TOÁN HỌC TUỔI TRẺ SỐ 456 THÁNG 6 NĂM 2015

Tạp chí Toán học và Tuổi trẻ Số 456 (Tháng 62015) gồm khoảng 29 bài viết trong các chuyên mục: Dành cho Trung học cơ sở, tin tức toán học, giải trí toán học, chuẩn bị cho kì thi THPT quốc gia, diễn đàn phương pháp giải toán, diễn đàn dạy học toán,... Mời các bạn cùng đón đọc.

DT Bi6n tap: (04) 35121607: DT - Fax Ph5t hdnh, Tri sLI: (04) 35121606 Email: toanhoctuoitrevietnam@gmail.com website: http://www.nxbgd.vn/toanhoctuoitre

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

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

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

t6 chriE-tqi He NQi trong hai ngity 25 vd261312015 Cdn

cu Quy ctr6 ttri chgn hgc sinh gioi cAp Qu6c gia hiQn

hdnh, BQ GD&DT dd triQu tdp 49 hoc sinh tham dp ki

thi tuy6n chgn n6i tr6n, g6m I hgc sinh dE tham dU IMO

2014vit 48 hgc sinh dat tu 21,50 tli6m trd 16n trong ki

thi chgn hqc sinh gi6i Qu6c gia m6n To6n THPT nim

2015 Trong m6i ngdy thi, m6i thi sinh dugc dC nghi

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).

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

sao cho tam gi6c ABC nhgn vd AB < AC Gqi H \d tr119

di6m clria AH vd BC Tia IH cdt (O) tai K, tia KD cit (O)

tai M Duimg thdng qua M vd vuong g6c voi Ag cit .lt

a) Chrmg minh rdng 1/ ch4y tr6n m6t ducrng trdn,pii dinh.

b) Euong trdn di qua diiim ff vd tit5p xric vdi dulng

thhng AK tai di6m ,,q, c1t,qS, AC l6,n lugt tai P., Q Gqi J

ld.trung di6m cira doan P9 Chrmg minh r[ng ducrng

thdng AJ di qua mQt di6m c6 tlinh.

Bni 3 (7 diA$ Sd nguyQn ducrng k duoc goi ld c6 tinh

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

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" ,

L

a) Tim tat ce c6c s6 nguyCn duong k c6 tinh ch6't T(20).

b) Tim s6eg@6n duongtnh6 *6tco tinh ch6t T(20t\

,\g,iy thi thft hai" 26l3l2t)15

nai + 1Z didm),.,::g6.1gO Jinh vien tham d1r mOt cu0c thi

v6n d6p $.an gi6m,khao:961n 25 thefi vi6n Mdi sinh

vi€n dugc h6i &jiboi mgt gi6m kh6o Bi6t rang.n6i sinh

tr6n.

a) Chimg minh rlng c6 th6 chqn ra7 gi6mkh6o sao cho

m6i sinh vi6n thich it nh6t mQt trongT gi6m kh6o ndy.

b) Chrmg minh ring c6 th6 sip xi5p cuQc thi sao cho m6i

sinh vi6n duo c h6i thi bdi gi6m k*r6o mlr minh thich vd

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

Bni 5 (7 di€m) Cho tam gi5c ABC nhon, kh6ng cAn vd

di6m P nim b6n trong tam gi6c sao cho

FB:;p-c: a voi a > 18oo -6ic c6c duong

AC vd AB tai E vd F Ldy di6m Q b6n trong tam gi6c

AEF sao cho AQE= AQF:a Gqi D ld di€m rl6i

ximg v6i Q qua ducrng thing EF' Ducrng phdn giSc

trong cira g6c EDF cdt PA tqi T.

a) Chimg mirljn DET: ABC, DFT: ACB.

b) Ducrng thing PA cfu Of , DF ldn lucrt.tai M, N Ggi I,

J 16n luot ld t6m c6c ducrng trdn nQi ti6p c6c tam giSc

v6i tdm ld di6m K Duong thlng DT cdt (&J tai di6m 1L

gi6c DMll

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

ii

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

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

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

,T&UNG c{, sd

(/rong di thi vdo THPT lu6n xudt hi€n bdi

?, bdn hAn quan ddn viQc dp dung hQ thttc

dqng dp dqng dagc hQ thuc Vidte, nhwng cfing

cd nhirng bdi,todn phdi rh khdo leo mdi thqtc

hiQn duqc di2u d6 vd no gdy kh6ng lt kh6 khdn

diii vai cdc em hgc sinh Sau ddy lit mdt sd

dqng togn nha vdy vd cach s* dryng h€ thac

Vidte d€ gidi chung

D4ng 1: Phuang trinh bflc hai c6 tham s6

Bii 1 Cho phtrcrng trinh:

8-t: 8x+m2+1:0 (*) ';

md xf -"tl =.ri-"t:'.

NhQn xdt Ta thdy h€ thric dA bai dua ra c6 v6 phfrc

tpp vd g6y kh6 khln khi dua vd xrt xz vd x1.x2

nhrmg ta c6 thrS bien dOi xy, x2 th6ng qua phucrng

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

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 phdn

biQt x1, x2 thda mdn

Ldi gidi A' : (* - 2)' + 2 > 0 lu6n rlirng vdi

mgi m, vfly phucrng trinh (1) lu6n c6 hai nghiQm

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

1

ph6n biQt x1, x2 YOi mQi m Khi d6:

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

!xf

Ding thirc xhy rakhi 4 t : 324xi,

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:2

Vfly 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+l 1r,.*, =2(m-l) ''

xe,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

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

1u6n c6 hai nghiQm phdn bigt.xr, xz li"hi d6

lx,+x,=k-l iY, =xi

lxr.x, = -4 l r= = ,j Vfly:yr + yz: yt.lz a x? + xl = 11.fi

€ (k- t;2 + 8: 16 e k: I +2J7,

a TOnN HQC

Z icruagw

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

Tt d6y tim du<yc nghiQm: x : 4, x : 0 .

VQy tQp nghiQm cira phuong trinh ld: S = {+;O}

Cfru 2 a) 8p -1,8p,8p + 1 ld ba s6 nguy6n li6n

ti6p, n6n c6 mQt si5 chia h6t cho 3 Md Bp - t,

kh6ng chia htit cho 3 Do v6y (Ap)i: Ua

(8; 3) : 1 nOn p i3 Dop ld s6 nguy6n t6 n€n

p :3; p: 3 thi 8p + I :25ldhqp s6.

Vay khdng c6 sd nguy6nt6 p thbamdn 8p - 1,

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

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

qB$} qefffl LE ToNG KET vA TRAo cr qr rmu'Gtic rrxr t.EoN{c 20ts

F{ST SOI{G T}IT E{A NOI

chric ngdy 22/312015 de thdnh c6xg t6t dqp Ngny 17i5l2015 tpi Truong THPT Chu Vdn An, He NQi, Ban chi <Ipo

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;

thuong cta kj,thi HOMC 2015 ld: 449 gibi g6m 45 girli Nh',t, 120 giitiNhj, 155 giiliBavd.lz9 giii Khuy6n khich Bdi thi duqc itt6m tneo thang ditim 15, ph6 Aidm cO tu 0 di6m ai5n i:,S ditim Iftdng c6.dir5m tuyet d6i 15/15 Ch6t

luqng bdi ldm cira thi sinh t6t hcm c6c nbm tru6c, ttr6 nien 0 di6m binh quOn f,uong d6i cao Thri khoa o hia tu6i

Junior c6 I em: Mdn Ddo Son Tirng,THCS Hoang Vdn Thp, Lqng Scrn, 13,5 di6m Thri khoa d lua tu6i Senior c6 3 em' Phqm Kim Anh, THPT chuy6n Hd NQi-Amsterdam, 13,5 di6m; Mai DQng Qudn Anh, THPT chuy6n Ha NQi - Amsterdam, 13,5 itirSm; Dodn bao KhA,THPT chuy€n LC Khi6t, Qu6ng Ngdi, 13,5 di6m Thi sinh dat gi6i dugc

nhQan Giiiy chilmg nhQn cira S0 GD - DT vi HQi To6n hgc Hd NQi, girii thu&ng cua S0 vir qud tflng cria HQi C6c tinh, thenh c6 hgc sinh tham dp lc, thi HOMC 2015 duqc tflng C] luu niQm cta Ban t6 chric Mgi ngucti d€u mong mu5n, nhu, ldi GS Nguydn Vdn Mdu, Cht tich H6i To6n hqc Hi NQi: "Srira md rgng cu\c thi HIMC ra cdc rurdc ASEAI\1'd6 hgc sinh Viet Nam dugc dua tdi cring hqc sinh ciic nu6c trong khu 1uc D6ng Nam A * Th6i Binh Duong.

Th6ng tin chi ti6t xin xem tr6n trang Web cira HQi To6n hqc Hd Ndiwww.hms.ors.vn

THAM NGQC KHUT (Hd Nr.i)

AtB+DKB =90'+90'=180" =ru giitc DTBK ndi

ti€p > ADB = HKB Ta co BAD = BHK (vi cung bir v6i BCD ) =MBD,n U{BK(e.E)

AD AD AB zAM AM

HB HK HB 2HN HN

AD Ai'MAM vd MHN c6: 'BAU = 91151.!! =!l!

li6n ti6p Vi ba dinh 1i6n ti6p U6t tcy c5c sii kh6ng

blng nhau n6n chi c6 hai lopi tich:

i .:

ba so ndy b1tng 2 Logi II: Ba s6 6 ba dinh 1i6n ti6p

c6 hai sii 2,tichbas5 ndy blng 4 .

50 -x Md s6 s6 2 6 50 tich c6 ld 30 3 : 90, ta c6

phuong trinh: x.1 + (50 - x).2 =90<+ .r = 10.

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

DP -rc =OE -U2 (l) Chrmg minh tuong tU

cfrng c6: W -W =OC -O,4 (2) vir

(oe -x:)+(n: -w)+(FN -Fr,)=o()

(4).VC OD'LBC tqiD' Cdn chimg minh D'=D

n& rm ruv*nr slffi v*rlr*p r o

Yrt$nrg Tl{trTshu;r€n L€ Auy S n -,T'lxhNinh,Shu$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

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

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

x'+7 Cflu 3 @ diefi Cho tlucrng trdn (O) tluong

kinh AB vd cludng trdn (Q di dQng lu6n ti6p

xric trong v6i nira tluong trdn (O) tai C vi ti6p

xric v6i doqn AB tai D; CA vd CB lin luqt cft

ACB vd CD diqua mQtiliOm c6 dinh E

c) Chung minh ring clu<rng trdn ngoai titip tam

gi6c ACD ti6p xric voi AE tqi A

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

n+5

ld s6 t.u nhi6n H6y tim tdt cb chc sO t.u nhi6n ,t

trong khoing tu I d6n 2015 sao cho phdn siip

, 1, "

CAO TRAN TtI TTAI

(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) DAt

vdo (3) tim duo c n:26 vd p :65 (4) Thay (4) vdo

YOi e: 18, g : 16 X6t hrcrng t.u h6n thi t6n t4i

ddi ximg ld tludng thlng di qua !6m c6c 6 trdn ghi

c6c s6 15, 10, 9 V4y bdi todn c6 th chb6n nghiQm.

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

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

flfrt nruGo;{n qenDg

qRoDG 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 lJ

U ;;;: 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.

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 g6c

nh6 nhat trong hai c{p g6c ddi dinh, vfy g6c

gifia hai cluong thing ld g6c c6 sd tlo kh6ng

vugt qu6 90"

3) Neu hai cludng thing a, b c6 vecto ph5p

tuyiin (VTPQ Dn luqt ld i,,i, tti

l;;l

/ ''\- | ' 'lcos(O'Dl =

l;m' | 'll 'l

a) N€u hai dudng thhng a, b c6 vecto chi

phuong (VTCP) lin luqt ni,,i, tni

OThi du l (Trich Cuu - rt'rl tg dt: tlti mitr t trrt

Bd GD&DT ndm 2015).T'rong mdt phdng v6'i

he truc tea clo Ox.v, cho tam gidc OAB c(t cdc

clrth 4 ra lJ rhuot clrrotrg thing

.\ : 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

\) 5)

Gqi d ld <lucrng trung tryc cria do4n CO th\ d di

qua 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

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 cila

i{o4tn AB, ll lu diint thuit' tlrsun AC scro t:ho

,4,V=3,VC L'iOt PT tltftt'tts lhing Ct), biit

rin,q M (l;2).,v(2: 1)

PhAn fich

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

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

ring HP, MP d6u tinh duoc /

theo a (a la do dai canh cua hinh

vu6ngIBCD)

Gqi P : MN O CD; Hlddidm

Dd6i xrmg cua M qua t0m cira

hinh vu6ng ABCD Taco

eb-0 holc '1 b=aa.

y6i b=0, chgn a:1 thi i=(t;O), cluong

a

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

4

ft t _!_+4t thdngCDc6PTlei"-3'-'

lY =-2+3t

SThi 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 ,',lnli

BC', 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 de cho phucrng trinh ducmg thing AN vir t1a d.Q di6m Mn6n ta nghi d6n viQc xemA= AM nAN

(g6c gita hai ducrng

thdngAN

ding ilinh li c6sin vi tinh dugc AM, AN, MN theo a.

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

Ldi gidi (h.3)Ggi a ld cpnh ciia hinh vu6ng, A

l^ ^\

ta c6 tan I DAN + BAM I

\/

tanfIfi + andifu 1- tan 6TN tandTfu D

khid6 a=friH+cos0 =HP MP =1.

Jto GSi i=(a;U)+O ld vecto chi phu<rng cta

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

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

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

Hinh 3

sti ase re-zorsl

-3s#8[ z

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

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) \3 ) n*

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 hai

ducrng thingAC,AD)

C6 phuong hinh iludng thdng BD ta tim duo c tga

t10 t0m I =AC aBD cua hinh chii nhft ABCD vir

tgad0tli6m D=ADaBD.

Tri d6 6p durrg c6ng thric

Ldi gidi (h.a)

Tqa d0 di6m A ld

nghiQm 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 vrPr

cira <ludng thing BD, ta c6 VTPT cua AD

,*=(t -t)non cosffi =S-=a .{aiq,[a 6

e5(a-b)' =2(a'+b') e3a2 -lUab+3bz =0

NOu b:0 thi a:0 (loai)

.Ybi b:3a, chQn a=l:)b:3, dudng thing

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

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 ABD

c6 trac tdm ld H (- 3;2) Tim tqa do cdc dinh C

vd D

PhAn fich

Trudc ti6n ta xem x6t tinh d4c biQt cria cilc tam

gi6c IBC,BHC

.Taxem D=ADaBD Vi6t phuong trinh duong thing AD qura A tqo vbi

AC mQtg6c bang fiE =fdi = 45'

AC,tac6 cos6;=]59 =+.

Jl+E.JI+l J5'

.r TOAN HOC

E icru,ufw

i&=4f Mat kh6c Hinh 5

BH L AD > BH L BC , suy ra tam gi6c BHC

vudng cAn t4i B, suy ra I ldtrung clirim ci,- HC

Dutrng thlng AC <li qua 11, vu6ng g6c v6i

duong thing BD, c6 phucrng tr\rth 2x-y + 8:0

Toa clO cua I ld nghiOm cira h0 phuong trinh

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

1

lx+2y-6 0 Iy=+

Tu 1 ld trung di6m cria dopn HC suy ra

C(-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 lit

IDE =idi =45" Suy ra

, E^ la +2bl la +zbl I

C^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)

ld toa d6 vecto'phfp tu5r(in cria AD, ndn AD co

phuong trinh 3x+y + 17 = 0.

[3x+r'+17 =O € I I x=-8 _ _- = 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 c6

phuong trinh x -3"y - 1 = 0.

Toa d6 cia D ld nghiQm cua h0 phucrng trinh

f6t tu4n Qua c6c thi du tr6n, chtng ta th6y vi6c

dQ.Oxy di qua mQt di6m cho tru6c vii tao voi rluOng

thlng cho tnr6c m6t g6c c[, c6 th6 khai th6c d6 gi6i

nhi6u bdi to6n chi cAn ta tinh rlugc g6c cr 116 ViQc

tim toa dO m6t di6m du6i dpng ld giao rti6m cira hai

di6m khri hiru ich Sau ddy xin gi6i thieu m6t s5 bai

d€ ban itgc luyQn tpp.

BAI LUYEN TAP

L @a thi DII tcnAi O ndm 2Ol4) Trong mdt

phing vdi hC tnlc tga d0 Oxy, cho tam gi5c

ABC co chAn du<rng phAn giSc trong cua goc A

ld 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 AB

vd PTTT tai A cua Clucrng trdn ngoai ti6p tam

gi6c ABC O6 tim toa clQ di6m A

Vi6t PT ducrng thing AD,tinh g6c @f;7

Vi6t PT dudng thing AC hqp v6i AD m6t g6c

; .* -bdng IAB,AD).

+

Tinh goc (an,ls)=(Ce,cn).

Viiit PT duong 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 ing v6i hd truc toa dd Oxy cho hinh chft nh6t

ABCD c6 diOn tich bing 15 Eucrng thing AB

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 BC

suy ra d0 dei canhAB G

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

9

THuslJc rpuoc xi mu

pEs6g

(Thdi gian ldm bdi: 1 80 Philt)

Cilu 1 (2 di6m) Cho hdm s6

!=mx3 -3mx2 +3(m-l)

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|tri

qia mae Z,qn'-(O,q' +OB2)=98

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

4sin'? | - J3 cos2x =2 - sin2x.

C6u "l (1 ai6m1 ZiaihQ phuong trinh

Cfiu 6 (1 diAd Cho hinh ch6p S.A,BC c6 d6y

ABCliLtam giric vu6ng t4i A, AB=3a, BC =5a; mdl

phing (SAC) rudng g6c v6i mat phing (ABC) Bi6t

ring SA =zali va SAC=30".Tinh,n* ?,T::l

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=0

ring m{t phing (a) cit mat cdu (.$1 theo mQt dutrng

trdn X6c dinh tim vd b6n kinh cria dudng tron d6.

CAu 9 (0,5 di€m).Tim si5 ha.rg kh6ng chta x trong

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

Ciu l0 Q diAfi Cho x y la c6c so thuc ducrng th6a

mdn x+y S 1 Tim gie ftl nho nhAt cua bieu thuc

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

"

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, ru6ng

g6c v6i AB, suy ra tga dQ B, suy ra tga dQ D (ddi

ximg v6i -B qua tdm 4 Cht lf gia thiCt di6m B c6

tung dQ lon hcrn 3.

3 Trong m[t phing v6i hQ truc tga dQ Oxy, cho

hinh vu6ng ABCD.Dicm Ff+;:'l 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

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

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 HOC

I 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

hodcy =7x+31.

CAu 2 a) Phucrng trinh d5 cho tuong du<rng v6i

sinx+ J3 cos* -2 erir[r*1) = r

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

Ggi cpnh hinh vu6ng ld a Trong tam gi6c

SISffiF$ frleffi cnfrr s0 xoc Dnc rRtrNE

Biitqof,nDnqwft

Oi " thwc ld mAt nAi dung riiy quan trpng,trong

U chwnng trinh todn hpc phii th6ng vd qat aiy

dagc gidng dqy trong chucsng trinh dqi s6 6 cdp

THCS Cdc bai tudn hAn quan d€n da th*c xudt hi€n

nhiiu trong cdc k) thi hpc sinh gi6i Quiic gia vd

Qu6c t€ Bdi todn v€ da thuc ld bdi todn dgi s6 tuy

nhiAn nhi6u bdi todn v€ da th*c bim chdt cila n6 lqi

ld bdi todn til ho" Niiu chting ta bih kiit ho p khai

:

thac mdi quan hQ giiia cdc tinh chdt dqi s6 vd s6 hpc

thi s€ gidi quy€t duqc nhiiu bdi todn vi da thuc.

tinh 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 thric c6 v6 s0 nghiQm ld da thric kh6ng

Da thirc c6 bflc nho hon ho{c bing n mi nh{n

ctngmQt gi5 tri tqi n + 1 gi6 tri.kh6c nhau cta

ddi s6 thi cla thric cl6 ld da thfc hing

Hai da thirc bflc nh6 hcrn ho{c bing r md nhfn

n + | giltri th6a mdn bing nhaul4i n +l giht4

Bdc cta tdng hai <la thirc khdng lcrn hcrn bQc

cria m5i <la thric <16.

BQc ctra tich hai da thric kh6c kh6ng bing t6ng

Hai da thirc f vd g thuQc IR.[r] trong d6 g kh6c

kh6ng (tla thric kh6ng) , khi tl6 c6 duy nh6t mQt

c[p cta thtc q, r e IR.["r] sao cho "f : qg + r

trong d6 hoic r: 0 hoflc deg r < deg g voi r

Du cia phdp chia cla thfcfix) cho x - c ldflc).

ctng tdn tai IJCLN ciafvdg vd LrCtN cl6 duy

nhdt.

N6u da thuc d h UCLN cria c5c da thtrc f vit

g, khi d6 tdn tpi hai da thitc u,v sao cho

fu + Sv : d Nguo c lai n6u da thirc d lit :uoc

chung cua cbc tla thric f vi, S vir th6a mdn

fu + Sv : d thl d le UCLN cinf vd g.

- LZ ^ TOnN - Gfi.rdiUa-gq-r*reeggl HgC

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, v

sao chofu+ gr: l.

nhau thi IJV)l'vd [g(x)]' s0 nguy6n t0 cring

Mqi nghiQm xs cta da thric

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

lr.l(lrrl -L)< A,A=maxlatl, k =r, ,n.

Da thirc pahttna quy khi vi chi khi mgi u6c

dqng ap vdi alithing s6 kh5c kh6ng

M6i da th1c f bAc 16n hcrn kh6ng U6t ty ACu

ph6n tich dugc thdnh tich cbc da thric bet khe

qly Ve sp phAn tich d6 ld duy nn6t ni5u kh6ng

k0 d€n thf"t.u c6c nhAn tu vd nhdn tu bdc khdng Tieu chuAn Eisenstein:

Cho P(-r) : a,{" * a,-t{-l+ + afi + aoe Zlx)

Ntiu c6 it nhil mQt c6ch chgn s5 nguy€n t6 p

*) a, khdng chia h6t cho p.

*) f6t cir cbc hQ s6 con 14i chia hi5t chop .

*) a6 chia h6t cho p nhmg khdng chia hi5t cho

p2 th\ P(x) kh6ng ph0n tich duqc thdnh tich c6c

da thuc c6 b{c thdp horl vcri c6c hp sO hiru r5i.

NQi dung cua cac bdi toan vi da thuc thudng xoay quanh viQc tim nghiQm vd xdc dinh si5 nghiQm ctia

mQt da thuc, xdc dinh m)t da thuc hofrc rdc luqnggia tri clta da thuc khi biA dq thuc thod mdn mil sd

Trong khu6n khd bdi vi€t nay , chung ta s€ dO cQp

ilnh chh s6 hp, drtc ffung thudng xuh hi€n 6 bdi

todn da thuc.

Bii to6n l Cho cac da thttc P(x), Q(x) e Zlx)

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.

phii 16, m6u thu6n

Bezout cring tinh chin, 16 cria s5 nguy6n ta suy ra

tli6u ph6i chimg minh (tlpcm).

Bii to6n 2 T6n tqi hay kh6ng da thtrc P(x),

deg(P(x)) : 2015 thda mdn P(*' - 2Ol4) chia

hAt cho P(x)?

Ldi girti Xdt da thric P(r) : (x + o)'o",ae IR

: l@+o)' - 2a(x + a) + a2 + a- 201412015 Da

th6y phuong trinh o' + o - 2Ol4: 0 lu6n c6 2

cho a2 + a * 2014: 0, tu c16 suy ra p(xz - 2ol4)

: (x + o)""(* - a)20rs chia h6t cho p(x)

Liti binh Tri y€u c6u P(x) c6 deg P(x):2015 vi

P(*' - 2014) phii chia hrit cho p(x) nen ta c6 ttr6 du

do6nngay P(x) c6 dpng (x +,)'0".

Biri toin 3 Cho & e N- Tim dt cd cdc da thuc

P(x) th6a mdn:

Ldi gidi Gi6 sir P(x) ld da thric th6a m6n di6u

kiQn trdn Tri gi6 thi6t ta thdy x : 2016 liL

nghiQm bQi bflc 2 k cl0la P(x)

hing s5.v6y p(*): a (x 2ot6)r Tht lpi ttfng.

Ldi binh Ddy ld bdi torin co b6n, sir dqng tinh ch6t

chia h6t, nghiQm bQi, cring tlinh lf Bezout cho ta loi

gihi

Bii toin 4 Cho P(x) vd Q@) ld hai da thilrc voi

hQ sd nguyAn Biiit ring da thuc xP(x3) + Q@t)

chia hih cho x2 * x * l Gpi d ld UCLN cria

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

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 vit

deg [xP(l) + g(1)] < 1 n6n tu (2) suy ra:

xP(t) + QQ) = 0 = P(1) : O(1) : 0 (3)

Bezout ta c6:

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

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

-_.-.A- r ^: +Li_ lr{c:otsl= 2014.Rr(2015)

-'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

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

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 li

ta: JU) = 0 (mod 2017), i : l, 2, 3 hayf(x) i2017

N6u (x1 - x2)(x2- r:)(x: - ,,) j 2017 thi bdi

N6u @1 - x2)(x2- xz)(xr x1) kh6ng chia hiit

cho 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)

fl*r): L axr2 + (b + 1)x1+ c] i 2017, suy ra

c : zol7

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

13

Ydy a + b + c + | :.2017.Theo cllnh ly Fermat

bd ta c6: o = o'\"(^od 2017), b = b2017(mod

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 li

Fermat 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)

dC bdi Khi d6 chgn trong tl6 mQt cflp ft, y) md.

y ld s6 lcm nh6t Ta sE chirng minh ntSu c6 cflp

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

sir dgng phuong ph6p clrc hpn, ph6i hqp vdi ph6p

to6n <ldng du trong qu5 trinh chimg minh phdn

Bdri to6n 7 Cho a, b, c td bo sd nguy€n phdnbi€t vd da thilrc P(x)e Zlxlsao cho P(a) : P(b): P(c) :2 Chilmg minh ring phuong trinh

P(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

d - b, d - c e {1,-1}, m6u thu6n v6i gi6 thitlt

a, b, c ddi mQt ph6n bi€t Tt d6 ta c6 cli6u ph6ichimg minh

Bii torln 8 Cho da thac fu)ez(x) Chilmg

minh rdng n€u da thilrc Q@) : J@ + 12 c6 it

nhiit 6 nghiQm nguyAn phdn biQt thi JU) kh6ngc6 nghiQm nguyAn

Ldi gidi Gi6 sir Q@) c6 6 nghiQm xt, x2 , ,

: (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

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 vdi

nguyAn 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

n6n: 31 :31.1 : (-1).1.(-31)

: (-1).(-31) : 31.(-1).(-1).

Do d6 31 kh6ng thd phdn tich thdnh tich cin 4

phucrng trinhflx) :2046 kh6ng th6 c6 nghiQm

nguyOn.

Ldi binh C6c bdi toin 7, 8, 9 cirng m6t th6 loai.

Pfuong ph6p chung il6 gi6i o ddy ld chimg minh

bdng phin chimg Sit dqng tinh chdt nghi6m, tinh

Ldi gidi Gi6 sir da thric P(x) khdng Uat nra

quy, suy ra P(x): F(x).G(;r) trong tl6 F(x) vd

G(x) le circ dathric bQc nguy6n ducmg v6i hQ s6

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 ila

thr?c P(x) U5t mra quy tr}n Zfx)

Biri toin ll (tMo-1s93) Chilmg minh riing da

quy ffAn Zlxl.

Ldi gi,fii OO th6y vbi n:2 thi flx): y2 + Jy -r

f U6t nra quy tr6n Zlxl vOi n ) 3, gii su

J@: s@).h(x) v6i g(x), fr(x) thuQc Z[x] vit c6

b6c ) 1 Vi deg g + deg h: n> 3 n6n suy ra

trong hai s6 deg g vd deg h co mQt sO > t tr,tpt

iirrilcfQ): 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

Nhdn c6c tling thr?c tr6n lai vd kilt hqp v6i (*)

ta dugc: l(a1 + 5)(a2 + 5) .(ap+ 5)l : 31 1x x)

lg(-s)l :l@r+ 5)(a2+ 5) (ap+ 5)l

vn 3 :l-s) : g(-s)ft(-s) ncnl(a1 + 5)(a2 + 5) (a1, + 5)l bing t ho[c bing 3.

Di6u ndy mAu thu6n v6i (**) vl k> 1 Tt d6

Bii to6n 12 (VMO 2013-20t4) Cho da thac

P(x) : (

"' - 7x + 6)2' + 13, ne N Chbng

minh riing P(x) kh6ng th€ phdn tich thdnh tich

cila n + I da thuc khdc hting s,i vdi h€ sti nguyAn.

Ldi gidi n5 thay deg P(r) : 4n vdP(.x) khdng

nghiQm 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 ld

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

,,.

gi6m tong qudt gi6 su P1(1) : 1 suy ra a: -b .

Khi d6 P,(6) : 36 - 5b > 0 vd P{6) + 13, suy

ra36 -5b: lhay b :7, a:-7 vdkhid6

Pr(x) : x' -7x + 7 c6 nghiCm thuc , m6u thuin!

Ldi binh Ca 3 bdi to6n 10, 11, 12 vA da thric b6t

chimg minh bing ph6n chimg Tuy nhi6n bdi to6n 10

ctrAt vC bflc cria da thirc ld cho k6t qu6 Bdi to6n I I d mric d0 cao hon, ta d6 sir dung tinh ctr6t ve b6c cria

da thric, tinh ch6t cira st5 nguy6n t6, s6 nguy6n, k6t

nhu bdi 11, tuy nhi6n tl6y ld bdi to6n m6i trong kj,

kh6 khln cho nhi6u hgc sinh Bdi to6n ndy c6 vdi c6ch gi6i tuy nhi6n d6u chimg minh bing phuong ph6p ph6n chimg ph6p phin chimg C6ch gi6i tr6n

dAy dga vdo viQc khai th6c b6c cta da thfc, tinh ch6t nghiQm ctng tinh ch6t cira s6 nguydn tii

=,, nr.,.-rorr, T?8I#EE 15

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 :

NGUYEN MINH HA

(GV THPT chuyAn EHSP Hd NQi)

l_ I _R( | _ 1 )

Jx ' J2x-t -"-( J6x1' J9x4 )

LAI THI HOA

(GV THPT L€ Quj,D6n, Thdi Binh)

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)

NUT UAI QUANG

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

cAc IqIp THPT

(GY THPT chcryAn Hd TInh)

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 minh

ring: 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 trinh

11

(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

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

TRAN xUAN DANG

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