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2015
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oAruu cHo rRUNG Hoc pnd rnOruc vA rRuruc Hoc
co s6
Tru s6: 187B GiSng Vo, Ha NOi.
DT Bi6n tap: (04) 35121607: DT - Fax Ph5t hdnh, Tri sLI: (04) 35121606
Email: website: />
HÃY ĐẶT MUA TC TH&TT TẠI BƯU CỤC HOẶC VỚI TOÀ SOẠN !
Vmo mq{ ry'x;,Y&wi w,u6g ffi{&
mtr TrHuil Hsfr$ E0E&
KX v,e{x c,{{ryN &d@o s,xNM
NGUYEN KHAC MINH
(Cuc Khdo thi vd Ki€m dinh CLGD - BO GDADT)
(t(X i thi chqn hsc sinh vdo dQi tuy6n Qui5c gia
thi blympic roin rrqt Q"6" t5 grrabl
ll[ y dr.r
lan thu 56 n[m 2015 (IMO 2015) dd tluoc
t6 chriE-tqi
- He NQi. trong hai ngity 25 vd261312015. Cdn
cu Quy ctr6 ttri chgn hgc sinh gioi cAp Qu6c gia hiQn
hdnh, BQ GD&DT dd triQu tdp 49 hoc sinh tham dp ki
thi tuy6n chgn n6i tr6n, g6m I hgc sinh dE tham dU IMO
2014vit 48 hgc sinh dat tu 21,50 tli6m trd 16n trong ki
thi chgn hqc sinh gi6i Qu6c gia m6n To6n THPT nim
2015. Trong m6i ngdy thi, m6i thi sinh dugc dC nghi
gi6i 3 bdi to5n trong th
m6i ngdy thilil2l di6m.
_
DE THI
nguydn kh6ng Am sao cho
co+ ctu + cra2 + ... + cdo = 2015 (*).
a) Chimg minh ring co+ q + q+ ...+ c,:2 (mod3).
r
thay tl6i
vh
co,
ct,...,cn thay d6i nhrmg
(x). Tim gi6 tri nh6 nh6t
co+ q + c2+...+ cn.
th6a mdn di6u kiQn
1u6n
ctra t6ng
Bdi 2. (1 di€m). Cho dudng trdn (O) vit d6y cung BC c6
dinh (BC khdc duong kinh). Di6m A thay aOl tr6n 1O;
sao cho tam gi6c ABC nhgn vd AB < AC. Gqi H \d tr119
tdm tam gi6i, t ld trung tlitim cua c4nh BC, D ld. giao
di6m clria AH vd BC. Tia IH cdt (O) tai K, tia KD cit (O)
tai M. Duimg thdng qua M vd vuong g6c voi Ag cit .lt
tai
di6m P nim b6n trong tam gi6c sao cho
FB:;p-c: a voi a > 18oo -6ic. c6c duong
trdn ngopi ti6p c6c tam gi6c APB vir APC l6n luqt cit
AC vd AB tai E vd F. Ldy di6m Q b6n trong tam gi6c
AEF sao cho AQE= AQF:a. Gqi D ld di€m rl6i
ximg v6i Q qua ducrng thing EF'. Ducrng phdn giSc
trong cira g6c EDF cdt PA tqi T.
a) Chimg mirljn
DET:
ABC,
DFT:
ACB.
b) Ducrng thing PA cfu Of , DF ldn lucrt.tai M, N. Ggi I,
16n luot ld t6m c6c ducrng trdn nQi ti6p c6c tam giSc
PEM, PFN v* (fl n ducrng tron ngopi ti€p tam giSc DIJ
v6i tdm ld di6m K. Duong thlng DT cdt (&J tai di6m 1L
Chrmg minh HK di qua tAm ducmg trdn nQi tii5p tam
J
lttgdy thi thth nhiit,25l3l20l5
Bni 1. (7 di€m). Cho a ld nghiQm duong cira phucrng
trinl I * x=5. Gii sir n vd c0,c1,...,c, ld c5c s6
b) Cho
Bni 5. (7 di€m). Cho tam gi5c ABC nhon, kh6ng cAn vd
1/.
ir
a) Chrmg minh rdng 1/ ch4y tr6n m6t ducrng trdn,pii dinh.
b) Euong trdn di qua diiim ff vd tit5p xric vdi dulng
thhng AK tai di6m ,,q, c1t,qS, AC l6,n lugt tai P., Q Gqi J
ld.trung di6m cira doan P9. Chrmg minh r[ng ducrng
gi6c
DMll.
BAi 6. (7 di6m). Tim s6 nguy6n duong r nh6 nh6t sao
cho t6n tqi n s6 thgc th6a mdn d6ng thdi c6c diAu kiQn
sau:
i) T6ng cira chring ld m6t s6 ducrng;
ii) T6ng cilclQp phucrng cira chfng ld mQt
s6 6m;
iii) T6ng c6c lfry thria bflc 5 cta chring ld mQt sd duong.
KET QUA
cf
k6t qui ch6m thi vd Quy ctr6 ttri chgn hgc sinh
gioi cap qu6c gia hiQn hdnh, B0 GD&DT d6 quy6t clinh
chon 6 hoc sinh c6 di6m thi cao ntr6t 1cO t6n du6i ilny)
vio DQi tuy6n Qu6c gia ftr thi IMO 2015:
Cdn
ii
l.
ltlguyin Tttin Hai Ddng, Ws 16p 12 Trudng THPT
chuyOn KHTN, DHQG
Hi
NOi, 32.50 di6m;
thdng AJ di qua mQt di6m c6 tlinh.
2. lVguydn Htrlt Hodng, h/s lcrp 12 Trudng PTNK,
DHQG TP. HO Chi Minh, 29.50 di6m;
3 lr,tgul,dn lh) Hoan, Us lop 12 Truong TIIPT chuy€n
Bni 3. (7 diA$.
KHTN, DHQG HA NO| 27.50 di6m;
Sd nguyQn ducrng k duoc goi ld c6 tinh
chiil T(m) n6u v6i mgi s6 nguydn a, tdn t4i s6 nguydn
ducrng n sao cho lk + 2k + ... +
.t.
,
tf = a (rwtn).
BQi Chdu, NghQ An, 25.00,diiim;
L
a) Tim tat ce
" c6c s6 nguyCn duong k c6 tinh ch6't T(20).
b) Tim s6eg@6n duongtnh6 *6tco tinh ch6t T(20t\.
.
,\g,iy thi thft hai" 26l3l2t)15
nai +. 1Z didm),.,::g6.1gO Jinh vien tham d1r mOt cu0c thi
v6n d6p. $.an gi6m,khao:961n 25 thefi vi6n. Mdi sinh
vi€n dugc h6i &jiboi mgt gi6m kh6o. Bi6t rang.n6i sinh
vi6n thich it nhetil0 gi6m khdo trong s6 cac thdnh vi6n
tr6n.
a) Chimg minh
4. Hodng Anh Titi,Ws lop 12 Tr,ucmg THPT chuy6n Phan
rlng c6 th6 chqn ra7 gi6mkh6o
sao cho
m6i sinh vi6n thich it nh6t mQt trongT gi6m kh6o ndy.
b) Chrmg minh ring c6 th6 sip xi5p cuQc thi sao cho m6i
sinh vi6n duo. c h6i thi bdi gi6m k*r6o mlr minh thich vd
m6i giam khao hoi thi kh6ng qu6 l0 sinh vi6n.
5.lr{guyAn Thi Viil Lla;,hls lop 12 Truong T}IPT chuy6n
Hi
Tinh,24.50 di6q
6. Vii Xudn Trtutg, h/s lcrp 1l Trucrng THPT
chuy6n
Thdi Binh, 24.50 rti6m.
B0 GD&DT dd triQu tfp 6 hgc sinh cira
tip hu6n chuy6n m6n
"C
chuAn bi cho IMO 2015. Trucrng EHSP He NOi duqc B0
GD&DT giao nhiQm vr; chri tri c6ng tdc tQ,p hu6n dQi
Ngiry
161412015,
DQi tuyi5n
Ha NOi tham dg lcrp
tuyiin, du6i sy gi6m s6t cua
BQ.
,:,.
IMO 2015 sE dugc t6 chr?e.'tu ngiry 4/7 di5n ngdy
1617l2O15 tai Chi6ng Mai, Ttrii Lan.
HÃY ĐẶT MUA TC TH&TT TẠI BƯU CỤC HOẶC VỚI TOÀ SOẠN !
,T&UNG
c{, sd
(/rong di thi vdo THPT lu6n xudt hi€n bdi Ldigidi.A'>0 o m- l>0<> la>lthiPT
?, bdn hAn quan ddn viQc dp dung hQ thttc
Vidte. C6 nhirng bdi todn c6 thO dd ddng dua vi
dqng dp dqng dagc hQ thuc Vidte, nhwng cfing
cd nhirng bdi,todn phdi rh khdo leo mdi thqtc
hiQn duqc di2u d6 vd no gdy kh6ng lt kh6 khdn
diii vai cdc em hgc sinh. Sau ddy lit mdt sd
dqng togn nha vdy vd cach s* dryng h€ thac
Vidte d€ gidi chung.
D4ng 1: Phuang trinh bflc hai c6 tham s6
Bii 1. Cho phtrcrng trinh:
8-t: 8x+m2+1:0
';
Tim m de phtrcrng trinh (*) c6 hai nghi€m
md xf
(*)
x1, x2
-"tl =.ri-"t:'.
NhQn xdt. Ta thdy h€ thric dA bai dua ra c6 v6 phfrc
tpp vd g6y kh6 khln khi dua vd xrt xz vd. x1.x2
nhrmg ta c6 thrS bien dOi xy, x2 th6ng qua phucrng
trinh (*) tl6 sir dlmg h6 thric Vidte.
: 8 - 8m2. DC PT (*) c6 hai
nghiQm thi A' >0c> -l
Ldi gi,rti. Ta c6 A'
hC
Vi
c6: xt+-x2=l; xr.xr=(m2 +1):8.
thuc Vidte
x1,
x2
(D
xl-xi=x?-x3
xl (8;f - 8-r, ) - xr2 (8 xl -8xr) = 0
Thay (I) vdo (1) ta dusc
xz)(xt + xzX-
€) xr- xz:0
m=
*7
*'-
11:
(vix1 + x2:lvd
I
Do d6 xt = -y2 =
r.*d
x1.xz =
110).
mz+l
?,
Bii 2. Cho PT *' - 2m, + m2 - m + 1 :
mdn:
pf
suy ra
Ldi gidi.A' > 0
e
m
>jt.l,nt
c6 hai nghiQm x1, x2,thi d6: x,
phuong trinh
* xz =2(m+l);
xl =Z(m+l)xr-m2 -4.
bii xl +2(m+l)x, <3m2 +16
e 2(m+1) (x1+ x2) - 4mz - 20 S 0
r ,
--2
2
l2(m+t)) -am -20<0.
(do (x1 a xr) :2(m+l))
Bii
m<2.
.0,
]2 < m < 2.
4. Cho phaong trinh:
(l)
.x' -21m - 1)x + 2m - 5: o
Tim m d€ phwong trinh (1) c6 hai nghiQm phdn
biQt x1, x2 thda mdn
0 (1).
c6 hai nhi€m phdn bi€t x1, x2 thoa
xl *2mx" =).
Bni 3. Cho phtcctng trinh:
x2 -21m+ 1)x + m2 + 4: o (m ld tham sd).
Tim m d€ phaang trinh co hai nghi€m x1, x2
thda mdn xi +2(m + 1)x, < 3m2 +16.
K6t hqp voi (*) tu
g
-*'*
* : :(chsn).
J
(loai),
o8m-16<0 e
(th6a min bdi to6n).
Tim m ae
e m:-2
(1)
(xl-xl)(-m'-1)=0
-
x1
4 *2*,+nf -m+l=lo4:2nx,-n? +m-1.
f0t hqp v6i tlAu bdi ta c6: xl +Zmx, =9
eZmxr-m2 +m*l+Zmxr=)
e 2m (x1+ x2) * m' + m - 10:0
o 3m2+m-10:0
a
e
(x,
thirc
Theo tl6
' ),.,.
e
hO
+xz=2m;xr.xr=m2 -m+1.
Vi x1 ld nghiQm cria PT(l) n6n :
Vidte c6:
xt.x2=m2 +4 vd
ld hai nghiOm cria PT ( *) n6n
[S*3-Ar, =-(m2 +l)
farj -8xr=-(m2 +l)
Tac6:
c6 hai nghiQm phdn biQt x1, x2 yd theo
xl -2mx1 + 2m - 1)(xi - 2mx2+ 2m - 1) < 0 (2)
Ldi gidi. A' : (* - 2)' + 2 > 0 lu6n rlirng vdi
(
mgi m, vfly phucrng trinh (1) lu6n c6 hai nghiQm
..
nu.,.-roru,
T?8I#E[
1
HÃY ĐẶT MUA TC TH&TT TẠI BƯU CỤC HOẶC VỚI TOÀ SOẠN !
ph6n biQt x1, x2 YOi mQi m. Khi d6:
xl + xz = 2(m -l); x.x, = 2m
trinh (1) c6 hai nghiQm phin biQt e
-
5'
m<
Yl xy x2li hai nghiQm PT(1) n6n
lxl -2(m- l)x, + 2m-5 =0
r? -2mx,
,_- [
I=4
-
-2x
- 2x) (4 -
(2) suy ra: (4
e
e
4xrxr-8(x, +xr)+
4(2m
Ta c6: xfiz (!t + Y2) : - 48
O xtxz (2x, - m * | * 2xz-
r
Bdi5.
Cho
e
2xz) < 0
1)
+ 16 < o <>
phuongtrinhrl+(n
*,
+.
i),r 6:0
(i).
Tim m dA phwong trinh co hai nghiAm phdn hidt
.\1,.\ rnit B
Ldi gi,rti.
:
Ta
( ri
-
9)
(-r,t 4) dqt GTLN'"
*dy (1) lu6n co hai nglriQm
ph6n biQt
xl,x2ybimojm vi c6 a. c:- 6 < 0. Theo
Vidte: xr.xz=-6<=' x2=
,
a
:
(x?
-
g)(x|
-
4)
:
j
x? . x',
hQ
thtc
ue x1+xz=l-m.
x,
(4 *? + 9 xZ) +36
-
:36- (4xi +44*1+X
xi
!xf
e xl :81
<+
xr:3
ho6c
L'lrts purctl'tol
lPl:
xi,
xr:-3.
.t - 1rt vi
(,,r,q 1"1'2)
-'
clmhtg
48.
Ldi gidi. Phuong trinh hoinh dQ giao
* m- 1 :0
|* -2x
(4
vd
(d)tir:
DC
(A
cat (P) tai hai
a
Z icruagw
: - 48
48
*2m*2):*48
e*'-6m-7:0.
e m: - 1 (chqn) ho$c m:7 (loai).
Bni 2. Cho parabot (P): t' : -- *' t'a drrd'ng,
rhang (ct):j!': (3 nr)r * 2 - 2m. Tim m d,:
1ct1
cat @\ tai htti diim phdn bier A
thoa mdn'. il., - r'r l :
(.u
l.)'t):
B(-rr; .vr)
z.
Ldi gidi. Phucrng trinh hoinh
(P) vd (d) td:
dQ giao
ditim ctra
x'+13-m)x+2-2m:0 (1)
Phuong trinh (1) c6 A' > 0 e m * - l,tathiy
xe,xoldhai nghiQm cira phu
Lpi c6:
e
thang ({t\: .)' '- 2x rtt } l. Tirn m di \A t:dt (P\
tui hui efient phan hi\t t'ri tQrt do: (rr;.r',) vri
(r:..i,r) suo c:ho.r1..11
-2m + 2): -
1)
a
324
. Khi x1 : 3 thi xz: - 2, suy ra: m:0
. Khi xr : - 3 thi xz:2, suy ra: m:2
Vfly minB : 0, khi m: 0 hodc m: 2.
I)4ng 2: 'fuo'ng giao cria parabol vh tlutrng
thang
Bii l.
1)t8
xz)
m+
I xo+ x, = m -3,,^ Iro = 73-m)xo +2-2m
"' lr, =\3-m)xr+2'2m
1ro.*, =2-2m
Do ct6 lyo -yrl =2
ltl-*){*o-*ul=z
(3
*u)'
<+ - m;2 l(x1-Y
- 4x1 . xsf : 4
B<36- 2 E;g-+36=0.
Ding thirc xhy rakhi 4 t :
(xp2)12(xt+
e2(m-
16 < 0
- s) -t.z1m-
Khi d6 xt xz ld. hai nghiQm PT ( I ), ta c6:
lx,+xr=4 ,,n
'' Iy,=2xt-m+l
\Y,=2*.,-m+I
- lri -2*r+2m-l=4-2xz
Tt
.
1r,.*, =2(m-l)
\xj-2(m-l)x, + 2m-5=0
+ 2m
3
A'> 0 <>
(1)
(m
-
3)'
(* + 1)' : 4. Tim dugc
m=lxJ6; m=7xJT.
Bni 3. Cho clu'o'ng thang lch. r : (/. I )r - 4 r'a
parcrbal(P):-y: xt. Ti,, k (tO ((t) r'r) (P) c,it nlturt
tcri hai dient phtin bi1r. Goi ttttt ,lo giatt diun la
( xr, -r'r) ; ( -r2,-y2) . Tim k tli 1'1+ lz:.vt..\'t.
Ldi gidi. Phucrng trinh hodnh d$ giao diiSm cua
(P) vd (d) ld: x2- (k * 1), - 4:0 (fr le hing
sO;. fnucrng trinh niy c6: a.c : - 4 < 0, n0n
1u6n c6 hai nghiQm phdn bigt.xr, xz. li"hi d6
lx,+x,=k-l
iY, =xi
l..r= = ,j
lxr.x, = -4
Vfly:yr + yz: yt.lz a x? + xl = 11.fi
e (xr + x:)2 - 2.r1. x2: xi.xi
€ (k- t;2 + 8: 16 e k: I +2J7,.
TOnN HQC
HÃY ĐẶT MUA TC TH&TT TẠI BƯU CỤC HOẶC VỚI TOÀ SOẠN !
tlff#ng d*n gifri uE rHr TuyEru srruH vno r#r ,l0 {Huy*ru ro*r*
rnuftruG hxsp rn rr$ cmi FfIHm NAlvr Hoc 2014 - 2ots
I
riu
tr.
a)
EKXD:
-r-].
x:2.Tu(*)c6
Khi d6
)
PT <+ (J3x++)-
: (t + J2x+ t )-
-.
e x+Z=2J2*q e(x+2)'
=4(zx+t)
<+.r'' -4r-=0.
Tt d6y tim du
VQy tQp nghiQm cira phuong trinh ld: S =
{+;O}.
b) DKXD: x > 0. Khi tl6
rr e (-r + JI +t)' = z(*' +2x + t) -z+x
e (x + t)' +2Ji (x +t) + x =2(x +I)' -Z+*
e (x+1)' -3sx -z{i (x+1) = 6
e (x + t)' -t Ji (x +r)+sG(.r + 1) -35-r = 0
e (,r + 1) (x + t -t "{i) + sJi (x +t -t Ji) = o
e (x + t-7 Ji)(x + r +sJx ) = 0.
Tim tluoc tdp nghi6m cua phu'crng trinh lir:
2Y
47-2tJs)
'
"=t 2 ,
3*
-t=r(r'o - t)+z=r(r- - !)+2
-
(8; 3) : 1 nOn p i3. Dop ld s6 nguy6n t6 n€n
p :3; p: 3 thi 8p + I :25ldhqp s6.
Vay khdng c6 sd nguy6nt6 p thbamdn 8p - 1,
8p + 1 ld c6c s6 nguy6n tii.
b) Ta c6 3* *2Y :1<> 3*
(*)
-I=zt
x:2k (fteN.), tu(*)tac6
(a-*r)(:* -r)=2,.
Do do
d6 a,beN vda>b.Tac6
[:-.1
LJ -t=Z
-ll):(9-1)) +2' chia cho 8 du 2
+2Y =2=y=1. Ta c6 3'-l=21 ex=1.
YQy chc cflp s6 nguyOn duong (x; y) cAn tim ld:
(2:3),(1; l).
{--6u 3. a) Ta c6 ("*u)'
Do cl6 (a+
b)'
=r(o'
Dod6
f* ,,
.2
1',-'-".
l?r
:)l
<=13(=3'
<>k=l.Khid6
, (o*u)'* (r-t)'=\;
*t7.
+a'). ruong tr;
(a'+u')' ,r(o* +uo). raco
(o' + u')'
(a + b)'
.
o(o' + u')(o^
+ ao
(a+b)' =4. Do vQy,tac6 a2 +b'
b) Do x, !, z > 0, xy + yz + 7x :
). ua
+bo.
1, 6pdgng b6t
t
t( t
I \
- ,tl +-r+r+* - 2[x+Y ' x+z)'
Dod6 2=!(-t*-j-)
t- ''
,ll+i 2(x+y x+z ) rrl
A:'-l-r-l
ruong tu ta c6:
,
Jt*r'
oong
+.1[-r*-r)
O,
,ll+Y" z\x+Y Y+z )
'
l(
-:.'l
taco:
-t =2e*(t' -1)=2. Nc,
12"-u=2'
la-l=l <><
la=2
{l2'-n-l=l<}<
<}<
-"
'
I
lb=
[tr= t
lb=l
l2o =2
T
8
du 2 (vi(er
J
=zou
chia cho
t * I )
.r(
-2[r+y'x+z)
Cfru 2. a) 8p -1,8p,8p + 1 ld ba s6 nguy6n li6n
ti6p, n6n c6 mQt si5 chia h6t cho 3. Md Bp t,
8p + 1 ld c6c sd nguy€n tO tcrn trcrn : n6n U sO
kh6ng chia htit cho 3. Do v6y (Ap)i:. Ua
N6u
ay=3,
=23
N6u x =2k+ I (k e N),ta c6
tl6ng thr?c Cauchy cho hai sd duong, ta c6:
. fqt+ztJi
.
=3'-le2Y
.
,
, \
"
'
*
\-/' rt
\-" (2)
\-/'r,d (3)
-- (l),
- Z\y+z' x+z)l (:).
-4--L--L
Jl+ x' ,lt* y'
-l( x
< _t _J_
x
y
L-_!_
'll+
y
L__Lr_f_
r'
z
z ) :
|
2\x+y x+z x+y y+z'y+z x+z)=_2'
C4u 4. a) Gi6 sir (dr),(dr),
(4)
d6ng quytaiO.
Do LOBD w6ng tai D,
D n€n
Pythagore ta
LOCD rndng tai
theo tlinh
ly
B
D
-1
Sti ese (6-2ors)
TOAN I,{Q(
-i-qrudifue
J^
HÃY ĐẶT MUA TC TH&TT TẠI BƯU CỤC HOẶC VỚI TOÀ SOẠN !
LE ToNG KET vA TRAo cr qr rmu'Gtic rrxr t.EoN{c 20ts
F{ST SOI{G T}IT E{A NOI
qB$} qefffl
Kj thi Olympic To6n Hd NOi ry0 rQng - giii To6n bing tiiSng enfr - tan thri 12 ndm 2015 (HOMC ZOIS; tO
chric ngdy 22/312015 de thdnh c6xg t6t dqp. Ngny 17i5l2015 tpi Truong THPT Chu Vdn An, He NQi, Ban chi
thi HOMC 2015 dd"ti6n hinh 16 tOng ti5t vi trao gi6i thuong cho c6c thi sinh du thi d hQi tl6ne thi He NOi.
L6n thi thri 12 niy dd c6 42 tirt , titann phi5 ttram dU vdi sO luqng 634 thi sinh: 320 thi sinh lua tu6i Junior
8
Qdp THCS) vir 314 thi sinh hia tudi Senior (lop 10 THPT) thi t4i ba HQi d6ng: Hn NQi: 487 thi sinh (237 lop 8;
250 lop l0); Dak Ldk 57 thi sinh (13 lop 8; aj lcrp 10); E6ng Th6p: 90 thi sinh (2516p 8; 65 lop 10). T6ng s6 gi6i
thuong cta kj,thi HOMC 2015 ld: 449 gibi g6m 45 girli Nh',t, 120 giitiNhj, 155 giiliBavd.lz9 giii Khuy6n khich.
Bdi thi duqc itt6m tneo thang ditim 15, ph6 Aidm cO tu 0 di6m ai5n i:,S ditim. Iftdng c6.dir5m tuyet d6i 15/15. Ch6t
luqng bdi ldm cira thi sinh t6t hcm c6c nbm tru6c, ttr6 nien 0 di6m binh quOn f,uong d6i cao. Thri khoa o hia tu6i
Junior c6 I em: Mdn Ddo Son Tirng,THCS Hoang Vdn Thp, Lqng Scrn, 13,5 di6m. Thri khoa d lua tu6i Senior c6 3
em'. Phqm Kim Anh, THPT chuy6n Hd NQi-Amsterdam, 13,5 di6m; Mai DQng Qudn Anh, THPT chuy6n Ha NQi Amsterdam, 13,5 itirSm; Dodn bao KhA,THPT chuy€n LC Khi6t, Qu6ng Ngdi, 13,5 di6m. Thi sinh dat gi6i dugc
nhQan Giiiy chilmg nhQn cira S0 GD - DT vi HQi To6n hgc Hd NQi, girii thu&ng cua S0 vir qud tflng cria HQi. C6c
tinh, thenh c6 hgc sinh tham dp lc, thi HOMC 2015 duqc tflng C] luu niQm cta Ban t6 chric. Mgi ngucti d€u mong
mu5n, nhu, ldi GS. Nguydn Vdn Mdu, Cht tich H6i To6n hqc Hi NQi: "Srira md rgng cu\c thi HIMC ra cdc rurdc
ASEAI\1'd6 hgc sinh Viet Nam dugc dua tdi cring hqc sinh ciic nu6c trong khu 1uc D6ng Nam A * Th6i Binh Duong.
Th6ng tin chi ti6t xin xem tr6n trang Web cira HQi To6n hqc Hd Ndiwww.hms.ors.vn
THAM NGQC KHUT (Hd Nr.i)
c6
DB2 +ODz =O82, DC2 +OD2
DP
-rc
cfrng c6:
FA2
=OE
=OC2. Do d6
-U2 (l). Chrmg minh tuong tU
W -W
=OC -O,4 (2) vir
-FB2 =OA2 -OB2 (3). Tri
(oe
-x:)+(n:
(l), (2) vd (3) ta c6
-w)+(FN -Fr,)=o()
b) cia sri c6 (*). Ggi O ld giao di6m cria (dr) vit
(4).VC OD'LBC tqiD' . Cdn chimg minh D'=D.
Tir cdu a) ta c6
(o' w - o' c) +(rc" - a,+)+(raz -raz ) = s 1"x;.
Tir (*) vd (**) ta c6 DBz - DC2 = D' 82 - D'C2
o BCIDB - DC)= BC(D', B - D',C\
o
DB-DC =D'B-D'C
<+ (or + DC) -zDC=(o' a + D' c) -zD' c
e DC =D'C eD'=D (tlpcm).
- irr ::"
Md BAT=BCK (th gi6c ABCD
=BHI=BAT.
BIIT BCK , do d6"
"glcp)_l3 180"=> K,H,T
thdng hdng.
BHT + BHK =
Chrmg minh rucrng tU cfrng co P, Q,7 th6ng hdng.
Vpy c6c duong thdng AD, PQ, HK d6rg qtry.
ti€p
"
-blli
c6:
rytc_ITBK
AtB+DKB =90'+90'=180" =ru giitc DTBK ndi
ti€p > ADB = HKB. Ta co BAD = BHK (vi cung
bir v6i BCD )
=MBD,n U{BK(e.E)
AD AD AB zAM AM
HB HK HB 2HN HN
MAM vd MHN c6:
6
= MAM
'BAU
AD
Ai'
=!l!
HB^HN
= 91151.!!
^
trpHN {c.g.c) =>BMA=BNH
gi6c BNMTnliti6p
=BNM+BLM=IKP.
BNM =180. -BTM =180" -90" =90".
Vfly n4lru6ng g6c v6i -A/8.
1S
Cfiu 6. C6 50 dinh n6n c6 50 tich ba s6 tr€n ba dinh
li6n ti6p. Vi ba dinh 1i6n ti6p U6t tcy c5c sii kh6ng
blng nhau n6n chi c6 hai lopi tich:
Logi I: Ba s6 0 ba tlinh li6n ti6p chi c6 mQt s6 2, tich
.
i. . .:
ba so ndy b1tng 2. Logi II: Ba s6 6 ba dinh 1i6n ti6p
c6 hai sii 2,tichbas5 ndy blng
4.
.
Gqi s6 tich 1o4i I Id x (x e N) thi s6 tich loai ll ld
50 -x. Md s6 s6 2 6 50 tich c6 ld 30 . 3 : 90, ta c6
phuong trinh: x.1 + (50 - x).2 =90<+ .r = 10.
Vf,y c6 10 tich loai I vd 40 tich loai II. Do vf,y t6ng
a)
Yd Et LAD. Ta c6 BHC=BKC=W
=
ru
t6t cd cdc tich ba sO tr6n ba dinh 1i6n ti6p cira da giSc
"si6c le:2.ro+4.40=l8o.
BHKCnditidp <+ BHK+BCK =180". Md
ttr giitc ATBH nQi
ATB + AHB - 90" +90" = 180'
=
T$ffiru
E{#{
- rflaaeffis*Sd
"
ase
4
NGUYEN PIIC TAX
(TP H6 Chi Minh)
(6-2015)
HÃY ĐẶT MUA TC TH&TT TẠI BƯU CỤC HOẶC VỚI TOÀ SOẠN !
n&
rm ruv*nr slffi
v*rlr*p
r
o
S n -,T'lxhNinh,Shu$n
ruAnnrrcca6rq-aots
Yrt$nrg Tl{trTshu;r€n L€ Auy
(Thoi gian ldm bdi 120 phrtt)
Cflu 1 p aia6. Cho phuong trinh:
x' -Zx + m' -2m +l= 0 (1), voi m ldtham s6.
a) Gi6i phuong trinh (1) khi m=J, .
b) Chung minh rdng nliu phucrng trinh (1)
hai nghiQm
x'x,
th\
lxr-
-^
c6
xrl<2.
Cdu 2 (2 die@. Tim gi6 tri nh6 ntr6t va gi6 tri
thtic: D =4{*3 .
x'+7
Cflu 3 @ diefi. Cho tlucrng trdn (O) tluong
kinh AB vd cludng trdn (Q di dQng lu6n ti6p
xric trong v6i nira tluong trdn (O) tai C vi ti6p
xric v6i doqn AB tai D; CA vd CB lin luqt cft
lcrn nh6t cua bi6u
duong trdn
(q
a) Xhc dinh tdm O'crta
chimg mkthAB ll MN.
b) Chung minh CD ld tia phdn gi6c cira g6c
ACB vd CD diqua mQtiliOm c6 dinh E.
c) Chung minh ring clu
gi6c
ACD ti6p xric voi AE tqi A.
Cflu 4 (2 diAm).Cho phdn s5 p =
t:+,v[i
n+5
n
ld s6 t.u nhi6n. H6y tim tdt cb chc sO t.u nhi6n ,t
trong khoing tu I d6n 2015 sao cho phdn siip
,
.1, ."
cnua tol glan.
CAO TRAN TtI TTAI
@i M, N.
Qtlinh ThuQn) gidri thiQu
6dt drp
ortt{sbvtu*ix**oa
(Di ildng tr4nTH&TT sd
qSO
thdng 12 ndm2014)
PHr PHr (rrd N1i)
G,.t,1,,."0ffio.
Ki hi6u c6c s6 trong
c6c 6 trdn nhu 6 hinh 1.
. YOi e: 16, g = 18 vd h: 17, k: 14, suy ra c * c'
= 14 : 6 + 8, d + d' : l7 : 4 + 13 : 5 + 12, a t a' :
17 : 4+ 13 : 5 + 12, b + b' : 11 + 7.K6thqp v6i
(U, (4) c6 nghiQm (e, g, h, k, c, c', d, d', a, a', b, b')
bdng (16, 18, 17, 14, 6, 8, 4, 13, 5, 12, ll,7)
(hinh 2)vd (16, 18, 17, 14, 6, 8, 5, 12, 4, 13, ll, 7).
Hinh I
a' + b'
d':
* c' +
m:
40 (l). DAt
n,= q + b + c + d vd p e + g + h + k.X6t t6ng c6c
s6 n5m h6n s6u vdng trdn ta c6 3(m + 10) + 2(n + 9)
+ (? + 15) 50.6 hay 3m + 2n + p:237 (2). Tn (l)
vd (2) suy ra 2n -t
ll7 (3). Tdng c6c s6 dugc
tli6n ld 4 + 5 +...+ 17 + 18: 165 m * n + p + 15
+ 10 + 9, krit hqp v6i (l) suy ra n + p 91. Thay
vdo (3) tim duo. c n:26 vd p :65 (4). Thay (4) vdo
(2) duqc m + 2(a' + b' + a + b) + 2(c + c' + h) +
2(d + d' + k) + e + g 231 + h + k.Tri d6 c6 40 +
2.35 + 2.31 + 2.31 + e + g 237 + h + k, t&cld e +
s 3 + h + k (5).Tt (5) vd p e * g * h + k = 65
suyra e +
l8 + 16 vd h+ k:31= 17 + 14.
Tri gi6 thi6t ta c6
:
:
p:
:
:
g:34:
:
:
:
:
Hinh2
. Y6i e:16, g: 18 vd h:14, k:17, suy ra d+
d' : 14: 6* g, c * c' : 17 : 5 + 12, a_t a, :20:
7 + 13, b + b'
:
15 = 4
+
ll,
kh6ng th6a m6n (1).
YOi e: 18, g : 16. X6t hrcrng t.u h6n thi t6n t4i
hinh d6i ximg v6i hinh c6 nghiQm n6u h6n qua t4rc
ddi ximg ld tludng thlng di qua !6m c6c 6 trdn ghi
.
c6c s6 15, 10, 9. V4y bdi todn c6
trnr.,.-roro
th chb6n
nghiQm.
T?lI#E!,
5
HÃY ĐẶT MUA TC TH&TT TẠI BƯU CỤC HOẶC VỚI TOÀ SOẠN !
cHuir
vAN DUNc a6coE arAr
B!
flfrt nruGo;{n qenDg
I{* THI
Tnur0 HQc
rxd rxOxc
OHO
qRoDG rnfrq
Su6c GrA
pnfrnc
TRINH BA
(GV THPT chuyAn Hodng L€ Kha, Tdy Ninh)
lJ
Dqi tli';m
t:: c tti. httttrrttt
t
tlo hun'c -\' 'int tot do ''ir'
U ;;;: ir"ia"ri t*d nay sei rd Ki thi i;;i
Quiic gia) co m\t cdu h6i ri Uai bdn hinh hqc tltnh ,4, B.
vQn &1ng phuong phdp tpg d0 trong mfit phdng. phin tich.
-i;
Ddy ld bdi todn tuong aai ma di rhdn tooi th! :
,;; aO aicm A tru6c.. A tit siao diilm cua
sinh; md phdn co bdn nhdt ld vid,t phaong trinh duong trung tr.uc cria doaa COvd
1j/rong
dd thi tuy€n sinh vdo cdc trudng
daonq minq
vidt ndy.chilltF t:: :;,;",;;6 il;;,;;1.,
ft)
.rlo"s,bdi
drct ra mQt s6 thi du v€ cdc bdi todn vi4t PT
dudng thdng li qua mQt di€m cho tru6c vd
vdi dudng thdng cho trubc m6t gdc a.
Ldi gi'fii' (h'l)
THUyEl'
r. co so r,.i
1) G6c gifia hai yecto
K(6;6)
a vd b kh6c 0 ilugc tinh
-l
/- :\
qua cos(a,,
tqo
ura- *a-ducrn! thing
A vditucrng thhngOB.ViZ,pr
o.l)
)=w.
2) G6c.gita hai
nh6 nhat trong hai. c{p g6c ddi dinh, vfy g6c
gifia hai cluong thing ld g6c c6 sd tlo kh6ng
vugt qu6 90".
3) Neu hai cludng thing a, b c6 vecto ph5p
tuyiin (VTPQ Dn luqt ld i,,i, tti
C
l;;l
H)nh
/
cos(O'Dl
''\-= | ' 'l
l;m'
| 'll 'l
a) N€u hai dudng thhng a, b c6 vecto
phuong (VTCP) lin luqt ni,,i, tni
l+
chi
vi c thuOc A non c(4,*)
\.) r . on, ,
)
trung cli6m cua do4n CO th\
+t
cos(a,b)=#.
,(+,-2)
5)
qua Mvitvu6ng g6c v6i CO:
l',ll",l
d: 2x-Y
rr. MQT so THi DL;r
OThi du l. (Trich Cuu - rt'rl tg dt: tlti mitr t trrt
Bd GD&DT ndm 2015).T'rong mdt phdng v6'i
he truc tea clo Ox.v, cho tam gidc OAB c(t cdc
clrth 4 ra lJ rhuot clrrotrg thing
: 0 ya ttietn K l(;6)
tu
\)
Gqi d ld
lu,.url
.\ : 4r-t- 3.v - l2
I
ltr rcitn
.:
chitrg tn)n ltittg tiitt goc O. Goi C lrt tli['m tt,tnt
/r1n \ sao cho .AC = AO yd c:cic' diim C, B
: ,,,
ttuttt kltat' phiu ttltutr trt tr'r'i tlir;nt l. Bitit t'tittg
x+2y:0
n6n
-6:0.
Do AC = AO vd,4 thuQc A n6n I ld giao cli€m
ci:a d vd A, tqa d0 clioa A ld nghiQm cria hQ
12*-u-6=0
lx=3">A{3:0).
" <>1"'
l-^
\''
l4x+3y-12=0 [y=0
Ta co
o-_r.=
cos,I6k =o-4
oA.oK
-{dk
=45o
- J'6t0t_L
Je JOTO
)frfi
=
- €2
=90'.
. TOAN HQC
6 'cfndiU@
HÃY ĐẶT MUA TC TH&TT TẠI BƯU CỤC HOẶC VỚI TOÀ SOẠN !
a
t
Eudng th5;ng OB w6ng g6c vdi OA n€n c6 PT
eb-0 holc
'1 b=aa.
. y6i b=0, chgn a:1 thi i=(t;O),
ld.r=0.
Tqa d0 cua B ld nghi€m cira hC PT
{*=o
ol'=o=+r(0,+).
-"
f4x+3y -12=O
l, - 4- \"'
"''
DH Khii A ntim 2014).
Trong mdl phdng t,ri'i h€ trut' loct tl6 Or;,, t'ho
OThi du 2 (Trit'h
cD c6 Pr
thang
te{- =!*
dA
hinh tttottg ABCD c'6
lt't
lY
. Vdi
trutrc dicm cila
AB, ll lu diint thuit' tlrsun AC scro t:ho
,4,V=3,VC. L'iOt PT tltftt'tts lhing Ct), biit
rin,q M (l;2).,v(2: 1)
'
uN
=t.w,
v6i
k
b
a=4 thi i=(+,5),
=la,chgn
4
lY
=Y+=*=t.
NP NC
=frFH vd coso. =ffi,
ring HP, MP d6u tinh duoc /
cr
Gei P(x:y), tu he thuc rr€n ta .O
a,W
=!,gp
- = "[MII2+
6'-'- =!3-""
HPz
=oq
3
phu
n6n
_ la_3hl
Jto JF;P Jm
l;llMNl
til
IrMwl
I
t
Jaz
+bz
o
4b2
iort
7'int
dr.t
iii,,:nt
iltin.1
4
-3ab =0
'frifr
tinh g6c friil c6 thri thsc hiQn b[ng cSch
ding ilinh li c6sin vi tinh dugc AM, AN, MN theo a.
Nhrmg trong bii ndy ta tinh dugc gOc ffiF *ro
/ , \ tana+tanb
tatrlo+bl-*^\*
' "/
l_tana.tanb.
cons Inuc
.
Ldi gidi. (h.3)
Ggi a ld cpnh ciia hinh vu6ng, A
l^
^\
ta c6 tan I DAN + BAM I
\/
tanfIfi + andifu
Jto
=la-ZAl=
r'r./ tltr,,'rt,g
ViQc
\r
a=friH+cos0 =HP
MP=1.
nncd:
,
(g6c gita hai ducrng
N6u ta tinh dugc g6c
thing AM,IA), thi ta vi6t tluqc phuong trinh du
thdngAN.
r[],-z)')'
Ggi cr ld g6c gitia hai ducrng thbng CD vit MP,
=(t;-:)
, ,l
'
li
thingCD,v\ Mfr
-l
r
' Vi dC bdi y6u ciu tim tqa dQ diCm A trong khi de
cho phucrng trinh ducmg thing AN vir t1a d.Q di6m M
n6n ta nghi d6n viQc xemA= AM nAN.
d0 ddi canh cira hinh vudng ABCD, tir
hC thfc (1) ta c6:
ducrng
:oP.I: 2r ,
thr,
Phdn tich.
NP PC NC
GSi i=(a;U)+O ld vecto chi
Oru Khdi .4 ndnt )Al2).
Trong mcil phiing tri'i he tr.te dO O\,, t'hr, hinh
vttong ABCD. Goi .\,1 la trtrrr,4 tiii)rtt ,tr.t ,',lnli
.,1N
Ldi gi,rti. (h.2)
Gqi P : MN O CD; Hlddidm
...-l'....-."'..N^
D
H P L
d6i xrmg cua M qua t0m cira
hinh vu6ng ABCD. Taco
Hinh 2
MN AM AN .
\'' -../ '- ntl =ZtttF .
khid6
SThi du 3 (Trlch-ai
(',ry l.\l). Giu ttt' 14'il+.1
ari,, V
vu6ngIBCD).
MH =a,PC
=-2+3t
BC', lV td die:m trOn cunh.CD suo
theo a (a la do dai canh cua hinh
Ggi a
dudng
thdngCDc6PTlei"-3'-'
Euong thing'CD di qua tli6m P vd tpo vdi clulng
thing MN g6c
=-2
ft
t.._!_+4t
PhAn fich.
' Tim tga ttQ di6m P ld giao
thuc
t
a
i{o4tn
hQ
cluong
1- tan 6TN
t1
').+-a
',-13'2I
.tandTfu
D
= 1. Suv ra
6IN *6Iu
Hinh
3
= frtrfr
= 45'.
MAN ld g6c giira hai ducrng thdngAMvdAN.
= 45o
Eudng thingAN c6 VTPT ld
sti
ase re-zorsl
i=(z;-t).
-3s#8[
z
HÃY ĐẶT MUA TC TH&TT TẠI BƯU CỤC HOẶC VỚI TOÀ SOẠN !
cqi i=(a;b)*O n VTPT
A
M
thi
"orfrTN
=
ctra tlucrng thing
JL=
J5.Ja2 +b2
cos45o
o thi
(,)*,(;I-'(;)-r=,
. YOi a =3b,
chqn b = I
DB,
6E=6k.
,*=(t
e3a2 -8ab-3b2:0 (1)
b=0 thitu (1) c6 a=0 (loai).
N6u b +
68
ld g6c gita hai tlucrng thhng AD vit
Gqi
i =(":b)*6 a vrPr
cira
= JI6.J7TF =zlz"-bl
Ni5u
G6c
-t)non cosffi
=S-=a
.{aiq,[a 6
e5(a-b)' =2(a'+b')
*:L
> a = 3 thl |, = (Z;t),
e3a2 -lUab+3bz =0.
NOu
b:0
Ni5u b
thi
a:0
* 0 thi
3a2
(loai)
-IAab+3bz =0
.=',(g)'
" -\b) - ro[g)+:
-\b) - - o-
thingAMc6 PT le 3x+y-17 =0.
c>
la =3b .
lb=3a
Tri A= AMoAN tatimduqc A(a;5).
.Yot a=3b, chqn b=l=a=3, PT c'ia
. YOi b - -3a
rtudng thbngBDqw
tlucrng
, chgn
c:1=b=-3 thi |1=(t;:),
thhngAMc6 PT
Tt A= AMaAN
li -r-3y -4=0.
tatimduqc A(1;-1).
o Thi dq 4. (Trich ai pa kh6i D ndm 2012).
Gqi t h giao di6m cua AC vd BD thi tga d0 cria
1ld nghiQm cria h0 PT
{lx
dadng thting BD di qua aiam
'
)
toa d0 cac dinh cira hinh chrt nhqt ABCD.
Phdn ttch.
. Tru6c ti6n ta tim dugc tga d0 tli6m A = AC a AD.
. Vii5t dugc phucrng trinh tluong thing BD di qua M,
(g6c gita hai
t4o v6i AD mQt 96" ffi:6il
ducrng thingAC,AD).
. C6 phuong hinh iludng thdng BD ta tim duo. c tga
t10 t0m I =AC aBD cua hinh chii nhft ABCD vir
tgad0tli6m D=ADaBD.
. Tri d6 6p durrg c6ng thric
tim tga tlQ trung tli6m tl6
tim c6c ilinh cdn lai.
Ldi gidi. (h.a)
Tqa d0 di6m A
(
-:
("
Vi 1 le trung
ci.r.
1
BD n6n tga d0 di6m
r(t;-:).
.Ybi b:3a,
a=l:)b:3, dudng thing
BD c6 VTPT ld i:(1;:), t*O"g hqp ndy thi
chQn
,BD song song ho{c trung v6i AC (loqi).
OThi du s. (PA thi DH khdi B ndm
2013).
Trong mfit phdng vdi h€ t7a dQ Oxy, cho hinh
thang cdn ABCD cd hai &rdng chdo vu6ng g6c
vdi nhau vd AD :38C. Dwdng thdng BD c6
phactng trinh x + 2y - 6 :0 vd tam gidc ABD
c6 trac tdm ld H (- 3;2). Tim tqa do cdc dinh C
PhAn fich.
Hinh 4
)'
{::ur'*:oo=o
G6c 6k ld g6c gifia hai cludng thlngAD vit
AC,tac6
=o
[3x+v=o
1-^'' -", <+]^ -.'=o(-t:3).
[x-Y=-+ [Y=J
.
a
o.= I'
vd D.
ld
nghiQm ctra HPT
+
Y=
Tqa d0 di6m D ld nghiQm ctra hO PT
ru(--I,t). n*
\3
+
\' ''
= l(o:o).
lx+3Y=g LY=Q
,J
phdng voi h€ tga dQ Oxy, cho hinh
Trong mQt
chir nhqt ABCD. Cdc dtrdng thl:ng AC vd AD
ldn laqt c6 PT ld x + 3y : 0 vd x - y + 4 : 0;
u(-*;r) u rr*y=0.
\3 /
;1
cos6;=]59
=+.
Jl+E.JI+l J5'
Trudc ti6n ta xem x6t tinh d4c biQt cria cilc tam
gi6c IBC,BHC.
. Vi6t phuonc trinh iludng thing AC di
qua
vd ru6ng g6c voi ,BD. Suy ra tga dQ di€m I, C, A.
.Taxem D=ADaBD
. Vi6t phuong trinh duong thing AD
AC mQtg6c bang
qura
A tqo vbi
fiE =fdi = 45'.
HOC
.r TOAN
icru,ufw
E
HÃY ĐẶT MUA TC TH&TT TẠI BƯU CỤC HOẶC VỚI TOÀ SOẠN !
Loi gi,rti. (h.5)
[*-"-r
=o
lx+2,r,-6=0
.
Gqi -I ld giao di€m cia AC
vd BD, suy ru IB =1C. Md
IB
IIC
n6n tam giSc IBC
w6ng cdn t4i
{
suy
ra
A
i&=4f
. Mat kh6c
Hinh 5
BH L AD > BH L BC , suy ra tam gi6c BHC
vudng cAn t4i B, suy ra I ldtrung clirim ci,- HC.
. Dutrng thlng AC
duong thing BD, c6 phucrng tr\rth 2x-y + 8:0.
Toa clO cua
I ld nghiOm
*
{;= f =o(+:r).
f6t tu4n. Qua c6c thi du tr6n, chtng ta th6y vi6c
vi6t phuong trinh tludng thing trong m5t phSng tqa
dQ.Oxy di qua mQt di6m cho tru6c vii tao voi rluOng
thlng cho tnr6c m6t g6c c[, c6 th6 khai th6c d6 gi6i
nhi6u bdi to6n chi cAn ta tinh rlugc g6c cr 116. ViQc
tim toa dO m6t di6m du6i dpng ld giao rti6m cira hai
tluong thing cfrng li m6t hudng gini bdi to6n rim
di6m khri hiru ich. Sau ddy xin gi6i thieu m6t s5 bai
d€ ban itgc luyQn tpp.
BAI LUYEN TAP
cira h0 phuong trinh
l2x-v+8=0.'-i lx=-2
L @a thi DII tcnAi O ndm 2Ol4). Trong mdt
phing vdi hC tnlc tga d0 Oxy, cho tam gi5c
Tu 1 ld trung di6m cria dopn HC suy ra
C(-t;O); Hldtrungditim cua doanAC, suy ra
ABC co chAn du
ld di6m o(t;-t). Eudng thing AB c6 pT
1
lx+2y-6--0-- Iy=+
e(-s-z).
GSi i=(a;O)*O ld vectcr ph6p tuyiin cria
thingAD.
Ta c6 g6c gita hai tlucrng thhng DB vd AD lit
IDE =idi =45". Suy ra
ducrng
E^
la +2bl
la +zbl
I
C^c4\o
=.:l-:L...'.:l:-Js.Ja, +b, ,15..t;t;V O
,
+l*)€3d
N6u b=0 thi a=0 (loai).
NOu b*0 thikhid6
>z{a+2b)'z =5(a,
-3U
-kb=g
lg-.
z r2
*r(rs)'
r-r
-s f, -:=o<+.,2=',
. v6i a:3b>n-(zt;
lb3
u)=b(3; i)
(*)
3x +2y
-9
ngoai ti6p
VlCt
Pf
= 0 , ti6p tuyr5n
hm giitc ABC
ducrng thing BC.
Hwdng ddn. . ciei hO g6m PT ducrng thlng AB
vd PTTT tai A cua Clucrng trdn ngoai ti6p tam
gi6c ABC O6 tim toa clQ di6m A.
. Vi6t PT ducrng thing AD,tinh g6c
@f;7
. Vi6t PT dudng thing AC hqp v6i AD m6t g6c
. ;--.*---
bdng IAB,AD).
. Tinh goc
+
(an,ls)=(Ce,cn).
. Viiit PT duong
f
Ai A cin dudng trdn
co PT x+2y -7 =0 .
thbng
BC qra D, hqp v\i AC
m6t g6c Aing (ce,Cn).
a=3b
lb=1a
2. (DA thi.th* ndm 2015 cia Trudng THPT
ftte fiinh - Hd NOi). Trong m6t pt ing
v6i hd truc toa dd Oxy cho hinh chft nh6t
ABCD c6 diOn tich bing 15. Eucrng thing AB
c6 phucrng tr\nh x-2y = 0. Trgng t6m cira tam
Lr,rong
=(:;
t)
ld toa d6 vecto'phfp tu5r(in cria AD, ndn AD co
phuong trinh 3x+y + 17 = 0.
Toa dQ cua D ld nghi€m cua h6 phucrng trinh
=O€ I x=-8 D\-9:7
.. _ _- =
).
lx+2y-6=0 Itr-/
. v 6i b =
_:)
-3a *i = (";-t") = o(t;_Z)
= (r;
[3x+r'+17
ld toa dO vecto ph6p tuy6n ctra AD, n€n AD c6
phuong trinh x -3"y - 1 = 0.
Toa d6 cia D ld nghiQm cua h0 phucrng trinh
si6c
BCDrd di6m
"(-*,+).
rim
tsa d0 b6n
dinh ctra hinh cht nhft bitit ring di6m B c6 tung
d0 lon hcrn 3.
Hunrg ddn.. Tinh dlG;ABl suy ra
alc ; enl =
.afc; an1=
oB7= ac .
).ayc,
. Tir di6n tich hinh cht nh6t vi dQ ddi c4th BC
ffi
suy ra d0 dei
canhAB.
*rnu.,.-roru,
G
T?3I#S
9
HÃY ĐẶT MUA TC TH&TT TẠI BƯU CỤC HOẶC VỚI TOÀ SOẠN !
THuslJc rpuoc xi mu
pEs6g
(Thdi gian ldm bdi: 1 80 Philt)
Cilu 1 (2 di6m). Cho hdm s6
!=mx3 -3mx2 +3(m-l)
{mldthamsO; cO aO thi (C.).
a) Kh6o s6t
thi
bi6n thi6n vd vC d6
s1r
(C) cira him
ld'i m=1.
b) Chrmg minh ring dO thi (C.) lu6n c6 hai iliiim
cgc tri Avd B v6i moi m+O,ldti d6 tim cdc gi|tri
qia mae Z,qn'-(O,q' +OB2)=98.
s6
t
Cn
{.iu
2
(l
didm).a)Cho g6c
a
th6aman
cota=1'
Tinh gi6 tn cua bieu thuc
'_
sin2 a
b) Tim s6 phric
-sina.cosa -cos2 a
m6dun nh6 nh6t th6a mdn di6u
z c6
kien liz-31=V-z-'1.
Ciu
di1@. GiniPhucrng trinh
3 (0,5
-
^-t
4sin'?
|
-
F
J3 cos2x =2
-
sin2x.
lrs
t (t"+
l8.r.+25vL
^- )
l3+./9,r2-4=:l
-9[*
='I'
-2Y+2 -' J (r'Ye1R1'
lt "^
Ciiu
5 (1 di€m).
.--
oc
I
.12
,uu(z*-]D(-)I t**ol.
\
Ciu l0 Q diAfi. Cho x. y la c6c so thuc ducrng th6a
mdn x+y S 1. Tim gie ftl nho nhAt cua bieu thuc
I
I
ABCbing 8 .
Cfiu 8 0 diAd. Trong kh6ng gian v6i he toa d0
Oxyz cho mdt ciu (S): "r2 +f +22 -2x+6y+k-D=0
vd mflt phing (a) :2x -2y - z-t2 = 0 . Chimg minh
ring m{t phing (a) cit mat cdu (.$1 theo mQt dutrng
trdn. X6c dinh tim vd b6n kinh cria dudng tron d6.
CAu 9 (0,5 di€m).Tim si5 ha.rg kh6ng chta x trong
khai trien Newton
Y2
-
Q diA@. Trong m{t phing vdi h0 tqa d0
Ory, cho dudng trdn (C):(x-2)2 +(y*2)2 =5 vh
duirng th6ng (A):r+y+1=0. Tu diem I thuQc
(A) ke hai duong thing l6n lucrt ti6p xric v6i (C) tai
B vir C. Tim tga d0 di6m I bi6t ring diQn tich tam
Cdu 7
.
C6u "l (1 ai6m1. ZiaihQ phuong trinh
f7"c +y: +3.0,(x-y)
cua kh5i ch6p S.ABC vd khodng c5ch tu di6m,
m4t phing (SBC).
gi6c
2015
.r
Cfiu 6 (1 diAd. Cho hinh ch6p S.A,BC c6 d6y
ABCliLtam giric vu6ng t4i A, AB=3a, BC =5a; mdl
phing (SAC) rudng g6c v6i mat phing (ABC). Bi6t
ring SA =zali va SAC=30".Tinh,n* ?,T::l
2x2 +6x =1
Tinhtich Phdn
(x' - x -2)e^ JTTT|
(x + l)J3 +2x -
-? +l ^
.
PHAM TRQNG THI.I
x2
IGV THPT chuyin Nguyin Quang Diiu, DdngThdp\
2
\lAN DUNG ffi ... ttiip the o)
. Tt d6 tinh dugc c6sin cira g6c BAC, tl6y ld g6c
thing EK c6 phucrng trinh ld 19x - 8y - 18 = 0. Tim
toa d0 di6m C cira hinh vu6rry ABCD bi6t ring di6m
giira hai ducrng thing AB, AC.
"
I vit5t pt oong- ttirftauot g thing
AC
E c6 hodnh
di
qua G, tqo
dulng thilngABmQt g6c bilg 6k.
Tim tga dQ A= ABaAC, suy ra tqa d0 C
vcri
.
?-..''AC =+ AG ), trung di6m
. Viiit
2
(tr'r
/ cia AC.
phuong trinh ctudng thtng BC qua C, ru6ng
g6c v6i AB, suy ra tga dQ B, suy ra tga dQ D (ddi
ximg v6i -B qua tdm 4. Cht lf gia thiCt di6m B c6
tung dQ lon hcrn 3.
3. Trong m[t phing v6i
hQ
truc tga dQ Oxy, cho
hinh vu6ng ABCD.Dicm
Ff+;:'l
\z
)
ra trung di6m
cfa c4nh AD, di€mE ld tmng di6m cta
di6m K thu$c cqnh DC sao cho
HOC
- ^ TOHN
- ctudiga
IU
56
canh
AB vit
KD = 3KC . Duong
rJQ
nho hcrn 3.
Hutng ddn.. Tinh dusc
thtc
tan (a +
u)=
tn{iF, tanfr,
ding
tirn
r]ut1c
fl##k
turiit ,oy .u "nrFD, vot ifr
cong
1i g6c giira hai
dudng thing FE vi EK.
. Vi6I phuong trinh duong th5ng .El- di qua -F, hgp
v6iEl(mQt gbcbang fit
. Tim E =FEIEK, tt d6 tinh dLrq'rc d0 dai canh
hinh ru6ng. Til <16 tinh duqr: cosFEC vtvi FEC ld
goc giira hai dtrtrng thirrg f/:-vd IC"
IviSt pt uorg trinh duorrg thing EC hqp vtri FZ mQt
.
glcbdng FEC.
. Tim tea ttQ di6m C t* C
e EC vir E(-' = FC.
456 (6-2015)
HÃY ĐẶT MUA TC TH&TT TẠI BƯU CỤC HOẶC VỚI TOÀ SOẠN !
.. ,..
\rr
?
t
,\'
}IUONIG DAN GIAI OE SO
Ciu 1. a) Bpn ilgc tg gi6i.
b) Gqi M(xo:to)ta ti6p di6m cira ti6p tuy6n.
H6 so g6c cua ti6p tuy6n:
ru
1,
=
--
7
-
Ciu5.DAt t=Ji t P-x+Ztdt=dx;
x =l
= t =l,x = 4 > t =2. Ta
.
=2t--!-dt
=2(i+*-+)o,
t'z(t +t)
t
(xo+2)'
I
--J- . [-])=
-,
(xu+2)' \ t )
€ Jo = - t hoic Jo = - 3; suy f&yo=- 4 holc
/e = 10. Phucrng trinh tit5p tuy5n: ! = 7x + 3
gia thi6t suy ,u,
CAu 6.
S*.
lt
^z
=t;".
hodcy =7x+31.
Ta c6
CAu 2. a) Phucrng trinh d5 cho tuong du
Gqi D ld trung tli€m
AB, th\ OD L AB
sinx+ J3 cos*
€
B
-2 erir[r*1)
\. 3,/ = r
x = !+ k2rc (k
6
b) Ta co z:
6
Ciu
16o=600.
oD = lrr='fi,
-
eZ).
=5-l2i.z+i=6
5-t2i s
14'=-=--'21
6
3. Ddt t
3-
-. . lW; =
',l
=2' ,
A
=A,DLAB
(t > 0 ), ta c6 phuong trinh
ft-c
1
5
t'
[+-=;
<+l . 1 ; t=2e x-l= lex=2;
t
L
ll=-
t)
6
A'O = Oo.tan[io
=l;
v*r.o,r,r, = Stac.A'o
=+
cc'
= d (AB,cc')
I
arll
V^*r=.S*r.A'O=
II
(ABB'A')
b5t phucrng trinh da cho trd thdnh
2ab < a2 -4(l-b)
(a-Z)(a+2-2b)>0
e(J?;zxt
o
soo, =
|e'o.ea
d
=;
(AB,cc'
+
o
=#.u^,
o"
3?'
So'*= 4'
) =3v=o'
C0u 7. Ggi S - AB n DE. Theo dinh
o
-z)(,17 +rx-t+z.r)>
a(c,(ABn' a')),
=
24
1
l= r€.r-1=-1<)x=0.
ca,i+. DK: x2 +Zx*t>0<> x<-t-Ji toa"
x >-1+O. DlLt JV+2x-l =a) 0,1 -x =b,
(clvtt). Ta c6
1t;
ta c6
n6n (1) tuong ctuong
ysl Jx'z +2x-l-2>O
Ir -1+J6
<>l
-=x>-t+J6.
<
>
[x -1-J6
TH2 x<-l-J1.Khi d6 JV +2x-I+2x<0,
n6n (l) trothdnh JV +2x-l -2
<+-1-J6
Kiit hqp di6u kiQn cira trudng hqp ndy ta dugc
-1-J6
<
x<-1-J1.
Vfy nghiCm cinbitphuong trinh d5 cho ld
-1-J6
-r-JI ;
Thales
!4 =BE =r
d6 d(c,
\_- DE)
54 AD6-#=1.ro
SM 7
X6t 2 trucrng hcrn sau:
THl: x>-1+J2. Khi 65 J*2,r2*j+2x>0,
li
# =ffi
= 5d(8. DE) =s.+d(M, DE)=+
Ggi cpnh hinh vu6ng ld a. Trong tam
gi6c
ru6ng DCE ta c6
11113661
(dc,rD)--= co'* ce'=7+
=d(c,
vi
DE)=
25r,,=
25o.,
5?
=-Y.
J6r. Suv ,aE
J6t J6l0 -o=2,11g.
( :,:::
to-zzr\
D e DE nan D[/:
r > -1+J6.
).
raco:
(Xem
tiip trang 27)
T?3I#S
"rnu.,.-roro
11
HÃY ĐẶT MUA TC TH&TT TẠI BƯU CỤC HOẶC VỚI TOÀ SOẠN !
SISffiF$
cnfrr s0 xoc Dnc rRtrNE
frleffi
Biitqof,nDnqwft
(Gf rruPf
Oi thwc ld mAt nAi dung riiy quan trpng,trong
U "chwnng trinh todn hpc phii th6ng vd qat aiy
dagc gidng dqy trong chucsng trinh dqi s6 6 cdp
THCS. Cdc bai tudn hAn quan d€n da th*c xudt hi€n
nhiiu trong cdc k) .thi hpc sinh gi6i Quiic gia vd
Qu6c t€. Bdi todn v€ da thuc ld bdi todn dgi s6 tuy
nhiAn nhi6u bdi todn v€ da th*c bim chdt cila n6 lqi
ld bdi todn til ho". Niiu chting ta bih kiit ho. p khai
:.
thac mdi quan hQ giiia cdc tinh chdt dqi s6 vd s6 hpc
thi s€ gidi quy€t duqc nhiiu bdi todn vi da thuc.
Tinh chdt s6 hgc dfic trtmg thudng s* dang d6 ld
tinh chia h€t, t{nh chdn li, tinh chiit cfia cdc sd
^1,:;t6, hqp s6, s6 nguyAn,...Cdc dinh ly v€ s6
nguy1n
hpc thudng s* dung d6 ld dinh li, Bezout, Fermat,
Eisenstein,...
Tru6c h6t ta nhic lpi mQt sO tOt qui co b6n:
. Da thtc bQrc n c6 kh6ng q:oh n nghiQm th1rc.
.Da thric c6 v6 s0 nghiQm ld da thric kh6ng.
. Da thirc c6 bflc nho hon ho{c bing n mi nh{n
ctngmQt gi5 tri tqi n + 1 gi6 tri.kh6c nhau cta
ddi s6 thi cla thric cl6 ld da thfc hing.
. Hai da thirc bflc nh6 hcrn ho{c bing r md nhfn
|
n + giltri th6a. mdn bing nhaul4i n +l giht4
kh6c nhau cria cl6i s6 thi cl6ng nh6t blng nhau.
. Bdc cta tdng hai
cria m5i
<16.
. BQc ctra tich hai da thric kh6c kh6ng bing t6ng
cdc bflc ctra hai cla thfc d6.
. Hai da thirc f vd g thuQc IR.[r] trong d6 g kh6c
kh6ng (tla thric kh6ng) , khi tl6 c6 duy nh6t mQt
c[p cta thtc q, e IR.["r] sao cho "f qg + r
r
trong d6 hoic r:
:
0 hoflc deg r
<
deg
g voi r
kh6c 0.
cla thfcfix) cho x - c ldflc).
. V6i b6t k! hai da thric -f, g e ZlxT bao gid
ctng tdn tai IJCLN ciafvdg vd LrCtN cl6 duy
. Du cia phdp chia
nhdt.
f
. N6u da thuc d h UCLN cria c5c da thtrc vit
g, khi d6 tdn tpi hai da thitc u,v sao cho
fu + Sv d Nguo. c lai n6u da thirc d lit :uoc
:
chung cua cbc tla thric f vi, S vir th6a mdn
fu + Sv : d thl d le UCLN cinf vd g.
NGUYEN LTIU
chuyan Hd Tinh)
. Hai da thtrcf vdg nguy6n td ctng nhau tuc ld
g):1
khi vd chi khi tOn t4i hai tla thhc u, v
chofu+ gr: l.
. N6u c6c
nhau thi IJV)l'vd [g(x)]' s0 nguy6n t0 cring
nhau v6i mgi m,n nguy€n ducrng.
. Mqi nghiQm xs cta da thric
(f,
sao
tl6u
:
*...ta,sx
?rt'
th6a mdn b6t d6ng thric
.J@
aox" +
+ a,(as
*
0).
:
lr.l(lrrl -L)< A,A=maxlatl, k =r,...,n.
. Da thirc pahttna quy khi vi chi khi mgi u6c
cria n6 cl6u ld cta thirc bQc 0 hoic ld tla thric c6
dqng ap vdi alithing s6 kh5c kh6ng.
. M6i da th1c f
bAc 16n hcrn kh6ng U6t ty ACu
ph6n tich dugc thdnh tich cbc da thric bet khe
qly..Ve sp phAn tich d6 ld duy nn6t ni5u kh6ng
k0 d€n thf"t.u c6c nhAn tu vd nhdn tu bdc khdng.
. Tieu chuAn Eisenstein:
Cho P(-r) : a,{" * a,-t{-l+ ...+ afi + aoe Zlx)
Ntiu c6 it nhil mQt c6ch chgn s5 nguy€n t6 p
th6a mdn cl6ng thoi c5c di6u kiQn:
*) a, khdng chia h6t cho p.
*) f6t cir cbc hQ s6 con 14i chia hi5t chop .
*) a6 chia h6t cho p nhmg khdng chia hi5t cho
p2 th\ P(x) kh6ng ph0n tich duqc thdnh tich c6c
da thuc c6 b{c thdp horl vcri c6c hp sO hiru r5i.
NQi dung cua cac bdi toan vi da thuc thudng xoay
quanh viQc tim nghiQm vd xdc dinh si5 nghiQm ctia
mQt da thuc, xdc dinh m)t da thuc hofrc rdc luqng
gia tri clta da thuc khi biA dq thuc thod mdn mil sd
tinh chdt cho trudc hodc ld ch*ng minh cdc da thuc
biit kha quy...
Trong khu6n khd bdi vi€t nay , chung ta s€ dO cQp
diin m1t sii mi ndn didn hinh nhiim minh hea cdc
ilnh chh s6 hp, drtc ffung thudng xuh hi€n 6 bdi
todn da thuc.
Bii to6n l. Cho cac da thttc P(x), Q(x) e Zlx)
vd a e Z thod mdn P(a) : P(a +2015) : 0 ;
QQOI4): 2016 . Chilmg minh rdng phaong
trinh Q(P(x)): I kh6ng c6 nghiQm nguyAn
.
HgC
- ^ TOnN
- Gfi.rdiUa-gq-r*reeggl
LZ
HÃY ĐẶT MUA TC TH&TT TẠI BƯU CỤC HOẶC VỚI TOÀ SOẠN !
Ldi
gi,rti.
P(x)
: (x - a)(x - a - 2015). s@)
Gii sir lxse ZdC
Q(P(xr)): 1 + Q(") = lx - P(xdl.h(x) + I
:
= QQ0l4) 12014 - P(xs)l.h(xs) + l.
=P(x) chin voi
Yx e Z.
Do P(x6) chin vd h(xs)e
phii
Z,
n€nv6 trai chin, v6
m6u thu6n.
Ldi binh. Ddy ld bdi to6n tlon gi6n, sir dqng phucrng
ph6p chimg minh ph6n chimg ptr6i frqp dinh li
16,
Bezout cring tinh chin, 16 cria s5 nguy6n ta suy ra
tli6u ph6i chimg minh (tlpcm).
Bii
to6n 2. T6n tqi hay kh6ng da thtrc P(x),
deg(P(x)) : 2015 thda mdn P(*' - 2Ol4) chia
hAt cho P(x)?
Ldi girti. Xdt da thric P(r) : (x + o)'o",ae IR .
Khi d6 P(i - 2014) : (* + a - zot4)2ots
: l@+o)' - 2a(x + a) + a2 + a- 201412015. Da
th6y phuong trinh o' + o - 2Ol4: 0 lu6n c6 2
nghiQm phdn biQt. Tric ld ta chgn dugc a sao
cho a2 + a * 2014: 0, tu c16 suy ra p(xz - 2ol4)
: (x + o)""(* - a)20rs chia h6t cho p(x).
Liti binh. Tri y€u c6u P(x) c6 deg P(x):2015 vi
P(*' - 2014) phii chia hrit cho p(x) nen ta c6 ttr6 du
do6nngay P(x) c6 dpng (x
+,)'0".
Biri toin 3. Cho & e N- . Tim dt cd cdc da thuc
P(x) th6a mdn:
(x - 2015)k P(x) : (x -20t6)k P(r +1) (*)
Ldi gidi. Gi6 sir P(x) ld da thric th6a m6n di6u
kiQn trdn. Tri gi6 thi6t ta thdy x : 2016 liL
nghiQm bQi bflc 2 k cl0la P(x).
Theo tlinh lf Bezout ta c6:
P(x): (x -z}rc)k.Q@),
thay vdo (*) ta dugc:
(x - 20 t 5)k .(x -20 rc)k .8@)
: (x- Zlrc)k1x-2015)k.e(x+
1).
Tt
d6 suy ra Q@): Q@ + 1), hay Q@): a
hing s5.v6y p(*): a (x 2ot6)r. Tht lpi ttfng.
Ldi binh. Ddy ld bdi torin co b6n, sir dqng tinh ch6t
chia h6t, nghiQm bQi, cring tlinh lf Bezout cho ta loi
gihi.
Bii toin
4. Cho P(x) vd Q@) ld hai da thilrc voi
hQ sd nguyAn. Biiit ring da thuc xP(x3) + Q@t)
chia hih cho x2 * x * l. Gpi d ld UCLN cria
P(2015) vd QQOI|). Chilmg minh rins d >2014.
Ldi gi,fii. Ta c6: xP(x3) + Q@\: lQ@\ - QOI
+
xlP(x3)-P(1)l + kP(l)
+0(i)l (l).
o6 trr6y e@\ - ee) chiahiit cho 13 - 1 suy ra
Q@') - QQ) chiahiSt cho ,'+ * + 1. Tuong tu:
P(r') - P(1) chia h6t cho x2 + x + 1. Tri (1) dsa
vdo gi6 thi6t ta suy ra t P(1) + Q\)1chia hi5t
cho x2 *.r * I (2). Do deg (.x2 + x +1) : 2 vit
deg [xP(l) + g(1)] < 1 n6n tu (2) suy ra:
xP(t) + QQ) = 0
P(1) : O(1) : 0 (3).
=
V6y 1 ld nghiCm cria P(x) yiL Q@). Theo
Bezout ta
c6:
{59=l;_i,,X;$
Do P(x)
vd Q@) li c6c tla thfc voi hQ st5 nguy6n n6n
vi R2(x) cfing li chc da thric vdi h€ s6
R1(x)
r ^: +Li_.. lr{c:otsl= 2014.Rr(2015)
-_.-.Ansuven.
-'a--J
-"' Lal
-a- tnav: <
"^-r
lee}ts)=20t4.Rz(2015)'
suy ra P(2015) vd QQ0l5) chia hiit cho 2014.
Vi d: (P(2015), QQ0l5)), suy ra d > 2014.
Ldi binh. Bdi to6n ndy
cla
k{r6 tle.p vC tinh chia hi5t cria
-
thric. Sri dtmg tinh ch6t 1a
b1l(P@)
- P(b))
,
vd ktit hqp gifia ttinh lf Bezout vdi tinh ch6t vd b6c
cria da thric tl6 tlem l4i loi gi6i.
Bii toin 5. Cho da thuc flx) : x20t7 + axz + bx
-t c vdi a, b, c e Z cd ba nghiQm nguyAn
x2,4. Chrhng minh rdng:
(o'0"
+ b2017 + c'0"
+ry1x1
-
xz)( xz- x3)(x3
chia hiit cho 2017.
Ldi gi,rti. X6t phuong trinh:
x2ot7 - x + loi + (b + l)x + cl
:
-
x1,
x1)
o.
D1t J@) : ax' + (b + l)x * c. Theo dinh li
Fermat bd ta c6: xz0t7 - x = 0 (mod 2017), suy
ta: JU) = 0 (mod
2017),
f(x) i2017.
i : l, 2, 3 hay
j
x2)(x2- r:)(x: - ,,) 2017 thi bdi
to6n chimg minh xong.
. N6u @1 x2)(x2- xz)(xr x1) kh6ng chia hiit
. N6u
(x1
-
-
cho 2017 ta c6:
flx) *J@)
:. 2017, suy ra
= (x1- xy) fa(xy +x2) + b +l I i2017
* axr+ axz* b+ 1 i 2017
\r: axza oxz+ b + 1 i2017
Tri ( 1) vir (2) suy ra: a( 4 - x) i ZAtl
Tuong
a :.2017,
(1).
(2).
+ li20l7,
fl*r): L axr2 + (b + 1)x1+ c] i 2017, suy ra
c :. zol7.
hay
suy ra b
nu.,.-rrru,
T?ll#ff
13
"s
HÃY ĐẶT MUA TC TH&TT TẠI BƯU CỤC HOẶC VỚI TOÀ SOẠN !
Ydy a + b + c + | :.2017.Theo cllnh ly Fermat
bd ta c6: o = o'\"(^od 2017), b = b2017(mod
suy ra a-t b-r c *
2017), c =
"""(^od20l7),
+
+
+
(mod
1
zolT).
l: a2017 b2017 c201'7
Tri d6 c6: a2011 + b2017 + tott + lizol7 (dpcm).
Ldi binh.Yl2017ld sti nguy6n t6, n6n trong loi gi6i
hr da thtc d6 cho ta th6m bort x d€ sir dgng dinh li
Fermat b6 ld mQt tli6u t.u nhi6n.
Biri toin 6. Cho s6 nguyAn n > 2 vd da
cdc
hQ
thr.rc cd
s6 nguyAn daong
P(*): x' + a,-tx'-| *...* ap * l.
Gid sir ak: an-k vdi mpi k: 1,2,..., n - l.
Chtimg minh riing tin tqi vd sii cdc cdp s6
nguyAn daong x, y sao cho xlPO) vd ylP(x).
Ldi gi,rti. nO tn6y it nhat cip (1, P(1)) th6a man
bdi to6n. Ta gii sir chi c6 hiru hpn c{p thoa m6n
dC bdi. Khi d6 chgn trong tl6 mQt cflp ft, y) md.
y ld s6 lcm nh6t. Ta sE chirng minh ntSu c6 cflp
s6 nguy€n ducrng (x, y) sao cho xlP(y) vd ylP(x)
- (I y, vP(v))I cfrng th6a mdn tinh chet d6,
thi cip
\JI
\/
to6n 7. Cho a, b, c td bo sd nguy€n phdn
bi€t vd da thilrc P(x)e Zlxlsao cho P(a) : P(b)
: P(c) :2. Chilmg minh ring phuong trinh
Bdri
P(x)
*3 :0
kh6ng co nghiQm nguyAn.
-2
Ldi gidi. Tri gin thi6t ta suy ra cla thric P(x)
c6 ba nghiQm nguy6n ph6n biQt ld a, b, c. Do d6
t6n tai Q@)e ZLxT sao cho:
(x - a)(x - b)(x - c)Q@).
P(x) c6 nghiQm
Gi6 sir phucrng trinh P(x) 3
dngryAn. Khi d6:
1 : P(A - 2 -- (d - a )(d - b)(d - c)Q(fl.
ndn d a,
Do d a, d b, d * c, Q@)
gi6 thitlt
v6i
m6u
thu6n
d - b, d - c e {1,-1},
a, b, c ddi mQt ph6n bi€t. Tt d6 ta c6 cli6u ph6i
2:
- :0
x:
eZ
-
-
-
chimg minh.
Bii torln
8. Cho da thac fu)ez(x) . Chilmg
minh rdng n€u da thilrc Q@) : J@ + 12 c6 it
nhiit 6 nghiQm nguyAn phdn biQt thi JU) kh6ng
c6 nghiQm nguyAn
Ldi gidi. Gi6 sir Q@) c6 6 nghiQm xt, x2 ,...,
xe € Z suy ra Q@):JU) + 12
: (x - x1)(x - x2)...(x - xu).g(x) v6i g:(x) eZlxl.
Gi6 sir t6n tai xs
12
:
Q@) :
€Zmdfixs):0
ta suy ra
:
(xo- xrXxo- xz)...@o- xo) . g(xo)
l*,-r,l lro -rrl ...lro *rul ls(ro)1.
aQ2;r1yr
r[401]
16o
+ 12:
x ' ta
\ x )'
Do re-x1, x0_ x2,...:q0_ xsld c6c s5 nguyen
di6u hi6n nhiOn). Thflt vfly tu gin thi6t at : ark
nghiald
taco: yf
''
"
-
mQt kh6c nhau nOn trong c6c s6
ta
\ x ) -rP(v))''[-l)
\, )'[P6)J
c6: P(,IO2)
Po)
= *,r((x)' l=1r1ry1,."t'=+.)
\r0)/
/\
tv\
Y) = r (x): o(mod Y)
= ,[
,Ur.,},mod
(doP@) =1(mody)).
Mat kh6c do xl Pb),n6n (x,
lai (x, y) :
d > 1 thi
y): t vi n6u ngusc
r(y)=
Suy ra
'
\.x)
"f4g)=o(mody).
^ P(v) y2
xx
1
-x,l,...,lro -rul kh6ng th6 c6 3 so trd lcn
bing nhau, suy ra
hon
lro -r,llxo -xrl...lxo -rul) 12.22.32 =24 ,
lro
(mod
d), vd ly.
Hon
nfta
y>x
tdn tai v6 s6 c[p sti nguy6n ducrng x, y thba mdn
dc bei.
didru then ch6t h bi6t
sir dgng phuong ph6p clrc hpn, ph6i hqp vdi ph6p
Ldi binh. Ddy ld bdi to6n kh6,
to6n
chrmg dd tim ra loi girii.
nta ls(xo)l> 1= vd ly.
Bii to{n 9. Cho JV) ld mQt da thtbc bQc 5 vdi
hQ sri nguyAn, nhQn gid tri 2015 vd'i 4 gid.tri
nguyAn khdc nhau cila bi€n x. Chrng minh rdng
2046 kh6ng th€ co nghiQm
phuong trinh flx)
:
nguyAn.
Ldi gi,rti. Theo gi6 thii5t phuong trinh J@ 2Ol5 0 c6 it *6t + nghiQm nguy0n. Ta c6
/(") - 2015 : (x- ,,)( x- x2)(x * x:Xx - xq).g@)
v6i x1 < xz l xt < .x+ vh g(x) ld m6t tla thric vdi
;,^
hQ so nguyen. GiA sir t6n tpi s6 nguy6n rs sao
choflxs) :2046 thl:
3l : (xo- x1)@s - x2)(xs- x3xxe - x4).g(xg),
v6i rs - xt) xo - xz; x0 xt ) xo - xa, Yd c6c sd
'-.:.A^
ndy
ddu ld so nguy€n. Vi 31 ld sd nguy6n td
HgC
L TORN
14 -cludi@
HÃY ĐẶT MUA TC TH&TT TẠI BƯU CỤC HOẶC VỚI TOÀ SOẠN !
n6n: 31 :31.1 : (-1).1.(-31)
: (-1).(-31) : 31.(-1).(-1).
Do d6 31 kh6ng thd phdn tich thdnh tich cin 4
s0 nguyOn kh6c nhau. Di6u ndy chimg t6 ring
phucrng
trinhflx) :2046 kh6ng th6 c6 nghiQm
nguyOn.
Ldi binh.
C6c bdi toin 7, 8, 9 cirng m6t th6 loai.
Pfuong ph6p chung il6 gi6i o ddy ld chimg minh
bdng phin chimg. Sit dqng tinh chdt nghi6m, tinh
,t
;.
i
i
^ to,
ch6t s6 nguy6n
s6 nguy6n, ph6i hqp v6i tlinh
lf
Bezout.
Blri to6n 10. Chilmg minh riing da ththc
P(r): (r-1)(x -2)...(* -201s) - 1
biit khd quy tr€n Zlx),
Ldi gidi. Gi6 sir da thric P(x) khdng Uat nra
quy, suy ra P(x): F(x).G(;r) trong tl6 F(x) vd
G(x) le circ dathric bQc nguy6n ducmg v6i hQ s6
nguy0n. Kh6ng gi6m t6ng qrilt giir su c5c hQ sO
bfc cao nh6t cira F(x) vd G(r) bing 1.
Videg P(x):2015n6ndegG(x) < 2014.
Ta c6:
-i : F(x)G(x).
Do
F(k).G(k): -1 hay F(k): - G(k): t 1,
Vk,l
Vk,I< k <2015 . Ddt Q@) : F(x) + G(x) thi
P(x):
(x
-1)(-r- 2)...(x-20t5)
<16
<
des Q@)
Suy
ra:
2014 vd Q(k)
Q@)
=
:
0Y
0 hay F(x)
k,l < k < 2015.
: - G(x), Vx e IR..
Khi d6 c6c hQ sO tac cao nhAt cin F(x) vd G(x)
cl5i nhau, m0u thu6n v6i gi6 thiCt 0 tr6n. Vfy ila
thr?c P(x) U5t mra quy tr}n Zfx).
Biri toin ll (tMo-1s93). Chilmg minh riing da
thilrc flx) : xn + 5l-t + 3 vbi n eN. bh khd
quy ffAn Zlxl.
Ldi gi,fii. OO th6y vbi n:2 thi flx): y2 + Jy -r
f U6t nra quy tr6n Zlxl. vOi n ) 3, gii su
J@: s@).h(x) v6i g(x), fr(x) thuQc Z[x] vit c6
b6c ) 1. Vi deg g + deg h: n> 3 n6n suy ra
trong hai s6 deg g vd deg h co mQt sO > t. tr,tpt
iirrilcfQ):
lrtol:l.
(fr
3 ld s6 nguyCn td ndn
Gie sir g(x)
:
ls(0ll=
: -
ho{c
*u + alxk-l +...1
> 1) va ls(o)l:1. Gei at, a2,...,
nghiQm cira g(x) ta c6:
g(x) (x ar)(x
t
- or)...(* -
ar,
aL ld" cdc
ou).
Vi ls(O)l=1 ndn la1a2...a1,l:1 (*)
Do g(a;) : 0 n€n Jla):0 v6i moi
1,...,k.
Tri cl6 ta c6: all-t(a,+ 5) : -3.
Nhdn c6c tling thr?c tr6n lai vd kilt hqp v6i (*)
ta dugc: l(a1 + 5)(a2 + 5) . . .(ap+ 5)l : 31 1x x)
i:
Mit kh6c ta c6:
lg(-s)l :l@r+ 5)(a2+ 5)...(ap+ 5)l
vn 3 :l-s) : g(-s)ft(-s) ncn
l(a1 + 5)(a2 + 5)...(a1, + 5)l bing t ho[c bing 3.
Di6u ndy mAu thu6n v6i (**) vl k> 1. Tt d6
suy ra dpcm.
Bii to6n 12 (VMO 2013-20t4). Cho da thac
P(x) : (
- 7x + 6)2' + 13, ne N..Chbng
"'
minh riing P(x) kh6ng th€ phdn tich thdnh tich
cila n + I da thuc khdc hting s,i vdi h€ sti
nguyAn.
Ldi gidi. n5 thay deg P(r) : 4n vdP(.x) khdng
c6 nghiCm thuc. Tam thirc xz - 7x + 6 c6 hai
li 1 vd 6;vi 13 ld s0 nguy6n t6.
Gi6 su P(*): Pk)...Pn*r(x), thi P;(x)
nghiQm
c6 b{c
Vi t6ng c5c bfc cira cbc Pi(x) Ld 4n n}n
, i. ^ phdi c6 it nhAt 2
da thfic c6 b{c ld 2 gii su ld
ch8n.
P1(x) vd P2(x). Do hQ s5 cao nh6t cira P(x) ld 1
+
n6n clflt Pr(x) : x2 + ax + b > O, Pdx):
"' "":
+ d > 0 Vr. Ta c6 13 : Pr(l).Pr(l)...P,*r(l)
P{6).Pz$)...P*r(6). Tu d6 suy ra trong hai sd
Pr(l) c6 it nh6t m6t sti bing 1. Khdng
P1(1) vd
gi6m tong qudt gi6 su P1(1) : 1 suy ra a: -b .
Khi d6 P,(6) : 36 - 5b > 0 vd P{6) + 13, suy
,,.
ra36 -5b: lhay b :7, a:-7 vdkhid6
Pr(x) : x' -7x + 7 c6 nghiCm thuc , m6u thuin!
Ldi binh. Ca 3 bdi to6n 10, 11, 12 vA da thric b6t
Lh6 quy d5 n6u
chimg minh bing ph6n chimg. Tuy nhi6n bdi to6n 10
thi d mric trung binh vd kinh tli6n, chi stl dUng tinh
ctrAt vC bflc cria da thirc ld cho k6t qu6. Bdi to6n I I d
mric d0 cao hon, ta d6 sir dung tinh ctr6t ve b6c cria
da thric, tinh ch6t cira st5 nguy6n t6, s6 nguy6n, k6t
hqp v6i dinh l)? Bezout. Bdi to6n 12 cing d mirc tlQ
nhu bdi 11, tuy nhi6n tl6y ld bdi to6n m6i trong kj,
thi hqc sinh gi6i Qu6c gia ndm qua, gAy kh6ng it
kh6 khln cho nhi6u hgc sinh. Bdi to6n ndy c6 vdi
c6ch gi6i tuy nhi6n d6u chimg minh bing phuong
ph6p ph6n chimg ph6p phin chimg. C6ch gi6i tr6n
dAy dga vdo viQc khai th6c b6c cta da thfc, tinh ch6t
ctng tinh ch6t cira s6 nguydn tii.
D6 luyQn tfp phin ndy c6c bpn hdy ldm mQt sd
(Xem fiAp fiung 26\
bdi tpp sau:
nghiQm
nr.,.-rorr,
=,,
T?8I#EE
15
HÃY ĐẶT MUA TC TH&TT TẠI BƯU CỤC HOẶC VỚI TOÀ SOẠN !
-
-lr
cAc IqIp THPT
fl{
B*i'fb/456. Giai
hC
phuong trinh
fx+v+z=3
#i4
lx2y+y2z+z2x=4
t^
|.*'*]' + z2 =5
ff{;
$M\,,
TR.AN QUOC LUAT
(GY THPT chcryAn Hd TInh)
cac r,op TI{CS
Bli
Ti1455 il,rirp 6')" Tim t5t ci c6c b0 sd
nguyOn t5 sao cho tich cria chimg bing 10 Dn
t6ng cira chring.
Bei T71456" Cho hinh ch6p SABCD c6 d6y
ABCD le hinh chir nh4t, Sl vu6ng g6c v6i mflt
phang @BCD). Goi G ld trong t6m tam giSc
,SBC vd kho6ng c6ch tu G d6n mpt phing (SBD)
ln d. Ddt SB : a, BD : b, SD : c. Chimg minh
ring: a? +b2 +c2 >-162d2.
lt
TRIIONG QUANG AN
(GV THCS Ngh\a Thdng, Qudng Ngdi)
Eni I"21456 (Lop 7). Cho tam
gi6c ABC cdn tai
Ac6
Etheothf
BAC =800.C6cdi6mD,
thuQc c6c cqnh tsC, C4 sao cho
BAD = ABE = 300. Tinh
sO
eua,Nc nAo
(GV THPT chuy€n Huinh Mdn Eqt, Ki€n Giang)
Bdri T8/,455. Chimg minh
ring phuong trinh
11
(x+1)'*t =1;
t.u
c6 nghiOm duy nh5t.
:
NGUYfN vAN xA
ilo BED.
(GV THPT
YAn
Phong 2, Y€n Phong, Bdc Ninh)
NGUYEN MINH HA
(GV THPT chuyAn EHSP Hd NQi)
BAi T31456. Gi6i phuong trinh
BAi T9/456. Cho
l_
I _R( | _ 1 )
'
Jx J2x-t -"-( J6x1' J9x4 )
LAI THI HOA
(GV THPT L€ Quj,D6n, Thdi Binh)
Bei T4/4S{i" Cho hinh vudng ABCD cqnh a.
TrOn c4nh AB, BC l6n
TIdN TOI oI,YMPIC TOAN
luqt l6y cbc di€m M, N
a,ar,...,ar, ldc6c s5 nguy€n
duong th6a m6n:
a) at
b) Vdi m6i
n6u
bo
sO
nguy6n duong k,
ki hiQu b.li udc
1an,th\
Chimg minh
bt > b2>
mdn'. xa +yz
+l3y+lS(y-2)x2
+8;.1.
...) br,
BSi 'f lEi456. Cho rla thric
lQ)=x3
Hdi trong dopn
nguy€n
a
+3x2 +6x+1975.
[1;3201s
]
cd tAr cA bao nhi0u s6
sao cho./(a) chia h6t cho 3r0rs ?
UAI QUANG
TRAN xUAN DANG
(GV TNCS Ydn Lang, Vi€t Tri, Phri Thp)
(GV THPT chuyAn LO H6ng Phong:, Nam Dlnh)
NUT
F.IQC
* -- TOHN
t cfu66q_9eggq
I6
nhSt cria ao sao cho
TRi,N NGQC THtur\rG
{GV THFT chry,,6n l/inh Philc)
(GV THCS LA Lqi, TP. Quy Nhon, Binh Dinh)
Eei Tg1456. Tim c6c s6 nguydn duong x, y thoh
1,..., 15,
ring a,r>2015.
MDN = 450. Tim vi tri cua M, N de d0
ddi tlopn thing MN ngin nh6t.
sao cho
uilr vAN cHr
sO tcrn
k:
-zotl)
_
HÃY ĐẶT MUA TC TH&TT TẠI BƯU CỤC HOẶC VỚI TOÀ SOẠN !
Bir;
/:
"il
[*ai g,lil4$tl" Thi nghiQm giao thoa s6ng trdn
mdt nu6c vdi hai nguOn dao
ttr: uz: acoso/. Coi ning lugng truy€n s6ng
456. Tim t6t cd chc hdm clcrn 6nh
IR -+ R. th6a mdn
n 1i
f(x')+ f(y') = ("+y)[/o(x)- fr(x)f(y)
+f'(x)f'(y)-f
v6i mqi 'r'Y e
kh6ng dOi, bu6c s6ng tlo duoc l"
(x)f3 (y) + /*0)]
KIru oixu M,NH
}*:ii
Cho tam gi6c ABC virdidm M
nim trong tam gi6c kh6ng trung v6i trgng t6m
G. N6i AM, BM, CMldnluqt cit BC, CA,AB
t1i A,r,Bu,C'. Ke A0A1 llCA, AoA2llABvoi
c6
8,82,
Ct,C2. Goi G,,G, lAn lucrt ld trong t6m tam
gi6c
A,B.C'
:
M tr€n m{t nu6c thuQc dudng cong
cyc d4i kC tu ducrng trung tryc cira,SlS2, M dao
(GV THPT chuy€n Hilng Yaong, Phti Thp)
"E'nlii'4iifl.
. Tucrng tg, ta
cm,SrSz
8 cm. Di6m
R
A,,d thu6c BuCo
:4
A2BzCz. Chimg minh rdng
cl6ng c6 phucmg tr\nh uy
Di6m
M'
:
2acos(at
nim tr0n dudng trung tryc
-*).
'2
Sr,Sz
vd
ctng pha voi M. Di6m M giln
Sr nh6t. Tinh khoing crlch tu M dtSn M nhb
nh6t.
r,t rAN nr
(GV TIIPT Hu)nh Thuc Khdc, TP. Ui Chi Uinnl
dao clQng t1i
M
$;.]i Lli'$56. Cho mpch diQn nhu hinh b0n. Bi6t
3 O i /i+ : 2 Q.Tirn
rliqn tr& tuong iluong cua dopn mqchAB.
a) ArB,ll Bp,llCi2.
Rr
b) MG di qua trung cliiim cua G,Gr.
:Rz :.R: :
()
1
; Rs
:
DINH THAT QUVNH
(Hd N1i)
PHAN VAN NAM
(.GV THPT chuy€n DHSP LId iVQi)
pffi&ffi&ffiffiffi &ffi ffiffi Kffiffiwffi
andy such that
F.$R. $EiC GI BAK V SC M{.}CIL
-N
Froirlem'i'$1456 1$'or 6tr'gr*rle)" Find all finite
sets of primes such that for each set, the product
of its elements is l0 times the sum of its
elements.
Prrrlrle&! 'X214*6 (F or T'i' gradc). Let ABC be
an isosceles triangle with the vertex angle
6ii
x4 +y2 +13y+
FOR IIf
Protrriern'f
rr-(r
r)
'
'
-'-[J6r-t
\ti Jzx-t
ien-T)
I _r__
- -/\
___
!
-r_
Frolrleru '84i45#. Let ABCD be a square and
let abe the length each side. On the sides lB
and BC, choose M and N respectively such that
frnN = 450. Find the positions of Mand l/
so
that the length MN is minimal.
Solve the following systern
r
lx+y+z=3
l*'y* y2z+z2x=4.
l,l*'*y'tz2 =5
=800. Choose D and E on the sicles BC
:'r'riirl;rn 'gJ1456. Solve the equatiorr
G t.} 5CX-I{ }{-}{,
of equations
and CA respectively such that
6iD=GE=300. Find theangte 6Ei.
61,1.56.
1"'-(y-Z)x2 +8.ry.
Frohelemr T7i;$${r. Given a quadrilateral
pyramid S.ABCD with the following
properties: the base ABCD is a rectangle anrl SA
is perpendiculal to the plane (ABCD). Suppose
that G is the centroid of the triangle ^!BC and let
dbe the distance from G to the plane (SBD).
Let SB : a. BD: b. and SD: c. Prove that
a2 +ltz +c2 ) l62ct2 .
Ilrohicm
'l'1t1"[56.
Prove that the following
1l
equation: (x + 1)'*t =
xi
has a unique solution.
tXr:rrc
Froi:leru 'l'51456. Find all positive integers x
Sti ase (6-2015)
tiitt trwng 2l)
T0ffiru
Hsfl * *
q
-:gfrffiffie
/
HÃY ĐẶT MUA TC TH&TT TẠI BƯU CỤC HOẶC VỚI TOÀ SOẠN !
(c.g.c), suy ra AM: BP (l) vd AM ll BP. Mdt
kh6c theo c6ch dlmg vd giri thi6t ta c6 Ol/ ld
trung tryc cua MP, nln MN : PN (2).
.|
6rnx
tsKX
xY
Bii
rnuoc
y ld t'oc s6 tu nhiAn khdc'0,
tirrt giti tri trhtt tthit ,rtrt hieu thirc.4 - 36' - 5' '.
Ldi gidi. n6 tnay ring 36' lu6n c6 cht s6 tQn
cung bAng 6, cdn 5/ lu6n c6 cht s6 tdn ctng
bing 5, do d6 nr5u 36* > 5v thi 36' - 5/ c6 chft
s6 tfln cirng bing 1, n6u 36' < 5v thi 5r - 36' c6
i .^
.
, J
chir so tin cung bdng 9. Vdi x : I, ! : 2 thi A
:36 - 52: 11. Ta chi cdn xdt xem A c6 th6 6y
grhtrib[ng t ho{c t hay khdng.
. Gi6 sir co 36* - 5v : t hay 36' - I : 5v thi v6
trdi chiahrit cho 36 - 1 : 35 : 5.7 n6n chia h6t
cho 7, trong khi vti ph6i 5r kh6ng chia h6t cho
7, do d6 khdng t6n tai c6c s6 r, y nhu th6.
. Gi6 su c6 5/ - 3€:9 hay 3€ + 9 : 5Y thi v6
tr6i chia h6t cho 3, trong khi v6 phbi 5v kh6ng
chia h5t cho 3, do d6 kh6ng tdn tai cic s6 x, y
ret irqp (1) vd (2) ta c6
nhu th6.
VQy giltri nh6 nh6t cira bi6u
BitiT3l452. Giai hC bdt phtrong trinh
f 1 frl,...,.+\j
<(.r+y)r (l)
I.l I
T1/452. Cho
-r,
thfc A ld fl. A
YNhQn xdt. MQt vdi b4n kh6ng x6t trudng hqp
5! 36". C6c bpn sau c6 ldi giii tltng: Vinh Phtic:
Nguy1n Th! Ngpc Mai,6A4, THCS Y6n Lpc; Qu.ang
Ngfli:-Zd Tudn KiQt,6A, THCS Ph4m VIn D6ng,
|tlguy€n Hodng Kim.Anh, 6A, THCS }Iirnh Trung,
Nghia Hdnh; TP. CAn Tha: NguyAn Hodng Octnh,
6A7' THCS
AM + BN = BP + BN > NP =
HAI
'IET
BdiTZl452. Goi O li trung tlieim ctru tloatr rhdng
AB. TrAn mdt ru,ra mdt phdng bd AB, vi hai tia
Or, Wo vtt6ng goc vo'i nhau. Trtn Ox, Oy ldn luot
tatt hai di€tn M, I'l kh6ng trirng vo'i O. Chimg
minh rcing: AM + BN > MN
.
Ldi gi,rti. TrCn tia d6i cta tia Ox 16y di6m P sao
cho OP : OM. X6t hai tam gi6c OAM vd OBP,
c6 OA : OB (gt), OM : CP (c6ch dlmg),
AOM = BOP (ddi dinh), do d6 LOAM = LOBP
-^
l8';4i,iiEry
YQ'y 1u6n c6
J
xit.
Day ld bdi to6n d5, mQt s6 bpn sri dung
tinh ch5t cua duong trung binh trong tam gi6c nhrmg
kh6ng chimg minh. C6c bpn s.au c6 ldi gi6i t6t:
NghQ An: Nguydn Dinh Tuiin,7C, THPT L)t Nhat
Quang, Dd Lucrng; Quing Ngfli: Zd Thdnh Hy,7A,
THCS Hanh Trung; Nguy€n Thi Ki€u Mdn, 78,
THPT Nguy6n Kim Yang; Bili Thi LA Giang,7C,
D6 fhi Mi Lan, Nguy€n LA Hodng DuyAn, 74,
THCS Ph4m V5n D6ng, Nghia Hdnh; Hu)nh Dqng
DiQu Huy€n, 7C, VA Thi.H6ng Ki€u, 7A, THCS
Nghia M!, Tu Nghia; CAn Thtr: Nguy€n Hodng
YNhQn
Nhi,1A6, THCS Th6tN6t, Q. Th6tN6t.
NGUYEN XUAN BINH
l-v..
1--
.
IJt-t.r*r')r =l-.r'
-
ThotN6t'
MN.
MN. DAu bing xity ru khi N, B, P
th[ng hdng (B nim gita N vd P), tuc ld khi
AM II BN.
AM + BN >
Loi gidi.DK: lx+yl< 1 vd x <
\21
1.
Ta xdt hai trudng hqp:
o
Voi
.t +
)
:0,
thay vdo (2) ta dugc
x:
0, suy ra
y):
y:0.
(0, 0) th6a m6n (1) n6n ld
Cap s6 (*,
nghiQm cira hQ b6t phucrng trinh.
o
voi -r*) r 0, thi lr*yl ,0.
x' -
Ta c6:
r
-ry
t y' :4tx + l)'.,+ 3. - l)').,
|(x
.)tr+y)2
4'
>0
)2,[x'-xy+y'>lr*yl
* (x+y)2 t lr*yl (theo (1))
= lr+yl(lr+yl-1)
>o
= lr+yl > t.
TOAN HOC
HÃY ĐẶT MUA TC TH&TT TẠI BƯU CỤC HOẶC VỚI TOÀ SOẠN !
f6t
hqp v6i DK suy ra
l;r+yl= 1
= x *y: t ho[c x + !: -1. Thay vdo (2) ta
tim dugc (x,y): (1, 0) ho[c (*,y): (1, -2), cl6u
kh6ng th6a mdn (1).
Vfly h0 b6t phuong trinh c6 nghiQm duy nh6t
(x;y): (0;0). tr
Y NhQn xit. Mqt s6 bpn ldm bdi thi6u thpn trqng
n6n dd xdt thi6u truong hqp ho4c kh6ng thr.rc hiQn
bu6c thir lai n6n kh6ng lo4i dugc nghiQm kh6ng
loi giii dirng ld: NghQ An:
Nguy€n Dinh Tudn,7c, THCS Lti Nhat Quang, D6
Lucrng; Vinh Phric: Nguy€n Minh Hi€u,9D, THCS
Vinh Y€n, TP. VTnh Y6n; Hii Duong: D6ng Xudn
Ludn, 98, THCS H-qp Ti6n, Nam S6ch; Hi NQi:
Nguydn Vdn Cao,gB, THCS Nguy6n Thugng Hi6n,
Ung Hda.
th6a min. Cilc ban c6
Bili
NGUYEN ANH QUAN
T41452. Cho tctm gidc ABC c:iin toti A, noi
ilAp dr.rc)ttg trt)n (O), AK ta tludng kinh, I ta
tliint bit ki trOn rtmg nhd AB (t khac.1, B). KI
t'tit c'ttnh BC' toi llt. Dud'ng lrung lnrc cttu J,U,
t'dt t:cit' cqnh AB, AC' thir try ttti I), E. i{ la tnng
cliint r'ila DE. C'hintg minh rdng ba cliem ..1, M,
Slty TaAIDE ld hinh thang cin, ddn t6i
AE: rD: MD
(4)
Tt (3) vd (a) suy ra ADME ld hinh binh hdnh.
Do cl6 A, N, Mthdng hdng. tr
YNhQn xdr M$ sO ba, ngQ nh{n khi kt5t lufn viQc
chimg minh ME ll AB vd chimg nxtu MD ll AC ld
tuong tir nhau _C6c bpn du6'i tldy c6 lcri gi6i t6t :
Hi NQi: Nguy€n Vdn Cao, Ngry? Thdnh Long, DSng
Thanh Tilng,gB, THCS Nguy6n Thugng Hi6n, Ung
Hod; Bic Ninh: Nguydn Thi Bich Hdng,9A,.THCS
YOn Phong, Y6n Phong; Phrfi Thg: Trdn Qu6c LQp,
8A3, THCS Ldm Thao; Vinh Phrflc: NguyOn Minh
HiA,t,gD, Hodng Vdn Hidu,gE, THCS Vinh YOq TP.
Hd Chi Minh: Nguydn Dinh Duy,9A6, THPT chuy6n
TranD4iNgtr'a
NGUYEN THANH HoNG
Bii T5/452. Tim .so ngtn'itr rn tli phtrttrrg trinh
-rr +(ir+ l).r) -(2m-l)r-(2nt) +rr+4):0 (l )
t'6 nghiAm ngul:1p.
Ldi gidi. Cach
(1)
e
l. Ta c6
x3 + (1n+l)x2
- (Zm-l)x
-
(2m-l)m-(2m-1)-5=0
o
e
l,l thong hctng.
x2
(x + m + l)
-
(2m
- l)(x
(x+m+l)(x2 -2m+
+ m a' 7)
1) =
-5: 0
5
(2)
Do m, x ld cics6 nguy6n n0n x * m +l vd
x' -2m +1 ld udc ctra 5.
Ta c6 5 : 1.5:(-1).(-5). NhQn thiy x + m +1 vit
f - Zm +1 phdi ta so le n6n r vd z ld s6 ch8n,
suy ra x' - 2m +l chia cho 4 du 1, n€n x2- 2m +
Loi gi,rti.
A
1
t
bing
hodc 5. Do tl6 tu (2) xhy ra hai kh6
ndng:
{*+mll'=l
lm=-x
l)
'' < -2m+l =5 <>{
[*,
ltx+t)2 =5 (*)
Vi
K
Gqi F ld giao tli6m cta DE vit BI.
Ti DE L MI (vl DE ld trung tryc cua lrfi) vit
AI L MI (do AK ld cluong kinh), sluy raAI ll DE.
Tac6
IFE=180o-FIA=ACB:ABC
(1)
(2)
Tt (l) vd(2)tac6 DFM
= DBM
,.,{
^. nep.
la tu glac nQl
Suy
ra 6fr8
,n€nDFBM
=iFi:frE = MD tt AC
lMdtI&6C IDE = EDM
:
IBC =I80"
Zn€n (x + l)2 le s6 chinh phuong, vi v6y
PT (*) kh6ng c6 nghiQm nguy6n.
lm=-x+4
'
r' -2m+ 1=
[(x+ li'z = 9 (**)
PT (**) c6 hai nghiCm nguy6n x : 2 vit x : - 4.
2t
-'
f
m:2 vit m: 8.
VQy khi m:2ho{c m:8 th\ PT (1) c6 nghiQm
Tucrng img
Met kh6c, do EF ld tmng tryc cria IMn€n
IFE = EFM
-rre
(3)
- IAC : AED .
tim dugc
nguyOn.
Cdch 2. Viet pr (1) thdnh d4ng PT b$chai dn m
(coi x ld tham s6):
2m2 -(x2 -2x -l)m-(x3 + x2 + x -4) = 0 (3)
Ta c6 A = (x2
-2x-1)2 +8(r3
.e nr.,.-roru,
+ x2 + x
-4)
T?ll,.,HS
19
HÃY ĐẶT MUA TC TH&TT TẠI BƯU CỤC HOẶC VỚI TOÀ SOẠN !
:(x2 +2x+3)2 -40.
De PT (3) c6 nghiQm nguy6n thi A phni
chinh phucrng.
Edt (x2 +2x+3)2 -40= ft'z (k e N)
e
(xz +2x +3 + k)(x2 +2x +3
h
- k) = 40
= c(ab+1)>c(a+b)
sd
t)
lx2
(4)
Z,
+2x+3+k=10
lk=3
otfr+l)z
fx, +2x+3-k=4
xiy
trulng hqp ndy.
l*, +2x+3-k =Z
-
nhi6u tinh ch6t cira c6c thrla s6 nguy6n vC trSi cria
(2) hoac (4) thi xdt.cang it trudng hqp. Da s6 c6c ban
tham gia gi6i bdi d€u ldm theo hai c6ch tr6n.
2) Tuy6n duong cic b?n sau c6 loi gi6i t6t:
Hn NQi: Dfing Thanh Tilng, 98, THCS Nguy6n
Thuqng Hi6n, tlng Hoa; Phri Thgz Nguydn Thiry
Duong,7A3, THCS Ldm Thao; YinhPhic: Phqm
Xudn Cudng,6A4, Bili Thi Lidu Duong,8A4, THCS
Y6n Lpc; Nguy€n Minh Hi€u,9D, THCS VTnh YOn;
Thanh H6r: Efing Quang Anh,8A, THCS Nguy6n
Chich, D6ng Scm; NghQ An: Thdi Bd Bdo,7C, Trin
L€ HiQp, 8A, Nguy€n Thai Hi€p,8C, Bili Thi Nhdt
Linh, Einh Vi€t 74, 8D, THCS Li Nhat Quang;
Quflng Ngni: Nguy€n L€ Hodng Duy€n,8A, THCS
6m, ta dugc
(a+bc+b+ca)z
+(c + ab)'
2
>
Jl.(a+bc +b+
(a+bc+b+ca\'
e \---Z--------!Tt
+
ca\(c + ab)
. + ab)'.a
lc
J2.(a+ u)(c +t)(c
+
ab)
f:1.
(2), (3) suy ra
+ ca)' + (c + ab)'
>$.(a+u)(c+t)(c+ab)
f+t
Ding thfc xity rakhi vi chi khi
fa+bc = b+ ca
\a+Ac+b+ca=J2.(c+aO)
e
a+bc
=b+ca=#
ring
(c +t)(c + ab)>(b+ c)(c
Ta chimg minh
Thflt
v{y (5) ec2
+ abc + c +
ab>
+
a)
(s).
bc + ab + cz + ac
eabc+c)bc+ac
e
c(ab+I)>-c(a+b).
n6t ding thric thing, do (1). Tt (4), (5) suy ra
U6t Aang thric trong tlAu bdi tlugc chimg minh.
Ding thr?c xhy rakhi vd chi khi
[c=0
I
la=l
Phqm VEn Ddng, Ngh?a Hdnh.
vd
c+ab (6)
.a+bc=b+ca=T
It =t
PHAM THI BACH NGQC
sO
(2).
U6t Oing thirc Cauchy cho hai si5 khong
(a + bc)' + (u
l1x+ l)2 = 9 1**;
PT 1*x) c6 hai nghiQm nguyOn x :2 vd x: - 4,
tuong img tim dugc m : 2 vd m : 8. J
YNhQn xit. l) Bdi to6n thuQc d4ng giii phucrng
trinh tim nghiQm nguy6n, n6n.viQc nh6n,xdt cdng
BitiT61452. Cho ba
+b+ca)'
=51r';
lk=9
- <(x2+2x+3+k=2O
.-5 I
Zl
''
Ap dung
>
PT (*) kh6ng c6 nghiQm nguy6n n6n khdng
ra
Ding thr?c xhy rakhi c : 0 hoac a : I hodc b : l.
Laic6 (a+bc)'z +(t+ca)' r(a+bc
keNn6n i + 2x + 3 + kvdx2 + 2x +
3 - klilu6c sii cria 40. Hon nfta, do
(i * u+ 3 + tr) - (i + 2x + 3 - k) : Zktd s6
chin, vi I + 2x + 3 + k> o, nen f + x + 3 + k
vd x2 + 2x + 3 - kld cbc si5 nguy6n du
Ta c6 40 : 10. 4 :20.2. Do d6 tu (4) x6y ra hai
khi ning:
Do xe
(1).
khdrrg dm a, b, c . Chimg
minh ring
(a
+ bc)' +(b + ca)' +lc +
ab).
Didu ndy kh6ng xity ra
u)(t + c)(c + a).
Ldi gi,fii.O€ y ring trong ba s6 o - \ b * \ c - t
,-
1u6n c6 hai sd kh6ng
Gi6 su
^^
20
J1.(o
+
trbidilt v6i
nhau.
(a*t)(U-t)> o = ab+l> a+b
Tucrng Wc6
N6u
b*1.
vl
bc > 0
= a+I
.
c=o: (6)=a:b=o'=o[o= ?=9
JZ
la=b=0
Vpy dSng thric xdy ra khi vd chi khi
';4"3i@
TOAN HOC
HÃY ĐẶT MUA TC TH&TT TẠI BƯU CỤC HOẶC VỚI TOÀ SOẠN !
a-b-c-O ho{c a=b=J1; c=0 vd chc AEll BH+OHLBH. GqiKligiaodi6mcira
BH voi tluong trdn (O) (f * n) thi Il ld trung
hoSn vi tucrng img ctra chring. D
)Nh$n *(r:.??v
Jdbdi
h6t dirim cria
^BK. Tt
Di6u + . . ,,Drrht
to6n tuons d6i kh6. Hdu
c6c ban gui bei v6 tda so4n ldm theo c6ch tr6n.
then ch5t cra bdi to6n rd gia
sr
(a-rXa-r)';
suy ra b6t d5ng thirc (5). C6c bpn sau dAy c6 bdi
t6t:
;
gi6i
ctdy, k6t hqp
H::#rHX',fl;;ir$T#
v6i AE ll BK ta
flHliffii}l')
th6y tir gi6c EBFK ld tu gidc AiCu trOa. Do d6
tjl, tuy6n.t4i B', K cila dudng trdn (o) vd EF
Hn NQi: Vfi Drbc Vdn, l0 Toiln l, THPT chuy6"
pni",
iri"ia"'uii uiZr,'gi-, ddng quy (l). Ctng ti EBFKId tu gi6c rli6u hda
DHSr He N6i; vinh
THCS Vinh YCn, TP. Vinh Y6n; HungYiln:,Nguydn . , KA BE KE BA ----. _.
vi€t E*c, Duong H6r; iry:, tac6 urv=ffi= KF-= BC: "uv raABCKlitttr
10A9, THPT Duong Quing Hlim, Vdn Giang; BIc
*.r
girlc tli€u hda' do vfy ti6p tuy6n t4i B' K clfi;a
Ninh: rd Huy cuong,ll To6n, Turr cnuycn IIJ
Ninh; Thanh H6a: Efing euang ern, gA,, iHCi ducrng trdn (O) vd AC d6ng quy (2). Tt (1), (2)
Nguy6n Chich, D6ng Son; Hi finh: Nguydn Vdn vd gi6 thi$t G =EFaAC ta co GB ld ti€p
ti,v:"Jryhg Nhat' ttrl' THPT chuYcn,Hd tuytin
rluong trdn (o) (dpcm). D
--'J --- cua
!d:
Tinh; Binh Dinhz Trdn Vdn ThiAn, 10 To6n, THPT
chuy6n LC Quf D6n, TP. Quy Nhcrn; Quflng Binh: YNhQn xit. S6loi gi6i gui ve toa sopn kh6 nhi6u
Hodng Thanh ViQt,.ll To6n, THPT chuy6n V6 theoc6chu6ng: Sridungtinh:Ptclia-hiqi6:qieu
NguyEn Gi6p, TP. D6ng Hdi; Long An: Nguydn LQc hda, hdng.dii5m -. chtm tli6u hda, tinh chAt duong
pi,ii, plrq* Qudc Thdig,l0T1, THPT chuy6n Long tt6i trung ho{c bi6n ddi g6c, ... Xin n6u t6n nhirng
iii" uri"i i;;;i"
An.
b4n c61
NGUYEN ANH D TNG
BiliT71452. Cho tam gidc ABC cdn tai A, n6i
ti6p &.rdng trdn (O). D ld trung didm AB, tia
CD cdt drdng trdn (O) tqi E. Kd CF // AE vdi F
thu6c (O), tia EF ch AC qi G. Chtrng minh
rdng BG ld ti€p tuy€n cua dadng trdn (O).
Loi gidi.
Hn NQi: Hodng LA NhQt Titng, 11 ToSn 2, THPT
chuy6n KHTN, DHQG Hd NQi, Vil Dtic Vdn, 10
To6n 1, Phgm Nggc Khdnh, l0 Toan 2,_THPT
chuyCn EHSP He NQi; Phri Thgz Nguy€n Dric
ThuQn, l0 To6n, THPT chuy6n Hing Vuong; Vinh
Phircz Hodng Vdn Hi€u,gE, THCS Vinh Y6n, Dd
Vdn Quy€t, Hd Hftu Linh,l0Al, THPT chuydn Vinh
Phirc; BIc Ninhz NguyAn Vdn Tdm,10 To5n, THPT
chuyCn Bic Ninh; Hii Duong: Hd Minh Hodng, L0
Torlrr, THPT chuy6n Nguy6n TrSi; Hung YCn: TriQu
Ninh Ngdn,10A9, THPT Duong Qudng Hdm; Thii
Blnhz Trdn Quang Minh,l0Al, THPT D6ng Thpy
Anh, Th6i Thlry; Thanh H6a: Vfi Duy Mqnh,l0T,
THPT chuy6n Lam Son; NghQ An: Nguy€n H6ng
Qu6c Khanh, 10A1, THPT chuy6n Phan B6i Chdu,
TP. Vinh, Nguydn Philng Thdi Crdng,l0Al, THPT
Th6i Hoa, TX. Th6i Hoa; Hdr Tinh: Phan NhQt Duy,
NS"yAn Quang Dilng, l0 To6n 1, Nguydn Vdn Thi|,
LA Vdn Trudng Nhqt,I/d Duy Khanh,llTl, THPT
chuy6n Hd finh; Quing Blnhz Trdn Nam Quang
Trung,l l To6n, THPT chuy6n V6 Nguy€n Gi6p; Dn
G
Gqi H ld giao tlitim ctra AF vd CE. Ta thi'y
AEFC ld hinh thang c6n, suy ra ba di€m O, H,
G thdng hing vd OG L AE.
M[t khSc AB : AC: EF, n€n BE ll AF. Tn d6
MDE = LADH (g.c.g) BE = AH, d6n d6n
=
tft gi6c BEAHId hinh binh hinh, suy ra
Ning: L!, Phudc C6ng, Trin Nhdn Trung,.10A2,
THPT chuy6n LC Quf Edn; Binh Dinh: Trdn Vdn
ThiAn, 10 To6n, THPT chuy6n LC Quf D6n; Kon
Tum: Nguydn Hodng Lan, llAl, THPT chuy6n
Nguy6n Tdt Thinh; Long An: Phqm Qudc Thdng,
10T1, THPT chuy6n Long An; CAn Tho: Nguydn
Trdn H{bu Thinh, l0Al, Trin Minh Nghia, 1141,
THPT chuy6n L;f Ty Trgng; Ddng Nai: Cao Dinh
Huy,ll To6n, THPT chuy6n Lucrng Th6 Vinh; TP.
HO Chi Minh: Nguy€n Dinh Duy, 9,{6, THPT
or
rr.,.-roru,
T?3I#S
21
HÃY ĐẶT MUA TC TH&TT TẠI BƯU CỤC HOẶC VỚI TOÀ SOẠN !
chuyOn TrAn Epi Nghia; Ci Mau: Trin Xudn Sdc,
10 Todn l, THPT chuy6n Phan Ng-oc Hi6n.
HO QUANG VINH
Bii
T8/452. II[i1' xtit' ciinh giit
hittn s6 ,f (.r)
:
tri nhd nhat c't)a
,(rn -,-+ ta, "r + Jcos -r + crltr
Jlt*E(l
-F;4 + J[*t {n j.i-,]i.
ld hdm tuAn hodn chu kj, 2t kh6ng m6t
tinh t6ng qu5t, <16 tim gi6 tri nh6 nh6t cira hdm
( :n)
!
.
sd ta x6t x el t|
j. Eat sinx : - a, cosx: - b
Viftr)
\
o)
* b: m thi a,b e (0,1) vd 1< m<$ Oau
.r
1J
,.-"rZ,^,,.--5n
" rr. a=D=
DangxayraKnl
nay x=
laco
2
4.
vd a
/(,)=
w-q.E(,-,).
1
f'-
tx) =,-l-
at)
-
+ b)
+2,!t
+ ab
- A + b1
-ti- I, -(a+b\+2l1la+b-r)2
-2)tm+tt
+(zO +z)@-rt]-(:JZ +r)m++-nD rg.
dr,rng U6t Oing
thric Cauchy cho ba s6 tro.rg
m6c vudng, ta dugc
I
+(zl1-2)trz+ l)+( zJi +z\ m -tt
-4,
m'-1
*t;l't i\l2
f'(x)>4-2O hay
fQ)>-Ji-iE. Ki5t ludn: min.f(;r) =J41O
vio
(1), ta thu duoc
-2) w + tt(zl1 +z) <,t -L)
D6u ding thuc xdy ra
d4t ilugc 14ri
nrc ld x
, a.hay
=!!4
q=b=1,
z
*21rn. k
sin.r
o
=msl
2'
tl
eZ.
DNhQn xit.Da s6 c6c b4n chi x6t trudng hgp
sinn > 0, cosx > 0 n6n chi nhQn dugc d5p s6
min/(x) =J4iJT.
Cdcbal sau d6y c6 loi gi6i clung.
Vinh Phrfic: Od fan Quy€t, 10A1, THPT chuyOn
VInh Phric; Hung YGn: Vfi Minh Thdnh, llTl,
THPT chuy6n Hrmg Ydn; Nam Dinh: Dodn Thi
Nhdi, l0Tl, THPT chuy6n L€ H6ng Phong, Th6i
Binh: Trdn Quang Minh,l0Al, THPT D6ng Tlrty
Anh, Th6i Thuy; Hi frnh: LA Vdn Trudng NhQt,
I lT1, THPT chuy6n Hd Tinh; Quing Binh: Hodng
Thanh Vi€t, llT, THPT chuy6n Vd Nguy6n Gi5p;
Binh D!nh: Trdn Vdn ThiAn, I I To6n, THPT
chuy6n LC Quli D6n; S6c Tring: Vaong Hodi
Thanh, 11T2, THPT chuy6n Nguy6n Thi Minh
Khai; B6n Trez Trdn Thanh Ducrng, 10T, THPT
Bii
=-4,*(e-r)m-A
m, _I \
-2*(2"D
'
lm'-l
rhc (2) vd (3)
"t91452. TrOn ntat phang tQa dO Ox1t,
a61
tap ho'p M c'ac di€rn c6 t()a d0 $; 1t) v6'i
,r..)'e N'' r'd .r ( 12,-v ( 12. Moi di€rn trong lll
dvo'c: lo bo'i mot lrong ba mdu'. ntdu dr), mdu
.)
fo
T
NGUYEN VAN MAU
=J;-\a+b)+Jl@+b-t)
al)
=l
t4
a=b=
chuy6n B6n Tre.
1
Ap
bing xby rakhi
.
-: so cric bqn)
Loi gioi. (Theo da
Di6u kiQn dd cdc bitfu thtc ciaflx) c6 nghia ld
sinx, cosx kh5c 0 vd ctng d6u. Nh6n x6t ring
khi sinx, cosx kh6c 0 vd cirng d6u thi
f (x) =fian x(1 + cos r) + Jcot.xtt + sinxl
>
D6u
=
g
121
t;
khi a = O = lf
trcing hoat' mdu ranh. Chu'ng minh rdng tort ttti
mot hinh c'hti' nhdt co cac' c'anh song ,song vo'i
cdc lruc tr2o dQ mit tdt c'a cac' dinh c'tia n6 thuoc
M vd dtrct'c 16 cilng mdu.
Ldi gidi. M gdm 12 x 12: 144 dii5m vd tlugc
.).
t6 b6i ba miru nen tdn tpi it nhAt 48 di6m tluoc
t6 ctng mdu, ching han mdu tl6. Chgn 48 di6m
clla M dugc td boi mdu do vd trong c6c di6m
ndy ki hiQu a, U sO di€m t6 mdu il6 c6 hodnh
.
dO
M{t kh5c, do 1< m<
J,
-(:J7+ t\m+a-Jz
>
-(:J2+t) ,[1+a-JT
=
-2-2J2
I (, :
12
1,.. .,12).
Ta co
n€n
la
=48. 56 c[p
i=1
(3)
di,3m mdu d6 c6 hodnh dq
a,(a, -l)
, ie C; _
2
Chi€u c6c ili6m mdu tl6 l6n tryc tung thi m5i
- diem
-.;.
mdu tlo c6 hoinh d0 I sC thinh mQt
c{p
HQC
-crua@
^ ^ TORN
ZZ
HÃY ĐẶT MUA TC TH&TT TẠI BƯU CỤC HOẶC VỚI TOÀ SOẠN !
thing [x, y] v6i ,r,ye N., x1l2,y<12.
Sti c6c dopn thing [.r,y] (x,y e N*.x <12. y <12)
= x?
clo4n
ld
C?2
=66. T6ng
chi6u rd
s6 c6c
c[p di6m mdu
= -10.2065671, x, = 4.206567t , x+ =-2065671.
YAy t:4 ld s6 nh6 nhet cAn tim. E
x2
cl6 dugc
YNhQn xit. Bitito6n kh6 o ch6 cAn chi ra s1r t6n tai
c6c s6 nguyCn x1,x2,...,x. th6a mdn tling thtc (1).
=**,,' -+2,, =iL": -ro
Ze
/ t2
lF"
Chi c6 hai b4n tham gia gidi bdi ndy vd ddu c6 loi
gi6i dring. T6n cria hai bpn ld: Hir NQi: Zfi Bd Sang,
l1 To6n l, THPT chuy6n Nguy6n Hu6. HA finh:
\)
I
)
,t\A-''
-2 12
_24=,
.o&.t
2
finh
NAM
HIIU
TRAN
NguyAn Vdn Th€,11T1, THPT chuy6n I{d
_r4=rr.
12
Bii 'tl
12
Vi IC; >72>66
n6n t6n tpi hai cflp clopn
.t,i ( t
thing, mQt c[p c6 hodnh dQ i, mQt c{p c6 hodnh
dO j Q <7) rluoc chiiiu 1€n cirng m6t tlo4n thing
fk, tl trdn tryc tung. Khi d6 hinh cht nhat
ABCD v6i A =(i,k), B =(i,D,C =(j,l), D --(.i,k)
ddu duoc t6 mdu d6.
chuy6n
To5n, TIIPT chuy6n Hr)ng Vuong. Vinh Phric: Dd
Vdn Quy\4 10T, THPT chuy6n Vinh Phirc; Nam
Dinh: Phqm H6ng Trudng, l0T, THPT chuy6n LC
Hdng Phong; Quing Tri: Nguydn Vdn Tu'dng,1lT,
THPT chuyCn L6 Quf D6n; Binh Dinhz Trdn Vdn
Thi€n,l0 To6n, THPT chuy6n L6 Qujr D6n.
DANG HUNG THANG
Bni T10/452. Xtic tlinh si ngut'Ort duottg t nho
nhat .sao cho rin tai t so ngttl'An ,rr ,.r,,...,11
.rf
-!
+
r] - ...+(-l)'*'-q :1)65:(i1-1 (l).
Ldi gidi. O6 th6y: ;3 = 0, +1 (mod 9), Yx eZ
Mar khec:
20652011
:
42011
(mod g) :43.611+t (mod g)
.r,: I
lr)
--u1 +')ir + l'
vri e \1"'
gkii lrun hfitr hun.
Ldi girti. (Theo bqn Nguydn rui Uinn Phudc,
11T1, THPT chuy6n QuOc Hqc Hu6).
Ta c6 x, = ^{4o a2 -.12, *r' -2o*, +2a +l> 0
Nguy6n HuQ; Hung Ydn Vil Minh Thdnh,l I To6n 1,
THPT chuy6n Hrmg Y6n; Phri Thqz LA Bao Anh,10
;
rtic dinh bd'i
Xac tlinh u de dd.v s, 1-r,,) t:o
xit.
lhicundrt
)
- l'\'
tr
Hn NQi: Vfi Ba Sang, 11 ToSn 1, THPT
11452. C'ho u ld so ngtr.t'tn tltro'ng vu dcil'
+2u+l
Trong s6 c6c b4n tham gia giai bdi ndy
chi c6 mQt b4n gi6i kh6ng dirng. Cbcb4n sau dAy c6
loi gi6i t6t.
YNhQn
- x; + xl - xl v6i x, =10.2065671,
.
e
I,l+a + z -
e
4a+2-2J1.'f{a +z
Ji)' -2{JTi +2 - O) a + 2a + r> 0
+Z
-2a.[4a +2 +2Ji.a +2a+ 1 > 0
o F = 2a(J4a +z -z - "D)+ 4J2o 4-5 < 0.
'N6u a>5 thi
JAiA -3-J-2>Jn-z-J2
vd 4J2a.+l
>
4,s-3-1,5 =o
*5> 4Jn -5 > 0. Suy ra F > 0.
.N6u a =4, F = s(Jts-: -J1)++.{o
-s
=rco-17 >t6.1,4-17 >0.
.N6u a =3, F =a(Ju -z-Jl)++J1 -s
=zJ1(tJ1+z)-zz-60
=4"64671(mod 9)= 4 (mod 9).
Do d6 vbt t : l, 2, 3
- 23 - 6.1,5
=5,2"6,2-32=A,24>0.
Nhu vfly vti a e N*, d ) 3, x, kh6ng x6c dinh.
V6i r - 4, ta co:
Xet ae {1,2}. Tac6
=
20652014
=
2065 = 103 + 103 + 43 +
> 2.2, 6.(3.I, 4 + 2)
13
(206567 )3 .2065
1
x2
=(2065671)3.(103 +103 +43 +13)
-Zax+2aa1=(x-a)2
Ki hieu
= (10'2065671)3 + (10'2065671)3
f (x) = ^t7 + rax
+ (4.2065a" y + (1.206561
|
)3
= "!(* +
- ^l-7 -rax + ra+ |
a2 -.,!u - of +2a +l- a-
+2a + |
af +2a +l-
.
+Za+l-i >2a+l-a2 >0.
TOflN HQC
s6 456(6-2015)
&
GTudiVil
..
Z5
HÃY ĐẶT MUA TC TH&TT TẠI BƯU CỤC HOẶC VỚI TOÀ SOẠN !