THTT So 456 Thang 06 Nam 2015

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rnp cxi RA HANG rmArue - ruAm rHtJ SzoAruu cHo rRUNG Hoc pnd rnOruc vA rRuruc Hoc co s6Tru s6: 187B GiSng Vo, Ha NOi.DT Bi6n tap: (04) 35121607: DT - Fax Ph5t hdnh, Tri sLI: (04) 35121606Email: [email protected] website: http://www.nxbgd.vn/toanhoctuoitre

HÃY ĐẶT MUA TC TH&TT TẠI BƯU CỤC HOẶC VỚI TOÀ SOẠN !

KX v,e{x c,{{ryN &d@o s,xNM Vmo mq{ ry'x;,Y&wi w,u6g ffi{&mtr TrHuil Hsfr$ E0E&

(t(X i thi chqn hsc sinh vdo dQi tuy6n Qui5c gia

ll[ dr.r thi blympic roin rrqt Q"6" t5 grrabl

- y lan thu 56 n[m 2015 (IMO 2015) dd tluoc

t6 chriE-tqi He NQi. trong hai ngity 25 vd261312015. Cdncu Quy ctr6 ttri chgn hgc sinh gioi cAp Qu6c gia hiQn

hdnh, BQ GD&DT dd triQu tdp 49 hoc sinh tham dp kithi tuy6n chgn n6i tr6n, g6m I hgc sinh dE tham dU IMO2014vit 48 hgc sinh dat tu 21,50 tli6m trd 16n trong kithi chgn hqc sinh gi6i Qu6c gia m6n To6n THPT nim2015. Trong m6i ngdy thi, m6i thi sinh dugc dC nghigi6i 3 bdi to5n trong th<ri gian 270 phit; di6m tOi tla ctra

m6i ngdy thilil2l di6m. _

DE THIlttgdy 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

nguydn kh6ng Am sao cho

co+ ctu + cra2 + ... + cdo = 2015 (*).

a) Chimg minh ring co+ q + q+ ...+ c,:2 (mod3).

b) Cho r thay tl6i vh co, ct,...,cn thay d6i nhrmg 1u6n

th6a mdn di6u kiQn (x). Tim gi6 tri nh6 nh6t ctra t6ng

co+ q + c2+...+ cn.

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. giaodi6m 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 .lttai 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 dulngthhng AK tai di6m ,,q, c1t,qS, AC l6,n lugt tai P., Q Gqi Jld.trung di6m cira doan P9. Chrmg minh r[ng ducrngthdng AJ di qua mQt di6m c6 tlinh.

Bni 3. (7 diA$. Sd nguyQn ducrng k duoc goi ld c6 tinhchiil T(m) n6u v6i mgi s6 nguydn a, tdn t4i s6 nguydn

ducrng n sao cho lk + 2k + ... + tf = a (rwtn)..t.

" , 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 thiv6n d6p. $.an gi6m,khao:961n 25 thefi vi6n. Mdi sinhvi€n dugc h6i &jiboi mgt gi6m kh6o. Bi6t rang.n6i sinhvi6n thich it nhetil0 gi6m khdo trong s6 cac thdnh vi6ntr6n.

a) Chimg minh rlng c6 th6 chqn ra7 gi6mkh6o sao chom6i sinh vi6n thich it nh6t mQt trongT gi6m kh6o ndy.

b) Chrmg minh ring c6 th6 sip xi5p cuQc thi sao cho m6isinh vi6n duo. c h6i thi bdi gi6m k*r6o mlr minh thich vdm6i giam khao hoi thi kh6ng qu6 l0 sinh vi6n.

NGUYEN KHAC MINH(Cuc Khdo thi vd Ki€m dinh CLGD - BO GDADT)

Bni 5. (7 di€m). Cho tam gi5c ABC nhon, kh6ng cAn vddi6m 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 citAC 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,J 16n luot ld t6m c6c ducrng trdn nQi ti6p c6c tam giSc

PEM, PFN v* (fl n ducrng tron ngopi ti€p tam giSc DIJv6i tdm ld di6m K. Duong thlng DT cdt (&J tai di6m 1L

Chrmg minh HK di qua tAm ducmg trdn nQi tii5p tamgi6c 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 QUACdn cf k6t qui ch6m thi vd Quy ctr6 ttri chgn hgc sinh

iigioi 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:

l. ltlguyin Tttin Hai Ddng, Ws 16p 12 Trudng THPT

chuyOn KHTN, DHQG Hi NOi, 32.50 di6m;

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

KHTN, DHQG HA NO| 27.50 di6m;

4. Hodng Anh Titi,Ws lop 12 Tr,ucmg THPT chuy6n Phan

BQi Chdu, NghQ An, 25.00,diiim;

5.lr{guyAn Thi Viil Lla;,hls lop 12 Truong T}IPT chuy6n HiTinh,24.50 di6q

6. Vii Xudn Trtutg, h/s lcrp 1l Trucrng THPT chuy6n

Thdi Binh, 24.50 rti6m.

Ngiry 161412015, B0 GD&DT dd triQu tfp 6 hgc sinh cira

DQi tuyi5n "C

Ha NOi tham dg lcrp tip hu6n chuy6n m6n

chuAn bi cho IMO 2015. Trucrng EHSP He NOi duqc B0

GD&DT giao nhiQm vr; chri tri c6ng tdc tQ,p hu6n dQi

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?, bdn hAn quan ddn viQc dp dung hQ thttcVidte. C6 nhirng bdi todn c6 thO dd ddng dua vidqng dp dqng dagc hQ thuc Vidte, nhwng cfingcd nhirng bdi,todn phdi rh khdo leo mdi thqtchiQn duqc di2u d6 vd no gdy kh6ng lt kh6 khdndiii vai cdc em hgc sinh. Sau ddy lit mdt sddqng togn nha vdy vd cach s* dryng h€ thacVidte 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 x1, x2

md xf -"tl =.ri-"t:'.NhQn xdt. Ta thdy h€ thric dA bai dua ra c6 v6 phfrctpp vd g6y kh6 khln khi dua vd xrt xz vd. x1.x2nhrmg ta c6 thrS bien dOi xy, x2 th6ng qua phucrngtrinh (*) tl6 sir dlmg h6 thric Vidte.

Ldi gi,rti. Ta c6 A' : 8 - 8m2. DC PT (*) c6 hai

nghiQm thi A' >0c> -l<m<l. Khi cl6 theo

hC thuc Vidte c6: xt+-x2=l; xr.xr=(m2 +1):8.

Vi x1, x2 ld hai nghiOm cria PT ( *) n6n

[S*3-Ar, =-(m2 +l)' ),.,. (Dfarj -8xr=-(m2 +l)

Tac6: xl-xi=x?-x3e xl (8;f - 8-r, ) - xr2 (8 xl -8xr) = 0 (1)

Thay (I) vdo (1) ta dusc(xl-xl)(-m'-1)=0

e (x, - xz)(xt + xzX- *'- 11: g€) xr- xz:0 (vix1 + x2:lvd -*'* 110).

I mz+lDo d6 xt = -y2 = r.*d x1.xz = ?, suy ra

m = *7 (th6a min bdi to6n).

Bii 2. Cho PT *' - 2m, + m2 - m + 1 : 0 (1).

Tim m ae pf c6 hai nhi€m phdn bi€t x1, x2 thoa

mdn: xl *2mx" =).

Ldigidi.A'>0 o m- l>0<> la>lthiPTc6 hai nghiQm phdn biQt x1, x2 yd theo hO thirc

Vidte c6: x1 +xz=2m;xr.xr=m2 -m+1.Vi x1 ld nghiQm cria PT(l) n6n :

4 *2*,+nf -m+l=lo4:2nx,-n? +m-1.f0t hqp v6i tlAu bdi ta c6: xl +Zmx, =9eZmxr-m2 +m*l+Zmxr=)e 2m (x1+ x2) * m' + m - 10:0o 3m2+m-10:0

e m:-2 (loai), * : :(chsn).J

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.

Ldi gidi.A' > 0 e m >jt.l,nt phuong trinh

c6 hai nghiQm x1, x2,thi d6: x, * xz =2(m+l);xt.x2=m2 +4 vd xl =Z(m+l)xr-m2 -4.Theo tl6 bii xl +2(m+l)x, <3m2 +16

e 2(m+1) (x1+ x2) - 4mz - 20 S 0r , --2a l2(m+t)) -am 2 -20<0.(do (x1 a xr) :2(m+l))

o8m-16<0 e m<2.

K6t hqp voi (*) tu .0, ] < m < 2.2

Bii 4. Cho phaong trinh:

.x' -21m - 1)x + 2m - 5: o (l)Tim m d€ phwong trinh (1) c6 hai nghiQm phdnbiQt x1, x2 thda mdn( xl -2mx1 + 2m - 1)(xi - 2mx2+ 2m - 1) < 0 (2)

Ldi gidi. A' : (* - 2)' + 2 > 0 lu6n rlirng vdimgi m, vfly phucrng trinh (1) lu6n c6 hai nghiQm

.. nu.,.-roru, T?8I#E[ 1


ph6n biQt x1, x2 YOi mQi m. Khi d6:

xl + xz = 2(m -l); x.x, = 2m - 5'

Yl xy x2li hai nghiQm PT(1) n6n

lxl -2(m- l)x, + 2m-5 =0

\xj-2(m-l)x, + 2m-5=0

,_- [ r? -2mx, + 2m - I = 4 -2x r

- lri -2*r+2m-l=4-2xzTt (2) suy ra: (4 - 2x) (4 - 2xz) < 0

e 4xrxr-8(x, +xr)+ 16 < 0

e 4(2m - s) -t.z1m- 1) + 16 < o <> *, +.

Bdi5. Cho phuongtrinhrl+(n i),r 6:0 (i).

Tim m dA phwong trinh co hai nghiAm phdn hidt

.\1,.\ rnit B : ( ri - 9) (-r,t 4) dqt GTLN'"

Ldi gi,rti. Ta *dy (1) lu6n co hai nglriQm ph6n biQt

xl,x2ybimojm vi c6 a. c:- 6 < 0. Theo hQ thtc

Vidte: xr.xz=-6<=' x2= j ue x1+xz=l-m., x,

a : (x? - g)(x| - 4) : x? . x', - (4 *? + 9 xZ) +36

:36- (4xi +44*1+Xxi

B<36- 2 E;g-+36=0.!xf

Ding thirc xhy rakhi 4 t : 324,xi

e xl :81 <+ xr:3 ho6c xr:-3.. Khi x1 : 3 thi xz: - 2, suy ra: m:0. Khi xr : - 3 thi xz:2, suy ra: m:2Vfly minB : 0, khi m: 0 hodc m: 2.

I)4ng 2: 'fuo'ng giao cria parabol vh tlutrng

thang

Bii l. L'lrts purctl'tol lPl: .t - 1rt vi clmhtg

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 (,,r,q 1"1'2) -' 48.

Ldi gidi. Phuong trinh hoinh dQ giao <lii5m cua

(4 vd (d)tir: |* -2x * m- 1 :0 (1)

DC (A cat (P) tai hai <ti€m phan biet thi phuong

trinh (1) c6 hai nghiQm phin biQt e A'> 0 <>

m < 3 . Khi d6 xt xz ld. hai nghiQm PT ( I ), ta c6:

lx,+xr=4 ,,n Iy,=2xt-m+l1r,.*, =2(m-l) '' \Y,=2*.,-m+I

Ta c6: xfiz (!t + Y2) : - 48

O xtxz (2x, - m * | * 2xz- m+ 1) : - 48

e (xp2)12(xt+ xz) -2m + 2): - 48

e2(m- 1)t8 *2m*2):*48e*'-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 (.u l.)'t): B(-rr; .vr)

thoa mdn'. il., - r'r l : z.

Ldi gidi. Phucrng trinh hoinh dQ giao ditim ctra

(P) vd (d) td:x'+13-m)x+2-2m:0 (1)

Phuong trinh (1) c6 A' > 0 e m * - l,tathiyxe,xoldhai nghiQm cira phu<rng trinh (2)'

Lpi c6:

I xo+ x, = m -3,,^ Iro = 73-m)xo +2-2m

1ro.*, =2-2m "' lr, =\3-m)xr+2'2m

Do ct6 lyo -yrl =2 a ltl-*){*o-*ul=z<+ (3 - m;2 l(x1-Y *u)' - 4x1 . xsf : 4

e (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, =xilxr.x, = -4 l..r= = ,j

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,.

a TOnN HQCZ icruagwHÃY 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-]. 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<yc nghiQm: x : 4, x : 0. .

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:

. fqt+ztJi 47-2tJs)"=t 2 ' , JCfru 2. a) 8p -1,8p,8p + 1 ld ba s6 nguy6n li6nti6p, 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

(8; 3) : 1 nOn p i3. Dop ld s6 nguy6n t6 n€np :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 (*). N6u x:2k (fteN.), tu(*)tac6

(a-*r)(:* -r)=2,. Do do [:-.1 =zou oong

LJ -t=Z

x:2.Tu(*)c6 2Y =3'-le2Y =23 ay=3,. N6u x =2k+ I (k e N),ta c6

3* -t=r(r'o - t)+z=r(r- - !)+2 chia cho 8

du 2 (vi(er -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)' , (o*u)'* (r-t)'=\; *t7.

Do cl6 (a+ b)' =r(o' +a'). ruong tr;

(a'+u')' ,r(o* +uo). raco

(a + b)' (o' + u')' . o(o' + u')(o^ + ao ). ua

(a+b)' =4. Do vQy,tac6 a2 +b'<ao +bo.

b) Do x, !, z > 0, xy + yz + 7x : 1, 6pdgng b6t

tl6ng thr?c Cauchy cho hai sd duong, ta c6:

.r( t * I )-2[r+y'x+z)t t( t I \A:'-l-r-l- ,tl +-r+r+* - 2[x+Y ' x+z)'

Dod6 2=!(-t*-j-) rrl,ll+i 2(x+y x+z ) t- ''

ruong tu ta c6: +.1[-r*-r) O,,ll+Y" z\x+Y Y+z ) '

, l( , , \-:.'l " * ' l (:). rt (l), (2) r,d (3)Jt*r' - Z\y+z' x+z) \-/' -- \-" \-/'

taco: -4--L--LJl+ x' ,lt* y'

'll+ r'-l( x x y y z z ) :< _t _J_ L-_!_ L__Lr_f_ |

=_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,LOCD rndng tai D n€ntheo tlinh ly Pythagore ta

B

d6 a,beN vda>b.Tac6T -t =2e*(t' -1)=2. Nc,

l2'-n-l=l 12"-u=2' la-l=l la=2{ <}< <}< <><l2o =2 lb=l "

lb= I -- [tr= t

'

f* ,, .2Dod6 1',-'-". <=13(=3' <>k=l.Khid6

l?r -1 :)lD

Sti ese (6-2ors)TOAN I,{Q( ^-i-qrudifue J


qB$} qefffl LE ToNG KET vA TRAo cr qr rmu'Gtic rrxr t.EoN{c 20tsF{ST SOI{G T}IT E{A NOI

Kj thi Olympic To6n Hd NOi ry0 rQng - giii To6n bing tiiSng enfr - tan thri 12 ndm 2015 (HOMC ZOIS; tOchric ngdy 22/312015 de thdnh c6xg t6t dqp. Ngny 17i5l2015 tpi Truong THPT Chu Vdn An, He NQi, Ban chi <Ipo

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

Qdp 8 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 tu6iJunior 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 dugcnhQan 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)

ti€p =BHI=BAT. Md BAT=BCK (th gi6c ABCD

"glcp)_l3 BIIT = BCK , do d6"

BHT + BHK = 180" > K,H,T thdng hdng. "

Chrmg minh rucrng tU cfrng co P, Q,7 th6ng hdng.

Vpy c6c duong thdng AD, PQ, HK d6rg qtry.

-blli rytc_ITBK c6:

AtB+DKB =90'+90'=180" =ru giitc DTBK nditi€p > ADB = HKB. Ta co BAD = BHK (vi cung

bir v6i BCD ) =MBD,n U{BK(e.E)AD AD AB zAM AMHB HK HB 2HN HN

AD Ai'MAM vd MHN c6:

'BAU = 91151.!! =!l!

^ HB^HN

= MAM 6 trpHN {c.g.c) =>BMA=BNH

1S gi6c BNMTnliti6p =BNM+BLM=IKP.BNM =180. -BTM =180" -90" =90".Vfly n4lru6ng g6c v6i -A/8.

a) Yd Et LAD. Ta c6 BHC=BKC=W =BHKCnditidp <+ BHK+BCK =180". Md

Cfiu 6. C6 50 dinh n6n c6 50 tich ba s6 tr€n ba dinhli6n ti6p. Vi ba dinh 1i6n ti6p U6t tcy c5c sii kh6ngblng 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 ti6pc6 hai sii 2,tichbas5 ndy blng 4. .Gqi s6 tich 1o4i I Id x (x e N) thi s6 tich loai ll ld50 -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

ru si6c t6t cd cdc tich ba sO tr6n ba dinh 1i6n ti6p cira da giSc" le:2.ro+4.40=l8o.

ATB + AHB - 90" +90" = 180' = ttr giitc ATBH nQi NGUYEN PIIC TAX(TP H6 Chi Minh)

c6 DB2 +ODz =O82, DC2 +OD2 =OC2. Do d6DP -rc =OE -U2 (l). Chrmg minh tuong tU

cfrng c6: W -W =OC -O,4 (2) virFA2 -FB2 =OA2 -OB2 (3). Tri (l), (2) vd (3) ta c6

(oe -x:)+(n: -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' ce DC =D'C eD'=D (tlpcm).

- irr ::"

" T$ffiru E{#{4 - rflaaeffis*Sd ase (6-2015)


n& rm ruv*nr slffi v*rlr*p r oYrt$nrg Tl{trTshu;r€n L€ Auy S n -,T'lxhNinh,Shu$n

ruAnnrrcca6rq-aots(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) c6

hai nghiQm x'x, th\ lxr- xrl<2.

Cdu 2 (2 die@. Tim gi6 tri nh6 ntr6t va gi6 tri

lcrn nh6t cua bi6u thtic: D =4{*3 .

x'+7Cflu 3 @ diefi. Cho tlucrng trdn (O) tluongkinh AB vd cludng trdn (Q di dQng lu6n ti6pxric trong v6i nira tluong trdn (O) tai C vi ti6pxric v6i doqn AB tai D; CA vd CB lin luqt cftduong trdn (q @i M, N.

a) Xhc dinh tdm O'crta <lucrng tron (Q vdchimg 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<rng trdn ngoai titip tamgi6c ACD ti6p xric voi AE tqi A.

Cflu 4 (2 diAm).Cho phdn s5 p = t:+,v[i nn+5

ld s6 t.u nhi6n. H6y tim tdt cb chc sO t.u nhi6n ,ttrong khoing tu I d6n 2015 sao cho phdn siip, .1, ."

cnua tol glan.

CAO TRAN TtI TTAI

Qtlinh ThuQn) gidri thiQu

6dt drp

ortt{sbvtu*ix**oa(Di ildng tr4nTH&TT sd qSO thdng 12 ndm2014)

PHr PHr (rrd N1i)

. 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).

Ki hi6u c6c s6 trong c6c 6 trdn nhu 6 hinh 1.

Hinh ITri gi6 thi6t ta c6 m: a' + b' * c' + d': 40 (l). DAtn,= q + b + c + d vd p : e + g + h + k.X6t t6ng c6cs6 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 p: ll7 (3). Tdng c6c s6 dugctli6n ld 4 + 5 +...+ 17 + 18: 165 : m * n + p + 15+ 10 + 9, krit hqp v6i (l) suy ra n + p : 91. Thayvdo (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 = 65suyra e + g:34: l8 + 16 vd h+ k:31= 17 + 14.

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 t4ihinh d6i ximg v6i hinh c6 nghiQm n6u h6n qua t4rcddi ximg ld tludng thlng di qua !6m c6c 6 trdn ghic6c s6 15, 10, 9. V4y bdi todn c6 th chb6n nghiQm.

trnr.,.-roro T?lI#E!, 5

G,.t,1,,."0ffio.


flfrt nruGo;{n qenDgqRoDG rnfrq pnfrnc

TRINH BA(GV THPT chuyAn Hodng L€ Kha, Tdy Ninh)

1j/rong dd thi tuy€n sinh vdo cdc trudng Dqi t:: . t lJU ;;;: ir"ia"ri t*d nay sei rd Ki thi i;;i tli';m c tti httttrrttt tlo hun'c -\' 'int tot do ''ir'

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.Ddy ld bdi todn tuong aai ma di rhdn tooi th! :

-i; ,;; 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 <ludng thing A.

ft) daonq minq .rlo"s,bdi vidt ndy.chilltF t:: :;,;",;;6 il;;,;;1., ura- *a-ducrn! thingdrct ra mQt s6 thi du v€ cdc bdi todn vi4t PT A vditucrng thhngOB.ViZ,pr<tucmgthingbAq.u-dudng thdng li qua mQt di€m cho tru6c vd tqo ;,*;;;mQI g6c bing g6c gfiab.a vir oK.vdi dudng thdng cho trubc m6t gdc a.

r. co so r,.i THUyEl' Ldi gi'fii' (h'l)K(6;6)

1) G6c gifia hai yecto a vd b kh6c 0 ilugc tinh-l/- :\ o.l)

qua cos(a,, )=w.2) G6c.gita hai <ludng thing ,cht nhau li g6cnh6 nhat trong hai. c{p g6c ddi dinh, vfy g6cgifia hai cluong thing ld g6c c6 sd tlo kh6ngvugt qu6 90".3) Neu hai cludng thing a, b c6 vecto ph5p

tuyiin (VTPQ Dn luqt ld i,,i, ttil;;l/

''\- | ' 'lcos(O'Dl = l;m'| 'll 'la) N€u hai dudng thhng a, b c6 vecto chi

phuong (VTCP) lin luqt ni,,i, tnil+ +t

lu,.urlcos(a,b)=#.l',ll",l

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 cdcclrth 4 ra lJ rhuot clrrotrg thing.\ : 4r-t- 3.v - l2 : 0 ya ttietn K l(;6) 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

. TOAN HQC6 'cfndiU@

cHuir B!

OHO I{* THI

Tnur0 HQc

rxd rxOxc

Su6c GrA

vAN DUNc a6coE arAr

C

H)nh I

vi c thuOc A non c(4,*) . on, , tu\.) r )

trung cli6m cua do4n CO th\ ,(+,-2)\) 5)

Gqi d ld <lucrng trung tryc cria do4n CO th\ d diqua Mvitvu6ng g6c v6i CO: x+2y:0 n6n

d: 2x-Y -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=3l-^ " <>1"' ">A{3:0).l4x+3y-12=0 [y=0 \''

Ta co cos,I6k =o-4 o-_r.= J'6t0t_L = €oA.oK - Je JOTO - 2

-{dk =45o )frfi =90'.


Eudng th5;ng OB w6ng g6c vdi OA n€n c6 PTld.r=0.Tqa d0 cua B ld nghi€m cira hC PT

{*=o ol'=o=+r(0,+).f4x+3y -12=O

-- l, - 4- " \"'

"''OThi du 2 (Trit'h dA DH Khii A ntim 2014).

Trong mdl phdng t,ri'i h€ trut' loct tl6 Or;,, t'ho

hinh tttottg ABCD c'6 <jit:nt ,V lt't trutrc dicm cilai{o4tn AB, ll lu diint thuit' tlrsun AC scro t:ho

,4,V=3,VC. L'iOt PT tltftt'tts lhing Ct), biitrin,q M (l;2).,v(2: 1)

PhAn fich.' Tim tga ttQ di6m P ld giao <ti6m cua MN vd CD tir

hQ thuc uN =t.w, v6i k =Y+=*=t.NP NC' Euong thing'CD di qua tli6m P vd tpo vdi clulng

thing MN g6c cr =frFH vd coso. =ffi, ari,, V

ring HP, MP d6u tinh duoc /theo a (a la do dai canh cua hinhvu6ngIBCD).

Ldi gi,rti. (h.2)

Gqi P : MN O CD; Hlddidm Dd6i xrmg cua M qua t0m cirahinh vu6ng ABCD. TacoMN AM ANNP PC NC . \'' -../ '-

...-l'....-."'..N^

H P L

Hinh 2

ntl =ZtttF .

ateb-0 holc b=aa.'1. y6i b=0, chgn a:1 thi i=(t;O), cluong

a

. Vdi b =la,chgn a=4 thi i=(+,5), dudng4

ftt.._!_+4tthdngCDc6PTlei"-3'-'

lY =-2+3tSThi du 3 (Trlch-ai Oru Khdi .4 ndnt )Al2).Trong mcil phiing tri'i he tr.te dO O\,, t'hr, hinhvttong ABCD. Goi .\,1 la trtrrr,4 tiii)rtt ,tr.t ,',lnliBC', lV td die:m trOn cunh.CD suo thr,

'il r(',ry l.\l). Giu ttt' 14 +.1 , r'r./ tltr,,'rt,g iltin.1

.,1N :oP.I: 2r , -l , ,l 7'int iort dr.t iii,,:nt 4

Phdn tich.' Vi dC bdi y6u ciu tim tqa dQ diCm A trong khi decho phucrng trinh ducmg thing AN vir t1a d.Q di6m Mn6n ta nghi d6n viQc xemA= AM nAN.

' N6u ta tinh dugc g6c 'frifr

(g6c gita hai ducrng

thing AM,IA), thi ta vi6t tluqc phuong trinh du<mgthdngAN.

ViQc 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

cons Inuc tatrlo+bl--*^\* ' "/ l_tana.tanb.

Ldi gidi. (h.3)

Ggi a ld cpnh ciia hinh vu6ng, Al^ ^\

ta c6 tan I DAN + BAM I\/tanfIfi + andifu

1- tan 6TN .tandTfu D

t1+-'). a= 1. Suv ra

,-1 I' 3'2

thang cD c6 Pr te{- =!* t

lY =-2

Gei P(x:y), tu he thuc rr€n ta .O r[],-z)\r ')'

Ggi a li d0 ddi canh cira hinh vudng ABCD, tirhC thfc (1) ta c6:

MH =a,PC =!,gp =! a,W = "[MII2+ HPz =oq .6'-'- 3-"" - 3

Ggi cr ld g6c gitia hai ducrng thbng CD vit MP,

khid6 a=friH+cos0 =HP =1.MP JtoGSi i=(a;U)+O ld vecto chi phu<rng cta

ducrng thingCD,v\ Mfr =(t;-:) n6n

IrMwl _ la_3hl Inncd:

l;llMNl Jto JF;P Jmtil t

=la-ZAl= Jaz +bz o 4b2 -3ab =0

6IN *6Iu = 45o = frtrfr = 45'.

MAN ld g6c giira hai ducrng thdngAMvdAN.

Eudng thingAN c6 VTPT ld i=(z;-t).

Hinh 3

sti ase re-zorsl

-3s#8[ zHÃY ĐẶT MUA TC TH&TT TẠI BƯU CỤC HOẶC VỚI TOÀ SOẠN !

cqi i=(a;b)*O n VTPT ctra tlucrng thing

A M thi "orfrTN = JL= cos45o

J5.Ja2 +b2

= JI6.J7TF =zlz"-ble3a2 -8ab-3b2:0 (1)

Ni5u b=0 thitu (1) c6 a=0 (loai).

N6u b + o thi (,)*,(;I-'(;)-r=, *:L. YOi a =3b, chqn b = I > a = 3 thl |, = (Z;t),

<ludng thingAMc6 PT le 3x+y-17 =0.

Tri A= AMoAN tatimduqc A(a;5).

. YOi b - -3a , chgn c:1=b=-3 thi |1=(t;:),

tlucrng thhngAMc6 PT li -r-3y -4=0.Tt A= AMaAN tatimduqc A(1;-1).

o Thi dq 4. (Trich ai pa kh6i D ndm 2012).- ,J

Trong mQt phdng voi h€ tga dQ Oxy, cho hinh

chir nhqt ABCD. Cdc dtrdng thl:ng AC vd AD

ldn laqt c6 PT ld x + 3y : 0 vd x - y + 4 : 0;

dadng thting BD di qua aiam ru(--I,t). n*' \3 )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,

t4o v6i AD mQt 96" ffi:6il (g6c gita haiducrng thingAC,AD).. C6 phuong hinh iludng thdng BD ta tim duo. c tga

t10 t0m I =AC aBD cua hinh chii nhft ABCD virtgad0tli6m D=ADaBD.. Tri d6 6p durrg c6ng thrictim tga tlQ trung tli6m tl6tim c6c ilinh cdn lai.Ldi gidi. (h.a)Tqa d0 di6m A ldnghiQm ctra HPT

{::ur' *:oo=o + a ( -: ; 1 )' Hinh 4

G6c 68 ld g6c gita hai tlucrng thhng AD vit

DB, 6E=6k. Gqi i =(":b)*6 a vrPrcira <ludng thing BD, ta c6 VTPT. cua AD

,*=(t -t)non cosffi =S-=a.{aiq,[a 6e5(a-b)' =2(a'+b')e3a2 -lUab+3bz =0.NOu b:0 thi a:0 (loai)

Ni5u b * 0 thi 3a2 -IAab+3bz =0

.=',(g)' - ro[g)+: - o c> la =3b ." -\b) -\b) - -

lb=3a.Yot a=3b, chqn b=l=a=3, PT c'ia

rtudng thbngBDqw u(-*;r) u rr*y=0.\3 /

Gqi t h giao di6m cua AC vd BD thi tga d0 cria

1ld nghiQm cria h0 PT

{lx + Y= o.= I' = o = l(o:o).

lx+3Y=g LY=Q \' ''

Tqa d0 di6m D ld nghiQm ctra hO PT

[3x+v=o (" 1

1-^'' -", <+]^ -.'=o(-t:3).[x-Y=-+ [Y=J

Vi 1 le trung <ti6m ci.r. BD n6n tga d0 di6m

r(t;-:)..Ybi b:3a, chQn a=l:)b:3, dudng thing

BD c6 VTPT ld i:(1;:), t*O"g hqp ndy thi

,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 hinhthang 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 ABDc6 trac tdm ld H (- 3;2). Tim tqa do cdc dinh Cvd D.

PhAn fich.. Trudc ti6n ta xem x6t tinh d4c biQt cria cilc tamgi6c IBC,BHC.. Vi6t phuonc trinh iludng thing AC di qua <li6m,I1

vd ru6ng g6c voi ,BD. Suy ra tga dQ di€m I, C, A..Taxem D=ADaBD. Vi6t phuong trinh duong thing AD qura A tqo vbi

AC mQtg6c bang fiE =fdi = 45'.

G6c 6k ld g6c gifia hai cludng thlngAD vit

AC,tac6 cos6;=]59 =+.Jl+E.JI+l J5'

.r TOAN HOCE icru,ufwHÃY ĐẶT MUA TC TH&TT TẠI BƯU CỤC HOẶC VỚI TOÀ SOẠN !

Loi gi,rti. (h.5). Gqi -I ld giao di€m cia ACvd BD, suy ru IB =1C. MdIB IIC n6n tam giSc IBCw6ng cdn t4i { suy ra A

i&=4f . Mat kh6c Hinh 5

BH L AD > BH L BC , suy ra tam gi6c BHCvudng cAn t4i B, suy ra I ldtrung clirim ci,- HC.. Dutrng thlng AC <li qua 11, vu6ng g6c v6iduong thing BD, c6 phucrng tr\rth 2x-y + 8:0.Toa clO cua I ld nghiOm cira h0 phuong trinh

l2x-v+8=0 lx=-2.'-i 1

lx+2y-6--0-- Iy=+Tu 1 ld trung di6m cria dopn HC suy raC(-t;O); Hldtrungditim cua doanAC, suy ra

e(-s-z).

GSi i=(a;O)*O ld vectcr ph6p tuyiin cria

ducrng thingAD.Ta c6 g6c gita hai tlucrng thhng DB vd AD litIDE =idi =45". Suy ra

, E^ la +2bl la +zbl IC^c4\o =.:l-:L...'.:l:- --Js.Ja, +b, ,15..t;t;V O

>z{a+2b)'z =5(a, +l*)€3d -3U -kb=g (*)

N6u b=0 thi a=0 (loai).

NOu b*0 thikhid6

f a=3b

lb=1a

. v6i a:3b>n-(zt; u)=b(3; i) =(:; t)ld toa d6 vecto'phfp tu5r(in cria AD, ndn AD cophuong trinh 3x+y + 17 = 0.

Toa dQ cua D ld nghi€m cua h6 phucrng trinh[3x+r'+17 =O I x=-8€ I .. _ _- = D\-9:7 ).lx+2y-6=0 tr-/

. v 6i b = -3a *i = (";-t") = o(t;_Z) = (r; _:)

ld toa dO vecto ph6p tuy6n ctra AD, n€n AD c6phuong trinh x -3"y - 1 = 0.

Toa d6 cia D ld nghiQm cua h0 phucrng trinh

z r2 lg-.r-r *r(rs)' -s f, -:=o<+.,2=',

lb3

[*-"-r =o * {;= f =o(+:r).lx+2,r,-6=0

f6t tu4n. Qua c6c thi du tr6n, chtng ta th6y vi6cvi6t phuong trinh tludng thing trong m5t phSng tqadQ.Oxy di qua mQt di6m cho tru6c vii tao voi rluOngthlng cho tnr6c m6t g6c c[, c6 th6 khai th6c d6 gi6inhi6u bdi to6n chi cAn ta tinh rlugc g6c cr 116. ViQctim toa dO m6t di6m du6i dpng ld giao rti6m cira haitluong thing cfrng li m6t hudng gini bdi to6n rimdi6m khri hiru ich. Sau ddy xin gi6i thieu m6t s5 baid€ ban itgc luyQn tpp.

BAI LUYEN TAP

L @a thi DII tcnAi O ndm 2Ol4). Trong mdtphing vdi hC tnlc tga d0 Oxy, cho tam gi5cABC co chAn du<rng phAn giSc trong cua goc Ald di6m o(t;-t). Eudng thing AB c6 pT

3x +2y -9 = 0 , ti6p tuyr5n Ai A cin dudng trdnngoai ti6p hm giitc ABC co PT x+2y -7 =0 .

VlCt Pf ducrng thing BC.

Hwdng ddn. . ciei hO g6m PT ducrng thlng ABvd PTTT tai A cua Clucrng trdn ngoai ti6p tamgi6c 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 thbng BC qra D, hqp v\i AC

m6t g6c Aing (ce,Cn).

2. (DA thi.th* ndm 2015 cia Trudng THPTLr,rong ftte fiinh - Hd NOi). Trong m6t pt ingv6i hd truc toa dd Oxy cho hinh chft nh6tABCD c6 diOn tich bing 15. Eucrng thing ABc6 phucrng tr\nh x-2y = 0. Trgng t6m cira tam

si6c BCDrd di6m "(-*,+).

rim tsa d0 b6n

dinh ctra hinh cht nhft bitit ring di6m B c6 tungd0 lon hcrn 3.

Hunrg ddn.. Tinh dlG;ABl suy ra

alc ; enl = ffi .afc; an1 = ).ayc, oB7 = ac .

. Tir di6n tich hinh cht nh6t vi dQ ddi c4th BCsuy ra d0 dei canhAB. G

*rnu.,.-roru, T?3I#S 9


THuslJc rpuoc xi mupEs6g

(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 s1r bi6n thi6n vd vC d6 thi (C) cira him

s6 t Cn 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|triqia mae Z,qn'-(O,q' +OB2)=98.

{.iu 2 (l didm).a)Cho g6c a th6aman cota=1'

Tinh gi6 tn cua bieu thuc

.r _ 2015' - sin2 a -sina.cosa -cos2 a

b) Tim s6 phric z c6 m6dun nh6 nh6t th6a mdn di6u

kien liz-31=V-z-'1.Ciu 3 (0,5 di1@. GiniPhucrng trinh

- ^-t F4sin'? | - J3 cos2x =2 - sin2x.

C6u "l (1 ai6m1. ZiaihQ phuong trinh

lrs t (t l8.r ^- )l3+./9,r2-4=:l "+ ='I' .+25vLlt "^ - -9[*

Y2 -2Y+2 -' J (r'Ye1R1'I

f7"c +y: +3.0,(x-y) - I 2x2 +6x =1

Ciiu 5 (1 di€m). Tinhtich Phdn

. oc (x' - x -2)e^ JTTT| -? +l ^ .--

I (x + l)J3 +2x - x2

2

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::lcua kh5i ch6p S.ABC vd khodng c5ch tu di6m,

m4t phing (SBC).

Cdu 7 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

gi6c 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=0vd 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. .12

khai trien Newton ,uu(z*-] I t**ol.\ D(-)

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

PHAM TRQNG THI.I

IGV THPT chuyin Nguyin Quang Diiu, DdngThdp\

\lAN DUNG ffi ... ttiip the o)

. Tt d6 tinh dugc c6sin cira g6c BAC, tl6y ld g6c

giira hai ducrng thing AB, AC. "I vit5t pt oong- ttirftauot g thing AC di qua G, tqo

vcri dulng thilngABmQt g6c bilg 6k.. Tim tga dQ A= ABaAC, suy ra tqa d0 C (tr'r

?-..''-AC =+ AG ), trung di6m / cia AC.

2. Viiit phuong trinh ctudng thtng BC qua C, ru6ngg6c 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 ra trung di6m\z )cfa c4nh AD, di€mE ld tmng di6m cta canh AB vit

di6m K thu$c cqnh DC sao cho KD = 3KC . Duong

thing EK c6 phucrng trinh ld 19x - 8y - 18 = 0. Tim

toa d0 di6m C cira hinh vu6rry ABCD bi6t ring di6m

E c6 hodnh rJQ nho hcrn 3.

Hutng ddn.. Tinh dusc tn{iF, tanfr, ding

cong thtc tan (a + u)= fl##k tirn r]ut1c

turiit ,oy .u "nrFD,

vot ifr 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.

- ^ TOHN HOCI U - ctudiga 56 456 (6-2015)


.. ,.. t ? \rr ,\'}IUONIG DAN GIAI OE SO B

Ciu 1. a) Bpn ilgc tg gi6i.b) Gqi M(xo:to)ta ti6p di6m cira ti6p tuy6n.

H6 so g6c cua ti6p tuy6n: 1, = -- 7

- .

(xo+2)'

ru gia thi6t suy ,u, --J- . [-])= -,(xu+2)' \ t )

€ Jo = - t hoic Jo = - 3; suy f&yo=- 4 holc/e = 10. Phucrng trinh tit5p tuy5n: ! = 7x + 3

hodcy =7x+31.CAu 2. a) Phucrng trinh d5 cho tuong du<rng v6i

sinx+ J3 cos* -2 erir[r*1) = r\. 3,/

€ x = !+ k2rc (k eZ).6

b) Ta co z: =5-l2i.z+i=65-t2i s -.14'=-=--'21 . lW; =6 6 ',l

Ciu 3. Ddt t =2' , (t > 0 ), ta c6 phuong trinhft-c1 5 t'[+-=; <+l . 1 ; t=2e x-l= lex=2;

t L ll=-t)1l= 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,b5t phucrng trinh da cho trd thdnh2ab < a2 -4(l-b) o (a-Z)(a+2-2b)>0e(J?;zxt -z)(,17 +rx-t+z.r)> o 1t;X6t 2 trucrng hcrn sau:

THl: x>-1+J2. Khi 65 J*2,r2*j+2x>0,n6n (1) tuong ctuong ysl Jx'z +2x-l-2>O

Ir > -1+J6<>l -=x>-t+J6.[x < -1-J6TH2 x<-l-J1.Khi d6 JV +2x-I+2x<0,n6n (l) trothdnh JV +2x-l -2<O<+-1-J6 <x<-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 < x < -r-JI ; r > -1+J6.

Ciu5.DAt t=Ji t P-x+Ztdt=dx;x =l = t =l,x = 4 > t =2. Ta<lugc

t =2t--!-dt =2(i+*-+)o,I t'z(t +t)

CAu 6.

^z ltTa c6 S*. =t;".Gqi D ld trung tli€mAB, th\ OD L AB

=A,DLAB A

- 16o=600.oD = lrr='fi,

3- 6

A'O = Oo.tan[io =l;v*r.o,r,r, = Stac.A'o =+ (clvtt). Ta c6

cc' I I (ABB'A') = d (AB,cc') = a(c,(ABn' a')),

I arllV^*r=.S*r.A'O= 24

soo, = |e'o.ea =; + o =#.u^,d (AB,cc' ) =3v=o'

o" = 3?'

So'* 4'C0u 7. Ggi S - AB n DE. Theo dinh li Thales

ta c6 !4 =BE =r -#=1.ro d6 d(c, DE)54 AD6 SM 7 \_-

= 5d(8. DE) =s.+d(M, DE)=+ # =ffiGgi cpnh hinh vu6ng ld a. Trong tam gi6cru6ng DCE ta c6

11113661(dc,rD)--= co'* ce'=7+ 25r,,= 25o.,

=d(c, DE)= 5? . Suv ,aE =-Y. -o=2,11g.J6r J6t J6l0( to-zzr\vi D e DE nan D[/: :,:::

). raco:

(Xem tiip trang 27)

"rnu.,.-roro T?3I#S

11


SISffiF$ frleffi cnfrr s0 xoc Dnc rRtrNE

Biitqof,nDnqwft

Oi " thwc ld mAt nAi dung riiy quan trpng,trongU chwnng trinh todn hpc phii th6ng vd qat aiydagc gidng dqy trong chucsng trinh dqi s6 6 cdpTHCS. Cdc bai tudn hAn quan d€n da th*c xudt hi€nnhiiu trong cdc k) .thi hpc sinh gi6i Quiic gia vd

Qu6c t€. Bdi todn v€ da thuc ld bdi todn dgi s6 tuynhiAn nhi6u bdi todn v€ da th*c bim chdt cila n6 lqild bdi todn til ho". Niiu chting ta bih kiit ho. p khai

:.thac mdi quan hQ giiia cdc tinh chdt dqi s6 vd s6 hpcthi s€ gidi quy€t duqc nhiiu bdi todn vi da thuc.Tinh chdt s6 hgc dfic trtmg thudng s* dang d6 ldtinh chia h€t, t{nh chdn li, tinh chiit cfia cdc sd

^1,:;nguy1n t6, hqp s6, s6 nguyAn,...Cdc dinh ly v€ s6

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{nctngmQt gi5 tri tqi n + 1 gi6 tri.kh6c nhau ctaddi s6 thi cla thric cl6 ld da thfc hing.. Hai da thirc bflc nh6 hcrn ho{c bing r md nhfnn + | giltri th6a. mdn bing nhaul4i n +l giht4kh6c nhau cria cl6i s6 thi cl6ng nh6t blng nhau.. Bdc cta tdng hai <la thirc khdng lcrn hcrn bQc

cria m5i <la thric <16.

. BQc ctra tich hai da thric kh6c kh6ng bing t6ngcdc 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 mQtc[p cta thtc q, r e IR.["r] sao cho "f : qg + rtrong d6 hoic r: 0 hoflc deg r < deg g voi rkh6c 0.. Du cia phdp chia cla thfcfix) cho x - c ldflc).. V6i b6t k! hai da thric -f, g e ZlxT bao gidctng tdn tai IJCLN ciafvdg vd LrCtN cl6 duynhdt.. N6u da thuc d h UCLN cria c5c da thtrc f vitg, 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.

- ^ TOnN HgCLZ - Gfi.rdiUa-gq-r*reeggl

NGUYEN LTIU(Gf rruPf chuyan Hd Tinh)

. Hai da thtrcf vdg nguy6n td ctng nhau tuc ld(f, g):1 khi vd chi khi tOn t4i hai tla thhc u, vsao chofu+ gr: l.. N6u c6c <la th(rcflx) vir g(x) nguy€n ti5 cringnhau thi IJV)l'vd [g(x)]' s0 nguy6n t0 cringnhau v6i mgi m,n nguy€n ducrng.. Mqi nghiQm xs cta da thric

.J@ : aox" + ?rt' *...ta,sx + a,(as * 0).

tl6u th6a mdn b6t d6ng thric :

lr.l(lrrl -L)< A,A=maxlatl, k =r,...,n.. Da thirc pahttna quy khi vi chi khi mgi u6ccria n6 cl6u ld cta thirc bQc 0 hoic ld tla thric c6dqng 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 kheqly..Ve sp phAn tich d6 ld duy nn6t ni5u kh6ngk0 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 pth6a 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 chop2 th\ P(x) kh6ng ph0n tich duqc thdnh tich c6cda thuc c6 b{c thdp horl vcri c6c hp sO hiru r5i.

NQi dung cua cac bdi toan vi da thuc thudng xoayquanh viQc tim nghiQm vd xdc dinh si5 nghiQm ctiamQt da thuc, xdc dinh m)t da thuc hofrc rdc luqnggia tri clta da thuc khi biA dq thuc thod mdn mil sdtinh chdt cho trudc hodc ld ch*ng minh cdc da thucbiit kha quy...Trong khu6n khd bdi vi€t nay , chung ta s€ dO cQp

diin m1t sii mi ndn didn hinh nhiim minh hea cdcilnh chh s6 hp, drtc ffung thudng xuh hi€n 6 bditodn 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 phaongtrinh Q(P(x)): I kh6ng c6 nghiQm nguyAn .


Ldi gi,rti. P(x) : (x - a)(x - a - 2015). s@)

=P(x) chin voi Yx e Z. Gii sir lxse ZdC

Q(P(xr)): 1 + Q(") = lx - P(xdl.h(x) + I

= QQ0l4) : 12014 - P(xs)l.h(xs) + l.Do P(x6) chin vd h(xs)e Z, n€nv6 trai chin, v6

phii 16, m6u thu6n.Ldi binh. Ddy ld bdi to6n tlon gi6n, sir dqng phucrngph6p chimg minh ph6n chimg ptr6i frqp dinh liBezout cring tinh chin, 16 cria s5 nguy6n ta suy ratli6u ph6i chimg minh (tlpcm).Bii to6n 2. T6n tqi hay kh6ng da thtrc P(x),deg(P(x)) : 2015 thda mdn P(*' - 2Ol4) chiahAt 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. Dath6y phuong trinh o' + o - 2Ol4: 0 lu6n c6 2nghiQm 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 viP(*' - 2014) phii chia hrit cho p(x) nen ta c6 ttr6 dudo6nngay P(x) c6 dpng (x +,)'0".Biri toin 3. Cho & e N- . Tim dt cd cdc da thucP(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 di6ukiQn trdn. Tri gi6 thi6t ta thdy x : 2016 liLnghiQm 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@): ahing s5.v6y p(*): a (x 2ot6)r. Tht lpi ttfng.Ldi binh. Ddy ld bdi torin co b6n, sir dqng tinh ch6tchia h6t, nghiQm bQi, cring tlinh lf Bezout cho ta loigihi.Bii toin 4. Cho P(x) vd Q@) ld hai da thilrc voihQ 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) dsavdo gi6 thi6t ta suy ra t P(1) + Q\)1chia hi5t

cho x2 *.r * I (2). Do deg (.x2 + x +1) : 2 vitdeg [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 <linh lfBezout ta c6:

{59=l;_i,,X;$ Do P(x)

vd Q@) li c6c tla thfc voi hQ st5 nguy6n n6n

R1(x) vi R2(x) cfing li chc da thric vdi h€ s6

-_.-.A- r ^: +Li_.. lr{c:otsl= 2014.Rr(2015)nsuven. Lal tnav: <-'a--J -"' -a- "^-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 k{r6 tle.p vC tinh chia hi5t cria

cla thric. Sri dtmg tinh ch6t 1a - b1l(P@) - P(b)) ,

vd ktit hqp gifia ttinh lf Bezout vdi tinh ch6t vd b6ccria 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 x1,

x2,4. Chrhng minh rdng:(o'0" + b2017 + c'0" +ry1x1 - xz)( xz- x3)(x3 - x1)

chia hiit cho 2017.

Ldi gi,rti. X6t phuong trinh:x2ot7 - x + loi + (b + l)x + cl : o.

D1t J@) : ax' + (b + l)x * c. Theo dinh liFermat bd ta c6: xz0t7 - x = 0 (mod 2017), suyta: JU) = 0 (mod 2017), i : l, 2, 3 hay

f(x) i2017.. N6u (x1 - x2)(x2- r:)(x: - ,,) j 2017 thi bdito6n chimg minh xong.. N6u @1 - x2)(x2- xz)(xr x1) kh6ng chia hiitcho 2017 ta c6: flx) *J@) :. 2017, suy ra

= (x1- xy) fa(xy +x2) + b +l I i2017* axr+ axz* b+ 1 i 2017 (1).

Tuong \r: axza oxz+ b + 1 i2017 (2).

Tri ( 1) vir (2) suy ra: a( 4 - x) i ZAtlhay a :.2017, suy ra b + li20l7,fl*r): L axr2 + (b + 1)x1+ c] i 2017, suy ra

c :. zol7.

"s nu.,.-rrru, T?ll#ff

13HÃY 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

2017), c = """(^od20l7), suy ra a-t b-r c *

l: a2017 + b2017 + c201'7 + 1 (mod zolT).

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 liFermat b6 ld mQt tli6u t.u nhi6n.

Biri toin 6. Cho s6 nguyAn n > 2 vd da thr.rc cd

cdc hQ 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)

- ( P(v))thi cip I y, v I cfrng th6a mdn tinh chet d6,

\JI\/

nghiald taco: yf r[401] 16o aQ2;r1yr

ta" '' \ x )' x ' -

di6u hi6n nhiOn). Thflt vfly tu gin thi6t at : ark

ta c6: P(,IO2) -rP(v))''[-l)\ x ) \, )'[P6)J

= *,r( Po)

l=1r1ry1,."t'=+.)(x)' \r0)//\tv\

= ,[ ,Ur.,},mod Y) = r (x): o(mod Y)

(doP@) =1(mody)).Mat kh6c do xl Pb),n6n (x, y): t vi n6u ngusc

lai (x, y) : d > 1 thi r(y)= 1 (mod d), vd ly.

Suy ra "f4g)=o(mody).

Hon nfta y > x' \.x)^ P(v) y2

xxtdn tai v6 s6 c[p sti nguy6n ducrng x, y thba mdndc bei.Ldi binh. Ddy ld bdi to6n kh6, didru then ch6t h bi6tsir dgng phuong ph6p clrc hpn, ph6i hqp vdi ph6p

to6n <ldng du trong qu5 trinh chimg minh phdn

chrmg dd tim ra loi girii.

Bdri 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 trinhP(x) *3 :0 kh6ng co nghiQm nguyAn.

Ldi gidi. Tri gin thi6t ta suy ra cla thric P(x) - 2

c6 ba nghiQm nguy6n ph6n biQt ld a, b, c. Do d6

t6n tai Q@)e ZLxT sao cho:P(x) - 2: (x - a)(x - b)(x - c)Q@).

Gi6 sir phucrng trinh P(x) - 3 :0 c6 nghiQm

x: dngryAn. Khi d6:1 : P(A - 2 -- (d - a )(d - b)(d - c)Q(fl.

Do d - a, d - b, d * c, Q@) eZ ndn d - a,

d - b, d - c e {1,-1}, m6u thu6n v6i gi6 thitlta, b, c ddi mQt ph6n bi€t. Tt d6 ta c6 cli6u ph6ichimg minh.Bii torln 8. Cho da thac fu)ez(x) . Chilmgminh rdng n€u da thilrc Q@) : J@ + 12 c6 itnhiit 6 nghiQm nguyAn phdn biQt thi JU) kh6ngc6 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 €Zmdfixs):0 ta suy ra :

12 : Q@) : (xo- xrXxo- xz)...@o- xo) . g(xo)

+ 12: l*,-r,l lro -rrl ...lro *rul ls(ro)1.

Do re-x1, x0_ x2,...:q0_ xsld c6c s5 nguyen <l6i

mQt kh6c nhau nOn trong c6c s6

lro -x,l,...,lro -rul kh6ng th6 c6 3 so trd lcn

bing nhau, suy ra

lro -r,llxo -xrl...lxo -rul) 12.22.32 =24 , hon

nta ls(xo)l> 1= vd ly.

Bii to{n 9. Cho JV) ld mQt da thtbc bQc 5 vdihQ sri nguyAn, nhQn gid tri 2015 vd'i 4 gid.tringuyAn khdc nhau cila bi€n x. Chrng minh rdngphuong trinh flx) : 2046 kh6ng th€ co nghiQm

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

L TORN HgC14 -cludi@


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 ringphucrng 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 minhbdng phin chimg. Sit dqng tinh chdt nghi6m, tinh

,t ;. ^ i ich6t s6 nguy6n to, s6 nguy6n, ph6i hqp v6i tlinh lfBezout.

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 nraquy, 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:

P(x): (x -1)(-r- 2)...(x-20t5) -i : F(x)G(x).Do <16 F(k).G(k): -1 hay F(k): - G(k): t 1,

Vk,l<k<20l5.Suy ra F(fr) + G(k) 0,

Vk,I< k <2015 . Ddt Q@) : F(x) + G(x) thi

des Q@) < 2014 vd Q(k) : 0Y k,l < k < 2015.

Suy ra: Q@) = 0 hay F(x) : - 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 ilathr?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 khdquy 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 ratrong hai s6 deg g vd deg h co mQt sO > t. tr,tpt

iirrilcfQ): 3 ld s6 nguyCn td ndn ls(0ll= t ho{c

lrtol:l. Gie sir g(x) : *u + alxk-l +...1 ar,

(fr > 1) va ls(o)l:1. Gei at, a2,..., aL ld" cdc

nghiQm cira g(x) ta c6:g(x) : (x - ar)(x - or)...(* - ou).

Vi ls(O)l=1 ndn la1a2...a1,l:1 (*)

Do g(a;) : 0 n€n Jla):0 v6i moi i: 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)

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 thacP(x) : ( "' - 7x + 6)2' + 13, ne N..Chbngminh riing P(x) kh6ng th€ phdn tich thdnh tichcila n + I da thuc khdc hting s,i vdi h€ sti

nguyAn.

Ldi gidi. n5 thay deg P(r) : 4n vdP(.x) khdngc6 nghiCm thuc. Tam thirc xz - 7x + 6 c6 hainghiQm li 1 vd 6;vi 13 ld s0 nguy6n t6.Gi6 su P(*): Pk)...Pn*r(x), thi P;(x) c6 b{cch8n. 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 ldP1(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

P1(1) vd Pr(l) c6 it nh6t m6t sti bing 1. Khdng,,.gi6m tong qudt gi6 su P1(1) : 1 suy ra a: -b .

Khi d6 P,(6) : 36 - 5b > 0 vd P{6) + 13, suyra36 -5b: lhay b :7, a:-7 vdkhid6Pr(x) : x' -7x + 7 c6 nghiCm thuc , m6u thuin!

Ldi binh. Ca 3 bdi to6n 10, 11, 12 vA da thric b6tLh6 quy d5 n6u <ldu c6 chung m6t hu6ng gi6i ld:chimg minh bing ph6n chimg. Tuy nhi6n bdi to6n 10

thi d mric trung binh vd kinh tli6n, chi stl dUng tinhctrAt vC bflc cria da thirc ld cho k6t qu6. Bdi to6n I I dmric d0 cao hon, ta d6 sir dung tinh ctr6t ve b6c criada thric, tinh ch6t cira st5 nguy6n t6, s6 nguy6n, k6thqp 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 itkh6 khln cho nhi6u hgc sinh. Bdi to6n ndy c6 vdic6ch gi6i tuy nhi6n d6u chimg minh bing phuongph6p ph6n chimg ph6p phin chimg. C6ch gi6i tr6ndAy dga vdo viQc khai th6c b6c cta da thfc, tinh ch6tnghiQm ctng tinh ch6t cira s6 nguydn tii.

D6 luyQn tfp phin ndy c6c bpn hdy ldm mQt sd

bdi tpp sau: (Xem fiAp fiung 26\

=,, nr.,.-rorr, T?8I#EE

15HÃY ĐẶT MUA TC TH&TT TẠI BƯU CỤC HOẶC VỚI TOÀ SOẠN !

#i4

ff{;$M\,,

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.

TRIIONG QUANG AN(GV THCS Ngh\a Thdng, Qudng Ngdi)

Eni I"21456 (Lop 7). Cho tam gi6c ABC cdn tai

Ac6 BAC =800.C6cdi6mD, Etheothf t.u

thuQc c6c cqnh tsC, C4 sao cho :

BAD = ABE = 300. Tinh sO ilo BED.

NGUYEN MINH HA(GV THPT chuyAn EHSP Hd NQi)

BAi T31456. Gi6i phuong trinh

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 luqt l6y cbc di€m M, N

sao cho MDN = 450. Tim vi tri cua M, N de d0

ddi tlopn thing MN ngin nh6t.

uilr vAN cHr(GV THCS LA Lqi, TP. Quy Nhon, Binh Dinh)

Eei Tg1456. Tim c6c s6 nguydn duong x, y thoh

mdn'. xa +yz +l3y+lS(y-2)x2 +8;.1.

NUT UAI QUANG(GV TNCS Ydn Lang, Vi€t Tri, Phri Thp)

cAc IqIp THPT

TR.AN QUOC LUAT(GY THPT chcryAn Hd TInh)

Bei T71456" Cho hinh ch6p SABCD c6 d6y

ABCD le hinh chir nh4t, Sl vu6ng g6c v6i mfltphang @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 minhring: a? +b2 +c2 >-162d2.

lt eua,Nc nAo(GV THPT chuy€n Huinh Mdn Eqt, Ki€n Giang)

Bdri T8/,455. Chimg minh ring phuong trinh11

(x+1)'*t =1;c6 nghiOm duy nh5t.

NGUYfN vAN xA(GV THPT YAn Phong 2, Y€n Phong, Bdc Ninh)

TIdN TOI oI,YMPIC TOAN

BAi T9/456. Cho a,ar,...,ar, ldc6c s5 nguy€n

duong th6a m6n:

a) at<a, 1...1ary

b) Vdi m6i sO nguy6n duong k, k: 1,..., 15,

n6u ki hiQu b.li udc sO tcrn nhSt cria ao sao cho

bo 1an,th\ bt > b2> ...) br,

Chimg minh ring a,r>2015.

TRi,N NGQC THtur\rG

{GV THFT chry,,6n l/inh Philc)

BSi 'f lEi456. Cho rla thric

lQ)=x3 +3x2 +6x+1975.

Hdi trong dopn [1;3201s ] cd tAr cA bao nhi0u s6

nguy€n a sao cho./(a) chia h6t cho 3r0rs ?

TRAN xUAN DANG(GV THPT chuyAn LO H6ng Phong:, Nam Dlnh)

- -lrfl{ B*i'fb/456. Giai hC phuong trinh

fx+v+z=3

lx2y+y2z+z2x=4t^

|.*'*]' + z2 =5

* -- TOHN F.IQCI 6 t cfu66q_9eggq -zotl) _


Bir; "il n 1i 456. Tim t6t cd chc hdm clcrn 6nh

/: IR -+ R. th6a mdn

f(x')+ f(y') = ("+y)[/o(x)- fr(x)f(y)

+f'(x)f'(y)-f (x)f3 (y) + /*0)]v6i mqi 'r'Y e R

KIru oixu M,NH(GV THPT chuy€n Hilng Yaong, Phti Thp)

}*:ii "E'nlii'4iifl. Cho tam gi6c ABC virdidm M

nim trong tam gi6c kh6ng trung v6i trgng t6mG. N6i AM, BM, CMldnluqt cit BC, CA,ABt1i A,r,Bu,C'. Ke A0A1 llCA, AoA2llABvoi

A,,d thu6c BuCo . Tucrng tg, ta c6 8,82,Ct,C2. Goi G,,G, lAn lucrt ld trong t6m tam

gi6c A,B.C' A2BzCz. Chimg minh rdng

a) ArB,ll Bp,llCi2.b) MG di qua trung cliiim cua G,Gr.

PHAN VAN NAM(.GV THPT chuy€n DHSP LId iVQi)

F.$R. $EiC GI -N BAK V SC M{.}CIL

Froirlem'i'$1456 1$'or 6tr'gr*rle)" Find all finitesets of primes such that for each set, the productof its elements is l0 times the sum of itselements.

Prrrlrle&! 'X214*6 (F or T'i' gradc). Let ABC bean isosceles triangle with the vertex angle

6ii =800. Choose D and E on the sicles BCand CA respectively such that

6iD=GE=300. Find theangte 6Ei.:'r'riirl;rn 'gJ1456. Solve the equatiorr

rr-(r r)-r_

- -/\ I _r__ ___ !

\ti ' Jzx-t -'-[J6r-t ' ien-T)Frolrleru '84i45#. Let ABCD be a square and

let abe the length each side. On the sides lBand BC, choose M and N respectively such that

frnN = 450. Find the positions of Mand l/ so

that the length MN is minimal.

Froi:leru 'l'51456. Find all positive integers x

[*ai g,lil4$tl" Thi nghiQm giao thoa s6ng trdn

mdt nu6c vdi hai nguOn dao <lQng t?i Sr vd Sz:

ttr: uz: acoso/. Coi ning lugng truy€n s6ng

kh6ng dOi, bu6c s6ng tlo duoc l" :4 cm,SrSz :8 cm. Di6m M tr€n m{t nu6c thuQc dudng cong

cyc d4i kC tu ducrng trung tryc cira,SlS2, M dao

cl6ng c6 phucmg tr\nh uy : 2acos(at -*).'2

Di6m M' nim tr0n dudng trung tryc Sr,Sz vd

dao clQng t1i M 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

$;.]i Lli'$56. Cho mpch diQn nhu hinh b0n. Bi6t

Rr :Rz :.R: : 1 () ; Rs : 3 O i /i+ : 2 Q.Tirn

rliqn tr& tuong iluong cua dopn mqchAB.

DINH THAT QUVNH(Hd N1i)

andy such that

x4 +y2 +13y+ 1"'-(y-Z)x2 +8.ry.

FOR IIf G t.} 5CX-I{ }{-}{,

Protrriern'f 61,1.56. Solve the following systern

of equationsrlx+y+z=3l*'y* y2z+z2x=4.l,-l*'*y'tz2 =5

Frohelemr T7i;$${r. Given a quadrilateralpyramid S.ABCD with the followingproperties: 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 letdbe 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 following1l

equation: (x + 1)'*t = xi has a unique solution.

tXr:rrc tiitt trwng 2l)

pffi&ffi&ffiffiffi &ffi ffiffi Kffiffiwffi

T0ffiru Hsfl * *-:gfrffiffie

q /Sti ase (6-2015)


.|

6rnxtsKX

xY rnuoc

Bii T1/452. Cho -r, 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 ctngbing 5, do d6 nr5u 36* > 5v thi 36' - 5/ c6 chft

s6 tfln cirng bing 1, n6u 36' < 5v thi 5r - 36' c6i .^ . , J

chir so tin cung bdng 9. Vdi x : I, ! : 2 thi A:36 - 52: 11. Ta chi cdn xdt xem A c6 th6 6ygrhtrib[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, ynhu th6.

VQy giltri nh6 nh6t cira bi6u thfc A ld fl. AYNhQn xdt. MQt vdi b4n kh6ng x6t trudng hqp5! - 36". C6c bpn sau c6 ldi giii tltng: Vinh Phtic:Nguy1n Th! Ngpc Mai,6A4, THCS Y6n Lpc; Qu.angNgfli:-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 ThotN6t'

'IET HAI

BdiTZl452. Goi O li trung tlieim ctru tloatr rhdng

AB. TrAn mdt ru,ra mdt phdng bd AB, vi hai tiaOr, Wo vtt6ng goc vo'i nhau. Trtn Ox, Oy ldn luottatt 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

(c.g.c), suy ra AM: BP (l) vd AM ll BP. Mdtkh6c theo c6ch dlmg vd giri thi6t ta c6 Ol/ ld

trung tryc cua MP, nln MN : PN (2).

ret irqp (1) vd (2) ta c6

AM + BN = BP + BN > NP = MN. YQ'y 1u6n c6

AM + BN > MN. DAu bing xity ru khi N, B, P

th[ng hdng (B nim gita N vd P), tuc ld khiAM II BN. JYNhQn 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 DqngDiQu Huy€n, 7C, VA Thi.H6ng Ki€u, 7A, THCSNghia M!, Tu Nghia; CAn Thtr: Nguy€n HodngNhi,1A6, THCS Th6tN6t, Q. Th6tN6t.

NGUYEN XUAN BINH

BitiT3l452. Giai hC bdt phtrong trinh

f 1 frl,...,.+\j <(.r+y)r (l)l-v.. I.l I .

1--

IJt-t.r*r')r =l-.r' \21

Loi gidi.DK: lx+yl< 1 vd x < 1.

Ta xdt hai trudng hqp:

o Voi .t + ) :0, thay vdo (2) ta dugc x: 0, suy ra

y:0. Cap s6 (*, y): (0, 0) th6a m6n (1) n6n ld

nghiQm cira hQ b6t phucrng trinh.

o voi -r*) r 0, thi lr*yl ,0. Ta c6:

r ., 3. .,x' - -ry t y' :4tx + l)' + |(x - l)')

.)tr+y)2 > 04')2,[x'-xy+y'>lr*yl* (x+y)2 t lr*yl (theo (1))

= lr+yl(lr+yl-1) > o = lr+yl > t.

- ^ TOAN HOCl8';4i,iiEryHÃY 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). trY NhQn xit. Mqt s6 bpn ldm bdi thi6u thpn trqngn6n dd xdt thi6u truong hqp ho4c kh6ng thr.rc hiQn

bu6c thir lai n6n kh6ng lo4i dugc nghiQm kh6ngth6a min. Cilc ban c6 loi giii dirng ld: NghQ An:Nguy€n Dinh Tudn,7c, THCS Lti Nhat Quang, D6Lucrng; Vinh Phric: Nguy€n Minh Hi€u,9D, THCSVinh Y€n, TP. VTnh Y6n; Hii Duong: D6ng XudnLudn, 98, THCS H-qp Ti6n, Nam S6ch; Hi NQi:Nguydn Vdn Cao,gB, THCS Nguy6n Thugng Hi6n,Ung Hda.

NGUYEN ANH QUANBili 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). KIt'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 tnngcliint r'ila DE. C'hintg minh rdng ba cliem ..1, M,

l,l thong hctng.

Loi gi,rti.A

K

Gqi F ld giao tli6m cta DE vit BI.

Ti DE L MI (vl DE ld trung tryc cua lrfi) vitAI L MI (do AK ld cluong kinh), sluy raAI ll DE.

Tac6 IFE=180o-FIA=ACB:ABC (1)

Met kh6c, do EF ld tmng tryc cria IMn€n

IFE = EFM (2)

Tt (l) vd(2)tac6 DFM = DBM ,n€nDFBM^. ,.,{la tu glac nQl nep.

Suy ra 6fr8 =iFi:frE = MD tt AC (3)

lMdtI&6C IDE = EDM : IBC =I80" - IAC : AED .

Slty TaAIDE ld hinh thang cin, ddn t6iAE: rD: MD (4)

Tt (3) vd (a) suy ra ADME ld hinh binh hdnh.

Do cl6 A, N, Mthdng hdng. trYNhQn xdr M$ sO ba, ngQ nh{n khi kt5t lufn viQc

chimg minh ME ll AB vd chimg nxtu MD ll AC ldtuong tir nhau

_C6c bpn du6'i tldy c6 lcri gi6i t6t :

Hi NQi: Nguy€n Vdn Cao, Ngry? Thdnh Long, DSngThanh Tilng,gB, THCS Nguy6n Thugng Hi6n, UngHod; Bic Ninh: Nguydn Thi Bich Hdng,9A,.THCSYOn Phong, Y6n Phong; Phrfi Thg: Trdn Qu6c LQp,

8A3, THCS Ldm Thao; Vinh Phrflc: NguyOn MinhHiA,t,gD, Hodng Vdn Hidu,gE, THCS Vinh YOq TP.Hd Chi Minh: Nguydn Dinh Duy,9A6, THPT chuy6nTranD4iNgtr'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 l. Ta c6

(1) e x3 + (1n+l)x2 - (Zm-l)x -(2m-l)m-(2m-1)-5=0

o x2 (x + m + l) - (2m - l)(x + m a' 7) - 5 : 0

e (x+m+l)(x2 -2m+ 1) = 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 vitf - 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 + 1

bing t hodc 5. Do tl6 tu (2) xhy ra hai kh6

ndng:

{*+mll'=l lm=-xl) < <>{'' [*, -2m+l =5 - ltx+t)2 =5 (*)

Vi -rre Zn€n (x + l)2 le s6 chinh phuong, vi v6yPT (*) kh6ng c6 nghiQm nguy6n.

lr+**l =5 lm=-x+42t < <)<-' f r' -2m+ 1= I '

[(x+ li'z = 9 (**)

PT (**) c6 hai nghiCm nguy6n x : 2 vit x : - 4.

Tucrng img tim dugc m:2 vit m: 8.

VQy khi m:2ho{c m:8 th\ PT (1) c6 nghiQm

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 + x2 + x -4)

.e nr.,.-roru, T?ll,.,HS

19


:(x2 +2x+3)2 -40.De PT (3) c6 nghiQm nguy6n thi A phni h sd

chinh phucrng.

Edt (x2 +2x+3)2 -40= ft'z (k e N)

e (xz +2x +3 + k)(x2 +2x +3 - k) = 40 (4)

Do xe Z, 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 + kvd x2 + 2x + 3 - kld cbc si5 nguy6n du<rng chin.Ta c6 40 : 10. 4 :20.2. Do d6 tu (4) x6y ra haikhi ning:

lx2 +2x+3+k=10 lk=3t) fx, +2x+3-k=4 otfr+l)z

=51r';PT (*) kh6ng c6 nghiQm nguy6n n6n khdng xiyra trulng hqp ndy.

- (x2+2x+3+k=2O lk=9Zl < .-5 I'' l*, +2x+3-k =Z - 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. JYNhQn xit. l) Bdi to6n thuQc d4ng giii phucrngtrinh tim nghiQm nguy6n, n6n.viQc nh6n,xdt cdngnhi6u tinh ch6t cira c6c thrla s6 nguy6n vC trSi cria(2) hoac (4) thi xdt.cang it trudng hqp. Da s6 c6c bantham 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 Nguy6nThuqng Hi6n, tlng Hoa; Phri Thgz Nguydn ThiryDuong,7A3, THCS Ldm Thao; YinhPhic: PhqmXudn Cudng,6A4, Bili Thi Lidu Duong,8A4, THCSY6n Lpc; Nguy€n Minh Hi€u,9D, THCS VTnh YOn;Thanh H6r: Efing Quang Anh,8A, THCS Nguy6nChich, D6ng Scm; NghQ An: Thdi Bd Bdo,7C, TrinL€ HiQp, 8A, Nguy€n Thai Hi€p,8C, Bili Thi NhdtLinh, Einh Vi€t 74, 8D, THCS Li Nhat Quang;Quflng Ngni: Nguy€n L€ Hodng Duy€n,8A, THCSPhqm VEn Ddng, Ngh?a Hdnh.

PHAM THI BACH NGQC

BitiT61452. Cho ba sO khdrrg dm a, b, c . Chimgminh ring

(a + bc)' +(b + ca)' +lc + ab).

,- J1.(o + u)(t + c)(c + a).

Ldi gi,fii.O€ y ring trong ba s6 o - \ b * \ c - t1u6n c6 hai sd kh6ng trbidilt v6i nhau.

Gi6 su (a*t)(U-t)> o = ab+l> a+b

= c(ab+1)>c(a+b)

6m, ta dugc

(a+bc+b+ca)z2

(1).

Ding thr?c xhy rakhi c : 0 hoac a : I hodc b : l.

Lai c6 (a+bc)'z +(t+ca)' r(a+bc +b+ca)'

(2).

Ap dung U6t Oing thirc Cauchy cho hai si5 khong

+(c + ab)'

> Jl.(a+bc +b+ ca\(c + ab)

(a+bc+b+ca\' . .ae \---Z--------!- + lc + ab)'

> J2.(a+ u)(c +t)(c + ab) f:1.

Tt (2), (3) suy ra

(a + bc)' + (u + 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=#Ta chimg minh ring

(c +t)(c + ab)>(b+ c)(c + a) (s).

Thflt v{y (5) ec2 + abc + c + ab> bc + ab + cz + ac

eabc+c)bc+ace c(ab+I)>-c(a+b).

n6t ding thric thing, do (1). Tt (4), (5) suy raU6t Aang thric trong tlAu bdi tlugc chimg minh.Ding thr?c xhy rakhi vd chi khi

[c=0I c+ab .-la=l vd a+bc=b+ca=T (6)

It =t

Didu ndy kh6ng xity ra vl bc > 0 = a + I .

Tucrng Wc6 b*1.

N6u c=o: (6)=a:b=o'=o[o= ?=9JZ la=b=0Vpy dSng thric xdy ra khi vd chi khi

^ ^ TOAN HOC20 ';4"3i@


a-b-c-O ho{c a=b=J1; c=0 vd chc AEll BH+OHLBH. GqiKligiaodi6mcira

hoSn vi tucrng img ctra chring. D BH voi tluong trdn (O) (f * n) thi Il ld trung

)Nh$n *(r:.??v Jdbdi

to6n tuons d6i kh6. Hdu h6t dirim cria ^BK. Tt ctdy, k6t hqp v6i AE ll BK tac6c ban gui bei v6 tda so4n ldm theo c6ch tr6n. Di6u + . . ,,Drrhtthen ch5t cra bdi to6n rd gia sr (a-rXa-r)'; ; H::#rHX',fl;;ir$T# flHliffii}l')suy ra b6t d5ng thirc (5). C6c bpn sau dAy c6 bdi gi6i th6y tir gi6c EBFK ld tu gidc AiCu trOa. Do d6t6t:

Hn NQi: Vfi Drbc Vdn, l0 Toiln l, THPT chuy6" tjl, tuy6n.t4i B', K cila dudng trdn (o) vd EF

DHSr He N6i; vinh pni", iri"ia"'uii uiZr,'gi-, ddng quy (l). Ctng ti EBFKId tu gi6c rli6u hda

THCS Vinh YCn, TP. Vinh Y6n; HungYiln:,Nguydn . , KA BE KE BA ----. _.iii" uri"i i;;;i" vi€t E*c, Duong H6r; iry:, tac6 urv=ffi= KF-= BC: "uv raABCKlitttr

10A9, THPT Duong Quing Hlim, Vdn Giang; BIc *.rNinh: rd Huy cuong,ll To6n, Turr cnuycn IIJ girlc tli€u hda' do vfy ti6p tuy6n t4i B' K clfi;a

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

!d: ti,v:"Jryhg Nhat' ttrl' THPT chuYcn,Hd tuytin cua rluong trdn (o) (dpcm). DTinh; Binh Dinhz Trdn Vdn ThiAn, 10 To6n, THPT --'J --- -

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:qieuNguyEn Gi6p, TP. D6ng Hdi; Long An: Nguydn LQc hda, hdng.dii5m -. chtm tli6u hda, tinh chAt duongpi,ii, plrq* Qudc Thdig,l0T1, THPT chuy6n Long tt6i trung ho{c bi6n ddi g6c, ... Xin n6u t6n nhirng

An. b4n c61<ri gi6i ggn hcrn c6:

NGUYEN ANH D TNG

BiliT71452. Cho tam gidc ABC cdn tai A, n6iti6p &.rdng trdn (O). D ld trung didm AB, tiaCD cdt drdng trdn (O) tqi E. Kd CF // AE vdi Fthu6c (O), tia EF ch AC qi G. Chtrng minh

rdng BG ld ti€p tuy€n cua dadng trdn (O).

Loi gidi.

G

Gqi H ld giao tlitim ctra AF vd CE. Ta thi'yAEFC 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

Hn NQi: Hodng LA NhQt Titng, 11 ToSn 2, THPTchuy6n KHTN, DHQG Hd NQi, Vil Dtic Vdn, 10

To6n 1, Phgm Nggc Khdnh, l0 Toan 2,_THPTchuyCn EHSP He NQi; Phri Thgz Nguy€n DricThuQn, l0 To6n, THPT chuy6n Hing Vuong; VinhPhircz Hodng Vdn Hi€u,gE, THCS Vinh Y6n, DdVdn Quy€t, Hd Hftu Linh,l0Al, THPT chuydn VinhPhirc; BIc Ninhz NguyAn Vdn Tdm,10 To5n, THPTchuyCn Bic Ninh; Hii Duong: Hd Minh Hodng, L0

Torlrr, THPT chuy6n Nguy6n TrSi; Hung YCn: TriQu

Ninh Ngdn,10A9, THPT Duong Qudng Hdm; ThiiBlnhz Trdn Quang Minh,l0Al, THPT D6ng ThpyAnh, Th6i Thlry; Thanh H6a: Vfi Duy Mqnh,l0T,THPT chuy6n Lam Son; NghQ An: Nguy€n H6ngQu6c Khanh, 10A1, THPT chuy6n Phan B6i Chdu,TP. Vinh, Nguydn Philng Thdi Crdng,l0Al, THPTTh6i 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, THPTchuy6n Hd finh; Quing Blnhz Trdn Nam QuangTrung,l l To6n, THPT chuy6n V6 Nguy€n Gi6p; DnNing: L!, Phudc C6ng, Trin Nhdn Trung,.10A2,THPT chuy6n LC Quf Edn; Binh Dinh: Trdn Vdn

ThiAn, 10 To6n, THPT chuy6n LC Quf D6n; KonTum: Nguydn Hodng Lan, llAl, THPT chuy6nNguy6n Tdt Thinh; Long An: Phqm Qudc Thdng,10T1, THPT chuy6n Long An; CAn Tho: NguydnTrdn H{bu Thinh, l0Al, Trin Minh Nghia, 1141,THPT chuy6n L;f Ty Trgng; Ddng Nai: Cao DinhHuy,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


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 tri nhd nhat c't)a

hittn s6 ,f (.r) : ,(rn -,-+ ta, "r + Jcos -r + crltr .

-:Loi gioi. (Theo da so cric bqn)Di6u kiQn dd cdc bitfu thtc ciaflx) c6 nghia ldsinx, cosx kh5c 0 vd ctng d6u. Nh6n x6t ringkhi sinx, cosx kh6c 0 vd cirng d6u thi

f (x) =fian x(1 + cos r) + Jcot.xtt + sinxl

> Jlt*E(l -F;4 + J[*t {n j.i-,]i.Viftr) ld hdm tuAn hodn chu kj, 2t kh6ng m6ttinh t6ng qu5t, <16 tim gi6 tri nh6 nh6t cira hdm

! . ( :n)sd ta x6t x el t| j. Eat sinx : - a, cosx: - b

\ o)

vd a * b: m thi a,b e (0,1) vd 1< m<$ Oau

1J " rr. ,.-"rZ,^,,.--5n .rDangxayraKnl a=D= 2 nay x= 4. laco

/(,)= w-q.E(,-,).1

f' tx) =,-l- - <a + b) +2,!t + ab - A + b1- at)

I -ti-- , -(a+b\+2l1la+b-r)21

=J;-\a+b)+Jl@+b-t)al)

.)

=-4,*(e-r)m-Am, _I \

fo=l -2*(2"D -2)tm+ttlm'-l '

+(zO +z)@-rt]-(:JZ +r)m++-nD rg.

Ap dr,rng U6t Oing thric Cauchy cho ba s6 tro.rgm6c vudng, ta dugc

I-4, +(zl1-2)trz+ l)+( zJi +z\ m -ttm'-1l't*t; i\l2 -2) w + tt(zl1 +z) <,t -L) = g

121

t;D6u ding thuc xdy ra khi a = O = lf .

M{t kh5c, do 1< m< J, n€n

-(:J7+ t\m+a-Jz > -(:J2+t) ,[1+a-JT

= -2-2J2 (3)

D6u bing xby rakhi a = b =t4

T rhc (2) vd (3)

vio (1), ta thu duoc f'(x)>4-2O hay

fQ)>-Ji-iE. Ki5t ludn: min.f(;r) =J41O, a. o

d4t ilugc 14ri q=b=1, hay sin.r =mslz 2'nrc ld x =!! *21rn. k eZ. tl

4DNhQn xit.Da s6 c6c b4n chi x6t trudng hgpsinn > 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 ThiNhdi, l0Tl, THPT chuy6n L€ H6ng Phong, Th6iBinh: Trdn Quang Minh,l0Al, THPT D6ng TlrtyAnh, Th6i Thuy; Hi frnh: LA Vdn Trudng NhQt,

I lT1, THPT chuy6n Hd Tinh; Quing Binh: HodngThanh Vi€t, llT, THPT chuy6n Vd Nguy6n Gi5p;Binh D!nh: Trdn Vdn ThiAn, I I To6n, THPTchuy6n LC Quli D6n; S6c Tring: Vaong HodiThanh, 11T2, THPT chuy6n Nguy6n Thi MinhKhai; B6n Trez Trdn Thanh Ducrng, 10T, THPTchuy6n B6n Tre.

NGUYEN VAN MAU

Bii "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 llldvo'c: lo bo'i mot lrong ba mdu'. ntdu dr), mdu

trcing hoat' mdu ranh. Chu'ng minh rdng tort tttimot 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

12

dO I (, : 1,.. .,12). Ta co la =48. 56 c[pi=1

di,3m mdu d6 c6 hodnh dq , ie C; _ a,(a, -l)2

Chi€u c6c ili6m mdu tl6 l6n tryc tung thi m5i- -.;.c{p diem mdu tlo c6 hoinh d0 I sC thinh mQt

^ ^ TORN HQCZZ -crua@HÃY ĐẶT MUA TC TH&TT TẠI BƯU CỤC HOẶC VỚI TOÀ SOẠN !

clo4n thing [x, y] v6i ,r,ye N., x1l2,y<12.

Sti c6c dopn thing [.r,y] (x,y e N*.x <12. y <12)

ld C?2 =66. T6ng s6 c6c c[p di6m mdu cl6 dugc

chi6u rd Ze =**,,' -+2,, =iL": -ro

/ t2 \)lF" I

,t\A-'' ) _24=, .o&.t _r4=rr.-2 12 2 12

12

Vi IC; >72>66 n6n t6n tpi hai cflp clopn

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. trYNhQn xit. Trong s6 c6c b4n tham gia giai bdi ndy

chi c6 mQt b4n gi6i kh6ng dirng. Cbcb4n sau dAy c6

loi gi6i t6t.Hn NQi: Vfi Ba Sang, 11 ToSn 1, THPT chuy6n

Nguy6n HuQ; Hung Ydn Vil Minh Thdnh,l I To6n 1,

THPT chuy6n Hrmg Y6n; Phri Thqz LA Bao Anh,10To5n, TIIPT chuy6n Hr)ng Vuong. Vinh Phric: DdVdn Quy\4 10T, THPT chuy6n Vinh Phirc; NamDinh: 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

lhicundrt ; .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)

=4"64671(mod 9)= 4 (mod 9).

Do d6 vbt t : l, 2, 3 <ldu khdng thoa min (1).

V6i r - 4, ta co: 2065 = 103 + 103 + 43 + 13

= 20652014 = (206567 1 )3 .2065

=(2065671)3.(103 +103 +43 +13)

= (10'2065671)3 + (10'2065671)3

+ (4.2065a" y + (1.206561 | )3

= x? - x; + xl - xl v6i x, =10.2065671,

x2 = -10.2065671, x, = 4.206567t , x+ =-2065671.

YAy t:4 ld s6 nh6 nhet cAn tim. EYNhQn xit. Bitito6n kh6 o ch6 cAn chi ra s1r t6n tai

c6c s6 nguyCn x1,x2,...,x. th6a mdn tling thtc (1).

Chi c6 hai b4n tham gia gidi bdi ndy vd ddu c6 loigi6i dring. T6n cria hai bpn ld: Hir NQi: Zfi Bd Sang,

l1 To6n l, THPT chuy6n Nguy6n Hu6. HA finh:NguyAn Vdn Th€,11T1, THPT chuy6n I{d finh

TRAN HIIU NAM

Bii 'tl 11452. C'ho u ld so ngtr.t'tn tltro'ng vu dcil'

.t,i ( t ) rtic dinh bd'i .r,: I lr)

- l'\' --u1 +')ir + l' vri e \1"'

Xac tlinh u de dd.v s, 1-r,,) t:o 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

e I,l+a + z - Ji)' -2{JTi +2 - O) a + 2a + r> 0

e 4a+2-2J1.'f{a +z +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 > 4,s-3-1,5 =ovd 4J2a.+l *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> 2.2, 6.(3.I, 4 + 2) - 23 - 6.1,5

=5,2"6,2-32=A,24>0.

Nhu vfly vti a e N*, d ) 3, x, kh6ng x6c dinh.

Xet ae {1,2}. Tac6

x2 -Zax+2aa1=(x-a)2 +Za+l-i >2a+l-a2 >0.

Ki hieu

f (x) = ^t7 + rax +2a + | - ^l-7 -rax + ra+ |

= "!(* + af +2a +l- a2 -.,!u - of +2a +l- a-

+2u+l

. TOflN HQC ..s6 456(6-2015) & GTudiVil Z5


Ta thdy ngay f (x) > 0 Vx e R*

tdt ch cdc sO thuc ducrng) vd

xwng cila A, qua AB, AC. A"ld trung di1m cua

AoA,.. Cac didm Bo, C, dactc xac dinh tLrong tl,t.

nhu di€m Ao. Ch*ng minh ring cac diem

Ao,Bb,C, thdng hang.

Lai girti.

Ta cAn c6 m6t bd de.

g6 06. Cho hai bQ ba di0m thing hdng Ar, Az,

A3Yd.81, Bz, Bssao cho 4o' =14=fr. N6u Cr,44 8,4

Cz, Cz theo thir tu ld trung <litim cria A1By, A282,

9=n.CrC.t.rat ocrn glan,

Tro lai gi6i bdi toinTl2l452 (h.1,h.2).Goi As, Bo, Co theo thf tU ld hinh chi6u cira l,B, C tr€n BC, CA, AB; B', C' theo thir ql lddiem dOi ximg ctra B, C quaAC,AB.Vi phdp dOi xung tryc b6o todn su thing hdng

;. _ ^ A^B A,B A^B'Va tl SO OOn nen ===! =* =

AOC' A'C A,CA

Hinh 2Tri d6, 6p dsng bO eC tr6n cho hai bd ba cli6mAb B, C' vd A", B' , C, chtr ;f ring A,, Bo, C6 theothir t.u ld trung di6m cta AtA,, BR', CC',

(1R.. lA tap hqp

x+af'(x)=

,l@+o)'+2n+l-az

V1

r-------.-----',lQ-a)'+2a+l-a2

tl

>0 V;elR*

17 +r;+G ,re IR* lir

hdm s6 tdng.

M[t kh6c Vx e ]R*

f (x) : ^{?;za:t + 2a + | -,1"? -2ax + 2a + |4a;1-=4a.

,lx2 +Zax+2a+l+^lx2 *20ca2a+l ' x

Ap dpng ta c6 xr=l<L.=^[4oa2-"12 @e{1,21),

J@) li, hdm s6 ting (do f'(x)> 0 Vx e IR.* ), suy

ra f(xr) < f(xr) > x2 < x3 ...

Nhuv{y 1:x, ( xzlxz<...<4a.Tri d6 suy ra lx,li=, h day s6 c6 gi6i hpn hiru han.

K6t luan: C6 hai s6 nguydn ducrng th6a m5n biito5nld aell,2l.Chf 1f: DEt c = lim -x,. Ta c6

f (r)=t oJTlz**zo+t-,[f -2or+2sq =,4ac

x-a

-L

Al83th\ Ct, Cz, Qthinghdng vd

Phdp chimg minh b6 dO tr€nkh6ng trinh bdy 6 ddy.

,l c2 +Zac +2a +! + rf c' -Zac +2a +le J7;2ot +2a +r +,[7 -zo, +zo +r : 4o

12^[] a2oc a2" a1 = c + 4a * 3cz = l6a2 -8a - 4/.----=-

= c - *tl4a- -2a-1. JVJ

YNhQn xit.Ddy ld bdi to6n gi6i hpn ddy s6 lp, hay.Ngoii bpn Phudc, hai bpn hoc sinh sau cfrng c6 ldigiii t6t:

Hi finh: L€ Vdn Trudng NhQt, Nguydn Vdn Th€,11T1, THPT chuy6n Hi TInh.

NGUYEN MINH DIIC

BdiTl2l452. Cho tam giac ABC. Dudng thdng

A kh6ng di qua A, B, C cdt BC, CA, AB tha twtqi At,Bt,Ct. Ab,A,theo thn tw ld di€m diii

. 2a +l- a2l*-__.._

Hinh 1

^ . TORN HOC24 ;*"3i[a se "'CI-"'oHÃY ĐẶT MUA TC TH&TT TẠI BƯU CỤC HOẶC VỚI TOÀ SOẠN !

o Bo ArBsuy ra A, e BnCni L =+=." A,Co ArC

Tuong W BueCoeo; W=4 ,aBu1{ BrA

l6n trong 6rg) . Vi lgc k6o rj'lQ v6i dQ cao hcira cQt nu6c n6n luc k6o trung binh li:T,o =01,P =*. uA, c6ng thgc hi€n <luoc khi'o22"kdo cQt nu6c c6 chi€u cao ft ld:

D1A, =!-.h =ld .S.h.h' z 2'

1

= :-. 10000. 0,01 . l0' l0 = 5000 (J).Z

Khi pittdng d?t khdi cQt nudc thi lUc kdo ltcndy cdn bdng v6i 6p lgc ctra khi quy6n t6c dpngl6n m{t tr6n cira pittdng. C6ng thgc hi6n tronggiai do4n ndy ld:

4 = po.S(H - h) = 105.0,01(15 - 10) = 5000 (J)

Vfy cong thgc hien tdng cdng ld:

A= 4+ 4.=5000+5000 = 10000 (J) = 10 (kJ) il)NhQn xit C6c ban c6 loi gi6i dirng:

Quing Ninh: I/# Hodng Y€n, LlA7, THPT U6ngBi; Thii Binh: l/96 Vdn Khoa, I lA2, THPT B[cD6ng Quan, H. D6ng Hrmg; Bic Ninh: Nguydn DdcNam,ll L)r, THPT chuy6n Bic Ninh.

NGUYEN XUAN QUANGBitiL2l452. Cho doqn mqch diQn xoay chiiugim di€n trd thudn R, cuQn cam thudn cd dQ tacdm L va tq diQn c6 diQn dung C *ii" ndi ti6p.

Trong d6.cdc dai lagng.(R, C. a\ kh6ng doi cdnL thay d6i duqc. Bi€t rdng khi L : Lr vd L : Lzth! thiiy diQn ap giira hai ddu cu6n cdm thudn Ud€u nhu nhau. Xdc dinh L theo Ll vd L2 d€ di€n

. -;dp hai ddu cuQn cam dgt cqrc dgi.

Ldi gi,rti. DiQn 6p gita hai diu cuQn cim thuAn

Uy: IZl : (chia ci tu vir

mSucho Zy).Tac6:UL: ,=, y:' = -

U

6 )UL^*=.,;'

R2 +Z? 22.f,i f,: ,,=.- .-1.

'ZiZc

Ntiu dat , : + thi hdm y co dpng tam thric b4c'zzhai y : ax2 + bx * c c6h9 s6 a : Rz + ZZ > 0

bi _l-in khi -ro =-;. Khi tl6:

C, eAnBn;4=Yc,Bo c,B

Chf )i ringA6 Br Ct thing hdng vd theo dinh liMenelaus, ta c6

48" !{" 9e=48.!t-.2=r.A,Co BbA\ C,Bo Ap BlA Cp

Tri tl6, lpi theo dinh li Menelaus, suy ra Au, Ba,

C"thbnghing. O

YNhQn xit. Ddy ld bdi to6n hay nhrmg kh6ng kh6,tuy nhi6n rdt itb4n tham gia gi6i, xin n6u t6n c6c

bpn c6 ldi gini dfng: Hi NQi: Hodng Anh Qudn,10T1, THPT chuy6n EHSP Ha N6i; Hrmg YGn:

Dmtng H6ng Son, 10A9, THPT Ducrng Qu6ng Hdm,Vdn Giang; Thanh IJo6: Vil Duy Mgnh,l0T, THPTchuy6n Lam Son; CAn Tho: Nguydn Trdn Hibu

Thinh,l0Al, THPT chuy6n Lf Tp Trsng.

NGUYEN MINH HA

Biti Lll452. M6t xylanh ddt thdng dtmg, ddudadi" chim trong nrdc, ddu ftAn th6ng vcti khiquy6, Trong xylanh c6 mQt pitting dfu nAm

tuAn mfit nadc. Kdo chQm pittdng l€n dQ cao11 : 15m. Tim c6ng kdo pitt6ng. Cho diQn tichpittdng ld S 1dm2, dp sudt khi quy€npo: 105 Pa. Bo qua khiii lwong pitt6ng.

Ldi gi,rti. Ggi D, ld kh6i lugng ri6ng cira nu6c,

P ld trqng lugng cira cQt nu6c trong 6ng khi k6opitt6ng tu ngang mit tho6ng cta nu6c l€n tdi c16

cao liL H: 75m.Khi k6o pittpng l€n tr6n, do t6c dpng ctra 5psuSt khi quy6n nudc sE d6ng l6n trong 6ng theopittdng cho tdi khi 5p su6t do cQt nu6c gdy ratai <liOm ngang m{t thoSng cdn bing v6i 5p khi;. ., .quy6n thi ngimg. Ta c6: po=pr (p,ld 6p su6t

do cQt nu6c gdy ra)

) po : d,,.h t h = t= *ffi = 1o (m)

YSy ta c6 th6 chi hirt dugc cQt nu6c c6 chiducao t6i da hbing I\m.Do cQt nu6c tbng dAu ndn lyc k6o cfing tlngddu tu 0 d6n P (P ld trqng lugng cQt nudc ding

te nr.,.-roro T?3I#S 25


R2 +ZAZ, =---=:+ Z;.Zs:.2,

Khi Urr : Urz€ IlZy: I2Zp

lirZ rzc - zZlrz rrzc + Z?(?z

: Zizzrzc -z12zLpc + Z?{?z

o 7?fl1 - 2trZrz : 7?rz, - zZlzzu

e zr(zf1 - tr) = 2271272(zLr - zL2)

o Zr(Zs + Z7) =2Zy1Zy2.

e (J 1^* *,, tC.+)=+=r=ffi a

YNhQn xdt,}/rgt s6 ban c6 loi gi6i dring vi ch[tchE: Bic Ninh: Nguydn Ddc Nam, 11 Lli, THPT

chuy6n Bic Ninh; Nam D!nh: Phqm Ngsc Nam,

ll Ly, THPT chuy6n L6 H6ng Phong; Th6iBinh: Ng6 Vdn Khoa, 11A2, THPT Bic E6ng

Quan, huyQn E6ng Hrmg; Bic Giang: fti Thanh

Titng, Ng6 Tudn Hitng,12 A6, THPT Tdn Y6n Sfi

1; NghQ An Hi Thanh Titng, Hodng Nghia

Vuctng, 12 Cl, THPT Kim Li6n, Nam Dhn.

DrNH rHAr QUi'Xn

R2 +Zf

U.2,,

^,lxz +27 - 2ZrrZr+ z2r,

U.2,,- 2ZrrZ, + Z2r,

2",Rz +Zl. - 2Z,Zr. + Z2r,

2",R2 +22, - zzr_2zc * Z'r,

Thay (1) vdo (2) = 2,.2" -2Zr.rZr+22r,

(2)

2",

2",ZrZ, -LZL2Z,*Zr,

TINII CH/4T SO HQ{ ". lTiep !ltcr; lruqg t,:\

tt^l I \Pl " Tim tAt cA cics6 nguyOn ducrng m, n saocho da

thric flx) : ,' - nJ '+

*, - ni c6 t6t ch cic,;.

ngnlem oeu nguyen.

2. Cho da thfc Q@): x3 + l9x2 + 99x + a, aeZ.Chrmg minh ring: V6i mqi s6 nguy6n t6p > 5, trong

c6c s6 nguy6n QQ), Q0), QQ),....,Q@ -1) kh6ng

th6 c6 hcrn 3 s6 chia hi5t chop.3. X5c dinh c5c tla thfc P(r) c6 bpc nh6 nh6t v6i c6c

hQ s6 nguy6n kh6ng 6m sao cho v6i m6i s6 nguy6n

ducrng r th\ 4' + P(n) chia h6t cho 27 .

4. Cho pt, pz, pt,pa Id b6n s6 nguyOn t6 ph6n biQt.

Chrmg minh ring ktr6ng t6n t4i da thric Q@) bdc 3,,^

^co ne so nguyen tnoa man:

lQ(p,)l : lQ(dl : 1Q(pt)l : I Q(p il : 3.

5. Cho n eZ*. Tim s6 tu nhi€n kb6 nhdt sao cho

t6n tpi da thirc dang P(x): xk + a,,xt-l + . ..*ar fi tat eZlxlchia hi5t cho z v6i mgi x eZ.6. Cho da thric P(r) : ax3 + bx2 * cx * d eZlxltrong d6 a, b, c, d eZ, a chia h6t cho 3, b chia h,5t

cho 3 vd c khdng chia h6t cho 3. Chimg minh ring,. i. tvoi mdi so tq nhien n 1u6n lu6n t6n tai s6 nguyCn a,

sao cho P(a,) chia h6t cho 3'.7. X6t da thric:

- - TOnN HQC/0 - sli,ldiuc s6 456(q-2o15)

P,(x) = C] + Clx + Clx2 + ...+ C3k+2xk ,

voi n ld mot so 1u nhi6n ,U r =l *)LJ]a) Chimg minh ring

P *t@) : 3 P,*2(x) - 3 P "*1@)

+ (x+ I )P,(x).b) Tim tdt cit cdc srl t.u nhi6n a sao cho P,(a3) chia

f,-tlh6t cho 3'Tl voi moi n>2.8. Tim tdt ci cilc gi6 tri cira n sao cho da thirc:

{ + 4 c6 th6 phdn tich thdnh tich cua hai da thirc

khSc hing s6 v6i hQ s6 nguy6n.

9. Cho p ld sO nguy6n t6. Chtng minh ring da thtc:

fr(r): f -t + f -2 +...+ x + I b6t khi quy trdn

zlx).LOI KET

Bdi todn hAn quan diin da thilrc v6 cilng phong phu

vd hiip ddn, ngodi bdn chtit sii hpc, da thuc cdn

mang bdn chiit hinh hqc vd dqi sii thudn tuy cila n6.

Stb dung tinh dctn diQu , gi6i hqn, dqo hdm vd dithi... ta c6 th€ gidi quy€t dugc nhiiu bdi todn vi da

thac. Tuy nhi€n trong phqm vi bdi ndy ta chi di theo

mQt huong nhu dd trinh bdy, mong bgn dpc trao dtii.

Xin chdn thimh cdm on GS.TSKH Phirng HO HdiTS Trdn Htru Nam dd gdp y aA ni hodn chinh bdi

vidt nay.


Pfr{}#LEM,$ .""lfiip r hea trang. 17}

r-()1l', \RnS \t.\THEM \ |'!CAI( }I,Y\,I PtrAt}

Prerbtrern l'ql4s$. Given positive integersa1,a2,... t a15 satisffing:

a) ar < a, 1...1arr)

b) for each k (k : 1 ,.. . , 1 5), if we denote bn the

largest divisor of ao such that bn<a* then

b, > b, > ... ) br, . Prove that ar, > 2015 .

Pmlrlem 'f l$/456. Given the followingpolynomial f (x)= x3 +3x2 +6x +19:75 .

In the interval [l;3,0,,], trow many are there

integers a such thxfla) is divisible by 320ts ?

Problem 'f l ll{56" Find all injections

/:1R -+ IR. satisfizing :

{fi'{;.\{;D,1.\'{.;1,41 l}1. -S/lI { f iri1, 1111,,t !ttttt! ll\

DNP = DA2 + AW=50 <+( r+3)' +.[f)' =so

It=2e6l0l +440,r-3320=0<)l t66

Lr=- O, (I-oar).

Suy ra D(2: -4).C6u 8. Di€m A e A, n6n A(2 + t;2 + t;3 + 2t).

lor + old(A.e))=r=. a(e,1r1)=6 <> t=2ho{c r = -4.VAy A(4;4;7) hodc A(-2; -2; -5).Ciu 9 Sd c6c s6 thuQc M co 3 cht sd ldA3 = 60. Sri c6c s6 thuQc M c6 4 cht s6 ld

A! =120. Si5 c5c sti thuOc M c6 5 cht s6 ld

A! = 120. Suy ra sd phAn ft cta Mlit60+120+120=300.

Circ tltp con cira E c6 t6ng c6c phAn ttr bing 10

gdm E, = {1,2,3,4}, E, = t2,3,5L 8, = {1,4,5} .

Gqi,4 ld tAp con cin M md m6i s6 thudc,4 c6

tdng c6c cht s5 bing 10. Tir Er lflp cluqc s6 c6c

s6 thuQc Ald 4l tumSi tdp E, vd E l4p rluqc

s5 c6c s6 thuQc Alit 3l

f (x')+ f (y') = (x+y)[,f o(x)- fr(x)f (y)

+ f' (x) f' (y) * f (x) f' (y) + /a g)] for

alt x,ye R.Frohlern Tl2/456. Given a triangle ABC andlet G be its centroid. Clhoose a point M,whichis different from G, inside the triangle. Suppose

that AM, BM, and CM intersect BC, CA, and ABat ,4o,Bo,C, respectively. Choose A*A, on

BoCo such that AoA, ll CA and AoA2 ll AB.We

choose four points Bt,B2, C,,C, similarly. Let

G,G, be the centroids of the triangles AB.C*4BrC, respectively. Prove that

a) A,B, ll BS2 ll Cl2.b) MG goes through the midpoint af GtG2.

Translated by NGUYEN PHU HOANG LAN(College oJ science-Vietnam Nation al Lln iversi ty, Hanoi)

Suy ra sO phAn ff cila A ld 4r.+2.31 =36. Do cl6

x6c sudt cin tinh le P: 36 =0.12.

300('Su t 0. Ta c6 b4 + 16 > 8b2 . Suy ra

o u(- tf \ a(- #\ o( tP)u +16: 16[

I - y *,u.,J t * [' - * .,i

:*-[' -a.,J

. b b( c2\tUonStUtaCo

->-l

l-- l.t'a+16 16[ 8)c c(- a')*.-:

- )-:i l- i. Suyraaa+76 16( 8)A.+-*(ab2 +bc2 +ca2) (l)16 128

Khdng m6t tinh tdng quSt ta gi6 su b nim gitaavdc. Khi d6 tac6 a(b-a)(b-c) < 0, hay

ab2 +bcz +ca2 1a2b+bc2 +abc =b(az +c2 +ac)3b(a+c)2 =b(3-b)2.

Xdt hdm sO /(f) = b(3 - b)2 ,0 < b < 3 ta duoc

J'@)<l(1)= 4 (D.Tn(l)vd (2)taco t2 t -

32'Khi a =0, b=!, c=2 taco d6u ding thfc xiy

5

32 PHANvA.{ rHAr

(GV THPT chuyAn Phan B)i Chdu, Ngh€ An)

se nu. r.-,orur --St##S ZZHÃY ĐẶT MUA TC TH&TT TẠI BƯU CỤC HOẶC VỚI TOÀ SOẠN !

Ta c6

r- J74

r+ Jj4

Ta c6:

(2) e 2cos' a-coso-1= 0 <=) cosd = Y{4 "--""* 4'

vi cosa < o nen "o"o=

t-fr I= 4 SSma-r-cosa

=1_l-fi=l*fi .r^.,-r- 4 - 4 ''qtsin2a=2sinacosa =).1*Jj .1-J1 =''4'4-

cos2a= 2cos' a-t =z ( | - lr\' -1= -f-1. 4 ) 4

^ sin2a 3

=tan)O=-=-cos2a J1'. Ldi giai cua Xudn:Tt gi6 tniet f <a <n

lsina>O 3n lsin2a<O={*ro.o 'u 4 'z'< zTc)

{cos2a > o'

sin2a -3Suyra: tanlo:-=-' cos'Za J1'So sdnh k6t qu6, E6ng vit Xudn tlm mdi kh6ngbi6t vi sao lpi c6 hai k6t qui kh6c nhau, sai dd6u? R6t may bpn Thanh- c6y toSn cria 16p t6icho loi gi6i.. Ldi gidi cila Thanh:

Tu gi6 thi6t: sina+*.r=* =(sina+cos o\' =!'2\4^l 3

= I +sin 2a = 1> sin2a = -:.44Tir si6 thi6t: fI < o < n-3n <2a <2n

2t-

= cos}a > 0 =cos2a =.,/l -sin' ,o =td^ sin2a -3)tanlo=-=-

cosZa J7'Thanh vd Xudn vui mfng, khing dinh loi gi6icira minh ching vi ilimg vdi d6p s6 cira sSch gi6okhoa. B4n D6ngkhing d6ng 1i.

Cilcbany6u to6n, hay girip mQt y kitin cho bi6tai thing? ai sai? Vi sao?

PHAM CONG MINH(TP. Hdi Duong)

(1)

2 a=l (2)

4'

DQC LAI CHO DaNG. Tr€n Tpp chi TH&TT s5 455, th6ng 5 nim 2015 trong mgc Di ra ki ndy. Xin dgc lai dC bdi T61455 nhu sau: Phuong trinh thf ba cira hQld: 323 + x3 = 4x2 -5x+92+8.. BdiT61454 d6t:lh sO tai tnann Bdi T61455, BiLiT71454 thenh Bai T71455,BiLiTl2l454 thdnh BaiT12t455.

Thanh that xin l5i ban <Ipc.

TH&TT

^,- TORN HoCZd -clirdiU@

(. 1

]srna+cosa=7l.z 2 .lsrn 4+Cos a=l


[}['IIFI *}t*,lt\[

n//iru qua, b6o di€n tu Vietnamnet c6 tl[ng bdiy to6n do m$t gi6o vi6n 0 huyQn B6o LOc

tinh L0m DOng ra trong dd 6n tfp lcrp 3:

Diin chin sii ttr nhiAn fu I ddng vdo cac 6 trbng:

Bii to5n dd lan sang tfln Chdu Au. Td bilo TheGuardian cira Anh, trong muc Khoa hqc ggi d6ld, *Bdi todn con rdn", dd tl6ng bdi ci'n AlexBellos_vA cilch giti bii to6n ndy. Bdi vii5t tlugcNguydn Thdo dich tr6n Vietnamnet nhu sau:

"Bdi todn ndy gi6ng nhu mQt bdi todn d6 ddnhcho Khoa hqc mdy t{nh nhiiu hcrn ld ddnh choTodn h7c.Tuy nhi1n, vbi hdu hAt chung ta - nhirng ngudicfia thdi dqi gidy vd bilt chi, tdi sd vid lqi bditodn dubi dang phurmg trinh nha sau:

13h ohtt *L+tl +l)c_ [ _ I l+ ^_ _ l0= 66.

CITrudc khi gidi phrong trinh ndy, hdy xem tiingsA aap dn.cfia bdi todn ndy: co tdi 362880 t6hqp cdc s6 tb I dAn 9 cd th€ di€n vdo 6 trdng.

Quay lqi vbi bdi toan, cfuing ta co thii rtit gc2n

phuong trinh thdnh

l3b olra+'"" +d+l2e-.f +a:87.CI

Ti ddy ta c6 thii gid dinh ring ! ,a 4 td caccl

; ^ :l3bs6 nguydn vd chung ta khong muon f qud lctn.

C6 nhiiu hon m\t ldi gidi nAn c6 nhiiu &t dodnkhac nhau ddn bi kiit qud dilng. Ldi gidi md t6icho ld tnTc quan nhdt thuQc vi dQc gid

Brollachain. Dii :nh6 nhdt, c6 th) anh dy

dd cho b:2, c : l. Tir dd ta duqcoh

a+ d - f +26+ l2e+* =87 hayI

oha+d-f+lZe+6" =61."iVQy cdc iin sd c.dn tqi sd ttr 3 ddn 9, trong d6 c63,5,7 ld cdc s6 nguyAn fi. Nha Brollachain.lQp.luQn thi loqi b6 chilng cdng sdm cdng t6t dO

khdng ldm phtc tqp thAm cdc s6 khdc.Cho a : 3, d: 5 vaf :7, ta c6

oh oh3 + 5 -7 + 12e + * = 61 hoy l2e + e- = 60.it

Cdc s6 cdn lqi ld 4,6,8,9. Lilc ndy ta thu cdc;:.,/an s6 vcti cac so tren thi ductc mQt k€t qud hqpli ld e : 4, g : 9, h : 8, i : 6 vd c6

7)48+-1=48+ 12=60.

6

C6 nhfrng bdi todn yAu cdu bqn phdi soi xdt thQtki, nhang cilng c6 nhftng bdi toan nhu bdi todnndy, chdng c6 cdch gidi ndo khdc ngodi viQc

thu, sai, lqi thir lqi.Cd hai dqng bdi toan diu cd thA khiiin chongudi ta th6a mdn khi gidi xong."cAcu GrAr DAY ou "sAr roAN coN RAN"

Nhirng nguoi y6u thich lap lu{n chflt che ch6ckh6ng th6a mdn v6i loi giii tr6n, vi t16 chi ldmQt d5p s6 chri chua phii tdt cd cbc cl5p s6. VdviQc cho b:2, c: l, a:3, d: 5,"f :7 mangtinh rlp cl4t.

Ta di tim lcri gini d6y dt cho bdi to6n tr6n, tricli tim tdt ci c6c giir t4 nguy€n khdc nhau tu I<l6n 9 cta a, b, c, d, e,.f, E, h, i th6a mdn

t3bc

o*l3b *d+12e-.f +4=87 (l).c"iTru6c frtit, ae chq bii to6n dd phjrc t4p,.ta bdsung th6m vdo dC bdi mQt y6u c6u ld k6t qu6

te nr.,.-roro T?8I#EE 29


c6c ph6p tinh trung gian cfrng ld so nguydn tl6

di d6n nhfn xdt l3b v' 4 46., ld sd nguy€n,CI

do d6 kh6ng phAi xdt c5c trudng hqp nhu:l3b sh l3.l 5.2

-T=l-

=--;-T-- -., C ,I J O

Kh6ng mit tinh t6ng qudt, gi6 str a < d vd h < g.

Do l3b ld s6 nguy6n, mdr (13, c): I nln b : c.

C

,hNeu : > 5 thi c: l, e>2. YOi e > 3 thi vO tr6i

C

cira (1) l6n hcrn 87. Vdi e:2 th\

o *73b * d *ru - r + 4) 3 + 13.5 +4+ tz.z*9 +t =88.c"i.bVay gia tri cura : chi co the ld 2, 3. 4.

C

,.A" Xet " : 2. Khi d6 ( I ) ffo thdnl

c

a+ d +12e - J' * 4 =87 - 26 = 6l (2).I

^. . A lb=2 lh=4 lh=6 lb=B"*""D "'"' lc=l lr:2 lc=3 lr=4.

I. Xet h:2, r: l^Do e*l,e*2 vd kh6ng

th() e>6 n6n eel3;4;51.1) Xdt e:3. Khi d6 (2) tr6 thdnh

oha + d - f +{: = 6l-36 =25. Chc cht nhfln gi6

l,tri 4, 5,6,7,8,9. Ta chri y rdng

oh* =25 - a - d + f >25 -8-9 + 4 = 12.i

Lai x6t ba trudng hqp cria l:

- Xdt i: 4 thi 4 ,t' the ld: $=l6.loai;44* =rr= a+d - f =13, cac cht nhpn gi6 tri4"5,7,g loai; Y =A>a+d-J'=11, c6c chfr

4

nhdn gid tri 5, 6. 9 lo4i: T=,t +a+d - f =7.4

c6c cht nhfn gi6 tri 5,6,7 loai.oh ., AA

Xdt l:6 thi !- c6 the ld: j- =6. lo?i;66T=tr=>a+d-f =13, cac chir nhAn gi6 tri6"4, 5,7 loai.

- xet ,: g rhi g!, =9!= 3. loai.884 Xa e: 4.Khi d6 (2) h0 thanh

lrha + d - f +!- = 61-48 = 13. C6c chti nh6n gi6

Itri 3, 5, 6, 7, 8, 9. Ta chri y rdng:

oh*=13-a-d+.f <13-3-5+9=14. Lqi xet

Iba trudng hqp cira i:

oh- xet l:3 thi s c6 the ld:

J6'5-rA+-r s-r-3 = ,, - a+ d - I =3, c5c cht nh0n gi5 tri

7, 8, g loAi; 4 =14 > a+d - l'= -1, c6c chtJ

nh4n gid tri 5, 8, 9 loqi; * = l6;T = l8;JJ

2l = fi;T - zt;T= 24 dou loai.

oh- Xet l:6 thi 4 co thd li:

6

9;7=+)a+d-f =9,c6c cht nh6n gi6 tri 5,6"7, 9 loqi; 4 = r, ) a + d - f =1, c1ccht nhfn

6gi6tri 3,5,7 th6a mdn voi a:3, d: 5 vdf :7.Ta tim dugc mQt tl6p sti, d6 1d d6p sd trong bdivi6t trdn td b6o cria Anh:

3 *13'2 *s +12.4 -i+ 9'8 = 87.t6

- xdt,:9 thi *=9=2=a+d-t'=ll,99c5c chfr nhfn gi6 ti 5,7,8 loai.3) Xdt e : 5. Kh6ng c6 cl6p s6, bp, dqc t.u gi6i.

II. X6t b - 4, c: 2. Khi d6oha+d-f+lZe+6:=61 (3). Do e*2,e*4I

vd kh6ng th€ c6 e> 6 n}n e e 1I;3;51.l) xdt e : 1. Khi d6 (3) tfo thanh

oha+d-f +*=61-12=49. Cdc chtnh4ngi6"itri3,5,6,'7,8,9. Ta chf y rdng

4=49-a-d+ f>49-8 -9+3=35. Lai xdtI

ba trudng hgrp cta l:

- xdr i:3 thi 4. 2.3 =24.roai.J.'

xdtl: 6thi 4 c6 th6 la: I =4, T=12,.666cl6u lo4i.

- xet , : 9 thi 8!' =!i=z toai.992) Xdt e:3.Khi d6 (3) trdr thdnh

oha + d - f +T =61-36 =25. Cdc chfi' nhQn gi6"itri 1,5, 6,7,8,9. Tachirljrring

- ^ TORN t-{OCJt, ' cT udigA_s(i 4scj6-2ors)


oh*=25-a-d + f <25- 1-5+9 =28. L?i xdt

Ihai trulng hqp cria l:

Xet l: I thi €4, 6'5 = 30. Ioai.ll

- Xet l:6 thi 4=ry=12+a+d-f =13,66c6c cht nhfn gi6 tri 1, 5, 7 loai.3) Xdt e : 5. Kh6ng c6 d5p sd, b4n dqc t.u gi6i.III" X6t b : 6. c: 3. Khi d6

oha+d+l2e-.f **=61 (4). Do e*3 vd

I

kh6ng th€ e> 6 nOn eell;2;4;51.-Yoi e: 1, lQn lucrt xdt i:2, i: 4, i:8.-Ybi e:Z,ldn lugt xdt i: l, i: 4.*Y6i e: 4, lAn lugt xdt i: l, i:2.-Y6i e: 5, lin lugt xdt i : l, i : 2, i - 4, i : 8.

GiAi theo c6ch tu<rng t.u nhu tr6n, kh6ng c6thdm drip s5 m6i.l\'. X6t & : 8" t' : 4:khdng c6 d6p s6, bpn dgctU girii.

hB. \et " - -1 Khi do ( l) tro thdnh

C

oha+d+lze-f+*=87-39=48 (5).

(r.=3 lb=6 lb=gCobatn.rcrnghqp, ]"-l ]"-: 1"-:.

[('= l [('= z L('= J

-Yoi b : 3, c : 1, do kh6ng thd e >5 n6n

e e{2;4} . Vdi e:2,ldnlucrt xdt i: 4, i: 6,

, : 8. V6i e : 4,Tirn lugt xdt i :2, i: 6.

- V6i b : 6, c : 2, do kh6ng th€ e25 nOn

e e{l;3;4]r.Yoi e : 1, lAn lugt xet i : 3, i : 4.Yoi e:3, lAn lugt xet i: l, i:4.Yoi e:4, lAn lugt xet i: l, i:3.-Vdi b:9,c:3,dokh6ngth6c6 e>5 n6ne e ll:'2;4\.Yoi e - 1, lAn luot xet i : 2, i : 4, i : 8.

Yoi e:2, lAn luCI xet i: l, i: 4, i:8.Yoi e:4, lAn luort xet i: l, i:2.Gi6i theo c6ch tuong t.u nhu tr6n, ta c6 th6m ba

d6p sd trong c6c trudng hqp sau:

Ybi b: 3, c: l, e:2, i: 4, chgng:8, h:7,ta dugc a : 6, d:9,f : 5, chgng:9, h: 8, ta dugc a : 5, d:7,.f : 6 vd c6:

6* I 3'3 * 9 + 12.2-5 + 8',7

= 87:l4

5*13'3 * i +12.2-6+ 9'8 =87.l4

VOi b : 9, c : 3, e : 2, i: 4, chgng : 8, h : 7,

ta dugc a: 5, d - 6,"f : I vh c6:

_ l3.g . a15 + :;: + 6 + 12.2- I +',' = 37.->4

C. X6t ?- == 4. Khi d6 (1) trO thdnh

C

oha+d+lZe- f +* = 81-52=35 (6).

4

. lb=A fn:zC6 haj truong hqp { IL('= t l('= z

- V6i b -- 4, c: 1 do kh6ng thti e > 4 n€n

e e {2;3 }. V6i e : Z,ldnlugt xdt i : 3, i : 6,

i:9.Yoi e:3, xet i:2.- Vdi b:8, c:2 do khdng th€ c6 e>4 n€n

e eIt;3). V6i e - 1,b,nlugt x6t i : 3, i : 6,

, : 9. V6i e:3,1dn luqt xdt i: l, i: 6.

Gi6i theo ciich tucrng tu nhu tr€n, ta c6 th€mi ,.

mot ddp s6 rnoi trong truong hqp b : 4, c: l,e : 2, i : 6, g : 8, h :3, do d6 a - 5, d : 9,

l'-'l vitco: 5+ 13'4 +9+12.2-7*ti=37."16KET LI.AN

Ta dugc ndm d5p 16 trCn v6i gih srt a < d vit/r < g. Ni5u COI gia tri cira a vd d, cua h vdg thi tirmQt.d6p s6 c6 th6m ba diry s6 nfia. Do d6 bdi to6nc6 tAt c6 hai muoi d6p s6.

Loi giii d6y dt.cta bdi torin nhu trinh bdy d tr6ntuy phAi x6t.nhidu trudng hqp nhtmg c6 ducrng l6irb rdng. MAu ch6t ctra c5ch gidi ld ph6t hiQn ra

bic vir g/zii.Saukhix6t c(pgiittri cria bvdc,tax6t e tl6 gi6m s6 trudng hqp cAn x6t, r6i x6t ; dtichon g vd /2, cu6i cing xric dinh a, d,f.RO rdng biri to6n ndy vdn thich hqp cho m6n To6nchu khdng phdi chi ddnh cho m6n Khoa hgc m6ytinh.B4n dgc niro c6 himg thri v6i bdi to6n d4ng ndy c6

th6 thu suc v6i "Bdi toltn con! ,...

ran mot :

Diin cdc s6 U nhi€n tir 1 d€n:

6 vdo cdc o trong (cac k,!r quatrung gian cilng ld si| ntnhi€n).Bdi to6n ndy tlcrn gi6n hcrn "Biic6 tat ca b6n <l5p s6.

con rin" vit

Sti ase (6-2015)TOAN HQ( * O

I trudiffA J IHÃY ĐẶT MUA TC TH&TT TẠI BƯU CỤC HOẶC VỚI TOÀ SOẠN !

lqprhi Iffit H0(vuIUdl Tnixufr nAr,t rtJ uo+

sd 456 (6.2015)Tia soan : I 878, ilo' Giens U0, Ha iloi

0I BiCn tap: 04.351216070T - Fax Ph6t hAnh, I1! su:04.35121606

Email: [email protected]*luthemttic md Youth ffiugurine

BAN CO VAN KHOA HOC

GS. T sKH.NcuvEN cAmi roaNGS. TSKH. TRAN VAN NHTING

TS.NGTI\€NVANVONG

cs.ooaNeuirrrPGS.TS. TR]{]{VANIII{O

Tdng bian Qp : TS. rnAN ntlu NllrnQr odxc nfiN r$e

IHIU rnAcu Nrururu xuAr n,4.N

Chri tich HOi d6ng Thinh vi6nNXB Gi6o duc Vi0t Nam

MACVANTItrENTdng Gi6m rl6c ki6m Tdng bi€n tap

NXB Gi6o duc Vict Nam

cs. rs. r,uvaN ulrNc

ThukiTda soan: ThS. ffd QUaNC VfNft

rs. rn AN oNn cnAu, ras. NcrrvEN aNn ofNG, rs. IRAN Neu DLTNG, rs. NcuvEN rrarNu ot]c, rs. Ncuv6,N rrarNn

HA, rS. NGUYEN VIET HAI, PGS. rS. LE QUOC HAN, fl,S. PHAM VAN HU{G, PGS. rS. VU THANH KHIET,GS.TSKH. NGL|\.EN vAN uAu, 6ng NGtryEN rHAc urNg, rs. PHAM THr BACH NGoc, pcs. zs. NcuyEN oANc psAr,pcs, rs. TA DUy pHrfNG, rns. NcuvEN rid rnecn, GS.TSKH. DANG HtrNG rHANc, pcs. zs. pnaN ooAN THoAI,rirs. vO rlrra ruuy, pcs. rs. vO oUONc rHUy, GS. TSKH. Nco vmr TRUNG.

TRONG SO NAY

@ uro,rg d6n giAi - Dd s6'8.

@ Oii5n din phttong ph6p giii to6n

Nguy,n Lxu - Tinh chdt sd hoc ddc tningtrong bdi to6n da thitc.

Problems in This Issue

T1/456, ..., T721456, LL/456, L21456.

@ ciai bai ki trtidcSolutions to Preuious Problems

@ s"i ldrn & dAu

@ ,U thi tuydn sinh vio l6p 10 Trtrdng THPT Phqm C6ng Minh - Ai drlng? Ai sai?

@ Oii5r, din d4y hoc to5.n

Vfi. Hilu Binh - V6"'Bei to6n con r6n".

Anh Bia 1: GS.TSKH Nguydn Van Md.u.

Chil tich HQi Tod.n hoc Hd. N6i ud.dng Ch*Xud.n Dfrng. Ph6 Gidm ddb Sd GD - DTHd N1i (Tht nhdt ud tht mtili mQt tt phd.iqua trdi) tai L6 Tdng hdt ki thi Todn hoc

Hd Ni,i rn6 r6ng ldn tht 12 nd.m 2015.

Vi sao?

@ Cfr.rd" bi cho ki thi THPT Qudc gia

Trinh Ba - V4n dung g6c dd gi6ri c6c bditoan toa d6 trong mdt phing.

@ oarrf, cho Trung hgc Co sd

For Louer Secondary School

Hodng Thudt - Linh hoqt dd dria bii to6n

v6'sii dung hO thiic Vidte.

@ HuOrrg dAn giii DO'thi tuydn sinh vdo l6p 10 @ ,A ra ki niychuy6n To6n Trrrdng DHSP TP. Hd Chi

@ ri" tfic to6n hgc

Thdm Ngoc KhuA - L6 Tdng kdt vd trao gi6ithrl6ng thi HOMC 2015, HOi ddng thi HdN0i.

chuy6n LO Quf DOn, Ninh Thu4n, n6m hoc

2074 - 2015.

Giai tri to6n hgc

Phi Phi - Giai d6p: Di6n so vi.o hinh hoa.

@ ffrrf srlc tni6c ki thi - Dd sd9.


THTT So 456 Thang 06 Nam 2015

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