Tạp chí toán học và tuổi trẻ tháng 10 năm 2015 số 460

xudT siil rUrgo+ 10 39; p, rnp cx[ nn uAruc rHAruc ryAM rnU 52 oAnru cHo rRUNG xoc pnd rxOruc vA rnuruc xoc co s6 Tru s6 1B7B Gi6ng Vo, Ha NOi DT Bi€n tdp (04) 35121607; DT Fax Ph6t hdnh, Trisu (04)35'121606 Email toanhoctuoitrevietnam@gmail com Website http //www nxbgd vn/toanhoctuoitre {;ffi*i l(j, ] ,/ * 8 ,& HÃY ĐẶT MUA TC TOÁN HỌC VÀ TUỔI TRẺ ! HÃY ĐẶT MUA TC TOÁN HỌC VÀ TUỔI TRẺ ! TRfING [f rongqua trinh gihitoin ngodi viQc n6m virng a, ki6n thuc co b6n chring ta cdn phii biOt vfln dUng mQ[.]

xudT siil rUrgo+

rnp cx[ nn uAruc rHAruc - ryAM rnU 52

oAnru cHo rRUNG xoc pnd rxOruc vA rnuruc xoc co s6

Tru s6: 1B7B Gi6ng Vo, Ha NOi.

DT Bi€n tdp: (04) 35121607; DT - Fax Ph6t hdnh, Trisu: (04)35'121606 Email: toanhoctuoitrevietnam@gmail.com Website: http://www.nxbgd.vn/toanhoctuoitre

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[f rongqua trinh gihitoin ngodi viQc n6m virng

a, ki6n thuc co b6n chring ta cdn phii biOt vfln

dUng mQt sO phuong ph6p gi6i toan <l{c biQt Vi du

nhu phuong ph6p phar.r chimg, phucrng ph6p OuV

nap to6n hgc, nguy6n tic Dirichlet, nguydn tdc qrc

h4n, 6 bdi vi6t niy chung t6i mudn trao tl6i th6m

vcri ciic b4n vC nguyCn t6c qrc han T6i n6i "th6m"

ld vi vdn il6 ndy ctng il6 c6 mQt s6 tac gi6 dC cQp

rl6n trong c6c sti T4p chi tru6c dAy C6 nguoi gqi

tl6 ld nguy6n tdc "Khoi ddu cqtc tri", nguyAn tdc

qrc hqn, cdn theo tdi thi c6 l€ goi ld quy 6c "bi6n"

ld hOp.li ! Vi quy t6c ndy khuy6n chung tg khi gi6i

mQt s6 c6c bdi torln thi hdy nhin c6c y6u t6 rQng ra

b6n ngodi (bi6n), vi dpd6i voi mQt dopn thlng ta

n6n quan tdm d0n hai rlAu mrit cira n6, ddi vtri mQt

t{p hqp s6 ta n6n quan tdm d6n c6c gi6 tri lon nhAt

hay nho nhat, .

Say ildy ta sE xem quy tic ndy rlyOc 5p dpng nhu

thd ndo th6ng qua cbc tht dg cr,r th6.

Thi dB l TrAy dudng thiing cho mQt tQp diem

M sao cho m6i diAm ctia M ld trung di€m c[.ta

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mQt doan thdng noi hai di€m khac thuQc M

Chrtng minh riing M ld m6t fip hW vd hqn.

Ldi gidi Gid su ducmg theng dA cho rtflt nim

ngang vir gid srt, M h t$p hqp hffu h?n

Khi d6 trong s6 c5c diOm ci;a Mphiti c6 mQt cli6m

:^

ndm t4n cung ben tr6i (bi6n tr6i), gqi di6m d6 ld,4

R5 rdng,a khOng tn6 U t *g ditim qia dopn thing

nOi hai dii5m kh6c oba M, suy ra Ac.M Di6u ndy

tr5i voi gi6 st Yay Mliltap hqp vd hqn.

Thi dU 2 TrAn mdt

phting cho m1t tQn dily

M sao cho m6i di€m

cria M la trung dihm

cila mQt dogn thdng n6i C

hai di€m khdc thuQc M

Chang minh rting M ld

mQt tQp hqp vd hqn.

Ldi sini ft.1) Giesu MliLt6a hgp nry han.v) ulit

tap htu hpn n6n sd c6c doan th5ng ndi hai di6m bdt

IUY IAE "BIEII"

NGUYfN XTTANTT NGUYTN(3l2gE,Dd Ning, Hdi Phdng)

ki thuQc M cfinghiru han, do d6 phii t6n tai m$t

cloqn thAng c6 ctQ dai l6n nh6t, gi6 su d6 lit doan AB.Theo dO Uai ttri B phhild trung <1ii5m cua cloan thdng

CD ndo d6 (C,D.M) D6 chrmg minh du-o.c:

N6u lB I CD thi AC: AD> AB;n5'tABLh6ng

rudng g6c vcri CD tlll IEc t;ho1rc FED tu, khicl6 ho[c AC > AB,ho{c AD > lB, di6u ndy tr6i voiviQc chgn doanAB.YQy MldIQp hqp vd hAn.

Thi du 3 Tr€n mdt phting cho n du'dng thiing

(n>3), trong dd kh6ng c6 hai furdng ndo songsong, kh6ng c6 ba dadng ndo d6ng quy, cac

dfing thdng ndy chia mQt phdng thdnh nhttng

phdn Chang minh rdng n€u ldy bdt ki du'dngthdng ndo tr.ong s6 do thi,bao gid cilng c6 phdn

mQt phang k€ voi drdng thdng dd c6 hinh tam gidc.Lin gidi Ggi c5c <luong theng da cho ld dt, d2, ,d,.Xetcluong thdngdl,*tatcacbc gtao diOm cua

c6c dunng thlng cdn lai, ta chen ra mQt giao ili6mc6 khoang c6ch tltin dlr1hb ntr6t Cia su d6ld giao cli6m cira hai dudng thing d2vd dt.DE dang chimgminh <lugc hinh tao boi ba cluong thlng dy, d2,il,ld

tamgi6c th6a mdn ddl bei.

Thi dlt 4 Tr1n mar ph,ang clo n dwrng thdng

(n 2 3), hai dadng thdng bdt ki trong n dadng

,: -, -; J ,

thdng do d€u cdt nhqu vd qua m6i giao di€m ddcdn co thAm ft nhAt mot daong rhang khdc diqua Chang minh ring tdt ca n dadng thdng ddd6ng qaty.

trong c6c rli6m d6 Trong c5c dudng thdng kh6ng

cli qua K, giit su ft ld cludng th5ng c6 khoang c6ch

d€n Kld nh6 nhAt Gid su quaKc6 ba tluong thlng

cit duong lJrrdng h tqi A, B, C (theo thi t.u d6) Theo

dC bdi thi qua B cdn c6 mQt cluong th5ng nfia, gi6

sir il6 ld tluong thing k, cluong th5ng ndy c\t t<C

(holc KA) tai D R6 ring D phni h diiSm fong cria

KQ (toac KA) vd.khi d6 dC th6y D eM dy*g

thlngh h*5- <ti6u ndy tr6i voi viQc chgn di6mK

vd cluong thing h,ti dbtac6 <li6u phai chimg minh.

Ti aF 5 Chilmg minh ring trong mQt ngti giac

l6i bdt ki lu6n tim duoc ba du'dng chdo cd dQ

ddi la ba canh cua m6t tam gidc

Ldi gidi (h.3)

B

E

Ilinh3

Giit stt AD ld duong ch6o 16n nh6t cta ngfl gi6c 16i

ABCDE,Ia sE chimg minhbatludng cheoAD,AC,

BD tAo thdnh mQt tam gSdc Nghi, ld ta chi cAn

"hgrg minh b6t dinglilrrttc AD < AC +BD ld xong.

DiAu ndy hodn todn hi6n nhi6n dlra vdo U5t Oeng

thic tam gi6c.

Thi dB 6 Tr€n mfit phdng cho n didm @ > 3),

bi€t rdng diQn tich cila tam gidc biit ki voi ba

dtnh trong n didm d6 kh6ng :: l6n hon l Chtrng

minh rdng tdt ca n di€m dd cd th€ chtra trong

m6t tam gidc cd diQn tich biing 4.

Ldi gi,fii (h.4) Xdt tilt ca

cilc tam gi6c c6 dinh tai n

di6m dA cho, rd rang s6

tam gr6c d6 h hfru han, do

d6 t6n tai mQt tam g;6c c6

diQn tich lon nh6t, gi6 sir

d6 ld tam giilc KLM Qua

K vE tluong thrtng d ll LM N6u N h ditim nim

kh6c phia Z so voi d th\ SNLM) Sxtu, do <16 dt cd

c5c diiSm d5 cho phdi nlm cirng mQt phia c16i v<ri

clu<nrg thdng d Tuong V qtra L vit M vEc6c tludng

thlng I il KM, m ll KL tlltth cd n di6m da cho phii

nim trong (ho{c f6n cpnh) cira tam gifuctao boi ba

tluong thing d,l, m.Rd rdng tam giSc nay c6 diQn

tich blng b6n hn diQn tich tzrrn gSbc KLM.Tir d6 ta

c6 didu ph6i chrmg minh.

Z clirdifuGtacotlo-2olo_

Thi dU 7 GiA sw O la mqt didm ndm trong m6t

da giac.l6i ndo d6, ti O hq cdc dudng vu6nggdc xu6ng cdc canh crta da giac Chung minhrdng lt nhdt c6 chdn cira mQt trong cdc dadngvu6ng gdc d6 ndm ftAn mQt cqnh cia da gidc(chu kh6ng phdi ndm tuAn phdn kdo ddi ctia no)

Ldi gidi (h.5) Trong c5ckhoang cSch tu O d€n cac Dcqnh cria da giirc, .gii sir c

khoang cSch tu O d€n czuth

AB ld r1h6 nh6t (d6 ld cluongvu6ng g6c OQ Ta sE chimgminh <li6m E phiti thu6c canh AB GliL sri E nim

ngodi c4nh AB,khi d6 OE phhi cht mQt trong c6c

canh cira dg *itc lai G DC thdy OF < OG < OE, ngtria ld di6m O gan canhBChon ld canh,4B Didundy tr6i voi viQc cho.n canh AB, ti clO ta cO diOu

ph6i chimg minh

Thi dU 8 TrAn mqt phdng cho n didm @ 2 3),

trong d6 kh6ng c6 ba di€m ndo thiing hdng Chilmg minh rdng t6n tai mQt dudng trdn di qua

3 di6m trong s6 cdc di6m dd cho md kh6ng chuatrong n6 di€m ndo trong sd coc di€m cdn lqi.Ldi girti Vi sd khoang c6ch gifia hai cli6m fong ndi6m dA cho ld htu han n6n ta chgn hai ili6m c6

khoang crich nh6 nhdt, gA sir hai rlitim d6ld A, B.Tri n - 2 di6m cdn lai ta nOi diin A vd B Trong s6

c6c g6c tao thdnh nhin <loqn AB, ta chgn ditim mi

tu d6 nhin do1nAB duoi g6c lcyn nh6t, gin su <16ld

di6m C (de V *g 7Cn <600 ) Oudng trdn cli quabadi€mA, B, Clddudng trdn phdi tim ThAt vdy:' N6u c6 mQt diiim D nim fong dudng trdn vdctngphiavcri Cfln IDE TFdE (v6lf)

' NCu D nim trong clucrng fdn vd kh6c phia voi Ctll1 ADB ld g6c tu, do 116 AB > AD vd AB > BD(vO lf) Vdy ac6 di6uphii chimg minh

Thi dtr 9 Cho bdy yO nguy€n daong khdc nhau

a1,, a2, , a7 cd tdng bdng 100 Chung minhrdng trong do cd ba s6 c6 t6ng kh6ng nhd hon 50.Ldi gini.Khdng m6t.tiot tdng qu6t gibsA q< a2< 1a1 Ta c6n xdt t6ng ba sO l6n nhAt vd sE chimgminha5* aa* q > 50 Th{tv4y:

' N6ua5 > 16 thi au) 17,a, ) l8,do d6 as * at> 76+17+ 18:51.

a61-'N6uas < 15thi aq*azaaz+at < 74+13+12+ 11 : 50, khi il6 ta cfing co a5 + a6 * a7 ) JQ.

Nhu vdy ta c6 cli6u ph6i chimg minh

Hinh 5

Hinh 4

Thi dq 10 TrAn mQt furrlng trdn cho n s6 ta

,.; : :

nhi1n, bi€t rdng moi s6 biing trung binh cQng

cua hai sii ki voi n6 Chang minh rdng ttit cd

,,:

cdc so tr0n dLrong tron ditt bung nhctu.

Ldi giili f6t ntri6n tongn sO t.u ohi€, tl6 phii c6

sO ntrO nh6t, gqi sO AO U m Gid sir hai s6 k€ vor m

ld a vd b, theo Ae UU h c6 a + b :2m (l) Vi

a) m,b) m, nOn n6u mQt trong hai bAt dang

thuc d6 thlrc sg lon hon thi sE c6 a + b > 2m, di€u

ndy tr6i voi (1) V{y phdi c6 a : b : m.

Li lu4n flrcvng t.u ta clugc tAt ce clc sd tr6n duong

,: ,:

rondeu oangm.

Thi dtr ll TrAn cdc 6 ctia bdn cd'hinh vu6ng n

x n dQt nhirng cai thap vdi diiu ki€n: Ndu c6

;mQt 6 nao do trong thi t6ng s6 thap ddt trAn

dadng ndm ngang va du'dng thdng ang qtra 6

d6 kh6ng nho ho'n n Cht?ng minh rdng tr€n bdn

2

cn do cd khdng ir ho'n \ thap.

Ldi gini Xdt mQt tongn ducrng ndm ngang vd n

rtuong thing climg md trong d6 c6 it th6p nhAt Gi6

su cl6 ld tludng nim ngang vd tr6n d6 c6 kthap

' Ntiu k > * ,2 tti s6 th6p tr€n bdn c<r kh6ng it hon

c6 n - fr 6 trdng vd m6i cQt cli qua c5c 6 tr6ng ndy

c6 khdng ithon n - k thdp (theo d0 bdi) n€n t5t ci

c6c cQt nhu vQy sE c6 kh6ng ithon (n-k)' Uap

Cdn lai k cQt, m5i cQt c6 khdng ithon k th6p Do

d6 At ca sO ttrap c6 fOn bdn co kh6ng it hon

("-k)' +k2 Ta chi cdn phii chimg minh rlng

2

(n-k)'+k'>+ \/,2 Rd rang BDT ndy ludn thing vi

I :2

n6 hrong duong voi b6t ddng " thric z[ [z * - t ) ) >0

- NOu n h s6 chBn thi ta c6 th6 ddt cic th6p hodn

todn trong c6c 6 den ho[c cdc 6 tring

-Niiu r ld sd le thi ta sE dflt clu-o c Ll tna,

bdng cSch: MQt th6p cl{t o mQt trong c5c g6c, cfrc

th6p cdn lpi clfi 0 trong c5c 6 ctng mdu.

Thi dg 12 Chilng minh riing khdng t6n tsi cdc

x' t y' c6 giri tri nho nh0t Gi6 su b6n sd <16 lir

a, b, c, d Tri ddng thitc ct2 + b2 :3(c2 + *1 suy

raa'+ b2 i 3 > oi3 vd bi3

Df;t a:3m, b :3n (m, n e Z* ), thay vdo dingthric tr6n ta duo c gmz + grf :3(i + *1tay C + d

- 3(m2 + n'),ngjfiald ta lai tim tlugc b0 bon s6 c, d,m,nth6amSn (1) d6ng thoi c'+ d < i + 02 Oi6undy tr6i voi viQc cho n c6c s6 a, b, c, d Ti d6 ta c6 diCuphni chtmg minh

O C6c ban thAn m6n! TrOn dAy 1d m6t s6 thi du minh hoa cho phu'ong ph6p v4n dgng ngrry6n tlc cgc han vdo vi6c giai m6t.s6 bdi toan md n6u kh6ng c6 phuong phdp

ndy chic chdn chirng ta sE glp kh6 lCrdn Cu6i ctutg rr,ong c6c ban 6p dmg phuong phSp ndy d6 luy6n tAp

gi6i c6c bAi to6n sau, chric.c6c b4n thdnh c6ng:

l.Cho MldtQ,p hgp c6c s6 thqc kh6c nhau, bi6t rang m6i

sd cda M ld trung binh c6ng cua hai s6 nio d6 ctng

thuoc M Chtmg minh ring MliLAp hqp v6 han.

2 TrOn mat phdng v6 han _c6 !e 6 w6ng nguoi ta vi6t

cric sO t.u nhi6n sao cho rn6i s6 ld trung binh cQng cta, : : ,^ ^ ;

bon so lan cin (so trcn, s6 du6i, sd tr6i, s6 phrii) Chung minh rdng tAt ci cdc s6 tr6n m{t phing d6u bdng nhau.

3 C6 2n + I .qua c6n, kh6i luong rn6i qud ld mQt s5

nguy6n ei6t rang fu 2n qud tuy y bao gio ctng c6 th6 chia duoc thanh hai nh6m c6 kh6i luong bing nhau, m6i nh6m.c6 ir qu6 Chirng minh r5ng tit ch circ quri can d6

c6 trrh6i lu'ong bing nhau.

4 Chrmg minh ring phuong trinh x2 + 1? + / - Lryz kh6ng c6 nghiQmnguy6n duong.

5 Chtmg minh rdng trong mQt luc gi6c 16i 1u6n t6n tai mQt dinh md ba ducrng ch6o xu6t ph6t tu dinh d6 ld d9 ddi ba oanh cua mQt tam gi6c.

6 Trdn mflt phdng cho r di6m (n > .3), bi6t rlng m6i ducrng thdng di qua hai di6m trong s6 d6 cdn di qua it, ; ^ ,.; ; ^

nhdt mQt diem trong so n - 2 di6m con lai Chimg minh

.*g At ca c6c di6m dd cho cirng ndm tr6n mQt duong thdng.

7 TrCn mflt phdng cho 2n di6m, kh6ng c6 ba di6m ndothlng hang, hong d6 c6 r di6m xanh vh n di6m do.

Chimg minh rdng c6 th6 n6i m6i di6m xanh vcri mQt

di6m do b[ng mQt do4n thdng sao cho trong r doqn

thing nh4n duo c.kh6ng c6 hai dogn ndo giao nhau.

8 Chrmg minh ring n6u m6t da giSc c6 nhi6u truc d6i ximg thi c6c truc ddi xfing d6 phai d6ng quy.

te n.o,ro-roru, T?[ilrHS

3

ioE THI ruyEru SINH vAo LOP 1o cHUyEu

e 5(m2 -3m+3)-(2m-4)2 -5:0 b) Ggi qu6ng ducrng AB ddild x (km), thoi gian

2 , A _ I * =2 (loai) tu ltc xe t6i xuSt ph6t d6n hic xe khSch xu6tem'+m-6=U v 'r' <>ll*:-3 (nh4n) phht.ldv (gid) (DK:x > 0,y > 0).

trinh * =y+i (1) Thdi gian thuc t6 xe

Ciu2 a) Bpn i19c t.u lim khrtch di: + +++.* (f,), ta c6 phucrng hinhb) Phuong trlnh hoinlr clQ giao cliem cua (P) vd 2 5U 2 b0 '

Vpy x : I ld gi|tri cAn tim.

L' =l+4n? >0 v6i mgi m <) (*) c6 hai nghiQm Tii (i) vd (2) ta c6 hQ phucrng trinh

l, L - u'r"'^ \/A,,^,,;-^,*,,) ^ tDAx:l;rrn,.*(thoamdn)'

LABC c6 BD, CEldhaitluong cao cltrrhaut4i H

=H ldtr.uct6mciratam giilcABC =AH LBC.

Ta c6: BEH+BIGI =90o+90"=180'= th gi6c

BEHK nQi tirip

=EEE =fu M[t kh6c6dq,=60)=90o = tb gi6c ABKD nQi tiiip

+ EBH = AKD Do <16 EKH = AKD (= EBH)

hay KA ldtiaphdn gi6ccuag6cEKD

b) AI, AJ ld cfuc titip tuytin cira dudng trdn (O)

)AILOI,AJLOJ vit AI: AJ Ta c6

AIO = AJO = AKO = 90o + I,J,K ctng thuQc

tluong trdn <luong kinh AO =I,J,K,A,O

cirng thuQc mQt tluong tdn Xdt <luong tdn tluong

l<rnhAocn AI = AJ =7i =D =Ffu =fu.

Md AKE = AI{D (chung minh t6n) v{y IIG=DKI

c) MEH cn MKB(gs) ,/ AK = 4+ = 0,4^AB

=AE.AB=AH.AK Xet LAIE vd MBI c6IAE (cfu,xry), AIE = ABI + N4IE @ L,4BI (g.g)

- l!- AB = A!= AI - AI2 = AE.AB Ta co

AH.AK = Ar2(= AE.AB) *#=ff=

MIH a MKI (c.g.c) =frt:aiR Tuong

w fu =IiR Md fik *ffr = 18oo (tu gi6c

AIKJ n}i tii5p), do d6 fri +frr =180" VQy

ba rliilm J, H,I thinghing.

d) Chrmg minh nrong t.u cdu a) Iht EC ld tia phdn

gShc g6c KED Ta c6 KSE = DEC (so le tong),

frE = fEs g diq 3 aKrs can tai K

= KS = KE MAtkh6cKQE + KSE = KEQ+ KES (= 90"), do tl6

@E = frO = LKQE cdn t4i K= KQ = KE.Yqy KQ= KS (= KE)

VONG II

Ciu 1 a) (3x+ 1)(4x+ 1)(6x+1)(L2x +l) = 2

a (36x2 +15x +L)(24x2 +lIx +l) = 2.

Dat ! =24x2+10.r+l +36x2+15x+l =1r-l

' V6i x =I-JZ thay vio (*) suy ra y - J1-t.

Thu I rrs lai ta th6v {x =l+ rs' Ji - {*:r-a ld

lY =-t-J2' U = J1-t

Pr da cho tro thhnh: (|r-;), =,

e3y'-y-4=0 Gi6i PT niy tim dugc y, tu

d6 tim ctuqc nghiQmx cria PT <16 cho ld:

C0u 2 a) 3x2 -2y2 -5xy+x-2y-7 =0

e3x2 -6*y+ xy-2y' + x-2y =J

e 3x(x - 2 y) + y(x - 2y) + (x - 2 y) = l

e (x-2y)(3x+ y+1)=1.

Do d6, ta c6x-zy

Yi x,y eZ ndnclic6x= 1 vdy::j th6am5n(*).Thir 14i, ta thdy nghiQm nguy6n cria PT ild cho

ld (1; -3)

b) Voi moi n e N*, ta c6: 20124" i4;20144":.4,

20134':(20134'-f')+l chia cho 4 du 1 (vi

(20134" -f\iQot3- 1) i a ) vn20t54' :lz}tso' -(-1)e]+1 chia cho 4 du I

1vi [zot so' - (-1)4'

nft rnr 1rryeu srNH yAo r-,dp +o NAMHeg20ls-20r6

rqtfdnrc pn6 rn0ruo NANG rynfu Dflmgc Qudc Gn Tp nd cni mnn

Ciu 4 Q diAm) a) Cho m6J tam giric vu6ng NOu ta

tnng d0 dai m6i canh q6c vudng th€m 3 cmttri diqn

tich t5ng th6m 33 cm'; n6u gram dO dai mQt cqnh

g6c vu6ng di2 crr vd ting d0dai canh g6c ru6ng

VONG rr {t50 ptuir)

cdn lqr them I cm thi diQn tich gram di 2 crrf .Uay

tinh d0 dai crlc c4nh cua tam giric vudng.

b) Ban An fu dinh tong khoang thoi gian hr ngiy

lB denngity 3014 sE gi6i m6i ngiy 3 bii torin Thuchien dung k€ hoqch du-o c mQt thoi gian, vdo khoangcu6i thring 3(thang 3 c6 31 ngdy) thi AnbibQnh,p!fi ,g1o giei (oan nhi6u ngdy liCn tifu Khih6i phqc, trorrg tu6n cldu A"

"Fgiai duqc 16 bdi; sau

d6 An c6 g5ng gtai 4 bdi m6i ngdy vi d6n 3014 rL:rt

An dmghoan thanh k0 hoach de dinh H6i An phii

nghi giai toag it nh6t bao nhi€u ngdy ?

COu 5 Q diAm).Innh binh hi.nh ABCD c6 tarn gritcADC nhq\ trDi =60" Euong tdn tAm O ngoqitiep tam grirc ADC clt cqnh AB t4i E (E + A), AC c1t

DEt4iI

a) Chrmgminh tffingi&Bc4tl6uvd IO I DC

b) Gqi K li fung dl€m BD, KO cit DC tqi M.Chnng minh A, D,,4d 1 cung thuQc mQt ducrng trdn c) Ggi.rh t6m duong trdn ngopi tiep tam grirc ABC

b) Cho circ sO a, b th6a man di6u kien

tJo +1[6 = Chrmgminhre"g -1 <a<0.

CAru2 Q diem) a)Tim c6c s6 nguy6n a, b, c sao cho

a * b * c : 0 vd ab -r bc + ca + 3 : 0.

b) Cho mld s6 nguy6n Chtmg minh r6ng n6u tdn

t4i cdc s6 nguy€n a, b, cldtilcO sao cho a * b * g : g

vd ab + bc * ca 't 4m :0 thi cflng t6n tai cdc s6

nguy€n a',b',c' khac 0 sao cho a'+b'+c' =0 vd

a'b'+b'c'+c'a'+m:0.

c) Voi k ld sd nguyfu duong, chrmg minh rlng

klr6ng t6n tai cric sO nguy6n a, b, cl<hfuc 0 sao cho

a* fi + s : 0 vd ab + bc + ca +2k = 0.

Cflu 3 (1 diem).Gibsriphuongtinh

2x2 +2ax+l-b:0 c6 2 nghiQm nguyfu (a, b lir

tlram sO) Chrmg minh r5ng a2 -bz +2 ld sO

nguy6n vd kh6ng chia h6t cho 3.

Cfiu 4 Q diem) Cho tam g16c ABC (AB < AQ c6

Mldt:urrrgcli6m cua cqnhBC,E ld di6m chinh gita

cira cung nh6 BC, F ld dilmAOi Xmg cin E qt:r M

a) Chrmgrninhreng EB2 =EF.EO

t) C-oj D ld giao dr€m cir AE vd BC.Chtmg minhcilc drdm A, D, O, F' cing thu6c mQt duong tdn

c) Gqilh tem dutrng tdn nQi tiep tam gi6c ABC vir

P ld di6m thay d6i t6n tludmg trdn ngoqr ti.p t *

g6c IBC sao cho P, O, F th}ngttrEng hang Chungminh reng tiep tuy0n taiP cuailuone trdn neoai ti6p

tam g)Ac POF di qua mQt tli&n cd dinh.

Cflu 5 Q di€m) EC khuy6n khich phong fdo hgc t4p, m6J fuong THCS <tE td chric 8 dqt thi cho cac

hgc sinh d m6i dgt thi, c6 dring 3 hgc sinh <lu-o cchon di5 fao giai Sau khi td chirc xong 8 dgt thi,

ngu<ri ta nh6n ra r5ng voi hai dqt thi b6t kj, ludn codung t hgc sinh dugc trao gii 6 ch hai <lqt thi d6.

cHUAn BlCHO KI THITRUilG HoC

pnd ruOrcoudc om

di todn ri t

"p nryiln va ai tni nam sA aa

du c gioi thiQu so qua tir sU A]loi dau caa

phdp t{nh dqo hdm khi hgc 6 chrong trinh l6p ll,

tidp sau do 6 lop 12 chilng ta dd dtto c hec trpn vgn

; , ,,:,

vA cdc th€ loqi cua nd thdng qua cdc c6ng a1 khdo

sdt hdm s6, trong bdi vid nay xin trinh bay lqi vAn

-; ^ ,J

d€ trAn, bdng mQt vdiphwngphap at th€.

Cho ham s6: y ="f(i (q vd di6m u(n; f@)).(Q.

Khi d6 ti6p tuy6n vdi d6 thi (q @i M c6

phuong trinh (PT) ld: y - f (x) = f'(x)(x- x).

Thi dt1 l Cha ham s6: ! = x2 -4.v+3 (C) ttiit

Pt- n,p tu1:sn t',;'i (C) rqi t'uc giut, diin cuu

{C) r,cri tmc hoinh

Ldi gi,rti PT hodnh <td giao <litim gita (Qvn

truc hodnh lit x2 - 4x +3 = o <> [' =

"f (x) (C) Vi6t PT ti6p tuytin

v6i (C) c6 hO s6 g6c k cho tru6c

Phuong phfp 1 (Tim ti€p tuytin bing cSch tim

hoinh dQ ti6p itiCm)

Budc l Giai PT f'(*r)=k d€ tim hodnh tl6

.,, ,.;

tlep olem xo.

Budc 2 Vitit pf ti6p tuy6n v6i tI6 thi hdm sd tai

di6mM(xo;f @)).

Phuong phip 2 (Dua vdo bi6u di6n hinh hgc)

Budc l ViCt Pf ti6p tuy6n dudi dpng:

y = g(x)= lac+b (7) trong d6 kldtring sO ea

bi6t vd b ld dnsi5 ph6i tim

$3i*4,

OOi voi phuong phbp 2 thi ta kh6ng tinh tr.uc ti6phoanh dQ tiep di6m md ta chi xdt f.uc ti6p PT ti6ptuy6n th6ng qua <1i6u kiQn tr6n.

Tiy theo thd loai hdm sd vd trong tring hoan canh

cU thi5 md ta c6 th6 ding phuong ph6p thich hqp.

hodc xo =2

Vdi ro = 0 ) lo = -1 n6n PT (d') ld:

y+1: -3(-r-0) hay y =-3x-1.

V6i xo =2= yo = 5 n6n PT (d') ld:

y-5=-3(x-2)e y=-3x+11 (ta 1o4i <luong

thing ndy do n6 trung vdi (d) ).

Vay PT ti6p tuytin cAn tim ld: y = -3x-1.

Luu !, Bdi to6n vi6t pt ti6p tuy6n (d') khi bitit

tru6c dugc PT mQt duorg thing (d) co cilctrucrng hqp sau:

(d) L(d') thi h6 s6 g6c ko.ko, = -1 .

' (d) ll(d') thi hQ sd g6c ko = ko,

Nhmg hQ s6 g6c bing nhau chua hin hai

<lucmg thlng song song vcri nhau (c6 thiS trung

re n.ooo-roru, T?8il#EE

7

nhau) Do d6 sau khi gi6i chung ta n6n ki6m tra

lai dC loai <tudng thing trung v6i tlucrng thing

cria gi6 thi6t, cg thi5 trong thi du tr6n ta lopi

ducrng thtng y:-3r+11.

Thi dr,1 3 Cho hitu s1i' ,:.fj*lr (C) r,ri

dtrilug rhting {d}:-v * 2x +i f'iir PT ridp tuttiri

{el'1vtii (C) vri sang song vril (r1).

Ldi gidi Titip tuytin (d') ll(d) thi c6 PT dpng:

! =2x+b (b *l) (d) ti1,p xric (Q n6n cAn c6:

Do cl6 PT ti6p tuy5n cAn tim ld: y =2x-2-$ .

2.'{iip uyin di qu* $ihw ,l(a;F) cho trrd'c

(ko$c phrii {inr)

Phuong ph6p 1.

Bubc l Titip tuytin v6i <16 thi tai M(xo,f@))

c6 PT d1ng: y-f(x)=f'(x)(x-xo), do tii5p

tuy6n qual n6n: B-/(xo)=/'(xo)(cr-xo) (*)

Babc 2 Giei (*) clii timxo r6i suy ra phucrng

trinh ti6p tuy6n

Phuong phfry2.

Brdc 1 Ti6p tuy6n qua A(a;il c6 PT:

y-F=k(x-u) (Q trong d6 k ld hQ s6 g6c

cin phii tim

Budc 2 Li lufn (f ti6p xric (C) <lC tim k, khi

tl6 tim ilugc PT ti6p tuy6n

7'h{ du 4 Cho hdm s6' .1' : .rt - 4.t + I ({i) I irz

PT tiip n.ren t.T) t,6'i tC) Ae tlr diiint A{2;_-6}.

Ldi gi,fii Ti6p tuy6n (I) di qua A(2;-6) c6 PT:

y +6- k(x-Z) e y = k-Zk-6 (I) tiiip xric

VOy c6 hai tirip tuy6n ld y=1J3x++Ji-A;

y =2Jix-4J1-a.

Th{ tty 5 Cho hdnr sit -r':.t: *2x+2 (C) vadurlngthdng (dj:x-1.Tiru tlilnt A thu6c td) sao

thct h)r A ke &rsc hai liep fi\'An vcri (C') va hai ti;ip

lit7,6n 6O vudng goc t;tti nhar.t.

Ldi gidi.Ta c6 y' :2x-2 vd gqi A(l;a) e(d).

Ti6p tuytin vdi (C) t1i (xn,f (xn))e(C)c6 PT

dqng: y - (fi - 2x o+2) = (Zxo - 2)(x - x) (T)

Do (D qua A(l;a) n€n:.

a-(fi -2x o+2) = (2xr-2)(1-xo) = 1fr + 4x, -2

D6 c6 hai tiep tuy6n voi (O ke tu I vd vu6ng g6c

voi nhau thi (*) phii c6 hai nghiQm xr,x, thbamdn: y' (x"r) y' (*r) = -l + (2\ - 2)(2x, - 2) + I = 0

e 44x, - 4(4 + xr) + 5 = 0 e 4a -3 = 0 e a =1 4'vqy di6m cin tim ld li l:+ l.

\'4)

Thi ttq 6" Cho ham s6 .]'= rr -3x2 -r x+2 iC).'i'im di€m A thu1c (.C1 sao cho tii A chi ki dw.t'c

tiu.t, nhdt m(st ti€p lu1:d11rifi ham sd (C)

Phfrn tich tim htbng gidi Ta <1o6n ring bdi

to6n c6 li€n quan il6n d6ng diQu hdm sd bfc ba,

dC th6y ring tr6n hdm si5 hdm s5 bfc ba n6 c6mQt vi tri rdt dac biQt d6 ld vi tri di€m udn

Bing,tn;c quan ta nhdn ra titip tuy6n xu6t ph6t

tu di6m u6n lu6n k6 dugc mQt ti6p tuy6n duynh6t, c6ng vi€c bAy gio li ldm nhu thi5 nio <16

nhQn dfnh tr6n ld dirng \r

LTS, Bdt diiu t* sa +AO ndy, Tda soqn Tqp chi TH&TT kh6i phltc lqi chuy1n muc Ti6ng Anh qua ctic bditodn nhdm giup cac bgn hqc sinh ndm thAm cdc t* vtlng fieng,4nh, qua d6 c6 thd doc ii€u cai cuiin sdch,

cdc bdi viiit tAn Tqp chi phd thbng tAn thd gi6i vi m6n To,in Tda soqi neu ftnvd tdng Tqp chi cho cdc bqn c6 bdn dich dt vd sa dng kiit khen thudng vdo cuiii ndm hpr.

.d

?xffiruffi &rw*€ sum GmG B&il x'@&il[

Bdi s6l

Frohlent Let A=lr!,2, ,n)i antl lei P(A) be

the set of all sttbse ts q/'A How, niany et'ement-r

cloes P(A) have?

Solution Let ,S be the set of all bit strings of

length n We construct an one-to-one

correspondence between P (A) and S as

follows For each A = {rr, ,1,} e P(A) where

l<4 < <i <U we assign B to the sting brbr b,

in which bi =l if and only if I e B For

example, in the case n = 4,A +) 0000,

{t,2,+\<+1101, I1 is not difficult to check

that this map is actually a bijection from P(l)

to S Therefore the number of elements in P(A)

is equal to the number of bit strings of length r

which is 2",

ff Ldi gidi Ta c6: !' =3x2 -6x+7; y' =6x-6.

Ta thdy (C) c6 cli€m u6n 1(1;1) Doi hQ truc

toa d6 Oxy v€ h€ truc tqa d0 IXY qtaphdp tinh

tiiin theo Ot(t;t)' x = X +l; y =Y +1

Trong h6lXlhdm sO da cho dugc vi6t lai :

y +t=(X+1)3 -3(X +t)2 +(X +t)+2

hay Y = X3 -2X (C), lirc d6 Y'=3Xz -2

Dat A(X = a,Y = a3 -Za1e1C1

PTti6ptuy6nv6i (C) tai (Xo,.f(X))e (C) c6

dqns: Y -(Xi -2x) :eX -2)(x -x) (7)

Ti6p tuy€n cli qua A n€n:

a' -za -1xl -zx) = (3x'o -z)(a - xo)

e rt -xl-2(a-x)=(3x'o -2)(a-xo)

e(a-X)(2Xi -axo-r')=O

eXo=ahay Xo=*2.

DC tu I chi kd dugc mdt tiOp tuy6n vdi hdm s6

(Cl thi phdi c6: o =-*o a=0= A: I.

iengthconstruct0ne-to-one correspondeirce

as followsassigncheckbijection

t8p hqptfrp conphdn tirmQt rlcxr vi thOng

0 (dr'urg, sai) trr6ng, driid0 ddi

xAy dqmg (r16ng tu)

song irnh (rianh tir)nhu sau

Sn.dinhki€rn tra (d6ng tu)song 6nh (danh til)

*t4n gri.i blri dich: Mut2n nhdt k) itai tlting -tau khi dang hdi.

NGUYflN PHU HOANG LAN(Trudng DHKHTN, DHQG Hd NAi)

B,\I TAT' Lt \ EN TAP

l Chohdmsii y=x3 -x2 +2x+1(C).

Chrmg minh tr6n (Q kh6ng t6n tpi hai diilm md titip

tuyi5n voi (C) tai hai diiSm d6 vu6ng g6c vcri nhau.

2 Cho hdm s6 y = x4 -5x2 -r+8 (C) Vi6tphuongtrinh tii5p"tuyi5n v6i (Q tai c6c giao di6m cria (C) voiduongthdng y=-x+4.

n2-n I

5 Cho hdm s5 , =r;# (c) viet Pr ti6p ruyr5n

vdi (C), bi6t ring ti6p tuy6n d6 vu6ng g6c vdi tiQm cdn xi6n cua (C).

6 Cho hdm s6 !=-x3 +zx2 -z(c) Tim tr6nduong thing td);y:,2 cdc ilitim md tu d6 c6 th6 ke

tlugc ba ti6p tuydn den hdm s6.

So eeorro-zorsr a cTudi&a 9

i-ruoNrG u,Au orAr.ot sd

lim f '(x)= -m, /'(1) = 13 > 0 n€n tdn t4i duy/r r-

t-l ,i

1

nhdt s6 a sao cho )<a<1, f'(a)=0 Nhd tinh

Id6ngbi6n ct.r- f'(x) tathdy f'(x)>}ex>a,

Bii 5 Dlt 1: cosx, tinh dugc

fr

t =o!12-z"os3-r)sin.rd-r =u-|n 0

Blri 6 Tinh ilugc SeBc =#,tO=+

,l

n'

)Vr.*, =f {dvtt) Gqi H li trung di6m BC,

ke AK l,SH tqi K, GI L SH tqi I Ta c6d(G,(SBC)) = IG ud + AKHA3= *= ] xdt tu*

si6c vudns SAH co -l =a*-L AK' SA' AH' =4.3d

Yar, 'Yr *\v dG.(SBctt=qf .

12 '

Bni 7 Gi6 st C(a;a+Z)= A(2-a;6-a) Mi

A thuQc <ludng thing 2x+y-1= 0 rr}n a=3 G

ro'?31#@

b) Vi6t phuong trinh ti6p tuytin cria AO tq 1q tpi

aicm a(0;-l).

Ciu 2 (1 di6m) a) Cho a ld g6c md cotr =2 Titlh

cosa

sin3 a+3cos3 a '

b) Trong m5t phdng Ory, tlmtap hSp c6c tti6m bi6u

di6n s6 phac z thoi man l, - (z - +i)l= t .

Ciu 3 (0,5 di€m) Giiiphuong trinh

irrtn(r+:)+f roe, (,-1)* =tosz4x.

Cflu 4 (l di€m) Cho 0 < x < y Chrmg minh ring

v6i moi s5 thgc a , ta c6

x2 -a2 *a2 - Y2 ,2(*-Y)

x2+a2 a2+y, x+y

CAu 5 (l di€m).Tinhtichphdn:, !p-r1ffi".

CAu 6 $ diA@ Cho hinh ch6p S.ABCD c6 d6y li

hinh w6ng cqnh a, mflt b6n SAB lit tam giilc

ddu, SC = SD = oJl.ti*th6 tich ttroi chop S.ABCD

vd cosin qia g6c gita hai mAt phlng (SlD) va (SBQ.

Ciu 7 (l di€m) Cho tam giitc ABC ruong t4i

A(l;2), cph BC c6 phuong hinh y+3=0 vddiem o( ;t) Gsi E,F lAn luqt ld trung tliiim cira c6c do4n BD,CD Tim toq ttd cria B,C ,bi6tducmg

trdn ngopi ti6p tam gi6c DEF tli qua tliiim

Tim ditim MthuQc d1, N thuQc d2 sao cho.44y' song

song vdi (P) ve duong thing MN c6ch (P) mQt

khoing bEng2 .

Cflu 9 (0,5 di€m) Gi6i"b6ng d6 nt v6 tlich D6ng Nam A ndm20l5 du-o c t6 chtc t4i thdnh ph6 H6 ChiMinh c6 8 tlQi b6ng tham dqr, trong d6 c6 hai ddiViQtNam vd Th6i Lan C6c ilQi b6ng tlugc chia ngSu

nhi6n thdnh 2 bingcO s6 eQl b6ng bing nhau Tinhx6c su6t sao cho hai tlQi ViQt Nam vd Th6i Lan nim

0 hai bdng kh6c nhau.

CAu l0 0 died Cho a ld s6 thgc duong Gi6i vd

bi6n ludn phuong trinh sau ^ ',' -2*= , = o .

x2 -1-,Jx2 -1

KIEU DINH NINH(GV THPT chuyAn Hirng Vaong, Phi Thp)P=

Ci- =A(-1;3),C(3;5) C6c <ti6m B, D ld giao di6m cua

dudng tron duong kinhlCvoi duong hung ft.uc cinAC

Tri d6 tim duo c B (0; 6), D (2; 2) ho{c B (2; 2), D (0; 6).

[iiri s, PT mflt cAu ln ("r+lf +(v-2Y+k-3Y=Z nep

aa* r(1 1 !'l

(3 3 3'

Bni 9 L6p s6 tu nhi6n c6 4 cht s6 d6i *6t ph6n biQt

oUra *c6c cht s6 O, t, 4,6 C6tAtca Z.Z.Z.I=tg

s6 nhu vdy Ta c6 n (fi)= 18 Sd abcct:.4etiiq.

ttrclit cd e{O+,+O,eoJe ,e+}

Trong f), s6 c6c s6 tdn ctng 04 hoic 40 hoic 60 ld

3.(2.1)=6, s6 c6c s,i t6, ctng ld 16 holc 64 ld

2.(l.l)=2 Goi A ldbi€n c6 cAn tinh x5c su6t thi

n(At = e +z= 8 vay P(A) =1-+ 189 =:

Bni l0.Tac6 xr8*y18 " 1718- | _1-1- 3-=:.

Tt BDT (xtee1 - yleel )(xta - ,ta; > 0 ta c6

x2ot5 + y20ts > x1997 y18 + x18y19e7

€ 2(fors + yzots ; 2 (ytsst + yteez )(ata u ytt )-2015 , ,,2015 -18 | ,r8

<? '' ^ ,.== -/,=- >n :- I ddng thric xdy ra khi

TONN HOC -

-SiSaeotro-2orsr i cTudiga I I

?l ,

GIfiI IO[1{

kgynnvnxn

nile6rofr1t+

OIAI TON IIIN ilIAY TII{II I{IIOA IIOC

h6c v6i c6c ndm tnr6c, dd thi ndm nay

g6m 3 bdi, t6ng ta :O Aiem NaSi Uai gdm

5 c6u, trong db Bdi 1(10 di6m) le bdi thi d4ng trdc

nghiQm nhi6u lUa chgn, m5i cdu 2 <tiOm; Bdi II (5

diCm) ld bei thi tr6c nghiQm d4ng dring-sai, m6i

c6u 1 tli6m; Bdi III 1tS ei6m; h bni thi tr'6c

nghiQm dpng tti6n tl6p s6 (kdm theo trinh biy v6n

tit toi gia1, m5i cau 3 di6m Thoi gian ldm bdi ld

60 phirt Du6i tl6y chring tdi gi6i thieu loi giei dC

thi cta Sd Gi6o duc vi Eio t4o Hn NOi 2015

(Trung hgc ph6 thdng) Bni gi6i <luqc trinh bdy

tr€n Casio fx-570 Yn PLUS, ld th6 hC m6y m6i

nh6t hiQn nay, b4n <tqc c6 th6 dC ding giii tuong

Ldi gi,rti Tinh tl4o hdm cria hdm si5

s(x):los.x+ e5 2' :lnx +2' tai x:2 vdsui vdo

Khai b6o bi6u thric f (x) ' :cosx * -}. I.-1 l4x+15 :

@ IALPHAI tr D E El El- m hLFHAI tr

xA ruQ1aor4 eor IHrrir.rn^ruu puoc (sd GD - DT Hd Noi\

DINH HUU LAM (GV THPT chuyAn NgayAn HUQ)

TA DUYPHUONG (W€n Todn Hoc)EoE E r E E t E r4lALPHAl E E ts

B6m phim plArq tl6p s6: -0.839020511

Ciu 2 Gia tri nho nhdt ctia hitm s,5

2x+3

f lr).= :=: + x' + x tr€n docttt l-t;Sln: r/r' + 20i5

M6y h6i: X? Khai bito giirtrf ban dAu: 0p

Gtri t6t que G0.s222814326) vdo 6 E,

lsHrFImE

Vi,f'(-l)x-0,9554<0vd /'(0)>0n6n

xo x -0,52228143261d tli6m cgc ti6u cira hdm s6

dd cho Tinh gi6 fi cira him s6 tai <li€m xo bing

A 4,5826; B 7,1414; C 8,77 50; D.10,344t.Ldi gidi Tu Hinh vE ta c6:

ED2 =ME2 +MD2 A

=AE2-AM2+NC2 M

=AE2 -AM2 +CE'-EN' p

= AE2 +CEz -(nw' + at'l')

= AE2 +CE2 - BE2 =62 +82 -72 =51

Ydy ED = .r/51

^, 7,1414

Bdi II (5 di6m) Thi sinh danh dAu X vdo 6 dunghodc sai trong cac cdu sau.

Ciu 1 Vdi mpi da thwc P(x) c6 he s6 ld sa

nguyan tu cd P(20rs)-P(-996) khong phdi ldsii nguyAn d

lQp phuong (c[ng ld tdm kh6i cdu ngo4i ti6p lQpphucrng) c6 c4nh bing a Tdm cria m{tddy ABCD ld H Khi 5y bdn kinh R cira kh6icAubing Olvd

rx2tx2^2

R2 oA2 =oH2 + HA2 =[ \z) l.Jr, g,| *l +1 =!- 4'

Suy ra

v =!nR, 3382 -4 n3Jia'=f n khi a:r.

Cdu 4 Chft sii thQp phdn thilr 2015 sau ddu phiiy

cita phdp chia 1357 cho 19 ld:

Ti6p tuc chia 8 cho 19 ta dugc chu ki titip theo.

Vi 2015: 18x111+17 n6n cht s6 thu 2015 sau d6u

phdy ld chfr s6 thri 17 trong chu ki, tuc ld cht sr5 6.

ts nrroo-roru, T?!I#S

13

{'itt r

;"*\ !,r i, .", itr; lrt,ti i \ -.'.

.,t ,!i,1,, ti: /t ! liti i \ (2) \,

ttirrlt :.it tiiq4 i)t i -"-j-= r

Liri gidl Hodnh dQ xn,x, cira hai dii5m cgc tri li

nghiQm cria phuong trinh .y' =3x2 -2x - 3 = 0.

Suy ra: (*r - *r)' =(*n + *u)' - 4*u*,

"'\ 8t) 9 8l 27'VQy d0 ddiABbing: AB =LJof* x 5,1373

27 Cdch 2 Tinh tluoc

Suy ra: .r+ I = y -l<> ! = x+2 Thay vho

phucrng trinh 3'-t = y * x * 4 +logr(2 + x)

ilting') tui [_i

Ldi gidi Vi /(x) =2" -x3 +sinx-3 ld hdm

li€ntuc vd f(-2)x5,21; f(0)=-2; f(10)x2l,l7

n6n phucrng trinh it nh6t hai nghiQm trong ciic

khodng x, e(-2;0) vd x, e (O;tO).

NhQn xit VC d6 thi nho phAn mdm Maple,hoic

Graph, ta c6 thiS khing rlinh phuong trinh c6

Bai ill {tr5 ili6m} T'hisinh rrirth !;i.v ld'igiii tigtin

,:ptt tii ghi liit quti.

Ldi gidi PhAn tich 20153vd 18593 ra thta s.5

cau may gini phuong trinh:

nguy€n ti5 nhd pti^F,qCf,tr6n Casio fx-570\DJ

- 2eKi5t qu6: -1,872130575

YQy ta c6 hai c{p nghiQm: .t =l;yr=3 vd

x, x -1,8721, yz = 0,1279.

Ciu 3" L'lto lt)nlt t'ltip .9.tl8('D t:6 dLi.r' .illt'il i,t

llinh lhttng v-uottg itti .l ti [] kldt b1n 5,1 D lit tun't

;,:itic r.litt t'ttnlt l,ritt,! 1 ,'r't nriut tt'in utt.it pitr'itr,-!

lrir)ng gric voi trttit ltltting (,.iBCD) Grir' gli?ri ^iJj

t'.t ttt,it 1tit,lt,: t lB( Di r'.ir, 1t)" ( /) J i ittlt r!rt

/itli ciln lihrii chip 5",.1P't'D.

LN gidi Trong tam gi6c SAD ke ,SH L AD

Do hai mflt phing (seo) ve (tacn)r,u6ng g6c

nen S11 L(ABCD).Yi ^sHle chi6u cao cira tam

gi6c tl6u c4nh bing AD = a =2n€n

,, ='f =6 vi sH L(ABCD)nen sBH

chinh le g6c gita SB v6i mAt phing

(,laco).Suy ra

HB : SH.cotSBH =SI1.cot300 =".6 6 = 3.

vay AB=JH* -Arf =JY 4 =2Jr DiQn

tich hinh thang ABCDbing:

sn,co =!.qo.(as + cD)=!.2.(zJz * q)

=2Ji + 4.

V{y the tich hinh ch6p S.ABCDbing:

DC tim nghiQm cdn lai ta sir dung phim

ISOLVERI nhu sau: Khai b6o phucmg trinh

{-f,u 5 711,,,1 jfi-t, ti (,,,, ) lhrru tnijti rr = l'

tt,,.j =.tt, '+{rr r i\t" r:t't'i ttrt.'i n .'i T)ttr 'tti ;Lr itltti'ri

phuong Vay chi c6 duy nh6t mQt gi6 tri

n =3 thbamdn tli6u kipn tl0u bdi.

Iiir tsfrn V6 co b6n, nQi dung tt6 thi kh6ng khrlc cdc ndm tru6c Theo qui ch6 mdi cira ky thi Gidi

ban ftAn mdy tinh, chi c6 ba bii tric nghiQm vddugc lim, tlugc ch6m tryc ti6p tr6n m5y tinh Bdi

thi He NOi 20i5 dugc ch6m tryc tiiip Nhu v$y,chi c6 c6ch ch6m thi li kh6c, cdn nQi dung tl6 thikhdng kh6c tru6c (v6n ld cric kii5n thiic co b6n criato6n ph6 thdng k6t hqp v6i m6y tinh d6 gi6i ra

,4.

dap so).

r.l9ese-,,"r-9,58E lS

(GV THCS Gidng V6, Ba Einh, Hd Nii)

Fidi T21460 (LOp 1) Tim c5c sO nguyOn duong

x, !, z sao cho:

(" - y)' + (y - z)3 +3lz - *l=zl .

NcuvEt{ KHANH ToAN

6f rUCS Aac Hdi, Tiin Hdi, Thdi Binh)

Bni T3/460 Cho a, b, c ld d0 ddi ba canh cira

m$t tam gi6c Chimg minh ring:

(GV THCS Hing Bdng, Q Hing Bdng, Hdi Phdng)

Bni T41460 Xdt tam gi6c ABC thay i16i, c6n tpi

A, nQi ti6p duong tron (O; R) cho truoc K& BH

vudng g6c vdi AC tai.Fl Tim gi6 tti lon nh6t

cta tlQ ddi tloqn thbng BH

PHAMVANBINH(GI/ THCS Vfi Hibu, Binh Giang, Hdi Duong)

Bni T5/460 Giai he Phucrng trinh:

Bii T61460 Chimg minh ring v6i mgi m)1,

phucrng trinh sau c6 nghiQm duy nh6t

*' -ltJT* +z* =2m .

CAO XUANNAM(GV THPT chuY€n Hd Giang, Hd Giang)

Bii T71460 Tim t6t cd cdc gi|tti cua tham s6

p vd q sao cho v6i c6c gi6 tti d6 hQ phucrngtrinh

(t)-),lr'+ y'+5=Q'+2x-4y

{fx2 +(12 -2p)x+ y2 =2py+12p-2pz -27c6 hai nghiQm (x1;"t/r) vd (x2;l) thoa mdn <1i6u

kiQn xf + y? = *3 * ytr

NGUYEN DE (Hdi Phdns)

Bni T8/460 Cho tam gi6c ABC c6 BC : a,

CA: b, AB : c Gqi R, r, p ldn lucvt ld b5n kinhclucrng tron ngo4i ti6p, b6n kinh duong tron nQi

tiiSp, ntra chu vi cua tam gi5c d6 Chimg minh

oabzrz b4c2a2 ,ao2b2 - 3

lu6n dung voi mgi sO tfrirc duong a, b, c vd abc :

^1.

LE VIET AN(Nhd 15, xom 2, Ngoc Anh, Phu Thactng, Phil Vang,

Thiia Thian - Hu€)

tsliti T12l46t!" Cho LABC vd di€m D chpy tr6n

canh BC Goi (1,),(1r) theo thu tu ld ducrng

tron n6i ti6p cira cdc tam gi6c ABD, ACD;

(I,)theo thf tU ti6p xric voi AB, BD tai E, X;

(Ir)theo thil t.u ti6p xric yoi AC, CD tqi F, Y.

N t, N 2 theo thir W cit EX, FY tai Z, T Chtng

minh ring

l) X, Y, Z, T cing thuQc mQt ducrng tron tAm K

2) K chqy tr6n m6t ducrng thing c6 dinh

NGUYEN MINH HA

(GV THPT chry'€n EHSP Hd N6i)

Bni L1l460 Cho mach diQn xoay chiAu nhu

hinh ve Chrmg minh ring chi s6 o ampe k6l

kh6ng phu thuQc O.

Tr€n TH&TT sil 459, *a,rc Di ra k) ndy ftn tdc gid Bdi T4l45g dd vi€t ld Vfi Qu6c Dfing, xin sua ld Vi Qudc Dting

I hanh Ihat xtn lot tac gto.

c2

VIET CUONG (Hd Ndi)tsiri t.2/460 MQt qui b6ng clugc ndm vdi v6n

tdc i, theo phucrng ngang, va ch4m rldn hOi v6i

m6t mdt nghiOng g6c cr so vdi phucrng ngang.Sau va cham qu6 b6ng nAy l6n, sau cl6 lqi vachpm v6i mqt nghi6ng Tai thoi cli6m va chamlAn thu 1y' + 1, vAn tdc qui b6ng c6 phuongvu6ng g6c v6i m{t nghi6ng

a) Xric rtinh g6c cr theo,A/.

b) X5c dinh kho6ng c6ch tu ditim va chpm thir Idlin di6m va ch4m tht 1/+ 1.

v0 rneNu KHIET (NXBGD viQt Nam)

I xz+yza5=q2+2x-4y

f xz +( I 2-2 p)r + yz =2 py +1 2 p -2 pz -21

has two solutions (x1;l) and (x2;!) satisfring

2222 xt+lt=x2*!2.

Prolrlern 781460" Given a triangle ABC with

BC: a, CA: b, and AB = c Let R, r, and p

respectively be the circumradius the inradius,

and the semiperimeter of ABC Prove lhat

ab!bc-lca

={ *n.n does the equality occur?

pt +9Rr

(Xem tidp trang 30')

PBOBI,ETIS III THTS ISfiI'E

FOR SECONDARY SCHOOL

Problem Tl/+60 (For 6th grade) Let a be a

natural number with all different digits and b is

another number obtained by using the all the digits

of a but in a different order Given a-b=lll l

D

(re digits 1 where r is a positive integer) Find the

maximum value of r

Problem T21460 (For 7th grade) Find

positive integers x, y, z such that

(x-y), +(y- z)z 1117 - xl:27 .

Protrlem T3l460.Let a, b, c be the lengths of

three sides of a triangle Prove that

2(t ab.*c\>e+h+g+2.

-\b'c'aJ- c'o'b'"'

Prohlem T41460 Let ABC be an isosceles

triangle (AB : lO inscribed in a given circle

(O; R) Draw BH perpendicular to AC at H

Find the maximal length of BH

Problem T5/460 Solve the system of equations

Bni Tl/455 '|in tdt ci t',tc'bQ so ngt4vi,, r,i tuo

cho tich t:ilcr c'hting bang 10 tin tdng cua chring

Ldi gi,rti Do tich c6c s6 nguy6n ti5 cin tim chia

hi5t cho 10 n€n phii c6 hai sti nguy6n t6 Z ve S.

, A ^ .A

Ggi c6c so nguydn t6 cdn lqiliL pt p2, , pn-1, pn

voi pr< pzA 1 pn-t< pn Tir gi6 thi6t c6

Ding thric xiry ra chi khipl :3 vit pz: 5.

.Yoin > 3, x6t hai truong hgrp sau:

- N6u p, :2 thl (1) trd thinh 2' :7 + 2n, di6t

ndy kh6ng xhy ravi sO Z le

Do p, > 3 vd s > 4 thi U6t Aang thric xiy ra chi khi

p,:3 vit s < 5, suy ra (pt ) pz) chi c6 thti ldQ ;2),

how Q; 3) Thay vdo (1) <l6u kh6ng thoa mdn.

VQy b0 s6 nguy6n t6 duy nh6t th6a man AC bai h

Q;5;3;s).tr

YNhQn xda NhiAu b4n tim tluqc ding thric (1) vd

cho k6t qui dfng nhrmg kh6ng lflp lupn chAt chC.

Chi c6 bqn Trdn Minh Huy, 6A, THCS Lf TU

Trgng, Huong Canh, Binh Xuy6n, Vinh Phric c6 lflp

lufln tldY tlu'

vIpT HAIBdi T2'1456 Cho tam giac ABC cdn tgi A cct

fAc =80" " C(lc' diim D E theo thtr tu' thudc

cac c'anh BC, C,4 suo cho'.

Tinh ,so do BliD

Loi gi,rti.

CAD =,48f, = 30".

Trdn BE 6y di6m F sao cho LDBF ctn t4i D.TrongA,DBF c6 DBF=DFB=?-ff, n€n BDF=LAff(1) Trong LDAB co DAB:DBA:50., n&t

frirao' (2) Tt (1) vd (2) suy ra 6F = 60" MAt kh6c, do DF: DB: DA,t}ntafriily tam gr6cADF ditu.Tt d6, chri f reng fu+t +ffE:3tr

ngtria li AE h ph6n gi6c cir' ilF, hay AE lit

cludrng trung f.uc oia tloqnDF Suy ra L,EDF {antai4 dend6n EDF = EFD =20'.

^

Dod6 BED=EDF+EFD =40o EYNhQn xit Trong bili T2'1456 tr6n chirng t6i dA

thay gih thi6t BAD = ABE = 30" (*) trong bii T21456

bdi gi6 thi6t CAD = ABE = 30' MFc dir voi gie thiet

(*), viQc tim s6 do 6ED c6 th6 gi6i theo hudng dulrg dlnh lli sin vd dinh li c6sin: D{t AB: AC: a,

bi6u di6n BE, BD, ED qua a theo hai tlinh $ ndy.

Cu6l cring sri {rng dinh li c6sin cho MDE dC rlm

"o" 6ED Tuy nhi6n cbch gihindy vuqt qu6 chu<rng

tdnh larp 7 Tda soqn hy vgng rAag c6 loi gi6i chi sridlrng ki6n thirc hinh hqc lop 7 ti citc.b4n dgcTH&TT grii v6 cho bdi to6n tr6n v6i gie thi6t (*)

Nguy6n TAt Thenh, TP Kon Tum.

NGI YfN AI\H QUANBdi T41456 Cho hinh trying ABCD cqtnh a.

'['rAn cgnh AB, BC: tin twqt tti1, cac diim tuL, ]V

sao r:ho MDN :45' Tint vi tri c:ilcr M N dA daddi dosn thdng MN ngtin nhdt

Ldi gidi

Ta c6: MDN = 45o, suy ra

ffii{ : trotr *ffie = 45'

Trtu tia d6i cua ta CB 16y dr6mK sao cho CK : AM.

W AD : Dc(canh hinh vu6ng) ; fu =fu =W

> LDAM = A,DCK (c.g.c)

IDK=DM

= { ^ ::::-_ :-=:_ (l)

IKDC = MDA= aN = MDN = 45o

LKDN vit A,MDN c6 c4nh Dllchung, t6t trqp(1) suy ra LKDN = A,MDN (c.g.c)

= KN = MN + MN + BN t MB =2AB =2a

(kh6ng AOi) OAt MN: x, BM: y, BN: z.

Ap dung ahtaingthttc 2(a2 +b')>(a+b)2 vit

dinh l)7 Pythagore vdo ABMN ta c62(y2 +22)> (y+r)' >2x2 > (2a-x)2

* J1'x > 2a - x= x )

# = (2a -z)a'

Ydy min MN = QJZ - Da d4t tlugc khi

! = z = * = aQ-JZ) mi do DM,DI/ thu tU

J2

ld phdn gi6c cia cbc g6c ADB, CDB.13YNhQn xdt.BiLitorin kh6ng kh6 vd c6 nhi6u cSch gi6i.

M6u ch6t h chimg m4rh du-o c MN :.AM + CN Cdc

bpn sau c6 lcri giii t6t: Yinh Phitcz Nguy€n MinhHi€u, 9D, Philng Vdn Nam,9E, THCS Vinh Y6n,

TP Vinh YEn, Phgm Nggc Hoa,8Al, THCS S6ng

L6, S6ng L6; Hn NQi: Dfing Thanh Tilng,gB, THCS

Jl6x 1.Jq;4 (J9, - + + JIo, - s )

l-x

"t-6 * - 1 ti * ( J5, + J6, :T )

- O6 th6yx = I ld mOt nghiQm cira PT(l)

- Voix +1, PT(2) tuong duong v6i:

,1

N€u *<x<1 thi 10x-5 <6x-l;9x-4<5x,

Z

suy ra VT(3) < VP(3) = PT(3) v6 nghiQm.

Vfy PT(l) c6 nghiQm duy nh6t x = 1 O

) NhQn xit Da s6 c6c ban gi6i PT nny blng c6ch binh

plo*g hai v6 r6i phdn tich da thric thenh nhan tu; MQt

s6 b4n tl[t 5x, lOx - 5, 6x - l,9x - 4 ho[c cln bflc hai

cria chring hn tuqt Ueng a, b, c,d(kfii d6 a + b : c + d

hoi,c I + b2 : cz + d1, did6n ki5t quli ab: c4 MQt s6

ban su dpng c6c tdt aZng thric Cauchy, Bunyakovsky

dC d6nh 916 ctng tli d6n k6t qu6 tlung MQt s6 bqn dd

d4tnh6mDKcuaxld x>|.

2

C6c ban dugc khen ki ndy ld: NghQ An: Dinh Yidt

\1, Nguydn Vdn Msnh, Nguydn Th! Nha Qu)nh A,

Nguydn Thi Nhr Qu)nh B, Tdng Vdn Minh Hilng,

84, Hodng Trdn_Duc,9D,.THCS Li NhAt Quang,

D6 Luong, Nguy€n Trong Bdng,8A2, THCS thi Trdn

Qurin Hdnh, Nghi LQc; Hi NOi: Od Uoai Phuong,

9C, THCS Tuy0t.Nghia, Qu6c Oai; Quing Ngiii:

Phan Thi M! H6ng, Cao Nit Xudn Lan, DO Thi

ItS.Lanr SA, THCS Ph4m VIn Ddng, Nguydn Thi

Ki4u Mdn,98, THCS Nguy6n Kim Vang, Ngtria Hanh,

I/A Thi Hing Kiiu,8A, THCS Nghia I\jI, Tu Nghia;

Phri Thg: Trdn Qu6c LQp,9A3, Nguy€n Hodng Phi,

8A3, THCS L6m Thao, Ldm Thao; Vinh Phric:

Nguydn Minh Hidu,9D, Phitng Vdn Nam,gE, THCS

Vinh Y6n, TP VTnh Ydn, Phqm Ngoc Hoa, 8Al,

THCS S6ng L6, S6ng L6; Thanh H6a: Nguydn Thi

Hodng Cuc,8D, THCS Nht BA Sy, thi tr6n Birt Scrn,

Dfing Quang Anh,8A, THCS Nguy6n Chich, D6ng

SorU Hii Ducrng: D6ng Xudn Ludn,9B, THCS Hgp

Ti6n, Nam S6ch; Kh6nh Hita: Trdn Minh Qudn,

8/1, THCS Chu Vdn An, Ninh Hda; Quing Tri:

(2)

TO6N HOC ,.

s6 aeo (ro-zors) ' cT.rdiEF t Y

Nguy6n Thugng Hi6n, tlng Hda, Einh Hodng NhQt

Minh, 7A5, THCS C6u Gi6y, Q Cdu Gi6y; Hni

Duong: D6ng Xudn Ludn, 9F., THCS Hqp Ti6n,

Nam S6ch; Thanh H6a: Nguydn Vdn Hirng, 8D,

Nguydn Th! Hodng Cuc,SD,THCS Nht 86 Sy, T.T.

Brit Son, Dfing Quang Anh, 8A, THCS Nguy6n

Chich, E6ng Scrn; NghQ An: Nguydn Trpng Bing,

8A2, THCS T.T Qu6n Hanh, Nghi L6c Hodng Trin

Duc,9D, Nglydn Thi I''lha Qu)nh A, Nguydn Vdn

Manh, Nguy€n Thu Qiang, 8A, THCS L)t Nhat

Quang, D6 Luong; CAn Tho: tti Cia Bdo, 9A6,

THCS Th6t N6t; Quing Ngii: Zd Th! Hing Kiiu,

9A, THCS Nghia M!, Tu Nghia, Nguy1n Th! Kiiu

Mdn,9B, THCS Nguy6n Kim Yang, D6 Thi W

Lan, 9A,THCS Pham Vdn D6ng, Nghia Hdnh

.

NGUYEN XUAN BINH

Bni T5/456 Tim cdc sd nguyAn &rong x, y

thda mdn *o + y' +l3y +l < (y -2)x2 +8xy

Ldi gidi Cdch l Ta co

,o + y'+I3y+l<(y-2)x2 +8ry

e (xo + 2x2 +l) - x' y + y' +l3y -8xy < 0

e (x' +l)z -2y@2 +1)+ y2 + yx' +75y -Sry < 0

e (x' - y +l)' + y(x' -8x+15) < 0

e (x' - y +1)' + y(x-3)(x-5) < 0 (*)

Vi (x2 +1- y)'> 0 suy ra y(x-3)(-r-5) < 0

Md y ld si5 nguy6n duong n6n (r - 3)(x - 5) < 0

<>3<x<5 Lpi c6 x ld s6 nguydn duong n6n

x e {3,4,5} .

' Voi.r: 3 thay vdo (*) ta c6 (10-y)'30<+y=19

V6i x : 4 thay vdo (*) ta c6

'Voix:5 thayvao (*) taco Q6-y)' <0ay=)6.

VQy c6 10 c[p si5 nguy6n duong (x,y)thbamdn dC

a x2 y +l5y -8xy <0 e (x-3)(x-5) < 0(do y eN-), md xe N* n6n x e{3,4,5}; r6i

ti6p trlc timy tuong t.u nhu cSch L D

YNhQn xit a) Ea sii c5c b4n gui bai tbu giai theo hai crich tr6n C6 ban dA hm c6ch kh6c bdng c6ch bi6n d6i BPT da cho thanhy2 + (13- f -3x)y+11 +l)'z <0 vd coi

ld BPT b6c haiiny,rOi d6n d6n

A, =(x-1Xx-3)(x-5)(3x+11)<0; tu d6 tim xnguy6n duong r6i tim y, tuy nhi6n cdch ldm ndy dii

vd dd 5p dung tlinh li vC d6u cria tam thirc bfc hai

c:fracip THPT.

b) Tuy6n duqC c6c bpn sau c6 loi gi6i t6t: PhriThgz Trdn Qu6c LQp,8A3, THCS Ldm Thao; VinhPhic: Nguy4n Minh Hi€u,9D, Philng Vdn Nam,9E,THCS Vinh Y€n1' Phqm Ngpc Hoa, 8Al, THCS

S6ng 16; Hn NQi: D(ng .Thanh Titng, 9B, THCSNg"yQn Thugng Hidn, Ung Hda; Thanh H6a:Nguy€n Thi Hodng Cuc,8D, THCS Nht 86 Sy.

PHAM THI BACH NGQC

BdiT6l456 Gidi hQ phaong trinh:

fa+b+c=0 la+b=-c

tt ja2b+ b2c+ c2a-l <>

1" - U + 3a2b = l.laz +b2 +cz =2 la2 + b2 + ab = |

ra c6: &+Uiab=t * ff,\ *$,*o)' =r

l2

l4=;coSl, tuc ld: ] J3

l;a=cosl

)L

Ir

6a+b=s],ntKhi d6 PT a*b+3db=l trd thinh:

fr cosr -sinr+fi .o.r ++r"r r(r*r -fr .or,) =,

.- ^TO6N l-lOC