. -.. -21v. 0.25. 0.25. vio (**). Ta co :. i -1 -l8r'-l=5.6-r'= -8y 4'' 36. 0.25. a. -J-.

<span class='text_page_counter'>(4)</span> rI \ - -,1 o Vti'i r = -I.--> 1tlrolr nrirr I)K\l)) z{+ l=., -l(;. Vo'i,r,' = + vo'i r = j. 5. =. r 015. l i l. ii. l. 1. ( thoa rnirn DKXD). (). 12. l-i. I I. t-]l' x+3y= l= 5. thi. I'. -r = 2.1,. - 1-5ay,. vao (*,F). 'l-a. l. 0.25 I. CO: I. 1. 1'. -) _i'. [,,, =. =. lt. i I. + -2t0 -. Vti-5. I I. 0.25. 1-a;*t L)'= 2 r0. I. i. I. I. V0i y =:* VCIi. $.tt,. = r - ,.. i.,tr. y=*-*J8s=.r-,-iJ". vay hq phuong rrinh c6 4 nghie," ,. ( thoa man DKXD) ( thoa mdn DKXD). 3. 0.2s. [?,+j, [*,-+),. 2 10 l(. s'. to'--). Tim nghidm nguydn ducrng cria phuong trinh:. 0.5. (3d). *' + y' -13(x -t)=0. *'+y'. -13(x. -/)=o. <+ (x + y)' + (*. - y)' -2.t3(x - y) +132 = 69 <+ (x + y)' + (13 -x -t y)' =169 =122 + 52 Vi x,yeZ* =0< x+y<13,0<13-x+y <13 1. | **!=lZ v{[ ,+y=5 =( fl3-x*y=5 [13-x+y=12 [x=10V<(x=3 -Jt -/1 lv=2 _ lv=2 IJ IJ. 4. 1.0. 0.5 0.5 0.5 0.5. cho hinh thang ABCD (AB ll cD), giao di€m hai duong cheo ta o. Duong thing qua o song song v6i AB cit ao vd BC lAn lucrt tai M vA N.. -4-. i I. I. (,. lJss, I. aGl,(, - !,6j,1'z- a*;). ( s. 0.25. (3d).

<span class='text_page_counter'>(5)</span> jl1A' ,d'l\/ -___ ___ +. .4t]. b) ili0t. S.,.,,,,. : n'rl;. ct). S(.r,, =. Sn,,,,. S,.,.,,,. S,\r(.r) liin lrrot. 'l'inlr. ltl. Ii. ,,\lr(,r) thcrl. nr r li n (r'tii. tanr giac AOI]. di0rr lfch. cliCn tfch. tam giiic COD. di.^n rfch ru.giiic AIICID).. I'linh. vC. ,\. l]. - ^\. ,'\ ->(-FL/t)\\. Nr. l"'. l/'. l). 4a). -'\\ -\ r. _ >\ (. . IVIO AM MO MD CD AD' AB AD o..- - MO MO AM + MD Suvra. 0.5. laC()-=-.-=--. - CD AB AD TNONO lu(rngtu,ta +--1. AD. =_:l ' AD. CD AB. /l\ \I'l. 0.5. t)t \-,. 0.25. MO+IVO /r\.,A,.\^.,. (r)\,'a(l)su-! MO+IVOq_-). CD. -. 0.5. S.ro, =oB . snoo oD'Scoo-oA. oc'. 0.25. oB oA. 0.25. Suou. oD -:-_. Srou_Sooo. oc. s.nro sro, = Sloo = m2 .n2 1 S ooo = m.n 5. Tuong tU Suo. = m.n. V4y Souc, = m, + n2 + 2ntn (m * n)2 = \-no ram glac nnon AIJU nQr ti€p trong dudng tron t6rn o vd. ^ ACB. 5a. 0.25. MN. _+__2 'CDAB. SU\,ra. 4b). MIV. AB. = 45o. Ke cric duong cao AA,, BB,. Goi H ld t.rc t6rn tam giitc ABC, M vaN tucrng irng ld trung di6m AB vd CH. a) Chfng minh r6ng A'MB,N ld hinh vu6ng. b) Chrine minh rins A'B'. MN OH r{Ano n,,, Hinh vE. -5-. 0.2s 0.25. (sd).

<span class='text_page_counter'>(6)</span> l---. ('. l i. l I. I. I I I. I I. I. I. TacoMA,=MB-:. !as. I. il). 2. I I. 0.,5. I I. NA'. :. NB'. : lcu. I I. (2). 0.5. 2 Niat klrac.. ,i FiB= 45" ncn LCAA' va ,\BA,l/. =. LCA'H = LAA'B(c.g.c) CH = AB = NA'= MA' (3). =. = ^ MA'N = AA'C = 900. n€n. I. 5h. (l),. (2), (3) va (4) suy ra A'MB'N la hinh vu6ns (*) Do O vd A' thuQc dudng trung truc cfia c4nh AC N0n 0l' L AC O,4'll B' (5) = Tucrng tu, ta co OB'll A' (6) Tt (5) vn (6) suy ra A'FIB'O ld hinh binh (**) Tir (*),(**) suy ra A'B', OH, MN clt nhau tqi trung di6m m6i duong.. henh.. -6-. I. 0,5. (4). H H. I. vu0.g cirr. Me ffd=m,_fffi,_@) Ti.r. I. I. 0.5 0.s 1.0. 0.5 0.5 0.5.

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