Bộ đề thi vào lớp 10 bộ môn toán trường chuyên năm 2017

Cho hai duong trdn t6m Q vd O, nimngodi nhau.. Drmgduong trdn tdm O ti€p xric ngodi vdi dudmg trdn Q tai O vd titip xric trong v6i duong trdn Or tqi C tli6m O khdng nim tren do4n QOr..

:r UBND TINH rHAt NcuvEN THI ruyrN sINH vAo Lop 10 THpr crruytx

' (Dnnh cho thi sinh thi viro chuy6n To6n)Thdri gian lim biri: 180 phft (kh1ng ke thdi gian giao di)

cau 1(1,0 ili6m) Kh6ng drrng m6y tinh cam tay h6y rut so.n A=

ffl';*)Ciu2(2,0 di6m) ciei he phuong trinh sau:

I*'*y'-xy+4y+l=0

Ir(r- *'- y'*r*)=z(*' +t)'

Cflu 3(1,0 di6m) Cho s0 tg nhi6n A =W - l8 + 2n vli n e N,z ) 2

n chilsdT

Chtmg minh ring A chia h6t cho 9.

CAu 4(1,5 ili6m) Cho a,b,c ld c6c sti duong th6a m6n a + b + c s Ji .

Tim giritrilonnh6t criabi6uthtc: P -+*L+*.

cffu 5(1,0 ili6m) vdi m6i sti nguy€n ducing n taky hiQu a, li si5 nguyEn g6n Ji *at.

Vf dU: o, =l;a, =l;az = 2;aq - 2;as = 2;ao = 2;o, = 3 Tinh gi6 tri cria t6ng:

^s=l*f*l* * 1* at a2 a3 dzon I

azots

CAu 6(1,0 ili6m) Cho hai duong trdn t6m Q vd O, nimngodi nhau DoAn tha"g eO, c*t

duong trdn (Q) ta, ei6* l Phan k6o ddi cta cloqn OrQ cdtdudrng trOn (q) @i B Drmgduong trdn tdm O ti€p xric ngodi vdi dudmg trdn (Q ) tai O vd titip xric trong v6i duong trdn

(Or) tqi C (tli6m O khdng nim tren do4n QOr) Chrmg minh rdng c6c dii:m A,B,C,D cingnim tr6n mQt duong trdn

Ciu 7Q,5 tliGm) Cho ba <ti€m A,B,C phen biQt thang,hang vd theo thf tp d6 sao cho

AB <BC Trong mOt nta mflt ph[ng c6 bd li cluong th6ng AC dggcdc hinh vudng ABDEvir BCFK Gqi / ld trung cti6m cria do4n thing EF, Dutng thlng qua ^I vu6ng gbc vfii EF

citcacducrngthdng BD vd AB ]ffilnlugtt4i M vd ff chungminhring:

a Tri giirc AEIN vi tri gi6c EMID nQi titip dugc trong dudng trdn

b Ba dii:m A,I,D thang hang vd c6c cti6m B,N,F,M,E nim tr6n cirng mQt dudng trdn

c Ba duong thdng AK, EF ,CD <tOng quy

UBND riNH rgar NcuypN

so crAo oqc vl oao r4.o

xAna Hec zotz.- 2ot8

wION THr: ToAN IreC

(Dlrnh cho thf sinh thi vlro chuy6n To6n)

( Bdn hudng ddn chiim gim c6 06 trang)

l'#t#?f,*"Xx?fviing ycu cAu cria hu6ng dan ch6m a6 arenn gi6 irung bdildm cria thi sinh Thf sinh ldm c6ch kh6c tl6p 6n niSu dring vin cho tli6m t6i tta.

- Khi v4n dpng ddp invi thang dir5m, gi6m kh6o c6n chri dQng, linh ho4t vdi tinh

thAn trdn trgng bdi lim cria hgc sinh.

" - Ni5u c6 viQc chi titit h6a cti6m chc y cdn phai d6m bio kh6ng sai lQch v6i t6ng

dii5m vd dugc th6ng nh6t trong todn hQi d6ng ch6m thi

- Dii6m toan bii li t6ng di6m cria c6c cdu h6i trong de thi, ch6m <tii5m 16 dln0,25

vi kh6ng ldm trdn

II Eip 6n vir thans di6m

HITOI{G oAN CHAM

THI TUYTX STNTT LOP 10 THPT CHUYTN

2

I

2 sdhang 4 sdhang 88 sti hpg 38 sdf hang

(tir a,rr, = 0ts82 = = aror, = 45)

ffi, -1800 - (m, *ffi)= 1800 -(ro' - *.e00 - ;)= ffe>

Tu(l) vir(2)suyra ffi=6)-a+B '

(kh6ng vE hinh khdng cho tli6m)

M

Ciu 7

215

a.Tac6:

fu =ffi = 900 n€n tf gi6c AEINnQi titip tluong trdn dudrng kfnh Ell

ffi -ffi = 900 n6n tu gi6c EMIDnQi titip dudrng trdn duln gkinh EM

b Ta c6 frE =fu - 450 n€n ADllBr (t)Gqi O li giao cli€m ci,r- BE vir AD Khi d6 O li trung rti6m ctr- BE vit AD

Ma 1ld trung cti6m cia EF nln OI ld ducrng trung binh cta tam gi6c EBFfmg vdi canh BF

Do d6 oI ll BF (2)Tir (l) vA (2) ta c6 A, I, D ctng nim tr6n dulng thdng qua O vd song song

v6i BFVdy A, I, D thinghdng

fu -ifu -450n€n tam gi6c NEF vu6ng cdn

fu =m = 45on6n tam gi6c MEFvuOng cdn

0.25

0.25 0.25

Vi vay tri giSc MENF le hinh vu6ng

Suy ra bi5n cti6m M, E,lf f ni* tr€n dudmg trdn dudrn gkinh MN

0.25

M4t kh6c tam gi6c MBN vuOng t4i B n€n .B cflng nim ffen dudmg trdn I

I

VAy 5 <ti6m B,N,F,M ,E nimtrOn ctng mQt dulng trdn ducm gkinh MN I O.ZS

c Chmg minh tuong tg ta c6 ba di6m C, I, Kthdng hdng I

I

D li trgc t6m tam gi6c AKC n€,n frd = 900(3) I O.rt

Trigi6c AEGDnQititip n€'nfu=ffi,=450(4) lO.r,

Tt giric CGKF nQi ti6p n€n GF =fu = 450(5)

rudt uBND T'iNH ruAlNeuyEr.r THI TUyEN srng YAo 16r t0 THpr cnuv.&N

sO cll.o DUC vA uAo r.So xAu Hec: zot6 - zot7

@irnh cho thi sinh thi vio chuy0n To{n)

Thli giau lim bii: 150 phrf,t (kh$nS kd thdi gian giaa dA)

C6u 4 (; ,A ciiA@ Tim tht cac6c s5 c6 5 chfr sd otni, sao cho |iffi = ah .

Cflu 5 (1,0 di6m) Cho ba s6 a, b, c nguy€n dlrong, nguy€n t5 cung nhau vd thdamdn diAu kiQn: 1o 1- l Ch,irg minh a + bli sd chinh phuong

abcCfiu 6 (.1,5 di€m) Cho duhng trdn t6m O vi ddy cung AB.Tii mQt di6m M b*ki tr6ndunng tritn (M kh6c A vit B), ke MH -t-AB tai 1{ Ggi E, F lAn luqt ld hinh chi6u

vu6ng g6c cua 11 tr6n MA, MB Qua &/ kd tluong th6ng vu6ng g6c v6i EF, cht ddy

cung AB l:aiD Chtmg minh ring: , W * AH .AD

MB2 BD- BHCffu 7 ()1,0 di€m) Cho tam gi6c ABC vudng tqi A, duong caa AH Gpi (O) ld ttudng

trdn ngo4i ti6p tam gi6c AHC 'llr6n cung rlhb AH cta (O) 16y di6m Mbdt ki kh6cl

vd H Tr,Otr ti;5p tuy6n @i M cua (O) lEy hai diOm D, E sao cho BD - BE =Bl Eudngthing BA4'c}it(O) t4i diem thf hai 1{"

a) Chrq; minh r6ng tri gi6c BDNE nQi ti6p dirgc trong mQt <luong trdn

b) Chrmg minh Cucrng trdn ngcpi tiep tt gi6c fiDlttlvirduong tr0n (C) ti6p xirc n-hau.

H6t uBND TiNH ruAlNcwBN

sO crAo DUC vAoAo rAo

@inh cho thi sinh thi viro chuy6n Todn)

( Bdn hudng ddn chtim gdm c6 04 trang)

- Gi6m f.frao rA" n?f"rrng y6u c6u cria huong ddn ch6m dC $6oh, gi6 etung bdilim cria thi sinh Thf sinh lem c6ch kttat Aep rin n6u U*g

"T:h: cti6m t6i tta'

- Khi van dud diry invi thang diem, gi6m khio cdn chri tlQng, linh ho?t voi tinh

tfran **' tran trgng bii ldm cria hgc sinh' ! !

: N.6; #rie"rrri tii5t h6a eti6m c6c j can phii rlim b6o k*rong sai lQch voi t6ngdi6m vi du-o c th6ng nh6ttrong toan hQi tl6ng chdm thi'

- Ei6m toan f,ai ln t6ng Ei6- cria c6c rau rroi trong d0 thi, chAm di6m 16 di5n 0,25

II Dfp 6n vi than

Noi d

;mu /-0 thi x-0=(O;O) ldnghiQmcuahQ'

+ NrSu y * 0 thi dAt * = tV(k > 0) ta dugc hQ phucmg trinh

D" tfnh;[fi|] * lrl ,lt*rl oau "-" xhy ra khi AB > 0' n6n

" = l,+ ,rl*F-.Fl = lr* J ,-s13 -.Fl = 6

D6u .,:,, xby ruUri (:+rFXr-.F) > oee-(r-q) >,0 <+x< 18

Vfy minP = 6 khi 9<x<18

,V+x+y 53 a'4xy +?*+2y =t6 e(Zx+l)(zv+l)=rct

Do 767 h s6 nguydn ti5 n€n 2x+l chi c6 tfre n mQt trong 4 sti

!1"+167

+ Vdi ?*+1-t ta dugc x=0,! = 83

+ Vdi ?s+!=-l ta dugc x = -1,! = -84.

+V6i ?*+l-767 tadugc x=83,Y=4.

+Vdi 2t+L !67 tadugc x=-84,Y 1.

Vfly PT dE cho c6 4 nghiQm nguy6n (x;y)li :

Cflu 4

1'0

CQng hai vi5 cua (1) v6i c2 ta dusc (a- c)(b - r)= "' (2)

Ni5u d li udc nguy6n ducrng oira a- c vir b - c thl c'i d'

Do d6 ci d, m{ a'- c vit b - c cingchia hi5t cho d ndn d li u6c cria

a, b, c.Theo gi6 ttrii5t a, b, c nguy6n td cung nhau n€n d -l

Yfly a-c vit b-c kh6ng c6 u6c lon hcrn 1, n6n (a -')(b-c) Us6 chinh phucrng, tumg sti a - c vd b -c li s6 chinh phucrng kh6c

Gqi E', F' lAn luqt ld hinh chiiiu vu6ng g6c ctra D tr6n 1/L4, MB.

Ta c6: ffiF =frF ( vi cung bir v6i g6c fuB)

Ta lpi c6: ffiE =ffi' ( hai g6c c6 canh tucrng img vu6ng g6c)

114d, ffi,=ffi,( cung ch5n cung DF)

0.25

Ciu 6

1r5

N'

a) Vi AH L BC *fu =900 + AC lilduong kinh cria tlucmg trdn

(b) n{a MBC vu6ng t$ A nln AB ln ti6p tuyiSn ctra (O)

X6t c6c tarr_ gi6c BAMvir BNA c6: ffi chung vd ffi =ffi vl

c6 si5 do bdng nria sii do cung fu YAy MAM vd tau,l d6ng

Mpt kh6c ffi ng6c chung crta LBDM vd IAWO .

Vfy n6n LBDM vd tauo ddng dang =ffi =ffi

Ta c6 BD=BE= LBDE cdntpi B +ffi =fri+ffi=ffi

+ Tri gi6c BDNE nQi ti6p dugc trong mQt cluong trdn.,

b) Ki hiQu ctucrng trdn ngopi ti6p fli gi6c BDNE ld (O)

Gii srl (O) vd (O ) kh6ng tiisp xirc nhau Vi (O) vd (O ) c6 di6m

chung l/n6n chring c6 cti€m chung thri hai la N' vd BN' cht On tai

Hai tam gi6c BMM'vdBN'Nc6 ffi' chung 9 ud %=Yr' BN' BN n€n

chring d6ng dang = ffi'= ffi =+ ffi + ffi = t8oo

=+ Tir gi6c M'MNN'nQi tii5p iludrng trdn

Dulng trdn ngopi ti€:p M'MNN' vd (O) c6 chung nhau ba di6m M,

if, if l n6n chring br)ng nhau ) M' h di6m chung thri hai ct.r- (O)

vd titip tuytin DE Ei6u niry v6 ly = DiAu gi6 su ld sai I

Vpy ducrng trdn ngopi tiilp tri giSc BDNEvi tludrng trdn (O) ti0p I o'zsxtic nhau

0.2s

-

ttift 4

0.25

UBND T1NH THAI NGUVEN

so ctAo DUC vA DAo rAo

pr cgiNn rntlc

THr ruYtx srNn Lo? 10 rHPT xAm Hec 2ot6 - 2ot7

Thli gian lim bii: 120 phfit (kh6ng kA thdi gian giao di)

Cflu 1 (r,0 dia@ Cho hdm sti ,=(J1-Ji)*+z c6 d6thi (d) Hdm sii dacho ldd6ngbirin hay nghich bi6n tr6n R ? Giai thich? Tim tga tlQ giao di6m crta @)vdi tryc tung.Cfru Z (t ,0 di€m) Khong dung m6y tfnh cam tay, rut gen bi6u thric , *(Z*il2[ -\E8-8

cnu 3 (1,0 di€m) cho bitiu thric r=(#,#)

w

Hdy rirt ggn B vd tinh giltri ctr Bkhi x = 3 +.6'

f**by-4C6u 4 (1,0 di€m).Xitcdinh c5c he sO avdb, Uititn9 phuong trinh L

[.0, - 2y 2

c6 nghiQm (*;y) = (z;-l)

Ciu 5 (1,0 di€m) Kh6ng dung m6y tfnh c6m tay, h5y gi6i phucrng trinh:

xz +6x-2016:0

Cffu 6 (t ,0 di€m) Cho phucrng trinh x' - Zmx * (*'- +):O 1t;, rn li tham si5.

a) Chung minh phucrng trinh ( 1) lu6n c6 2 nghiQm ph6n biQt vdi mqi gi6 t4 cliua m.

b) Goi x, xrlir2nghiQmcuaphuongtrinh(1) Tim mdd *r'*xz2 =26'

Ciu 7 (1,0 di6m) Kh6ng tinh ttmg gi6 tri cp th6, hiy sdp x6p c6c ti s5 lugng gi6c

cos 20' , sin 3 8' , cos 5 5' , tan 480 , sin 88' theo thri tg tSng dAn, gi6i thich?

Cffug (l,0diAm) Chotamgi6c ABC vu6ng tU A,c6 sin.B=l.16rtinhc6ctisiS luqng

gi6c cira g6c C

Cfiu 9 (1,0 di€m) Cho ttuong trdn tdm O vdmQt di6m Anim ngodi duong trdn K62

tirip tuytin AB, AC voi duong trdn (^B,C ld2ti€p diCm) Qua C k6 mQt duong thEng song

song voi OB, cgt OA taiH Chrmg minh rEng tu giac ABOC nQi tiep duo c trong mQt duong

trdn vi I/ ld @c.t6m cfia tam giac ABC.

Ciu l0 (1,0 diA@ Cho tf gi6c ABCD nQi ti6p tiong ilulng tron(O;R),c6 hai duong

ch6o vu6ng g6c v6i nhau vd ciltnhau tpi /

a) Chung minh.i,g, IA.DC - ID.AB; b) Tinh t6ng AB2 + CDz theo R

-

rudt S6 bao danh: .i!,.

NAvr Hec zot6- 2ot7

vrON THr: ToAN Hec

( Bdn hudng ddn chtim gim cd 04 trang)I.Iluring din chung

- Gi6m kh6o cdn ndm virng y6u cAu cria hudng d6n ch6m Oe Oantr gi6 rhing bdildm cta thf sinh Thi sinh ldm c6ch khric d6p dn n6u dring v6n cho di6m tOi rla.

- Khi vfln dUng dbp in vi thang di6m, girim kh6o cAn chri tlQng, Iinh hoat vdi tinh

thAn trdnlrgng bdi ldm Lta hgc sinh.

- NiSu c6 viQc chi ti6t h6a di6m cdc,y cAn ph6i ddm b6o kh6ng sai lQch vdi t6ng di6m vi <l-uOc thting nh6t trong todn hQi tl6ng ch6m thi

- Di6m todn bdi ld t6ng dii5m cria c6c c-6u h6i trong dC thi, chSm di6m 16 di:n 0,25

vd kh6ng ldm trdn

UBND TINH THAI NGUYEN

so GrAo pvc vA pAo r4.o

II Dip rin vi thang tli6m

HIIONG NAN CHAM THI TUYEN SINH VAO LOP 10 THPT

n

I

CAU I Cho hdm s6 y= (J1-Jr).+3 c6 Ao tni (d) Ham so da cho ld ddng

biiin hay nghich biiSn tr€n R ? Giai thich? Tim tga d6 giao cti6m

c:ioa (d)voi truc tung

0'5

0r25 0,25Cffu 3

Cho phucrng trinh x' -zmx *(*' - +):O 1t;, z ld tham si5.

a)Chung minh phuong trinh (1) lu6n c6 2 nghiQm phdn biQt vdi mgi

Thri lpi: Vdi m=3, ta c6 phuong trinh x'-6x+5:0 c6 hai

nghiQm: x, =1,x, = 5 =+ t * 4 - 26,thoa m6n dAu bdi

ygl_!! - -3 cfing th6a m6n itAu bdi Ydy m = +3

0r25 0r25

0r5

2

Khdng tinh tirng gi6 tr! cg thil, hay sip x6p cric ti sti lugng giitccos20', sin38', cos55', tan480, sin88' theo thti tg ting ddn, giii

thich?

Gidi:

Ta c6 bi6n cl6i sau: sin 380 = cos52o ,sin 88' = cos 20

Vdi 0 <d <900,n6u acdnglonthi cosa cdngnh6vd cosa<l

ndn 0<cos55'< cos52o <cos2Oo <cos20 <1

N6u 450 < P <900 thi tanp 21 = tan480 > I

Do d6 ta sEp xiip theo thri tU ting dAn nhu sau:

cos 55', sin 38", cos 20', sin88o, tan 480

Cdu 7

0r25

0,25

0r25 0r25

CAU 8

Chotam gi6c ABC vudng tqi A,c6 sin B -:.HEy tfnh citctisri luqng

3

Gidi: (Kh6ng cho diiim hinh vd)

0,25

Ciu 9 Cho dudmg trdn tdm O vi mQt di€:m AnAm ngoii duong trdn Kd 2

tiiip tuytin AB, AC voi ducrng tdn (B,C ld 2 n}p di6m) Qua C kd mQt

duong tlrang song song vu OB, cfut O,l t1i H Chung minh rang th gec

ABOC nQi ti6p duo c trong mQt duong tdn vd f/li trfc t6m cua tam gic

ABC

Grrii.' Flinh vE :

Xdt tu gdc ABOC c6:

trEd =frd =eoo (Theo

tinh ch6t cta ti6p tuyiSn).

Suy ra tu gi6c ABOC nQi

ti6p du-o c tong mQt ducrng

ch6o vu6n g g6c vdi nhau vd cit nhau t4i L

a) Chrmg minh ring: IA.DC - ID.AB;

b) Tinh t6ng AB2 + CD2 theo R.

Gidi: Hinh vE kh6ng cho tli6m

a) Xdt hai tam gi6c IAB vd IDC c6:

vd fri =i6Z (g6c nOi tii5p cung chiin D

cung BQ

Suy ra: NAB diing d4ng voi NDC do

=fi +IADC=ID.ABrc

b) Ke ducrng ktuh CE cta duong tdn (O)

Ta c6 ili =ide =90'(g6c nOi ti6p chan duong kfnh EC)

Suy ' ra {n:"!* IAC LAE AE t tBD+ ABDE le hinh thang cdn (hinh thang

nQi titip duong trdn)+ AB = DE

Do d6 AB2 +CDz = DE2 +CD2 = ECz =(zn)' =4R,(DoADEC

UBND TINH THAI NGUYEN

sd cra.o uuc va oAo rAo

Ciu 2(1,0 tli6m) Cho him siS Uac rirrdt y =(2m4)*+5m-l (m lirtham ,0,**)1 ,2,

a Tim m d6hdm siS nghich bi6n tr€n IR .

b Tim m d6OO ttri hnm s5 c6t trgc tung t4i.tliiSm c6 tung tIQ h -6.

Cfiu 3(1,0 tli6m)' Kh6ng dtng m6y tinh cAm tay, rut ggn bi6u thrlc:

cau 5(r,0 tli6nr) cho h€ phuong

a Khdng dtng m6y tfnh cim tay hay gi6i hg phuong trinh khi * !;n 2'3 -1.

b X6c itinh c6c tham sd m vi n biltring hQ phuong trinh c6 nghiQm li (-Lr6).

CAu 6(1,0 tli6m) Cho phuong trinh 2x2+3x-l=0 Ggi xr;xz li hai nghiQm phAn biQt criaphuong trinh Kh6ng gi6i phuong trinh h6y tinh gi6 tri cria bi6u thric:

p =z( !r*L-\.

[r, \)

Cflu 7(1,0 ili6m) MQt tam gi6c vu6ng c6 c4nh huyAn bing 5cm, di$n tich li 6cm' Tinh dO ddic6c c4nh g6c vu6ng cria tam gi6c vuOng d6.

Cfiu 8(1,0 tli6m) Hai dulng trOn (O) va (O') cit nhau t1i A vir B Gqi M ldtrung di'5m cta

OO' Qua ,,4 k6 tluong thang vu6ng g6c vdi AM cit c6c dudrng trOn (O) va (O') tan tugt O Cvir D Chung minh ring AC = AD

Cffu 9(1,0 tli6m) Cho dudmgtrdn (O), Auongkinh AB,ctxrg D ni*cirngphfa A6ivA ,lA

( D thuQc cung nh6 fu ).Gqi E ld giao eliiim ciua AC vir BD , F ldgiao diiSm ctn AD vd BC .

a Tinh g6c frE khi siS do cria

"*g D bing 800.

b Tinh st5 do r*g D khi g6c ffi ailne ss' ,

CAu 10(1,0 tli6m) Cho tam gi6c nhgn ABC (AB<AC) Euong trdn tdm O ttuong kinh BC cilt

cqrrh AC, AB hn luqt t4i D vir E H ld giao diiSm cria BD vit CE, K li giao itii5m ctn DE vd AH, F lit

giao cti€m cliua AH vir BC M tdtrung tli€m oiua AH Chtmg minh rang, MDz = MK -lfiF .

so crAo nuc vA DAo rAo rsAl ncuynm

TRI.TC}NG THPT GHUYEN

of rru rutl ruytN SINH vao r,Op to

NArr Hec 2ot6 -zot7

MOn: TOAN TIN

(Ddnh cho th{ sinlt thi vdo chuyAn Tin)

Thdi giun tdm bdi: 150 phtit, khilng kd thdi gian phtit di

Biri I (2,0 tli6m) cho bir6u thric n=( (V2" &*$).H +b +^,t2a-b "u,

^14a, -b, -2a+b ) ^t4a, -b,2a> b>0

a Rrit gen A

b Bi6t 2a-b=1 Tim gi6(_nh6 nh6t cria bi6u th{rc A

Bni 2 (1,0 di6m) Cho phucrng trinh (x+2)(x+3)(x+4Xx+s) =a Bi6t phuong trinh dd cho

c6 4 nghiQm ph6n biQt x,,x,x3,x4 Gqi r : xt.x2.x3.x4 H6y tinh P theo a?

Bni 3 (2,0 tli6m) Chrmg minh ring tu zAfi sii nguyEn duong U6t tci lu6n c6 th6 chgn radugc hai s6 md t6ng hofic hiQu cria chfng chia h6t cho 4030

Bni 4 (2,0 di6m) Trong mat phing cho 2016 itulng trdn trong d6 b6t kj,hai dudmg trdn

ndo cfing cit nhau t4i hai cti6m phdn bi6t vd kh6ng c6 ba tlucrng ndo c6 ili6mchung Tinh s6 phAn mAt phing md cilc tlulng trdn d6 chia ra (C6c phAn matpheng n€u 6 dAy khdng chdng 16n nhau Vi dg: n6u tr6n mAt phing c6 dring I

cludmg trdn thi tlucrng trdn d6 chia mflt phing thdnh hai phAn; n6u tr6n mflt phing

c6 dring2 dutrng trdn clt nhau t4i hai tli6m phdn biet thi 2 tluong trdn d6 chia mflt

phing thanh a phAn )

Bii 5 (3,0 tli6m) Cho n ld mQt sd nguy€n duong chin lcrn hcrn hoflc bing 4.Tat6 mdu

mdi sti trong c6c sti nguy€n duong tu I d6n ,,l sao cho

i sti trong chirng clugc t6'

mdu xanh ui, + 2 sO con lqi dugc t6 miu d6 Vdi m5i c6ch td nhu vfly, kf hiQu f ' le sii cdc s5 nguy6n duong b5t ki mi n6 c6 thti vitit dugc du6i d4ng t6ng hai s6

khSc mdu.

a Tim tdtcitcdc gi6t4ci,r- fo.

b Chrmg minh rdng l, <tgT t

Thi sinh khdng clugc sfr dgng tii liQu Cfn bQ coi thi kh6ng girii thfch gi th6m

Hg vi tOn thi sinh: ; Phong thi: ; Sti Uao danh: