BỘ đề THI HSG VÒNG TỈNH MÔN TOÁN lớp 12 năm 2013 2014 SGĐT tây NINH

Gqi Qr vd Qz lAn luqt li hinhchi6u vudng g6c cria Pr, Pz l6n dulng thing OrOz.. Chimg minh ring vdi ba da gi6c dE d{t trong hinh vu6ng, ludn tim dugc hai da gi6c midi€n tfch phin chung c

SO GIAO DUC VA DAO TAO TAY NII\H

Ky rrrr cHoN Hec srNH Gror Lop 12 THpr voNG riNn

NAvl Hec 2ot3 - zot4

Ngay thi: 25 thing 9 nlnr Z0l3Mdn thi: ToAx - Btroi ttri thri,ntrdtTho'i gian: 180 phut (khong kA thdi gian giao dA)

@A thi gont co 0 t

Bii 1 (4 dient)

Cho ba s6 ducvnga,b, c tho6 mdn cli6u kiqn a+b+c = 3.

Chirngminhr6ng: -L * l+bc l+ca l+ab b * t ,32

cho cludng thing d cri clua trtrc tam H cria ta,r gi6o ABC

dudng ttring dtii xung vdi d qua BC, CA, AB Chring

Gei ;,.

rninh

cho lrinh ch0'nh$t ABCD Tr0n c6c cluong thang BC va

cD, l6y lAn lugt c6c di6m di

fl:il*"x,,il;i?r:n" ffir = e00 Gqi H rd hinh chi6u vu6ng g6c cria A tr6n MN rirn qu!

tno't minh rf,ng trong I I s5 thuc khirc nhau thu6c croan

[l; 10001 , o6 the chon cluoc hai

so x vdy sao cho: 0 < x _y <l +:{.y

-socrAo DUC vAoAo TAo rAy NINH

xi, rnr cHeN Hec srNH crOr Lop 12 THpr voNc riNn

NApr Hec zots -2014 nucrrvii ;iiii'&iM ffi rilbil idfi{''i;;il il ;il nn6t)

X6t xo >28 thi f(*o)=f(xo -28+28):f(xo-28).f(28)=0.

t(28) = f(l 4 +14): f(l4).f(14) = 0 =+ f(l4) = 0 .

Lim tuong tU nhu trOn thi f(x) - 0, Vx > 14

LAp lupn tucrng tp thi co: f(x) - 0, vx >7; f(x) = 0, vx a!

Vay f(x) - 0, Vx > 0

trang I

rlei 3

g diem)

Cho hinh chfr nhQt ABCD TrAn cdc itudng thhng BC vd CD, ldy tdn lw,qt cdcifiAm di itQng M, N sao c/ro ffi = 900 Ggi H ld hinh chi6u vuAng gdc ctia Atr€n MN Tiim qu! tich ctic iti6m H

M(a; ma)

B(a; 0)

DUng hQ trpc toa d0 Oxy nhu hinh ve v6i A = O(0;0).

Gi6 sri B(a;0), D(0; b), & ) 0, b > 0 vd phuong trinh duong th[ng AM ld

y = mx thi M(a; ma)

2) NOu rD = 0 : Khi d6 M = B, N - D Suy ra H eBD.

Vfly quy tich cdc di6m H la dudrng thang BD

Gtti chfi: Ndu thl sinh s* dung ki6n ththc itudng thiing Simson diSi vdi tam

gidc CMN vd di€m A dd gidi thi:

- Phdn thudn: 2il

- PhAn ddo: 2d

trang 2

Bni 4

G dieln)

Cho itudng *dng d iti,qua trgc tdm H cfia tam gidc ABC Ggi d1 ,

ld gdc itudng thdng ddi x*ng: vt6,i d qua BC, CA; AB Ch*ng minh

thdng & , dz, dj cl6ng quy

dz, dj lhn lwgtrdng ba dwd'ng

Ggi A', B', I' lAn luqt ld c6c <li6m ctOi xrtng cria II qua BC, CA AB thi

A', B', C' nim trdn du&ng trdn (O) ngoai ti-6p tu- gia AnC.

vi d cft it nh6t mQt tr"ong cic dudng theng AB, BC, cA nOn gi& sir d c[t

BC t4i E Gqi I ld gi,ro eti0m cira d1 vd d3.

N6uI=A'thi Ie(tr).

Xit I I'hdp dOi xftng tqrc IIC hi6n Id thanh iJi, phip dOi xring

trpc AI3 bi6n tr ilrdnh IrCt

DAt (BC,BA)_ er O6c tam gi6c A'IJC', IBI2

A'BC' thi A'IC'=2a.

Suy ra b6n ditim A', B, C', I dAu thuQc dulng trdn (O)

vi B'HI, - BI-IF = BC'I = BB'I n6n duimg thlng c16i xrrng v6i d qua AC

,lTLll P'I.VAv {f: it,d, cliing quy t4i I

so GrAo DUC vA nAo rAo rAv NII.IH

xY rnr cHeN rrec sINH ctor Lop 12 THpr voNG rixn

NAna Hoc zot4 - 2ors

NSi,y thit 24 th6ng 9 nf,m 2014

MOn: TOAN ru6i thi th* nh6t

Thli gian: 180 phft (khong k6 thdi gian giao dA)

of cnixn rrnlc

pi tnt gim c6 0I trang, thi sinh khdng phdi chdp di vdo giAy thi)

Bii 1 6 aiam)

Cho c6c s6 duong x,y, zthoi mdn tli6u kiQn x + y * z=1.

Chimg minh ,ing e - , L- * -=-l- + -;-]- < 3

3xz +y +z 3y' +z+x 3z' + x + y

Beri 2 6 aieml

Cho hdm siS f thoe m6n di6u ki0n f(x) +2f (xy)= f(x +Zxy),Vx e i , Vy e i

Chimgminhreng f(x+y)=f(x)+f(y),Vxe 1 ,Vye i

BAri 3 6 arcm1

Cho hai dudng trdn (O1), (O) cilt nhau tpi hai tti€m A, B vi PrP2 h mgt ti6p tuy6nchung cira hai ttuimg trdn d6 (P1 thuQc (Or), Pz thuQc (Oz)) Gqi Qr vd Qz lAn luqt li hinhchi6u vudng g6c cria Pr, Pz l6n dulng thing OrOz Dulng theng AQr cdt (Or) tAi ili6m

thf hai M1, Dulng thing AQ2 cit (Oz) tai diem thri hai M2 Chimg minh reng ba cli6m M1, M2, B thing hang.

Bifi 4 6 aiaml

BCn trong hinh vudng c6 diQn tich bAng 6, d[t ba da gi6c c6 diQn tich cung b[ng 3.

Chimg minh ring vdi ba da gi6c dE d{t trong hinh vu6ng, ludn tim dugc hai da gi6c midi€n tfch phin chung cria chring kh6ng nh6 hon l.

.{

- HET

-so GIAo Drrc vADAo rAo rAyNIi\H

xi, rnr cHeN Hec srNH cror Lop tz rHpr voNG riNn

NAvr Hgc zot4 -zots

i\gny thi: 25 thfng 9 nIm 2AL4

Mdn thi: TOAX -Budi thi thfr hai

Thbi gian: 180 phrit (khdng kA thdi gian giao dA)

Chring minh ring vdi n nguy6n ducrng vi n > 3 thi phuong trinh 4xn + (x+1)2 =yz

kh6ng c6 nghiQm nguy6n duong (x; y).

Bli 4 6 adml

Cho tam gi6c ABC c6 ba g6c A, B, C deu nhgn vd AB > AC Ggi D, E, F hn luqt ld

chdn c6c ducrng cao cta tam gifucABC vE ttr A, B, C xuiSng c4nh di5i diQn.Ducrng thlng

EF cfu BC t4i P, dudng thang qua D vd song song vdi ef c6t c6c dudng tfrang AC va Ae

tuong img tpi Q vn R Gqi M ld trung ei6m cpnh BC Chr?ng minh ring b6n Oi6* P, Q, R,

{/ri,

SO GIAO DUC VA DAO TAO TAY NINH

xi Tm CHQN HQC SINH cror LOP 12 THPT voNG riNn

NAvt Hec 2or4 -zors

TTUONG oAN cnAu rm vrON roAN - su6r rnr rrrtl NnAr

trang I

i;(1) vd (2) suy ra: f(x + y) - f(x) + f(y), Vx e R, Vy e IR

Bili 3

Q diam)

Cho hai tlulng trdn (O1), (Oz) cit nhau t4i hai Oi6m A, B vi PlP2li mQt fi6ptuy6n chung cira hai dudng trdn d6 (P1 thuQc (Or), Pz thuQc (Or).Ggi Qr vi Qz

fa" foqt n trintr chi6u vu6ng g6c cria Pr, Pz l6n tlulng tneng OrOz.Dudng

tning AQr cit (Or) t+i aiOr" tfr1i hai M1, Dulngtneng AQz cit (O2) tli aiam

thrri hai M2 Chrfrng minh ring ba di6m Mr, Mz, B thing hing.

(hl)

(h2)

trang 2

trang 4

so cIAo DUC YADAo rAo rAvNIi\H

xV rHr cHeN Hgc sINH cr6r Lop 12 rHpr voNG riNn

NAna Hec zor4 -zots i\giy thi: 25 thfng 9 nlm 20L4

M6; thi: ToAll - Buoi thi thfr hai

Thli gian: 180 ph fit (kh6rg kA thdi gian giao ai)

Chrlng minh rang vdi n nguy6n duong vd n > 3 thi phucrng trinh 4xn + (x+l)2 =y2

kh6ng c6 nghiQm nguy6n ducrng (x; y).

Bni 4 6 aidml

Cho tam girlc ABC c6 ba g6c A, B, C ddu nhgn vd AB > AC Ggi D, B, F lan luqt ld

chdn c6c dudmg cao cta tam gifucABC ve tt A, B, C xuiSng c4nh e16i diQn.Duong thing

EF cat BC tai P, dudng tfring qua D vi song song v6i EF c6t c6c dudng thing AC vd ABtuong img t4i Q vi R Gqi M ld trung eiem cpnh BC Chrlng minh ring b6n AiCm P, Q, R,

SO GIAo DUC VA DAo TAo TAv NINH

KY THI CHQN HQC jsrNu cror Lop 12 THpr voNc riNu

Vdi y=t-1 thi t3 -3t-52=0<>t-4<+x=1 Suy ray- 3.

Vfly hQ d5 cho c6 nghiQrn (1;3)

v0y Iimxn I

2

Bei 3

Q diem)

chri'ng minh ring vrfi n nguyGn duong vi n > 3 thi phuorng trinh

4x' + (x +l)2 = y2 khdng c6 nghiQm nguyOn duons i*, v) "

Gi6 st v6i n nguyCn ducmg vi n.) 3, phucmg trinh d6 cho c6 nghiQm

nguyOndu<rng (x;y)- Khi d6: 4xn = y, -(*+1)2 =(y_*_1)(V+x+t)

Vi y-x-l vd yj*+l cirngchEnho6c16;4x".hEnrren y_x_l vd

y+x+l cirngch5n

Eat y-x-l=2a thi y+x+l-2(a+x+l), suyra xn =a(a+x+l)

Do (a,a +I + 1) ; I n0n ton tai citc so nguyen u, v sao cho

a-un, a+x+1-vnrx-uv

Do n>3 n6n

uv+l = x+l = vn -un =(v_u)(vn-l *rn-2, + +.rn-l) > l+uv +u2 .

v6 lf vfly phucrng trinh dd cho kh6ng c6 nghiQm nguyEn ducrng.

trang 2

cho tam gif,c ABC c6 ba g6c A, B, c tldu nhgn vi AB > AC Gqi D, E, r B"

luqt lir chffn cfc dud'ng cao ciia tam gi{c ABC vE tri A, B, C xuiing-cirnh tISi

djgr

Pf.ls thr_ng EF c_it tiC tgi P, d'u'd'ng thirng qua D vi song song vrii EF

crt cic dud'ng thrng AC vir AB tuo'ng ri.ng tqri e vn R Ggi rvr re truig di6;

c?nh BC chring minh rrng b6n tli6,r F, qln, Nr cnng thuQc mQt rludng trdn.

Gqi M la trung di6m BC

Do BEe = gFC = 900 non b6n di0m B, c, E, F cirng thuQc mQt dudrng

Khid6: PB=p+a; DB=a+d; pC-p-a; CD =a-d; Dp=p_d

Thay vdo (1) thi dugc: (p + a)(p - a) = (p - d)p Suy ra a2 = dp

so GrAo DUC vA nAo rAo rAv NII.IH

xY rnr cHeN rrec sINH ctor Lop 12 THpr voNG rixn

NAna Hoc zot4 - 2ors

NSi,y thit 24 th6ng 9 nf,m 2014

MOn: TOAN ru6i thi th* nh6t

Thli gian: 180 phft (khong k6 thdi gian giao dA)

of cnixn rrnlc

pi tnt gim c6 0I trang, thi sinh khdng phdi chdp di vdo giAy thi)

Bii 1 6 aiam)

Cho c6c s6 duong x,y, zthoi mdn tli6u kiQn x + y * z=1.

Chimg minh ,ing e - , L- * -=-l- + -;-]- < 3

3xz +y +z 3y' +z+x 3z' + x + y

Beri 2 6 aieml

Cho hdm siS f thoe m6n di6u ki0n f(x) +2f (xy)= f(x +Zxy),Vx e i , Vy e i

Chimgminhreng f(x+y)=f(x)+f(y),Vxe 1 ,Vye i

BAri 3 6 arcm1

Cho hai dudng trdn (O1), (O) cilt nhau tpi hai tti€m A, B vi PrP2 h mgt ti6p tuy6nchung cira hai ttuimg trdn d6 (P1 thuQc (Or), Pz thuQc (Oz)) Gqi Qr vd Qz lAn luqt li hinhchi6u vudng g6c cria Pr, Pz l6n dulng thing OrOz Dulng theng AQr cdt (Or) tAi ili6m

thf hai M1, Dulng thing AQ2 cit (Oz) tai diem thri hai M2 Chimg minh reng ba cli6m M1, M2, B thing hang.

Bifi 4 6 aiaml

BCn trong hinh vudng c6 diQn tich bAng 6, d[t ba da gi6c c6 diQn tich cung b[ng 3.

Chimg minh ring vdi ba da gi6c dE d{t trong hinh vu6ng, ludn tim dugc hai da gi6c midi€n tfch phin chung cria chring kh6ng nh6 hon l.

.{

- HET

-so GIAo Drrc vADAo rAo rAyNIi\H

xi, rnr cHeN Hec srNH cror Lop tz rHpr voNG riNn

NAvr Hgc zot4 -zots

i\gny thi: 25 thfng 9 nIm 2AL4

Mdn thi: TOAX -Budi thi thfr hai

Thbi gian: 180 phrit (khdng kA thdi gian giao dA)

Chring minh ring vdi n nguy6n ducrng vi n > 3 thi phuong trinh 4xn + (x+1)2 =yz

kh6ng c6 nghiQm nguy6n duong (x; y).

Bli 4 6 adml

Cho tam gi6c ABC c6 ba g6c A, B, C deu nhgn vd AB > AC Ggi D, E, F hn luqt ld

chdn c6c ducrng cao cta tam gifucABC vE ttr A, B, C xuiSng c4nh di5i diQn.Ducrng thlng

EF cfu BC t4i P, dudng thang qua D vd song song vdi ef c6t c6c dudng tfrang AC va Ae

tuong img tpi Q vn R Gqi M ld trung ei6m cpnh BC Chr?ng minh ring b6n Oi6* P, Q, R,

{/ri,

SO GIAO DUC VA DAO TAO TAY NINH

xi Tm CHQN HQC SINH cror LOP 12 THPT voNG riNn

NAvt Hec 2or4 -zors

TTUONG oAN cnAu rm vrON roAN - su6r rnr rrrtl NnAr

trang I

i;(1) vd (2) suy ra: f(x + y) - f(x) + f(y), Vx e R, Vy e IR

Bili 3

Q diam)

Cho hai tlulng trdn (O1), (Oz) cit nhau t4i hai Oi6m A, B vi PlP2li mQt fi6ptuy6n chung cira hai dudng trdn d6 (P1 thuQc (Or), Pz thuQc (Or).Ggi Qr vi Qz

fa" foqt n trintr chi6u vu6ng g6c cria Pr, Pz l6n tlulng tneng OrOz.Dudng

tning AQr cit (Or) t+i aiOr" tfr1i hai M1, Dulngtneng AQz cit (O2) tli aiam

thrri hai M2 Chrfrng minh ring ba di6m Mr, Mz, B thing hing.

(hl)

(h2)

trang 2

trang 4

so cIAo DUC YADAo rAo rAvNIi\H

xV rHr cHeN Hgc sINH cr6r Lop 12 rHpr voNG riNn

NAna Hec zor4 -zots i\giy thi: 25 thfng 9 nlm 20L4

M6; thi: ToAll - Buoi thi thfr hai

Thli gian: 180 ph fit (kh6rg kA thdi gian giao ai)

Chrlng minh rang vdi n nguy6n duong vd n > 3 thi phucrng trinh 4xn + (x+l)2 =y2

kh6ng c6 nghiQm nguy6n ducrng (x; y).

Bni 4 6 aidml

Cho tam girlc ABC c6 ba g6c A, B, C ddu nhgn vd AB > AC Ggi D, B, F lan luqt ld

chdn c6c dudmg cao cta tam gifucABC ve tt A, B, C xuiSng c4nh e16i diQn.Duong thing

EF cat BC tai P, dudng tfring qua D vi song song v6i EF c6t c6c dudng thing AC vd ABtuong img t4i Q vi R Gqi M ld trung eiem cpnh BC Chrlng minh ring b6n AiCm P, Q, R,

SO GIAo DUC VA DAo TAo TAv NINH

KY THI CHQN HQC jsrNu cror Lop 12 THpr voNc riNu

Vdi y=t-1 thi t3 -3t-52=0<>t-4<+x=1 Suy ray- 3.

Vfly hQ d5 cho c6 nghiQrn (1;3)

v0y Iimxn I

2

Bei 3

Q diem)

chri'ng minh ring vrfi n nguyGn duong vi n > 3 thi phuorng trinh

4x' + (x +l)2 = y2 khdng c6 nghiQm nguyOn duons i*, v) "

Gi6 st v6i n nguyCn ducmg vi n.) 3, phucmg trinh d6 cho c6 nghiQm

nguyOndu<rng (x;y)- Khi d6: 4xn = y, -(*+1)2 =(y_*_1)(V+x+t)

Vi y-x-l vd yj*+l cirngchEnho6c16;4x".hEnrren y_x_l vd

y+x+l cirngch5n

Eat y-x-l=2a thi y+x+l-2(a+x+l), suyra xn =a(a+x+l)

Do (a,a +I + 1) ; I n0n ton tai citc so nguyen u, v sao cho

a-un, a+x+1-vnrx-uv

Do n>3 n6n

uv+l = x+l = vn -un =(v_u)(vn-l *rn-2, + +.rn-l) > l+uv +u2 .

v6 lf vfly phucrng trinh dd cho kh6ng c6 nghiQm nguyEn ducrng.

trang 2

cho tam gif,c ABC c6 ba g6c A, B, c tldu nhgn vi AB > AC Gqi D, E, r B"

luqt lir chffn cfc dud'ng cao ciia tam gi{c ABC vE tri A, B, C xuiing-cirnh tISi

djgr

Pf.ls thr_ng EF c_it tiC tgi P, d'u'd'ng thirng qua D vi song song vrii EF

crt cic dud'ng thrng AC vir AB tuo'ng ri.ng tqri e vn R Ggi rvr re truig di6;

c?nh BC chring minh rrng b6n tli6,r F, qln, Nr cnng thuQc mQt rludng trdn.

Gqi M la trung di6m BC

Do BEe = gFC = 900 non b6n di0m B, c, E, F cirng thuQc mQt dudrng

Khid6: PB=p+a; DB=a+d; pC-p-a; CD =a-d; Dp=p_d

Thay vdo (1) thi dugc: (p + a)(p - a) = (p - d)p Suy ra a2 = dp

sO crAo DUC vA DAo TAo rAv Nrr\H

Ky rHr cHeN Hec srNH Gror Lo'p 12 THpr voNG rixu

NAtu Hoc zot3 -zot4 I\giy thi: 26 thftng 9 nf,m 2AI3

MOn thi: TOAN BtrOi ttri thir haiTIrd'i gian: 180 phut (khong ke tho,i gian giao dA)

Giai hQ phucmg trinh:

vol rnoi n nguyOn clucrng.

Tim limu,,

Tim t6t cA cdc s6 tq nhi0n m sao cho v6i n ld mOt s5 tU nhiOn nAo rlri, ta cci

mn =1(modn) thi m=1(modn).

Biri 5 g aie4

Cho tam gi6c dOu ABC canh a M vi N ld hai cli6m di dQng lAn lucrt tren hai canh AB vd

AC sao .ho AM * AN = l.

MB NC

a) chring minh ring MBCN la tf gi6c ngoai ti6p dugc m6t dudng trdn

b) Tim gi6tri lcyn nh6t cria diQn tich tam gi6c AMN theo a.

Chimg minh ring v6i moi sO nguyOn ducrng n Z 2 thi ,n , ,6.

Bei 4 & aiam)

s0crAo DUC vAEAo rAo rAy NrNH

rY rnr cHeN Hec jsrNH cr6r lop 12 Trrpr voNG riNn

EAt t- W.t>o thi x'=tz -2x-2.

Phucrng trinh dd cho tucrng ituong v6i t2 - (Z*-:)t - 4x + 2 = 0

Giai phucrng trinh tr6n tim dugc t - -2 (lopi) , t = 2x: l

Vdi t - 2x- I thi dugc phucrng trinh

W:2x-1<+ 3x2-6x-1-o (do 2x-1>o)

Giai phuong trinh trOn tim dugc nghiQm

3>0 n6n x>3=+y> -2=)x<3.V6 lli

<3.

at (3;-2) .

trang I

Bni 4

@ diem)

Tim tdt cd ctic sd 4r nhihn m sao cho vdri n ld mQt sa qt nhihn ndo itri, ta cd

mn = I (modn) thi m: I (modn)

N6u m =2,tac6 2" =1(modn) yoi -oi n >t.Z

Gi6 sti p ld u6c nguy0n t6 ntrO nhdt cria n, khi dO Z":1 (modp) (*) . 0,5

Tt (*) ta c6 p ld s0 16, suy ra n li sO le, do d6 (n, p-1) = 1 0,5 VO,y t6n tai cdc so nguy0n d,b sao cho an + b(p - l) - 1 0r5

Theo dinh lf Fermat nh6 thi: 2t 2nu.2@-r)b : I (modp).V6 lf. 0r5

N6u m > 2,ta"6 *(m-l)' =[1+(m-1;1('-rt2 = 1 (mod(m-t)r)

MAt khSc m/l (mod(m -t)') Bdi to6n kh6ng tho6 man.

a) Ch*ng minh rdng MBCN ld tir gtdc ngogi ti6p itwgc mQt itwdng trbn.

b) Ttm gid tr! td,n nhdt crta diQn tich tam gidc AMN theo a.

XA,

,A

B

a) Dat AM = x, AN = y Ggi E, F l6fl luqt ld trung rti6m cria AB, AC

ri, **g = 1 suy r" {Y:Y vay M, N ran ruqt thuQc c6c do4n

ngoqi.ti6p dugc Dudng trdn nQi ti€p tti gi6c MBCN ctng ld <tucrn! trdn

nQi tiep tam gi6c ABC