TOÁN HỌC TUỔI TRẺ THÁNG 3 NĂM 2014

CuQc thi Giei todn chdo m*ng 50 ndm Tqp chi TH&TT - DAi tuqng dq thi: Hqc sinh b?c THCS va THPT.. CuQc thi ViCt chuydn dO Todn chdo mwng 50 ndm Tap chi TH&TT - Doi twqng dq thi Gi6ro vi6

xuf,r eiu rUrgo+

2014

rnp cxi Ra xAruc rHAruc - NAM rx05l

oArux cHo rRUNG xoc pHd rHOruc vA rRuruc Hoc co s6Tru s6: 1B7B Gi6ng Vo, Ha NOi.

DT Bi6n t6p: (04) 35121607: DT - Fax Ph6t hdnh Tristr (04)35121606 Email: toanhoctuoitrevietnam@gmail.com Website: http:l/www.nxbgd.vnitoanhoctuoitre

Glliit[fl]t il0.G'll[, rudl ilt

xeww$@, ilP

\ff 6 chdro mllng 50 ndm thrinh lap Tap chi

Y Ed TH&TT (1s04-2014) vi h0 trE t6t hon cho

A.l phong trdo dqy vdr hec Toin & c;ic truong

b'n6 tf,Ong, trong ndm 2A#lrap chi sG t6 chuc hai

euOc thi:

1 CuQc thi Giei todn chdo m*ng 50 ndm Tqp

chi TH&TT

- DAi tuqng dq thi: Hqc sinh b?c THCS va THPT.

- Th6 te cuqc tt1i CuQc thi gOm 5 vong, d6 thi tu

Vong 1 d6n Vdng 5 l6n luot dwqc ding trdn TEp

chi TH&TT tu th6ng 2 tsb 44q d6n thang 6 (s6

44q De thi o m5i vong g0m 2 bdi toSn ddnh cho

hoc sinh THCS vir 2 biri to6n dinh cho hQc sinh

THPT" Hgc sinh THCS co th6 ldrm bdiTHPT nhung

hgc sinh TIIPT lirm bdri THCS s6 khOng duqc tinh

di6m D6p 5n ctia c:ic d6 thi sG lin luEt dwgc ddng

tu th6ng 6 ts6 444) d6n th6ng 10 (sO 44S) Kot

qud cuQc thi s6 duqc cong b6 tr6n Tqp chi TH&TT

thZrng 10 nhrn2014

- Thdi hqn gwi bdi dv fhl: MuOn nhit ta 2 thang kO

tu ngdy cu6i thang ra s6 tqp chI co dang drO thi

- Quy cAch bdidu fhl: M6i bai gidi c&a m6t biitodrn

phii viet tr6n mOt to,giSy (hoic mQt file) ri6ng Phia

tr6n m6i bii ph6i ghi 16: Ho t6n, dia chi truong,

lcrp, x5 {phuo'ng), huy6n (quan), tinh (thanh ph6),

i -,^ ., , i ,

s0 diQn thoai (n6u co) Tr0n phong bi hoic file ghi

16: 8di dtv thi GiAi todn chao mung 50 ndrn Tqp chi

TII&TT

2 CuQc thi ViCt chuydn dO Todn chdo mwng 50

ndm Tap chi TH&TT

- Doi twqng dq thi Gi6ro vi6n d6 hodc dang dqy &

THCS, THPT; gidng vi6n sinh vi6n cv cdc truong

DH, CD v;i ban dgc y6u thich To5n.

- NQi dunE bdi tltt ttti: LA c6c chuy6n de bOi du0ng

hqc sinh gidi Toin; cdc chuy6n dE on tQp, 6n thi

cuOl c6p; nhirng tlm toi, s6ng tEo tronE viQc dqy,

hoc To6n {bAc Ti{eS, THPT)

- ThA E cuQc tlti: M6i ngucri tham gia cuQc thi duEcg&i kh6ng qu5 3 bdi du thi kh6c nhau Cdrc biri dirthi co th6 liir c6c s6ng kien, kinh nghiem di ddng

ki, hoic doat gidi & Trucrng, Phong, Sd, nhungchua tirng duqc xuAt bdn thdrnh s6ch, biio vi c0ng

chua tham gia cu6c thi ndo khiic

K6t qud cuQc thi s6 duqc cOng b6 tren Tqp chiTH&TT thiing '11 ndm 2014 (s6 a+g) Cdrc bdri hay

co th6 duqc chen ding tren TEp chi, hoic in thirnh

s6ch, ciic tiic gi6 duqc hudng nhuin b*t theo quy

dinh c0a Tqp chi Bdri vi6t tham dur cuOc thi thuOc

l,

bdn quy6n cua Tap chi TH&TT

- Ttldi han gtri b;ii dtv thi: Trvoc ngdry 1 511012014.

- Quy cach bai dUr thi: Bdri du thi co the vi6t tr,On

giAy hoEc ddnh mdry vi tinh bdng chucrng trinhsoan th;io vin bdn word M6i bdi dLr thi phAi vi6ttren giSy (ho{c co file) ri6ng phia tr6n m6i nai pnal

ghi 16: Hq tOn, dta chi truong xd (phucrng), huyen(quan), tinh (thdnh pn61 so di6n thoai (neu co) B-ri dr=i thi kh6ng qudr 15 trang viet tay hodc 10 trang

d6rnh m6y Tr6n phong bi hodc file ghi 16' Bai dqthi Vi& cttuy\n dO Todn chdo mang 50 nam Tqp

chf TH&TT.

Cdch grhi biti dqr thi: B2ri dU thi c0a hai cuOc thi

tren gfri ve tap chi TH&TT bdng c6ch

+ Girifile word theo d!a chi email:

to a n h octu o itrev i etn a m @g m a i l -c o m

+ hoic g&i bai theo duro'ng bivu diOn Phong bi cod€rn tem, giri v6 dia chi:

Tap chi To6n hoc vi Tu6i tr6

1878 Giing V6, Di5ng Da, Hd NQi

.:

+ hodc d6n Toa soan Tap chi gtli truc ti6p

Alaf tnwdng; Gom gi6i c6 nh6n vd gi6itAp th6 cho

c6c trucrng vd dcvn vi co nhi6u cd nhAn tham gia" NhCrng c6r nhAn vdr tip th6 doat gidi cao s6 duoc

rncvi du vii trao gidi tronE L6 fi nienr 50 nim thdnhlQp Tap chiTH&TT t6 chuc vAo th6rng 1212A14.

Rdt mong cac bqn nhiQt tinh hu&ng ung hai cuQc thi tr6n !

f} am mO giai todn ld ti6n Ae *ri6t yeu OC tqc

lJ tOt mon To6n Long dam mC sE giirp cho

ngucri lim to6n c6 dugc sU ki6n tri, ph6t huy

tffi s6ng tpo vd tr0 n€n linh hopt khi gi6i to6n

Qua bdi vi0t ndy, t6i mu6n ctng voi bpn clgc

nhan dien sU linh hopt trong gihitoinhinh hqc,

drcctry cao Af{ vd phcn giit"c BD Giti I |t'r

ram chrdng trdn ruoi ti\p tctm giec AliC biit

( tii 4 ali rhi frtitt Ot:,i, ri tan rhir 5i))

Ldi gidl Gqi O ld giao di6m cua AH vd BD,ta

c6 Co ld phAn giiic cia triE -Id0=6d0.

HaOK LAC (K e AC)

Trudng hW l Nriu K tdng voi D tbt ABC lit

-;.

tam gi6c dOu vi co O vira ld tnrc t6m rua 1d

tAm dudng trdn nQi ti6p, suy ,u 6tre = 60o.

Y6ry BAC = 60"

hodtc 6k = 90o.

HHinh ]b

NhQn xit Bdi toan ndy thwc chAt h kh6ngkh6, chi cdn ki€n th*c THCS Vi€c dung OKvu6ng goc voi AC cfrng xuAt phdt tb y nghi n€u

tam giac ABC diu thi 6Di = 45o, itdy chinh

ld "chia khoa" d€ tim ra loi gidi don gidn cho

bdi toan

fi Blri toan 2 Clto {am giac ABC t'ri phnn gidc

AD; O lL't ciienr btir l;i tren AD (O khac A vd D).Tia tlo c'ot ccuth AC tai E, tia co ciit canh ABtai F Chirns ntinh ,'ring tc,n giac AIIC t:dn tai

:IIII

.4ttatt-;-= + .

.]81 AE2 AC:2 AT;2

Ldi sini Qua O ke OM ll AB; ON ll AC (M e AC;

nQi titip L4HC

H

Hinh lan6n.I e OC vit, OHI = CHI = 45o Suy ra hai

tam gi6c vu6ng OHC vit OKC bing nhau

-5fu =dfu = 45o = 6fu =6Di = 45o

s OKDIldtit glitc nQi tiOp

=COD=CKI =45o

=90o = EZe =90o.

Trudng hW 3 N6u KthuQc clo4n DC,ldtirc D

(hinh lb) thi tuong tu nhu trrdng hqp 2, ta

cfrng c6 ODKIliLtT giitc nQi titip, suy ra

DOC = CKI = 45o > BAC :90o.

Tuy nhi6n, khi d6 Eic ,67d = 9oo

',(pu uo.4 Euo.np

cgrE urel dgq

reoEu uo.4 8uo.np

():t!

)

r'i,J)\

ir i1 rilt1'

itltl tn!

,)ff

ilLli:)

;i;).:

! .-\rrt-'

-:)1')t

,', ti.) ':-

iltl(U II:f{

oa

gs

lbp Suop rcnp u?ot

!?q uztlu uttlu

Suiu ptpt

u?p

u?

(q ngc 'quts coq onc

coq qulq u?ot

7p13

Suoa ryqt

3uo,r7 ruoq qull Bupu

elq

pp

tt ttulp tpc Eunp

u?^

I

qrD)

'Lpry (, n?u rup ruq

Buo4 ,nnt1u

)?W

!?!3 Wry n?lqu

oc (o

nn3 yQx

ugt11r1

(o<z*,.,61 :te6:74

/

{r, ,)g i(i /

'y7 li.:l t,ili

Suonq

)o)

D.t

tpp ruoq quu

ug) ,uonb oqy

Bugqy

n7?

,tV,t nqu ntld Suenp

at Suanq

)?)

.Liy

:Cn

'(qV :gy) cg8

?uona

quoc doc )D)

hqd Bupnp

)D) or otlt

?p lq8u [ns Sugnq

:

gy',iy

:

gV u?u

pt (t)

+f

MB AB MC /17

MC.NC : AC.FC - 9Y MC-NC = 12'!3= AC.FC el

Tt (1) ve (2) suy ra dpcm

NhAn xdt W MAB = NAC vd ddng thitc cdn

chyn7 minh c6 dqng ti s6 cila tici cdc doqn

thdng n€n ta vd th€m dadng trdn ngoqi tidp

tam gidc AMN Khi gidi todn hinh hoc, hoc

sinh cdn daqc luyQn tQp nhi€u cdch vd dudng

phu khdc nhau, qua d6 s€ tich lu! &tqc kinh

nghiQm trong gidi todn hinh h7c.

fi ga; tohn 5 {-'l:r., ri:tst 1litir: ,11}{" t:i 1;iiriil ,2i1i,

tt'()t g.lD i{S ,,, {id-1 7 r't't:.JI) 1lir !t,ti tli, t \t ti

tttt) tlt(t iiti t A r 1! tt,trt, qiii.r : 1r' \i.

(gla thret) suy raLABK aLAMC (c.g.c)

Trong t& gi6c nQi titipBKCN cingc6 BKN = BCNsuy ra

feM =6dfi (dpcm)

NhAn xdt Lni gidi o cdch 1

tsni tip t Duong trdn t6m l nQi ti6p tam gi(c

ABC ti€p xirc vfi c6c cpnh BC, AB, AC Mnluqt tai D, E, l.Qua E ke ducrng thing song

song voi BC cdt AD, DF lin lugt tai M vit N.Chtmg minh M ldtrung diiSm ci:r- NE

Bni t4p 2 Cho hinh thoi ABCD canh a Gqi4

vd r lin luqt ld b6n kinh rlucrng tron ngo4i ti6p

cdc tam gi6c ABD vd, ABC

Ggi Mldtrung.diOm cin BC Chrmg minh OM

di qua trung tliOm cinAD.

Dinh chinh Biri T3 1440 - TH&TT thilng 212014 c6 in thir5u mQt do4n Xin d'qc lai nhu sau:

Chung minh rang na -5n3 -2n2 -lOn+ 4kh6ng chia hi5t cho 49 voi m5i sO tu ottl6n n kh6ng c6

du 3 trong ph6p chia cho 7 vdr chia h6t cho 49 trong truong hcr-p cdn 14i.

Cdch2 (h.5b) Ve duong frdn ngoai ti6p LBNC,

cit ducnrg thirng AD tai K (k]16c N)

Tri gi6c n6i ti6p BKCN c(t Gfi = ffi, md

ABM = CBN Gin thi6t)

=d?fr =78il > LABM cn LAKC (g.s)

nft nloutonn toP s

rixrr HArfrxn

NAvr rrQc zotz - 2ot3

Bii l a) Ap dung UiCn eOi

(o- b)t : a3 - b3 -3ab1a- b)tac6

b) Ta thAy I + x + l; * - x * | lu6n ducrng vd

x': 0 kh6ng h nghiQm cta phuong trinh dd cho'

Do d6 var t=x+L, phuong hinh de cho tr0

x

215

ft=2

Q (t -2)(5t+7)=0o1, L5 =_1 Ta tim tluo.c

nghiQm cua phuong hinh de cho ld x : 1.

nai Z a) Nhin cdhai v6 cira phucrng trinh thri

nhAt ctra hQ phuong trinh dA cho v6i 3 ta dugc

hai vd thu ggn ta tlugc *' i Yt : 3(*' - f)

e (x +y)( i' xy + f -3x + 3Y) : 0.

* V6i x-r !: 0 <> x: -!, thti vdo phuong

trinh thri nhAt ctra hQ ta tlu-o c Zi - +x + 3 : 0

e 2(x - l)' + 1 : 0, phu<rng trinh vO nghiQm

* V6i * - xy + f - 3x + 3y: 0, tni theo timg

vi5 voi phuong hinh thf nh6t cta hQ ta dugc ry

fx=3

-x-3y * 3 :0 <+ (x- 3Xy- 1): O oli=r.

Ta tim du-o c hai nghiQm (x; y) cila hQ phuong

Do a, b, c c6 vaitrd nhu nhau, kh6ng m6t tinh

t6ng qu6t ta giit thi}t a < b <c, suy ru a ) L;

) a+ b + c: (a-l)(b- 1X, -l)+2 (*)Ntlu (a - lxb - 1) > 4 thi voi a < b < c ta c6

3c> a + b + c > 3c > (a - l)(b- lXt - l) + 2

+ 3c> a@ - l) + 2 > 3c > 4c -2 + c 1 2,

tr6i voi di6u kiQn c > 3.

suyra b-t> 1,suy ra(a-l)(b- 1)chic6th6

nhQn gi5 frliLZ hoflc 3 Ta tim du-o c mQt b0 sd

(a: b; c) thod mdn ld (2; 3; 5).

Vqy c6c b0 sd tU nhi6n phdn bid1 @; b; c) thoitmln bdi to6n giim c6c ho6n vf ctra (2;3; 5)'Khi d6 P nhfln gi6 tri nguY6n ld 21

Bdi 3 Ycl a, b, c duong a + b * c: I ta c6

abc3abc3

o2(a*o*'{}-* I *l')=o v!\u,","/\b+c c+a a+b)

€ a: b: c e ABC li tam gi6c dOu.

Bii 4 Tu gii thli:t ta c6 AD ld trung tr.uc cria

BC vitla phdn gi6c ci.r 6k rt do ta cflng

nhQn thAy MNAP ld hinh vu6ng M[t kh6c do

fr@ *frFF =90" + 9oo = 18oo nan ANHP

ld tu gi6c nQi tiOp duong trdn ttucmg kinh NP'

, }

Suy ra ZruP =VNP = 45o (ctng chiln AP)

-m=1ED (:45o) > ABDH ld tu gi6c

noi ti6p >ZruA = ADB =90' hay AH L BH' C

il m 0$ Hec $ilt oil mil T0Ar rOt I n HAr DumG

BAi 4 g,S aiAml O).ui? tam gi6c BEF c6 mQt hinh vu6ng BMKN

a) Cho a., b, c lirbasO tiru ti thod mdn abc : I nQi,tiep (K e EF, M e BEvdN e BF) sao cho

a b c az U cz ti s6 gifra canh hinh vudng vcri b6n kinh duong

vit *+-+ ,r=;*;+7 chtmg minh rdng

?ig rinh c6c

it ,jat io, ior* ba s6 a, b, c ritbinh phucrng trdn nQi ti6p tam gi6c BEF ld 2 ' - "-'

-*-:r:?.r:J.:e.l*J1 .9::.*:i.:*3.i".T.913:.!r_!:,.

C- b) Theo phAn a) thl M, N, A, P,ll cung thuQc Bni 5 Ta nhpn th6y

Suyra AHM =90o = AHB ) H,M,Bthdng ' -€+yrwy)-6*;yr*4-

€+;\lr+x)

-cdc gi|tri nguyOn criax di5 A c6 giltri nguy6n

Bdi 2 (2 die@ Gihicdcphucrng toinh

{ J? 4x+z+",8T: =Jx1+J? +2x-l;

b) (4x+DJx+8 =3x2 +7x+8.

Bni 3 (1,5 diAm)

a) Cho f (*) : (x3 +l2x -3llzotz .

Tinh f(a) vli a =il6 -8G * il6* 8".6

ldtugi6cnQi ti6p, suyra BHI =BAI =45o (2)

Tt (1) vd (2) suy ra EEi =m MIt kh6c,

do cSch dpg ta c6 N vd 1 ctng nim vO mQt

phia cua BH suy ra H, N,lthing hdng

b) Cho a, b, c ld ba s6 ducrng c6 t6ng bing 3.

Rrit gon A, tim Chrmg minhrlng in.#.iA.}

Bii 5 Q diAm) Cho tluong trdn (O; R) vd haiduong kith AB, CD sao cho ti6p tuy}n tqi Acta (O;R) c6t c6c duong thtng BC, BD tqihaidi6m hrcmg img ld E, F G1i P, Q ldn luqt ldtrung di6m oia cfuc doqn thing AE, AF

a) Chrmg minh rEng tr.uc tdm H oiua tarn gi6cBPQliltrung <li6m cria dopn thing OA

b) Hai dudng kitlhAB, CD thoilmdn diAu kiQn

gi thi tam gi6c BPQ c6 diQn tich nh6 nh6t

Ap dung bstdingthkc i +t.ry (d[ng thric

C xiry ru khi vd chi khi a: b) ta c6

MQr sO pmudn{e_emAe qfu

(Trong nhimg ndm gdn d6y, m5i de thi t6t

'L rgttigp fftPf vd tuy0n sinh DH, CD

thunng co mQt ciu gi6i PT mf, hoic PT l6garit'

r.lfrirn"gitp

"a uai hqc sinh thi tot, bii vitit

ndy xin gioi tt i9,, mgt s0 phuong ph6p gi6i

thucrng gap de gidi hai loai PT tr6n

1 Phudng ph6p dda vG cung co s6

Cdch gidl Su dung c6c ph6p ti0n dOi tuong

duongv6i0<a+l,tac6

o qf(x) - os?) e f @): g(x).

logo f@)=1og,g(x)

^ {f t*l: B(x)

- i,r(rl > o (hoac g(x) > o)

*Thi dq l, Giai Tthucrng irinh

Cdch gidl D4t t-logof(1 Ddn d€n PT

mtz +nt+P =0, tim/suyra'r'

*Thi dv 4 Gioi phy-gj::L

2loga(-r2 -x) + :Jlogo1x - 1)2 - 2bga x = 4'

Loigi,rti DK: x >-2.PT cl6chotuongduongvoi

2logalx(x -llt + :.,0 fogot, + - 2loga x - 4 = 0

e Zloga@- r; +:JZrogo1, - r; - 4 = 0.

Ddt t = Jrt"tu@ -n (r > o), ta duoc pr:

t2+3t-4=Oet=1 ho6c t=*4 (loai)

vcri r : I, ra c6 Jz togo(, - t) : t

I

<> loga(x -1) :, e x - 3 (tho6 mdn DK)

*Thi dr1 5 Gidi phtrrmg trinh

Iogrr_,*7(z$r2 + i?"r+9)r-log2"-:((rr: +23x+ 2i) = -{.

log.t@)g(x) vd logr(,) /(x) thi ta c6 di6u

d'1 Phutrttg tritth dung

m.a-f G) 1r.6-f lx) * p:0, voi a.b =l

Cdch gidl Gia su a > l, ta df;t

1 - of G) (r > o), khi d6 6.f G) =!.

*Thi rlg 7 Giai lthtto"ng trinlt

(i - vB)'+ tr,.(:-'/5)' = 2'*r.Ldi gi,fiLChia hai v6 cria PT cho 2*, ta dugc

3 Phudng phep l6garit hoa, m0 hoa

Ldi girti Do hai v6 cira PT.da cho d6u duong

n6n l0y lOgarit co sO 3 hai v6, ta du-o c

ld hdm d6ng bi€n MAt khec, ac6 f (O) = 2.

Do d6, PT f(t)=2 c6 nghiQm duy nhAt r = 0.

Suyra log2x =0=x=1 tr

4 Phfidng phdp dda vd phttong trinh tfch

Cdch gi,fiL Khi gap PT d4ng

Vfly PT tld cho c6 hai nghiQm x :l; x = 2 a

*Thi dr; 11 Gidi phaong trinh

zlog?ex = log: r.toer(J, + 1 - t).Ldi gi,rtiDK: ;r > 0 PT dd cho htong duong voi

log?3 x = zlolz *.tog.1Jii -t1

<> 1og3 ,[tog, * -zlogr(Jz* +t- 1)] =

a) Phwong trinh ilu'a itwqc vi phnorug trinhdqns f(u)= f(v).

Cdch gi,fii Str dpng tinh ch6t: Cho hdm s6

y = f (x) don diqu tr6n t4p D Khi d6.f (u) = f (r) o Lt = v, vcri moi u,v e D

*Thi dB 12 Gidi phuong trinh

nh f (t) ld hdm sO eOng bitin tr6n lR Tt (5),tac6 f(xz -x)= f@x-A e *2 -x=4x-6

B -clirdiU@

b) Phu'ong trinh dgng logo f @) = lo96 g(x),

Ldi gi,rtL DK: x > 0 PT dd cho hrong duong

voi ro96 (J; * t/i) = )bgo *

<> rogu (G.t/i) = bgali.

D{t ro96 (J; * t/i) : bsali = t

ld hdm nghich bitin tr6n IR Lpi c6 /(1) = 1,

suy ra PT (6) co nghiQm duy nhAt t:1 Luc

do,ta c6 Jx = 4 (:) ir : 16 (thoi mdn DK) a

6 PhUdng phap danh gi6

Cdch gidi DOi khi dO giai PT mfl, l6garit dang

-f (x) = g(x), ta c6 th6 su bAt ding thric d6

d6nhgi6 f(x)<c<g(x),vcri ce IR.

MAtkh6c, tac6 2-(/ -02 <2,YxelR" Do d6

Tinh the tich kh6i ch6p S.ABCD vd diQn tich

m{t cdu ngo4i titlp hinh ch6p S ABD,theo a.

Cflu 6 (l di€m) Tim gi6 d 16n nh6t vi gi6 fi

nh6 nh6t cim P = *(*' +g)+ zy(+y2 +z),trong d6 x,y lirc6c s6 thlrc thoa mdn

xa +l6ya +(2*y+l)2 =2.

PHAN RIE,NG(Thi sinh chi ilugc chgn mQt trong hai phdn A hoqc B)

Cdu 7a (l diem) Trong mpt phlng voi he-truc fim ioa il di6m A vir B tr6n A di5 tam gi6c

tqa ilO Oxy, cho hinh ch0 nh$t ABCD c6

a 1r;r; rrqog tarn .t

" t r" eii )aC niim trcn oAB c6 oo :

+ vd c6 canh oB c[tduongtluong thhngd 3x- y -2 =0 OiOm N(4;6) tron (e tqiM sao cho MA: MB (vot Old g6c

ldtrungdii5mcria canhCD.Timtgad0 dinhl tqa dO).

Cflu 8b Q diAfi Trong kh6ng gian vcri hQ

tnlc tga dQ Oxyz, cho c6c di6m ,a(1;-1;0);

m{t cdu tdm I di qua ba di€m A,4 C sao cho

dO dei do4n thing OI nghnnnat lvcri O ld g6ctqa d0)

Giei he phucmg trinh

=2y-x J*y*y+2+"Jy'+x+2:4y.

NGUYEN TUAN LAM

2) Gqi A, B ldhai tli6m phin biQt h€n (Q sao

cho hai ti6ptuy6n.tqi A viL B song song voi

nhau Hai titip tuyr5n @i A vila cit t4rc tung

Dn luqt tui C, D sao cho CD : 4 Tim toa

dQ hai di6mA, B

CAu 2 (I di4@ Gi6i phuong trinh

Zcos4x+ cos 2x = 1 + rE sin 2x

Cflu 3 Q dii$ Giii phucrng trinh

Cflu 8a Q dii@ Trong kh6ng gian v6i hQ

tryc tea dO O*yr, cho hai mat phing

(r) : x -2y -3 = 0 vi (Q):x+Zy+ z+t:0.

Vitit phuong trinh duong thdng d qua dii5m

tt(l;O;2),vu6ng g6c voi dudng thtng OM

vd cht (p) t?i A, cfit (Q) tai,B sao cho oA: oB

(voi O ld g6c tea dQ).

Cflu 9a Q diA@.Cho s6 ph?c zthoa mdn didu Cflu

kiQn lz-1+il:lzl.rims6phric *=(I-i)., =,

sao cho s6 phric w c6 m6dun nh6 nh6t

B Theo chucrng trinh Nflng cao

Cflu 7b Q diA@ Trong mflt phing voi hg tr.uc

tqa d0 Ory, cho tluong trdn (Q c6 phuong trinh

Cflu 2 DK: cosx * 0; cos2x * 0.

PT dA cho tucrng ducrng vcri

4 s irrr s in2x co s2r c o s 3x : tary- .sin2-t co s 6x.

Tn he tr€n ta bitln tlOi ve d4ng sO phric nhu sau

(o' * b' +3b -3) + (2ab -3a +1)i = 0

o (a + br)2 -3i(a+ bi) + i - 3 = 0

Giai PT nity ta dugc z1= 1 + i; zz = -I + 2i

Tt BDT Cauchy ch.o 3 s5 duorrg (luu j'c6c

m[t cria tu diQn gin d6u ld c6c tam gi6c

(Xem ti€p trang 27)

".r *r,r-rorn, T?EilrHE5, rr

TOI

!}Irong ki thi VMO ndm20l4, ngiy thf nh6t

I o mOt bdi hinh hgc hay, d€ bdi iluo c ph6t

bi6u lai cho pht t q'p ,& beivirit nhu sau

OBili tofn 1 Cho tam giac: nhpn ABC nqi tiip

trong' dudttg trr)n (O) Goi I ld di€m chinlt

giti'a c'r)u cunpl EC khing chil'a A TrOn ,4C

l,i1; ff;in K khac C sao cho lK : tC Dr.rcng

thiing Rl( cdt (O) tqi D khac B TrAn DI tot'

rtiirn M sao cho Cl',{ song sottg vrsi AD iludztg

thdng Kh,f c:dt.dwdng thang BC tqi N Dwttg

tt"dn ngaoi ti€p tam gidc BKN cat (O) tqi P

ktruc B" Clwng winh rdng PK ali qtta trung

lfr: r8o'- ifu:180" - lER:leD (r)

Do 1 ld di6m chinh gita cung fr nen DI liL

tia phdn giirc ctla BDC (2)

rt (1), (2) suy ,u fiD : dD.

Y$y LKID: LCID (c.g.c) suy ra DK : DC

do d6 DI li trung t4rc cira KC Tucrng W AI

ld trung trpc cira BrK.

GqiIJ ldtluongkinhctra (O) tac6 AJIIKD

vd JDll AK Tit d6 tu gi6c AJDK ld hinh

Hfffi sJAMr rfu rffi rx0c ?hm

hdng Thpt vQry IPK : IPB + BPK

:6n +6frR (tu gi6c BKNP nQi titlp)

: BAI +(NKC + NCK)

:6h *frr *6dR QD liLtrung truc cua KQ

: BAI + CAD + BCK (doCM ll AD)

:fu *ffi+fu (do tu gi6c ABCD nQi tiep)

: IAK + AKB (tinh chdt g6c ngodi)

:90" (do,BK L AI)

: IPJ (do IJ ld cluong kinh cta(O)).

Tnd6 suy rabadi6mP,K,J thinghang Tti c6c

nhfln x6t tr6n ta c6 di6u phii chimg minh n

Bii to6n tr6n li mQt ktit qu6 tle.p vd ch{t ch6.

Trong loi gi6i tr6n ta co th6 tom lugc lpi y

chinh nim trong bdi to6n sau

&tsei irdn 2 Cho t(tt1l giac ABC n$i ti€pdw,)ng trdn (O) vo'i clLrmtg plrun gidt trottg'

AD Grli ti td di€nt doi xi'ng cua B quo drdrgAD: BE cat (O) r.Ji F khac ts Goi k! ld die:ut

CLch gibibdi toin ndy nim trong phAn ddu cira

ph6p chimg minh tr6n Ddy ld mQt k6t quA kh6c6 y nghia Bdi to6n 2 cing co th6 dugc nhinduoi mQt chchldthc nhu sau

Ofai tor{n 3 Cho tam giac ABC cdn tai A: Plci mot diint thu6c dtrrrng thang BC Chimg

DAy ld m6t ktit qui don gi6n li6n quan d6n tu

^ ,,4

gi6c nQi ti6p Tuy vfy n6u dC i ki c6c ban d€

thiy ld bdi to6n 3 thuc ch6t ctng li bdi to6n 2

6p dUng cho tam gi6c cdn ABE BiLi to6n c6

m6t mo rQng cho tam gi6c b6t ki nhu sau

OBAi tuin 4 Cho tam giitc ABC P ld m6t

di€m thay d6i tin dtrtng thiing BC Goi E,F

ti:n tryt lii cac di€m doi xtrng c{ra B,C qua

-.;

trung diem cua doatt thang AP

a) Chung minh riing furcrng trdn ngoai ti6p

tam gidc ACE,ABF ,dt nhou tgi mQt diem

Q nam tr€n BC

b) GOi AH ld dudng cao cua tam giac ABC

:

Cluiltg ntinh rang QP: HB + HC

Ldi gi,rtL (Ban dqc t.u vE hinh)

a)Tir gritc BCEF h hinh binh hdnh n6n

AQB:180" - AFB: AEC:t80" - AQC

suy ra tr@ *7& :180' n6n ethuQc BC.

b) Gqi R h di6m d6i xtmg yot Q qua trung

di6m 1 cua AF thi R thuQc EF vd tu gi6c

ARPQ ld hinh binh hanh ndn PQ: AR .

Tathdy fi gShc BREQ h hinh binh hanh, n0n

M[t kh6c AECQ H hinh thang n6i ti6p n6n

AECO h hinh thang cdn, do d6QC : QE (2)

Tt (1), (2) suy ra BR: AC, t0 gi6c ARCB lit

hinh thang cAn n6n R c6 dinh Tir d6

PO:,en lkhOrrg AOi) tvtit kh5c goi AH lit

duong cao cua tam gi6c ABC cing ld dudmg

cao ctra hinh thang cdn ARCB ta d6 chimg

minh AR: UA + ttC ftt d6 ta c6 di6u phdi

chimg minh o

N6u tam gi6c ABC cdn tqi A ta th.u dugc bdi

torin 3 Ntiu str dung c6ch ph6t bi6u dt5i ximg ta

c6 th6 d€ xuAt bdi to6n nhu sau tu bdi tohn2

SBdi tofn 5 Cho tam giac ABC n6i ti€p

dadng trdn (O) vd ngoai ti€p furnng trdn (I)

Duong trdn ngoai ti€p tam giac BIC cdt CA,

AB tdn luqlt tqi E, F khac B, C BE, CF lCtn

hrqt cat (O) tqi M, N khac B C Goi K L ldn

laqt ld trung di€m cua AM, AN Chmg ruinh.

riing EK vd l-L cdt nhau tr€n rludng trdn (O)

Qua bdi tofun 2, ta thdy EK,FL dAu di quadi6m chinh gita ctra .*g 6D chria l MQtc6ch tg nhi6n bdi toan 5 c6 th6 m0 rQng thdnh

bii to6n sau

OBlri tofn 6 Cho tunt giac ABC nQi tidpductng trdn (O) MQt dadng trdn (D) di qua B,

C cat CA, AB lin lwt tgi E, F khac B, C;

BE, CF lin luu ciit (O) tqi M, N khdc B, C.

Cac cti6m K, L ld:n fuqt thuQc AM, Al' sao cho

KLll EF; BE a CF : S; EK:FL=T.

Ch[mg ninlt rang ba di€m A, S,T thiing hdng

Bii to6n 5 cdn c6 mQt khai thic tl6ng chri j'

nhu sau

OBiri to6n 7 Cho tam giac ABC nQi ti€p

du'dug tron (O) r,d ngoai ti6p furong trdn (I).Dursng rt'ort ngoai ti6p mnt giac BIC cdt CA,

AB tan lmtt tai E, F khac B, C BE, CF ldnIrqt ccit tO) tqi M,I,,l khac B, C Goi K, L litt

Irot la tntng dietn ctia AM, AN

:a) Cltntg rtrirtlt rang EK vii FL cdt nhcnt tai

die:m T n'€n du'dng trdn (O)

b) Gia sti EK vd FL cat (O) tqi P, Q khac T;

PQ cot BC tc.ti .5 Ch{rng minh rangdtrong rhong AS ti6p ilc dadng tron (O).Bei toan 7 lqi cb mQt khai th6c kh6 t.u nhi€n

kh6c nhu sau

OBlri to:in 8, Cho tam gidc ABC nQi ti6p

du'ottg trdn (O) vd ngoai ti6p furdng trdn (I).Emntg tr"dn ngoai tidp nm giac BIC cdt CA,

AB lin ltrqt tgi E, F khdc B, C BE, CF tdnluqt cat (o) tqi M, N khdc B, C Goi K, L td:n

lr qt ld trung di€m cua AM, AN

'dg1 uQ,tu1 IpQ

lgru nqu tugl

19f Sugnq 9c

ug4 ugol Ipg

'2V nnry

iuo"^

q

onb 3upt11

3aru13 un1 1.on1 tpt lDt (O) )?tpl 1.'S "g ')

9fi3 *u,o.ng uo-ri lioBu c1211 tary tnr| 1dg W

(e

'g')

)pW O'd tbt

t?q Ot)

'(i

ugrt 7L4utp

dgp lno8u

?,r

(.O) uo.4 StLo.np

cl?U

lgu

)gv

)?!3 utDt

oq) 0I

ugot IEgg

e]

oqc

re1

Eueu Eugc uga Eug:

ulpl 4

eq enb

Sugql ugot lpq

??x a|qN

f,J

'rluru Eu4qc

reqd nglp

gc

BI 'iy

e+c 7,y

feH '

8uo.np

tgrlc qup oeq;

tv

P^

gs

ugu 'quPq qqq qqq

pt

ggsy

cVtA qq 9c

'quu

Eu4qc

reqd nglp 9c

eI

.g enb

Ip d7

dgUr

pp 9p

(O) u+c

gupt

Buenp

pt

^SO op)

lgul

cQnqt H' g' d'

Suo.t quttu 3unt13

qg

(q

6)

tlt inu

g'I'd'g

}up,r

t1u1tu

SofU) 'H lnt

g,

ag

')i'tlun

u2"r1tt/,tttt1t t7t

d?4

l/u )gf/

)?!3 utDt

?l

'npp u?g

offi ilu0to t t YIr milu rfrm 20$

r- so LUqc vE elAlrHudNo t-E vAtt rnlEut

Gi6o st"t LO Vdn Thi6m (1918 - 1991) la Chrj tich

ddu ti6n c0a HQiTodn hgc Viet Nam 6ng lir nhir

toSrn hgc n6i ti5ng, dd c6 nh0ng d6ng g6p ldn

trong nghi6n crJu vd rlng dr*rng to6n hgc Ong

cOng lir mQt trong nhfrng ngrtdi dqt n6n m6ng

cho n6n gi6o dr;c dai hgc 6 nudc ta, ld ngudi thAy

c0a nhi6u thd hC c6c nhir to6n hqc Viet Nam.

GS LA Vdn Thi6m lu6n giirnh sU quan tAm dic

bi6t d6n vigc giAng dqy to6n hgc 6 c6c trudng

pn6 tn6ng Ong lir mQt trong nhfng ngttdi sdng

lap H0 thSng phd thOng chuy6n todn vir b6o

To6n hgc vir Tudi tr6 Gi6o stJ L6 Vdn Thi6m dd

drjgc Nhe nudc t{ng HuAn chr-Idng Doc l4p hqng

NhAt va GiaithrJdng H6 ChiMinh Grril thudng LO

Vdn Thidm do HQi To6n hgc Vigt Nam dit ra

nhim g6p phdn ghi nhdn nhfng thirnh t(ch xudt

sic c0a nh0ng thAy gi6o vh hgc sinh ph6 th6ng

dd khic phr;c kh6 khdn d6 dqy toAn vd hgc to6n

gi6i, dOng vi6n hgc sinh di sdu vdo mdn hgc c6

vai trd dac biQt quan trgng trong sU ph6t tri6n lAu

diri cria ndn khoa hgc nudc nhd GiAi thudng LO

Vdn Thi1rn c0ng ld sU ghi nhQn c6ng lao cria

GS Le Vdn Thidm, mQt nhd to6n hgc l6n, mgt

ngrtdi thAy da h6t ldng vi sr; nghigp gi6o dqc.

u - crArrHUdNG lE vAN THICM 2013

Hoi To6n hgc Viet Nam quy6t dinh trao Gieithuang

L€ Vdn Thi€m ndm 2013 cho nh0ng thdy gi6o vit

hgc sinh sau ddy:

1 Thdy giAo Nguydn Vdn ThOng (THPT chuy6n

L6 Qu17 D6n, Dd NEng)

- Sinh nlm 1960.

- Tham gia dqy To6n hdn 30 ndm, trong d6 dqy

Chuy6n Todn 18 ndm.

- CUng vdi t6 todn c0a trudng, tham gia b6i

dttdng duqc nhi6u hgc sinh dat giAi cao, trong

d6 c6 hon 20 hqc sinh dat giAi cdp Qudc gia,

4 em dat huy chudng IMO Qu6c t6; nhi6u hqc

sinh dqt Huy chUdng Vdng 30/4.

- Nhi6u bdi vi6t chuy6n d6 trong HQi thAo gi6ng

dqy todn.

- 22 ndm lir gi6o vi6n dqy gi6i vir chi6n sT thi dua.

- DuOc phong danh hiQu Nhd gi6o uu t( ndm 2008.

HA HUY KHOAI (ViQn Todn hqc ViAt Nam)

2 Hoc sinh1) VO Anh DOc (THPT chuy6n Hd TInh), hiQnnay hgc t4i Khoa To6n - Tin, DHBK Ha NQi)

- Giai Nhi hec sinh gi6i Qu6c gia 2012

- Giai Nhi hqc sinh gi6i Qu6c gia 2013

- Huy chudng Vdng IMO 2013

2\ Phqm Tudn Huy (THPT chuy6n DHQG TP.

H6 Chi Minh)

- Giai Nhdt hgc sinh gi6i Qu6c gia 2013.

- Huy chudng Vdng IMO 20't3

3) Chu Th! Thu Hrdn (THPT chuy6n Long An)

Da vrjqt nhi6u kh6 khin, dat thdnh tich xudt sac

- Ldp 10: Huy chrtong B4c Olympic 30-4

- Ldp 11: Huy chLldng Virng Olympic 30-4, giAiNhi hqc sinh gi6i to6n tinh Long An; giAiKhuy6n khich hgc sinh gi6i Qu6c gia.

-Ldp 12: Giai Nhdt hgc sinh gi6i to6n t?nh Long

An; giAi Ba hgc sinh gi6i Qudc gia; Giai Nhi cuQc thi giAi b6o Toin hgc vir tu6i tr6.

4) LE Qu6c rnng (THPT chuyOn L6 Quf D6n, QuAng Tri)

- Ldp 10: Huy chudng Bqc m6n To6n D6ngbing B5c 89.

- Ldp 11: GiAi Ba m6n ToAn thi hqc sinh gi6i

Qu6c gia.

- Ldp 12: Giai Nhdt m6n To6n thi hgc sinh gi6i

Qudc gia.

L6 trao giAi d6 duqc t6 chrlc tqi cuQc G4p m{t

ddu xuAn c0a HQiTo6n hgc Viet Nam tai Hd NOi,ngiry 221212012 Ddn dg vd tham gia trao giAi c6

nhi6u nhir todn hgc, nhi6u thAy c6 gi6o, trong d6

c6 GS Ddo Trgng Thi, Ch() nhiQm 0y ban Vdn

h6a gi6o duc, thanh thi6u ni6n nhi d6ng c0a

Qu6c hOi, GS Irdn Vdn Nhung, T6ng thLI k( HQi d6ng Chrlc danh Gi6o su Nhd ntt6c, nguy6nThf trr.tdng BQ Gi6o dUc vd Ddo tao; c6c Gido stJ

Ch0 tich vir nguyOn Ch0 tich HQi To6n hqc Viet

Nam; D6 Long Vdn, Phqm Th6 Long, Ld Tudn

Hoa, Nguydn H1u DtJ.

Hai em Phqm Tudn Huy vit Chu Th! Thu Hi6n

kh6ng c6 di6u kiQn tham dU cuQc gap mat tai

TP Ha Nai s6 drJoc trao giai tai bu6i 16 ti6n hdnh

6 TP H6 Chi Minh.

T?[I#EE

rr

- M6i sd c6 2 bdi todn cho THCS (ki hieu T*/THCS) vd 2 bdi todn cho THPT (ki hiQu T*/THPT).

- Thcti han nhQn bdi muQn nhiit ld 2 thdng, tinh ti cu6i thdng cria sii Tqp chi c6 ddng di thi.

- Mdi bdi gidi vi€t t€n mQt td gitiy ri)ng, ghi rd h9 vd tAn, tnron[, lop (cd thZ gtbi chung phong bi, b1n ngodi ghi rd: Bdi dqt thi Gidi todn ilgc biQt ki niQm 50 ndm Tap chi_TH&TT, c6 ddn tem.

- K€t qud cuQc thi duqc c6ng b6 tr€n s6 448 (10.2014) vd trao gidi tdo L€ bi niem 50 ndm TH&TT.

;tliririr, (Vi6n C6ng ngh€ Th6ng tin)

l ;, Bhi T4ITHPT

M, N theo tht t.u ld trung tli6m cira BC', CB'

MH cit dugng trdn ngoai tii5p tam gi6c CHB'

tqi I; NH c5t tlucrng trdn ngoai ti6p tam gi6c

BHC' tai J GiA su P ld trung di6m cira cpnh

BC, chimgminh ring,4 P L IJ

Cho tam giitc ABC, (O), (I) vd 1, theo thir ru ld

::{ tim <luong tron bdng ti6p tl6i di6n dinh I cira

!fr tam gi6c Goi D ld ti6p ditim cria (I) va BC;

- D6y c6c so thuc duong {o,}7=o thod min

.,rs**n o,+r = - -?- voi n : 1,21 3, Chung minh

The sequence of positive real numbers {o,}7=o

satisfies o,,, = -1 for n: 1.2,3,

an + cln_I

Prove that there exists a pair of real numbers s, ;l

/suchthat sla,.-lforall n:0,1,2, '$,

T4lSenior

In a triangle ABC (AC > AB), the altitudes BB'

and CC' intersects at point H.Let M, Nbe themidpoints of BC', CB' respectively MHmeetsthe circumcircle of triangle CHB' at I; NH

meets the circumcircle of triangle BHC' at

point./, If P is the midpoint of segment BC,

provethatAP LIJ.

Translated by LE MINH HA

tz