TOÁN HỌC TUỔI TRẺ SỐ 455 THÁNG 5 NĂM 2015

Tạp chí Toán học và Tuổi trẻ Số 455 (Tháng 52015) gồm khoảng 31 bài viết trong các chuyên mục: Dành cho Trung học cơ sở, chuẩn bị cho kì thi THPT Quốc gia, diễn đàn phương pháp giải toán, đề ra kì này, giải bài kì trước, tin tức toán học, bạn có biết, diễn đàn dạy học toán, sai lầm ở đâu. Mời bạn đọc tham khảo.

xuflr siil rU r go+

2015

s6 455

==

r:-rap cni Ra HAruc rHAruc - ryAM rlnU 52

oi\rvH cHo rRUNG Hoc pu6 rxOruc vA rnuruc roc co s6

Tru s6: 187B Gi6ng V6, Ha NOi.

DT Bi6n tAp: (04) 35121607; DT - Fax Ph6t hdnh, Tri su: (04) 35121606 Email: toanhoctuoitrevietnam@gmail.com Website: http://www.nxbgd.vn/toanhoctuoitre

Gi6o sr.r,, ViSn sI Phqm Minh Hqc, Nguy6n Bo

tru&ng B0 Gi;io dUc - Dao tao, danh gia bQ

sach dqt ba ti6u chi: Khach quan, Chinh x6c,

"Quyiin uy" va vigc ph6rt hdnh Tu dien B6ch khoa

Britannica taiViet Nam co the duqc coi la mQt sr=r

ki€n lon trong doi song vdn ho5 - giao duc nuoc

nha TheerG.,:iAb.su, m6i trusng nen co mOt cu6n

c6c em hqc sinh tham khdo

1Nhu Loi nha xu*t kho khin, c6c dich tap vren da lam vrec

rthe cr:ng 'kho tr6rnh

ir.;il*

A BAN GIAO EIU'Cll

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tr^r oten Boch khoo Brit

']rfha xuAt ban Giao duc Viet Nam vua cho ra

5l m6t phien brin tieng Viet cOa cu6n Tu dien

Bach khoa Britannica c0a M!.

Ttr dien Bach khoa Britannica gOln$OO0 rnuc

tLr, 2.500 hinh minh hga vi beni:&.|51 lTnh vuc

khoa hoc vEr dcvi s6ng, gan 300$gc tu v6 Viet

Nam do cdc tic gid Viet Narn biCIn soqn theo

thda thuQn voi phia M!, C0ng ty tsacti.khoa.thu

Britannica lVly xet duyQt

Vigc chuy,nn dich sang tieng Vi0tdugc,lhEe triQn

r6t cOng phu, do 54 dlch gid, 52 chuyBngia{tr,AS$

do co iac chuyen gia t* oienl nigu,di!,!h$

d!nh, bien tap ducri sqr chi dAo ceq,H1$j.d&,891

soan - bi6n d!ch, do ong Ng0,TradAirlite,h$

HQi dong thanh vi6n - fon$,si6iii.a :*d6$x

B0 sach gom hai tQp, t6ng qg.,!rg $.$$ t ng

Hinh thuc trinh bdry cOng phu, tt&g.t$ngi:iin b6n

mdu toin b$, dong bia cung ch0'.d ,chim, 6p

nh0 vdng, co bia 5o cho tung cudn, d{t trong hQp

cung, ducvc phat hanh tu ngay 2011112014

Theo

bdn

l,9JQ,l

'rftl'\G r c{, s$

[/ rong cac bdi todn vi tfnh sd ilo gdc cfing

?, nhu trong cdc bdi todn chung minh hinh

hqc, c6 nhfrng trrdng hqp ta gqp khd khdn khi

chmg minh tryc ti€p bdi todn, khi d6 cd th€

dilng phmtns phdp chthng minh gidn fidp.

l Phrmtg phrip cht:rng minh hai iliAm tring

nhau

Thi du l Cho tam gidc ABC, BAC=60',

dudng phdn gidc CE TrAn canh AC liiy di€m D

sao cho CED =30' T[nh s6 do g6c BDE

A

Hinh I

Tim hrdng gidi.(h.l)

Bing c6ch do tr.uc ti6p, ta thiy 6DE = 30o, tric

ld NDE cdn (1ld giao tli6m cria BD vd CQ,

khi d6 DIC = 60' ndn B, + C, = 60" Ta lai c6

ABC + ACB =120", n€n B, * C, = 69o.

oo 4 =Q n€n A=A.Do d6 ta vE BD' lit

tludng phdn gi6c cria g6c AAC ri;i di chimg

minh di6mD'tn)ng voi di6mD

Ggi 1r(ld duong phdn gi6c cila L,BIC thi

I,=Io=60o, LBIE=LBIK (g.c,g) > IE:IK,

LCID' = LCIK (g.c.g)= ID' :,LtK suy ra

IE: ID' Tam gi6c IED' cdntai I c6 frD'=12ff

n€n IED'=ID'E=3U Do D'e K vd CED=3U

n€nD'trungD vi1v 6iE =66i,=30'.0

Thf du 2 Cho tam gidc ABC, 6tra = 115o,

ABC =40" TrAn nira mfit phdng bd AB khing

chtha C, ke tia Ax vu6ng g6c voi AB Liiy di1m

E tAn fia Ax sao cho AE : BC Tinh si| do g6c

AEB

Hinh 3

Tim habng gidi (h.3)

Bing c6ch do tryc ti6p, ta thiy fEE =25o, tftc

liL ABE = 65o Do tI6 ta 6y di6m E, tr€n tia Ax

sao cho fri'=65" r(ii chimg minh di6mE'

, -.},

trung vor dremz

sdnrr6-*ro T?8I#EE 1

Ldi gidl TrCn tia Ax lly di6m E' sao cho

GE'=65" Liiy di6m l tr6n dopn BE' sao cho

E'N =40" thi BA'I :90" -40" = 50o Suy ra

fii=180"-(65"+50")=65o, do d6 LNB

Tac6 LNE'= ABAC (g.c.g) nAnAE': BC.Ta

l4i c6 BC : AE n€n AE : AE' , suy ra E' tting E

Do 116 frE = 65" , frE = 9oo - 65o =25o .a

1'hf ctg 3 Cho tam giac ABC cd ABC =70",

IA : IB : IC, do tl6 ta vE 1' ld tAm tluong ftdn

ngopi titlp tam gi6c ABC rdi chimg minh di6m 1

tning voi di6m1'

Ldi gi,fiL Ggi 1'1d t6m cira cludng tron ngo4i

ti6p MBC Ta c6 fu=180-(7(),+50)=66n.

Theo m6i li6n hg giira g6c nQi titip vd g6c 6

tdm,ta c6 BI'C =2BAC = 120o n6n

fEE =fdn =30" v{y 1' trirng L Tam giiic

IAB citn c6 IBA = 70o - 30o = 40o ndn

iiE =40" Suy ,u itri = 600 -40o =20o.A

2 Phuurng phtip phiin cltil'ng

Tr0 lai thi du 3 n6i tr6n Ngodi c6ch gi6i bing

c6ch vE tdm I' ctra tludng trdn ngo4i titip tam

gi6c ABC tOi chtmg minh 1' trung lnhu tr6n,

cdn c6 th6 chimg minh bing phuong ph6p phan

^

Cz : ACB - Ct :50o -30o = 20o.

C6 the chimg minh bing phin chimg theo hai

c6ch:

o Cdch 1 GiA sir A, >40" thi 4<20' Xet

NAB, do Ar>40" =8, ndn IB > IA Xet NAC, do Ar<20o =Cz ndnIC <11 Suyra

IB> IA> IC,triLi voi (1)

Gi6 sri At<40o thi 4120' Chimg minhtuong t.u ta dugc IB < IC, tr6i voi (1)

YqY Ar=40', suyra 4=20'.

c Cdch 2 Giest IB: IC < IA.Xdt NAB, do

IB < IA n6n A, 1Bz = 40" ){':et NAC, do

IC < IA n€,n tr<ir=2O" Suy ,u tr+Tr.60',

^ trdivot BAC :60" Gia sir 18 : IC > IA Chungminh tuong t.u ta dugc ir*$> 60o, trhi vu

BAC = 60" Vay IB = IC: IA, suy ra

4 A=4oo' i:20"'J

Cdc b4n thr? dtrng phuong ph6p chli'ng minh gi6nti6p iI6 giii c6c bii tip sau:

Bni 1 Cho tam gi6c ABC, Gi =75",

frE:60", iti6m l nim trong tam gi6c d6 sao

cho tni=tCn=45' Tinh sti tlo c6c g6c IAB

vd goc IAC

Bdi 2 Cho tu giirc ABCD c6 ABD:35',

ADB =30', ACB = 60o, ACD =70' Gqi O ld

giao <li6m crta AC vd BD Tinh s6 do g6c AOB

A TONN HOC

A efurriUffi

Hrlilng df;n gifii uE THI {UYEr* s:rqrt unu lffr tu nnmru ronnl

2) Ta c6 i.reG-l ld udc cria 2 g6m:

+1,L2 Tt d6 tim rtugc x {O;+;e}.

2) Ta c6 A,: 4a2 +164-151 OC pf c6 nghiQm

nguyOn thi A: n2 v\in e N Khi d6

2) Ta chimg minh tlugc 'xyx+y 1*1t -!-,v*,y >0,

iting thirc x6y ra khi vd chi khi x = J Tt gi6thidt a + b - c > 0,b + c - a > 0,c + a - b > O.Ta c6

s=(b.=.*")*(#r.#-)

+t(_t r \-z t*0

y6+a-g- q+6-g)'7 b a

Mit 2c+b = abc o?u*l= a ncn

S>za+9>4J1 Eing thirc xiy ra khi vd chi

a

khi a= b=c:J5 VayminS:4J1.

C6.u,l

1) Theo tinh ch6t ctra ti6p tuy6n ta c6

NB = NC; OB =OC; ON h trung tr.uc cria

BC Gqi.Kld giao diiSm cira ONvitBcthi Kld

trung tli€m ctra BC Ta c6111111

Thdi gian ldm bdi: 150 phrtt

C6u 1 (2 diem) Gi6i c6c phucrng trinh sau:

a) Cho hai s6 thyc a, b th6a mdn a + b : 2.

Chtmg minh ring: a2 +b2 l aa +ba

b) Cho c5c sd ducrng x, !, z th6a mdn cli6u ki€n

x.v + yz + zx: 1 Chtmg minh ring:

JFF Jt+F'Jt+t-2'

CAu 4 0 diAd Cho tam gi6c ABC Trdn c6c

cqrilt BC, CA, AB l6n lugt lAry circ di€m D, E, F

Gsi (d1) ld ducrng thing qua D vd vudng g6c

voi BC, (dr)ldduong thing qua E vd vudng g6c

voi CA, (&)ld duong thing qua Fvd vu6ng g6cvot AB Chimg minh r[ng (dr), (dr) vd (d3) ddngquy khi vd chi khi c6 d[ng thirc sau:

(on, - ocr) + (nc, - a,+r) + (rt, - FBz) = s

CAu 5 (2 die@ Cho tu gi6c ABCD nQi tiiip

trong dudng trdn tdm O GSi 4 ld rli6m tr6ncung nh6 AB Goi H, K, P, Q li,n fuo.t ld hinhchi€u vu6ng g6c cua B l€n.AC, CD, AE, DE

Ggi M, Nl6n lugt li trung di€mciaAD, HK

a) Chtrng minh rdng AD, PQ,IIK d6ng quy.b) Chung minh r6ng,44/vudng g6c vcri NB

CAu 6 Q die@ Cho mQt da gi6c d6u 50 dinh

Nguoi ta ghi 16n mdi tlinh cua da gi6c sd I hoic

sd 2 Bi€t ring c6 20 dinh ghi sg 1, 30 clinh ghi

sd 2 vh c6c sd tr6n 3 dinh li€n ti6p bet k, ki.r6rgcl6ng thoi bing nhau Hiy tinh t,Ong ctra t6t ci

c6c tich ba sd trOn 3 dinh li€n ti€p cira da gi6ctr€n

NGUYEN DUC TAN gP HA Chi Minh)

(}- uat khdc, do AM llBC ndn AMCB ld hinh

thang cdn

= MC = AB,MB = AC (2)Tir ( l) 0\ '-,_ = AC ,AB.BP -CP .

3) Gqi Qld grao di6m cira AP vit BC.Ta chimg

Tir (3), (4), (5) suy ' ra ry=ry=BQ=gg AQAQ

>Q=/( Vay BC,ON,AP cl6ngquytaiK

gia thirit) Suy ra dpcm

2) N€u a,b chinthl a2 +b2 li hqp s6 Do d6

i^

ndu tdp con X cia A c6 hai phdn tu ph6n biQta,b md a2 +b2 ld m6t s6 nguy6n tO thi X

khdng th6 chi chta c6c s6 chin Suy ra k > 9

Ta chimg t6 ft = 9 ld gi6 tri nh6 nh6t cAn tim(nghia ld trong 9 phin * Uit tcy ci,a A 1u6n tiin

tai hai phdn tu phdn biQt a,b md a2 +b2 ld mQt

s5 nguyOn tO) That vdy, ta chia A thdnh 8 cap

pfran tu ph6n biQt (a, b) th6a mdn a2 + &2 ld mQt

so nguyen t6 nhu sau: (t;+),(z;3),(s;s),(0;tt), (z;to),(o;t0),(tz;t:), (t+;ts) rheo nguy;n ty

Dirichlet thi kong 9 phAn ff b6t ky cria X 1u6n

,;

c6 hai phdn hr cirng thuQc mQt cdp ndu trdn vd

ta c6 dPcm'

NGUYEN vAu xA

(Gl/ THPT Y€n Phong SA Z, Sdc Ninh) Swu im

1) Gie str O nlm ngodi mi6n tam giirc ABC

Kh6ng m6t tinh t6ng qu6t, gi6 su A vd O nim

vC hai phia cta cludng thing BC, doqn AO cit

:/N )

" n\ L

, TOnN HOC

+ -ctudiff@

cHUfu Bt

TRUilG H0Cpud rn0ro

ou6c on

.6/rong bdi vi6t ndy, chring tdi xin gioi thidu

v cdch ti6p cdn bdi torin tinh khoing c6ch

trong hinh hoc kh6ng gian nhd 6p dqng c6ng

thric tinh chidu cao cria kht5i ffi diQn w6ng

nhem gifp c6c ban hgc sinh chu6n bi cho ky thi

TIIPT Qutic gia sip toi

Bii tofn mO tIAu @di t7 trang 103, SGK Hinh

hoc Ndng cao lop tt) Cho hinh ta di€n OABC

c6 ba canh OA, OB, OC ddi m6t vu6ng gdc

a) Chilmg minh tam gidc ABC c6 ba g6c nhon.

b) Chimg minh riing hinh chi€u H ct)a di€m O

ffAn mdt phdng @Bq ffimg v6i truc tdm ct)a

tam gidc ABC

c'l Chmg minh ring #= oorl*#.#

D0 dai OH : h d bdi to6n tr6n chinh ld khoing

c6ch tu <1i6m O rI{in mflt phing (ABC), sri dqng

k6t qui tr6n ta c6 th6 tinh khoing c6ch ru m6t

tli6m di5n m[t phlng vd kho6ng c6ch gifia hai

dudng thing ch6o nhau m6t c6ch tu nhi6n vd

nhanh ch6ng

DANG 1 KHOANG CACH rrl urgr DrEM

DEN Mgr nAar psANG'

Bii toin l Cho hinh ldng trq ABC.A' B'C' v6i

AB=a,K=?tt,fu={t Hinh chiiu vuong g6c

cira A' len @BQ tritng vdi trpng tdm G cila

tam gidc ABC, g6c gira AA' vd mfit ddy bdng

60".Tinh Vo,o* vd khodng cach ti G ddn mqt

phdng (e'nc).

Ldi giQi (h.1, h.2) Ap dpng dinh li c6sin cho

MBC co:

AC2 =AB2 +BCz -2AB.BCcos60"

= az +4a2 -2.a.2a.| =3a2 = AC = ali.

Suy ra ACz + ABz = Bt, dfin d6n LABC

vu6ng taiA.Do A'G L(LSC) n6ntath6y

Eno DUNE TiN+t eflfir e[l

K+rdr rfl ol$x yu6xo

B

Hinh 1.Dlt d(G,(A',BC))=h

Ke GH ll AB, GK /l AC suy ra

GH rGK Ta c6 GH:|*:i Avir GK:Io, =+

Yilg sil rgnN TiN.fI I[+IgiNE EflE+I

NGUTEN NGQC XUAN

(GY THPT chuyAn Hodng Ydn Th4 , Hda Einh)

(ee' :(enc)) = ( ee' : tc1 = lQ! = 66" ;

A'G=AGtan6o" -Zalj 1 .uvw Dod6 v, - vA|AH -:ot

Bdri to6n 2 Cho hinh chdp S.ABCD co ddy

ABCD ld hinh vu6ng cqnh a Ilinh chi€u vu1ng

gdc cila S lAn (ABCD) tritng vdi trgng tdm G

tam gidc ABD, csnh SD tqo vdi ddy (ABCD)

m6t g6c 60 .T{nh th€ tich kh6i ch6p S.ABCD

vd khodng cdch t* A tdi 6Bq theo a.

Ldi gidi (h.3, h.4) Stnco=az Gqi G li trgng t6m LABD,

sG L(ABCD) = (so,(ancD))=.fr = 60

Ggilldtrungdi6ml Btac6 aC=lU 33 =of

vd sG = DGtan6o, - a'll5 -': -r-i

= +a(n.;(sen)) = +t(a > o).

Gqi E ld trung di6m cria AB, tac6

1-HE=ltB ,2 Isuyra AH LHE vit

an=1rc:+, ddn d6n H-SAE ld kh6i t'i

diQn vudng dinh 11 Ta c6

Ldi binh DC giai bdi to6n tr6n ta dd st dlmg

tinh chSt quen thuQc: N6;u d*d, ld hai ttudng

thing ch6o ntrau, (r)U m[t phing chira d, vdsong song vor d, khi d6 moi di,5m M e d, ta

c6 :d(d,;d,)=d(M;(P)) Nhu v6Y bdi to6nn?ry sau khi xdy dpg m{t phing (r) ta l4i quyvCUaito6ndqng 1.

Bii to6n 4 Cho hinh chop S.ABCD clt dav"

ABCD ld hinh binh hdnh t'6'i AR =2tr,

BC =uJ1, BD = aJ6 Hinh chi1u vu6ng git'

c'r)a dinh S l0n ncit phong t'4BCDl ltr tt'cttrg ttim

G ct.)q tunt gidt: BCD, bi1t SiG :Ztt 'linh tk1

rich V kh6i ch6p S.ABCD vd khoong ciich girt'cr

htti dru)'ng rhiing AC t'ti SB theo cr.

Ldi gidi (h.6, h.7)cY\ CD2 +BCz :4a2 +2a2 =6a2 = AC2 n€rr

DANG 2 KHOANG CACH Gr(rA HAr

DIIONG TIIANG CHfO NHAU

Blri to6n 3 Cho hinh chop S.ABCD c'6 dtil'

ABCD tir hinh thoi ;i)m I, AB =Zu;llD = rEAC ,

mqt bln SAB tit tcrrtt gitic' t',itr finh A Hinh

chiht vudng got: t't)q dinh S lAn ntqt phcing tlcil'

trirng v6'i tnrng diint H t:t)ct AI Tinh fiA ilc'h

khAi c'hop S.ABCD vd khodng c:at:h gii'a hai

dudng thcing SB va CD.

Ld gidi $.s)

o Tam gi6c SAB c6n tai A suy ra SA = AB =2a'

Ta c6 BD = {3AC = BI = J1,q = Jix v6i

x= N(x > o) Ma N2 +BI2 = AB2 n€n

x2 +3xz = 4a2 ,hay x = a.

Tu d6 so-ac, =

)ac.no =l.zo.zJ-lo =2Jia2.

SHz =SA2 -A1p =4a2 -+ E* Z- 4 ""t nuy st=$ "' 2'

- roHN !-loc

b';q"3it@

ABCD ldhinhchtnhat Vr.*ro=o{ o,.

^S

I

-::l "-D

ftinh 6

o K6 cluirng thing qua B

song song va AC clt oC vi,

DA tai I vd J

Qua G ke GH ttto(ru ent),

GK ll AD(K e B,I) suy ra I

GH2 ' GK2 4a2 ' 4a2 ' 2a2 02'

Vfy c6 h= a, hay a(eC:SA)= a.A

Eiri torin 5 {.'ir,t lrritit r.'hdp 5 18(-'D cd di)' lt\

hinh t'lti' rthtit t'iti t\B == a BC : uJ3 Hui mcit

i,hriii{ (ilf ') r,r} (S1J1)) tiutg t,uin51 got' t,ti'i dti.r,.

:;ii:lt ! liittr)t' ,Jttr.ur ,l(' ,rrro tho ,\(.':31C T'inh

gr",,, r'r:/l ilt;'itrr.q thirtg ,1i ri ,5tt hiOt t'ting ,41

''ilt',n!: r10i i'ii'l .t{.i.

oKO IM llSB (u enC)> SB ll(NM), suy ra

a (sa; el = a (sn;(u t r)) = a (n;(u r,r))

Ke rH llso (tt eoc) A

= IH L(eaco) ve

HC IC -=-=- (f SC 3 laco s

1-a(a,(uu))=u(c;(utt))

=2.!a@:(NM))=!n.re

AD tlinlt 7

1 Cho hinh hQp ABCD.A'B'C'D' c6 d6y

ABCD ld hinh thoi canh a , tdm O vd

ABC= 120" G6c gita c4nh b6n AA' vd mflt

day (,tSCo) Uing 60" Dinh A' c5ch tl6u c6c

dinh A,B,D M liL trung <Ii€m ct, CD Tinhth6 tich mroi tr6p ABCD.A'B'c'D' vd khoangcSch tu di6m M cltin mflt ph5ng (e'nO) .

2 Cho tctrOi Ung trU drmg ABC.A'B'C c6 ddy

ABC ld tam gi6c vuong tai B v6i

AR-q'M':2a;A'C=3a Gqi M ld trungdi6m c4nh CA', I li giao di6m cria hai duong

,rI)

./r" i\ t,, /, \

thing AM vir A,C Tinh theo a th,s tich kh6i DUng fr = HE > AH ll BI > AH tl(SBI).ch6p I.ABC vd khoing c6ch tu A d6nm[t phing Ta c6

(rBC)

3 Cho hinh ch6p S.ABCD c6 ddY la hinh

vudng v6i c4nh 2a , mf;t b6n (Saf ) wdng g6c

v6i m[t phing (A&CO) vd SA = a,SB = oJj

Hdy tinh th6 tich cta khoi ch6p S.ABCD vir

khoAng c6ch gita hai ducrng thtng AC vd SB

theo a

4 Cho hinh ch6p S.ABC c6 ABC ld tam gi6c

rudng tqi A; ZAC=BC=2a M4t Ph[ng

(sac) tpo v6i (eac) m6t g6c 60' Hinh

chit{u cira s lcn (ABC) ta trung diiim 11 cira

doqn BC Tinh th6 tich kh6i ch6p S.ABC vd

kho6ng c6ch gifra hai cluong thdng AH vd SB.

5 Cho hinh ch6p S.ABCD c6 ilSy ABCD ld hinh

thang ru6ng t?i A vd B vbi AB=fi,Q=s'

AD=Za , tam gi6c SAB c|,n tai dinh S nim

trong m{t phing ru6ng g6c vdi m[t phing d6y,

mflt phing (SCO) t4o v6i mat phing il6y g6c

60" Tinh theo a thc tich ttroi ctrop S.ABCD vit

khoing cilch gliraABvdSD ,.'

HI,ONG NAX CTAT

r.v *ro o, r, r, = + ; aQw:(t' no)l =*

a (en ; sn) = a (t n ;(sm)) = a(a; (sa)) : fr.O6 ttr6y AHBIliLtlinh binh hdnh

ndn H ld trung tliOm BC suy ra

HI ll AC.Ke HJ lt AB Q e rc) mdACLAB=HILHJ vit I

SH L(HIJ) d6n dtin H.SIJ tiL

5.

o Ggi 11 ld trung tli6m

AB, this[ t(esco).

= HK =1o, =l.ir"A =Uf

sH = HKtanUr =tJ fo vr.*ro ='{ or

o Trong (rcco) aung DF = E sry ra

L6y K ld trung di6m AC suy raACr(Sar)

> AC r sr( ncn ((sac);(arc))= ffi = oo".

.I TONN HOC

U tcruagw

iruoNro oAlr orfu.ot s6 z

C6u 1 a) Bpn tlqc t.u gi6i

b) Gqi M(r*yo)e (a)ta ticp di6m khi d6

y'(rr):-r-1 - Tt gia thi6t, suy ra:

Cdu2 DK: x e m.f {O;1} Phucrng trinh ttd cho

tuong tlucrng voi log, lxl+ log, lx - tl : 1sg, 2

<+ log, l"' - tl: logs 2 olxz - "l=2.

C0u 4 Si5 c6ch chsn 6 qui ciu b6t ki trong hQp

ld Cfr Chqn 3 qui c6u mdu tring trong 6 qud

miu ning c6 Cl c6ch Chgn 2 qui cdu mdu d6

trong 4 qui mdu tld c6 C| cich Chon I qui

ciu miu den trong 2 qui mdu den c6 Cl c6ch.

Cflu 6 Gqi K ld trung di€m A'B', vi tam gi6c

CA'B' c0n tai C n6n CK vu6ng g6c voi mlt

phing (ABB'A') vit eGx: 600 Ta c6

-Ggi S ld giao di6m.l/P vd BB', qua S vE tluong

thing song song v6i BK cht AB tqi T , AM t4i R

vit cit A'B' tai U; gqil li giao di6m AM vd A'B'.Hai tam gi6c ABM vd BB' Kbingnhau n6n

6trfu =dFE'-AM LBK = AM r (ac'r)

= rH, = uz + ,y= a(r,slu) =TH =+

= a(n,rus)= $a1r,ns )=+

CAu 7 Gqi d ld tluong thing qua M, song songvot AB; N : d I BD,I1ld trung di()m MN, I lit

t6m hinh cht nhdt ABCD vd P : IH I AB.Toa

d0 di6m B thda min hC

[x-v-1:0 lx=4 1; -;, *s lo' t; =, = B(ar3)' ra co

PT MN : x-3y+ 15 = O = lf (9;8), suy ra toa dQ

H tit H(0:s) =rr Ht:3x+y-5=o=l[],1). r J_v_,lz.z).

Do -B, D d6i ximg nhau qua l n6n O(_t;_Z).

Ta c6 P(t;2),vi P ld trung di6m cta AB n€n

e(-z;t) suy ra c(s;o).

b) Viet phuopg trinh ti,5p tuy6n cria eO tni 1q, Ui6t

rang ti6p tuy€n d6 w6ng g6c voi tlucrng thang

d:x+7Y=Q'C6u 2 (l di€m) a) Gi6i phuong hinh

/_ \

sinx+J3sinl \2 !-*l=2.

I b) Cho s6 phirc z:3 - 2i tinh -6 dro cua so phric

22

*=

ur'

Ciu 3 (0,5d1A@ Gibiphuong trhh 2"-t +2'-' =1 2'

Ciu 4 (1 die@ Gieib6t phuong trinh

2(l - x1'[x) +/r - 1 < x2 -2x - l

4.

Ciu 5 (1 di€m) Titthtich phAn ' t = l !L :

i x2+x,l x

Ciu 6 Q die@ Cho hinh lang hr,r ABC.A'B'C co

dily lir tam gi6c tt€u c4nh a, hinh chii5u w6ng g6c

c'iaA'l€nm4t phlng (ABQrrnngv6i tdm O c'batam

gi6c ABC, g6c gita m[t b6n (ABB'A') vd m[t d5y

bing 60" Tinh th6 tich kh6i 15ng try ABC.A'B'C vit

khoAng circh gl1ahai'<ludng th ng AB vir CC

CAu 7 Q diA@ Trong mflt phing vdi hQ tn4c Oyy,cho hinh tu6ng ABCD c6 M(-3;1) ld trung <li6m

ci.r AB, di6m E thuQc do4n thing BC sao cho

EC = 5EB Bi6t ring DE:23x + 9y - 10 = 0 vi dinh

D c6 hodnh dQ duong Tim tga d0 dinh D

CAu 8 (l did@ Trong kh6ng gian v6i h0 tqa d0Oxyz, cho mflt phdng (P): 2x + 2y + z - 5 = 0 vir

- ', r-1 "-'\

dudns th6ns L: a_=a=, _' =" _" Tim toa d0

-"o -o I I 2 didm A thuQc.dudng th.ang A sao cho khoing crich tu

,4 den mit phang (P) bdng6

CAu 9 (0,5 di€m) Cho tQp hSp E = {1,2,3, 4 5\.

Gqi Mlit tap hqp.t6t ci c5c s6 tu nhi6n c6 it nh6t 3

chfi s6, cdc cht i6 d6i m6t khilc nhau thuQc E Clor,r

ngiu nhi6n mQt s6 thu$c I/ finh x6c sudt d6 s6

dugc chgn c6 t6ng c6c cht s6 blng 10.

Cffu 10 (l di.Afi X6t c6c s6 thUc khdng 6m a, br 9

th6a mdn di6u kiQn a+b+c =3 Tim gi5 fi nh6

nhStcriabi6uthric l==! :+,b,=*

"

b4 +16 ' ca +16

-aa +16'PHAN VAN THAI(GV THPT chuy€n Phan B6i Chdu, Nghe An)

ff=

x3 _5x2 +l4x_4=6{V, _x+l

e (x+ 1)3 +3(x+ r) =(ax' -8x+8)

+:vsx, -sr+8 (x)oit f (t)=f +3t,telR., tac6

a, (o; - g;o), a (* t' 4 r($,a,), r'(* nn,,

o(,,i,,) = ru (aj;g), * (S,at)', (*-i")

.,y,u -E =l *,-+,"), *=[0,-",-;),

** =(+;,ol vay

.,/

lr- -r -l

l au.c,a ).ur,tl o$i

(11

(GY THPT chuy€n LA Qui'D6n, Dd NSng)

.uq(q-?,,)-ffisffiml s]affil nAroiuarnut'nffi

6rtroJt7 bai viit noy cfuing ta sd dua ya ry1t

?! s6 ap dqtng cua BET hodn vi, m6t bdt ddng.

thilrc don gidn nhmtg.dgp, du.ng n6 ta co th€

ch*ng minh tfuqc nhidu bdt ddng th*c khdc.

Xdt hai day (bO) cric s6 thgc a*a2, ,an yd

b1,b2, ,b^ N6u ta l6y tdt cit c6c ho6n vi

(x,x2, ,xn) ctra (bt,bz, ,b,) thi c6 t6t ca

nt = 1.2 n tOng c6 dang:

S = c,,t, + azxz+ ,+ anxn (*)

Cdu h6i ttugc d{t ra lir Trong cdc t6ng c6 dgng

(),, dng ndo ld ktn nhiit, t6ng ndo ld nhd

nhdt ?

Tru6c khi fie lcri ciu h6i ndy chring ta sE cAn

mOt vei khrii niqm

l KHAI NIENt Cho a, ,a2, ,a, vd bPbr, ,b,

ld hai d6y cric s6 thr,rc Hai d6y tluoc ggi ld sdp

cilng thtr tq ndL;- ci hai ddy ctmg ting (tuc ld

arlaz <an vd 4 < br3 <b ) ho[c cing

gi6m (tuc lit q>q> >-an vd 4 > br2 2b,).

Hai diy ttugc goi tiip nguq" tha tqr n6u mOt d6y

tdng vd mQt ddy giim

Thi du a) -2;3;5 vd l;2;4 ld hai dey sep

cirng thir tg, trong khi -2;3;5 vi 4;2;l lit

hai d6y sip nguoc thri t.u.

b) N€u 0<or<a23 <a, thi a1 ,a2, ,an yd

+,+, ,1 ,u hai ddy sip nguo c thri tu, trong

At AZ An

'*" "l '-'2""'-'n '* en_1*An' ' Az*h, Ar+a,

li hai.ddy sip cr)ng thf t.u.

c) NCu 0(o, ar< <sn vd m lit sO thgc

duong thl a,ar, ,a, vd af , af , ,ay lithai

ddy sip ctng thri tg, trong khi a, ,a2,.-.,an yd.

+,+, + ld hai ddy s6p ngusc thu r.u.

2 BAT DANG THU'C HOAN VI

Cho a,,ct., ,u,, t,it b,.b., b,, lit hui tlti.t, r'ac

so thqrr: vd (.r, .,\ , ,-r,, ) lir ntdt hoon vi lir.t,

tu thi

u,b, +u.h, + + {1,b,,(rl,,r, +rr i + +a,,-y,, ill)

Dittr ":" :t'ons (l) rd (ll) xcil, ro O q,:q.= -s,hoat' b' =ly = : 11, hoac (t5x2, x,,)=(b,,14, h,,') Chilmg minh.Xet hai ddy a1 ,a2, ,a,1 vd, b'b2,

,b,, cing tdng vd (x,x., ,x,) ld m6t hoSn vi

tuy f cta(b, ,b2, ,bn) Gi6 str x, ) xr

DAt S : atxt + a2x2 + a1x3 + + anx, vdS' = a{z + azxr + a3x3 + + anxn

(S'nhdn ttugc tu S bing c5ch rt6i vi tri cira

x, vd xr) Ta c6:

S'-S = atxz - atxt + aaxr - a2x2

= ar(x, - xr) + ar(x, - rr)

=(xz- xr)@r-ar)> 0

Do cl6 S'> S Nhu vdy khi d6i vi tri cira x, vd

r, thi gi6 fi cria ,S chi c6 thii tlng l6n Do d6

n6u chring ta cl6i ch6 tbt ch cic c{p (x,;,rr) v6i

,' ,:,

x,) x,,i <7 thi t6ng chi c6 th6 tdng l6n T6ng

ifat gi6 tri lon nfr6t mt (x'xr, ,x,,)=(4,b2, ,b,)

tuc ld khi S = arb, + arb, + + a,bn.

Khi q=,=sn hoqc h= =bn thi ddu ding thirc

cfing x6y ra LQp lufln tuong tU khi Q,a2,"',a,

vd br,b2, ,bn cirng gi6m, vflV (I) duoc chimg

minh (II) clugc chimg minh tuong t.u (I)

Tt BET hoSn vi ta c6 hai hQ qui sau:

HQ qud t, N€u a1,a2, ,a,, ld cdc sd thryc vd

(x, ,xr, ,x,) ld hoan vi cila (a,ar, ,a,) thi

al + al + + al, ) atx, + a2xz + "'+ anxn'

HQ qud 2 N€u a1,a2, ,an ld cac s6 thac

daong va (x.xr, ,xn) ld hoan vi c{'ra

\a,,a., a,) ' I thi 9L *lz* *L> r.

xr x2 xn NhQn xit l) CAu hoi o phAn ddu: Khi ndo t6ng dqng

(*) lon nhdt, nho nhdt d5 dugc tr6lcri qua BDT ho6n vi'

YOi n:3 thi he qui I tro thdnh BDT quen thuQc:

a2 + b2 + c2 > ab + bc + ca, Va,b,c e IR

HQ qu6 2 ld mQt bdi to6n hong cuQc thi KuschSk cita

Hnngary nlm 1935 vd ld bdi 6.2.9 -10.4 trong cuQc

thi Moscow Olympiad ctra Li6n X6 ndm 1940'

2) Trong chimg minh BDT vd ktri sri dpng BDT

ho6n vi ta hay dirng mQt ky thu4t saul

Ntiu /(a, ,or, ,o)) ld bi6u thirc il6i xrmg d6i voi

ay,a2, ,an(tuc ld f (a*ar, ,a,)= f (x.xr, ,x,)

vcri mgi ho5n vi (x.xr, ,x,) cin (a,ar, ,a,)) th\

di5 chtmg minh /(4, ,a2, ,a,)> 0 ta lu6n c6 th6 gi6

thitit ring et I az < anhoic a, 2 ar) > a^ .Li

do c6 th6 ldm dugc <li6u ney ld vi

f(a,ar, ,a-)kh6ng aOi voi mgi ho6n vi cria

(a,ar, ,an) .

Sau <l0y chirng ta sE n6u mOt s6 thi dl,l minh hoa

6p dpng BDT ho6n vi

3 MQr So rHi DU

Thi dg 1 Tim gia tr! nh6 nhdt ctia bidu thac

A=sin:'*-ry voi o< r.\.

Ldrt gidi Voi 0

".lthi 2 sinx > 0, cosr > 0 BET ddi xtmg d6i v6i sinx vd cosx n6n ta c6

th6 gi6 sir 0<sinx<cos-x Khi d6 hai ddy

,l , I vd sin3 x, cos3 x siP ngugc thu tu

slnJ cos,rnOn theo (II) ta c6:

sinjx, cossx-coslx, sin3x

BDT d6i xtmg ddi vbi a, b, c n6n c6 th6 giA sir

a> b2c > 0 Khi d6 hai ddY as,bs,cs vd

Hai ddy a2,b2,c2 t, *.#.* sip ngusc thri

tU nOn theo (II) ta c6:

a2 *b2 *c2 o, *L*L a3'b3 cl-c3 a3 b3

I I 1 ,a2 b2 c2

o!+ O+:<Yr*Y.* U Q)

Tir (1) vit(2) suy ra BDT c6n chimg minh I

Thi dU 3 Choa,b,cld d0 ddi ba cgnh ctia mQt

tam giac Chung minh riing

a2 (b + c - a) + b2 (c + a - b) + c2 (a + b - c) < 3abc'

Ldi gidl BDT cAn chimg minh tuong duong voi

a3 +b3 +c3 +3abc) a2b+ab2 +b2c+bc2

+ cza+ caz

e a(a2 +bc)+b(bz +ca)+c(c2 +ab)

> a(b2 +ca)+b(az +bc)+c(ac+bc) (l)

Do BDT le d6i ximg n6n c6 th6 gi6 sir

a>b>c Khi d6a2 +bc>-bz +ca Ap dpng (I)

cho hai ddy s6P cirng thil tU a, b vit

a2 +bc, b2 +ca tac6:

rz'?Eil,HB! **

a(az +bc)+b(bz +ca)>a(bz +ca)+b(a2 +bc) (2)

Lqi 6p ftrng (I) cho hai dAy sip cung thri \r b, c

vd, a, c ta c6: ba+ cz ) bc + ac

= ,(r, + ab)> c(ac + bc) (3) CQng tung vil

(2) vd (3) suy ra (1) <hing per, <c-rr

Theo dinh l1f c6sin trong tam gi6c ABC ta c6:

(4) o 2abc cos A + 2abc cos B + 2abc cos C < 3abc

<>cosA+costr+cosc<] (5)

2 O6 ttr6y (5) lu6n dtng, suy_ra di6u ph6i chimg minh.

2) K6t qun trong thi du 2 vdn thing khi a, b, c duong.

Tuy nhi6n Wi a, b, c duong thi ta kh6ng th6 v6n

dgng cilch dnng h9 thirc luqng trong tam gi6c nhu 6

nhdn xdt 1 dugc Lric ndy phuong phSp dirng BDT

hoan vi cho th6y 16 hiQu qui cria n6.

Thi dU 4 (IMO 1975) Xdt hai ddy cac sd thuc

21;22; ,2, ld mAt hodn vi cita !1,!2, ,/r.

.J-n

Chmg minh rang f {ri - y)2 {\{*, - r,)' .

Ldi gidl BDT cAn chimg minh tucmg rlucrng vcri

nnnn

oZ*,y,2\x,2, (*) @o Zt =|41.

Nhmg theo (I) ta c6 ngay BDT (*) O

Thi du 5 (IMO l97B) Cho a1,a2, ,an ld n sii

ngry\n duong phdn bieL Chffng minh rdng

a.all

a,+ + +-f >l+;+ +-.

'4n'2n

Ldi gidl Gqi bt,b2, ,bnld m6t hodn vi cria

a1ta2t ta, sao cho br<br< <bn Theo gi6

thi}t a,ar, ,a, ngly€n duong ph6n biQt n6n

h>i O),Yi=1,2, ,n Do l>1> >4 4n2 ndntheo

BDT (2) dOi xrmg COi vOi x, !, Z n6n c6 th€ gi6

sir: -r) y> z> 0 Suy ra x2)y2> z2 vd-f-l>-1 Theo(r)tac6:

Thi dU 8 (BD,r Nesbit) Chu'ng minh ring nAtt

a,b,t' ld ha sd duong thi

b+c' c+a a+b- a+c b+a' c+b'

CQng timg vi5 Z ePt trdn ta duoc

a b c l(a+b b+c c+a) 3

t=-b+c- c+a- a+b- 2\a+b' b+c' c+a ) 2'

DAu ":" xhy ra o a=b- c D

Thi d1.t 9 (nOr hoan vi dqng lfiv thira).

Cho u> l,b> 1,r') I Chirng minh rdng

51t l'rb ;c ) 61b lrc ru

Ldi gi,rti Cach 1 BDT khdng ttdi qua

ho6n vi vdng quanh n6n c6 th€ gia

a = max{a,b,c} N6ua > b> c2 1, ta c6:

(III) e sa-bfub-c ) ga-b 6b-c

BDT (1) ludn ching do ao b ) 6a-b,$b-c ) 6b-c

BAy gio xdt a> c2b> 1 Ta c6:

(III) e sa-csc-b ) 6a-cfic-b (2)

BDT (2) 1u6n clirng do a'-') sa-c,sc-b ) Sc-u

V4y trong mgi truong hqp ta dOu c6 (III)

Cach 2 BDT (IID tuong duong v6i

aln a + blnb + clnc > blna + clnb + aLnc (*)

N6u a > b> c> 1 thi (a,b,c) vd (1na,lnb,1nc)

li hai d6y s6p ctng thri t.u;N6u a > c> b > 1 thi

(a,c,b) vd (lna,lnc,lnb) lh hai d6y sip ctng

tht tu Trong c6 hai truong hqp theo (I) ta c6

(*).n

NhQn xdt BDT (IID v6n dung khi a, b, c ducrng vir

thi du 9 c6 th6 dugc md rQng nhu sau:

Vcri cdc sd duong a, )az, ,(tn, ta lu6n c6:

ai' a| ai" > o? oi' a|-rai' .

St dpng BDT ho6n vi ta c6 th6 chimg mfuh

nhi6u BDT cO dien kh6c

Thi dU l0 (BDT Chebyshev) Cho hai ddy cac s6

thltc ar,(12, ,a, vd b,b2, ,b,, Chilng minhriing: a) l,lJu hai ddy sdp cilng thtlr tu'thi

a,b, + arb, + + a,,b,,

nn

D lleu hai ddy tdp ,gu'P', thtt ta thi

arb, + a.b + + u,,h,,

t1

(IID CQng tung v6 c6c BDT tr6n ta dugc:

ph6p n(arb, + arb, + "'+ a,b,)2

ifrici' (a, + a, + + a^)(b, + b, + "'+ b),

(1)

b, + b, + + b,,

Ldi gi,fii a) Theo (I) ta c6:

arb, + arb, + + anb, = arb, + arb, + + anbn

arb, + arb, + + a,b, ) atbz + azbt + + anb,

arb, + arb, + + a,b, 2 arb, + arbo + + a nb,

arb, + arb, + + anbn ) arbn + arb, + + anbn-,

ta c6 BDT cin chimg minh

BDT 0 cdu b) duoc chimg minh tuong til D

NhQn xit DAu ":" hong BDT Chebyshev xhy ra

- O, = Az = = A, hOdC b, = b, = = bn'

Thi dq ll (BET Cauchy, cdn goi ld BET trung bin,h

cAng fttmg binh nhdn) Chlrng minh rdng n1u

,:

a | .a , a,t la cac so khong am thi

Ldi gi,rtL a) NOu it nhat mQt trong c5c s6a1,a2, ,an bing 0 thi BDT hi6n nhi6n dung

N6u c6c s6 a, Q =1, ,n) tldu duong, tlfltM=q8,0"-.a., r ' - *,=!1, a(to M X'= atoz"'*"

M,""'ln= A4, '

Khi d6 theo he qub2tac6:

M'?EI#BLe455 (5-2015)

DAu"-" xey ra <> Jr =, = xn e at = = ar.A

Thi du 12 (BDT Bunyukovslg) Chtimg minh ring

n€u a., ,ar, ,a,, va ht ,b), ,b, ltt 2n s6 thryc thi

(arb, + u.h + + ct,,b,,)2

< (ai + a! + + al)(bl + b] + + bl)

Dtiu "-" xdy ra €) :1 € lR : a- : tb, hodc b, = ta,.

Ldi gidi N6u a, = a2 = = an = 0 hoic

b, = b, = = bn = 0 thi BDT hi6n nhi6n dfng

N6u t6n tqi a, * O vd b, + 0,ta ddt

Suy ra BDT cAn chimg minh D6u ":" xdy ra

€ r; = Xn*i € a,N = b,M (i=|, ,n) A

Thi du 13 (BDT tntng binh c\ng - trung binh

binh phwong) Chu'ng minh rdng nAu o,az, ,un

a? + 4 + + a2, ) ara, + ara3 + + anal

al + al + + al ) ara, + aza4+ + ana2

al + aj + + a] ) ara, + a2at + + anln_l

CQng tung v6 cdc BDT tr6n ta duoc:

n(al + al + + a|) > (a, + a, + + a,)z .

Suy ra BDT cAn chimg minh D6u ":" xhy ra

€al=Az= =ar,J

'fiP"-(',,>- lll '+ + +

0t 4.2 d,Ldi gidi.D6t N = a[apr-.a,:x, = fi ,x, = #,

NhQn xit Ki hi6u Ar,4,4,Ao lAn luot ld trung

binh didu hda, trung binh nh6n, trung binh cQng vd

trung binh binh phucmg cta cdc sti duong

a, ,a2, ,a,thi tu c6c thi fir I 1, 13, 14 ta c6:

Ar34S4<A|

Qua c6c thi du trOn, ta dd thdy tinh hiQu qua cta

BDT ho6n vi trong chimg minh BDT, nhiAu

BDT kh6 nh.mg cl6 clugc chimg minh kh6 don

gi6n nho BDT ho6n vi De luy€n tdp, cdc b4nhay gi6i c5c bdi tflp sau.

2 Chimg minh ring n}u a,b,cld c6c s5 ducrng

th\ a) ab + bc + ca> albc + bJca + cJab .

a)t?,

Thi dtl 14 (BDr TB nhdnmlnh rAng neu U t.Q ) 11 tl

a'+b"+c, ' -'

^

J

- TB diiu hia) Chultg

ld c'cic s6 du'ang thi

n

4 (IMo tgs:) Biet ring a, b, c ld dO dei ba cpnh

ctra m6t tam gi6c Chrmg minh rlng

az b(a - b) + b2 c(b - c) + c2 a(c - a) > 0 .

5 (USA MO lgl4) Chrmg minh ring n6u a > 0,

b>O,c>0, neZ* th\ aafu,b6c >(abdt? .

Tii liQu tham khflo

1 Dragos Hrimiuc The Rearrangement Inequality

-n i-n the sky PISM, issrie 2, page 21 2312000.

2.Phan Huy Khdi vd Tr6n Hiru Nam Bdt ddng th*c

Sti ass 6-2015) TOAN HO(

-1 cruaga .d" 5

cAc r6p rHCS

Bni T1/455 (Lcrp.6) Tim s6 tu nhi6n c6 nhidu

hcrn 3 cht sO, UlCt ring ni5u ta b6 di 3 cht s6

cuoi cung cua so do thi ta dugc mQt sd m6i md

lpp phucrng cira n6 bing chinh s6 cAn tim

v0 HONG LTIqNG

(GV THPT Y€n Dactng, Tam Ddo, Wnh Philc)

Bni T2l455 (Lop.7) Cho hai si5 thuc du<rng a

vd 6 th6a *in,iici ki€n a20r5 - a 1 : 0 vd

b4o3o - b - 3a: 0 HEy so s6nh a vd^b?

Bni T4l455 Tr0n ducrng trdn tdm l cho tru6c,

6y hai di6m B, C cO Oinl vi di6m A chuy€n

ddng tr0n ducrng trdn sao cho tam gi6c ABC

nhon Trdn canh AC 6y di6m M sao cho W :

3MC, H ld hinh chi6u vudng.g6c ctra M tr€n

cqnh AB Chtmg minh r[ng di6m FI1u6n thuQc

TRAN VAN HANH

(GV DH Pham Vdn Eing, Qudng i{gai)

- CAO MINH QUANG

(GV THPT chuyAn Nguy€n Btnh Khi€m, Wnh Long)

BdiT71454 Cho tam giirc ABC GSlm*ntb,rll,

ld d0 ddi cdc dudng c6c trung tuy0n 16n lugtimg v6i c5c cpnh BC : a, CA : b, AB : c.

Bii T8/455 X6t tam gi6c nhqn ABC c6 c6c g6clit A, B, C Tim gi6 tq lon nh6t ctra bi6u thirc

tan2 A+tafi B tNP B+tan2 C tar2 C +tatf A

'" - wo A+tan| B ' larf B+tan4 C 4o C +tarf A'

KIEU DINH MINH(GV THPT chuy€n Hitng Vuctng, Phu Tho)TI6N T6I OLYMPIC TOAN

Bni T9l455 Tim he sO cta r' trong khai trii5n

(1 +r)(1 +2x)(1+4x) (1 + 2'ot.'*)

NGIIYEN TUANNGOC

(GV THPT chuy€n Tiin Giang)

Bii T10/455 Cho c6c sd ducrng a1, a2, , an

th6a man at+a2+ +an:1*l* +1 "

at ^az a,Tim giri tq nho nh6t cria ,q,=o,+7+ +

Lt vIET N gn*o Thien': Hue)Bni T11/455 Tim sd thuc klonnhdtth6a mdndi€u kiQn: Vdi 3 O tt U a, b, c sao cho

lrl*lal+l'l<r thi hc b6t phuorg trinh sau vd

nghi6m:

l*to + or'+ bxo + cx + 15 ( 01l*'o- l.llll rn+11+ l*o - r +tl<z '

TRAN TUAN ANH

(GV Khoa Todn Tin, DHKI{TN, TP HA Chi Minh)BitiTl2l454 Cho tam giSc ABC rthqn,n6i ti€p

ducrng trdn (O) v6i tludng cao AD Ti6p tuy6n

tai B, C cna (O) cdt nhau t4i 7 Tr0n dopn th[ng

AD l6y di6m K sao cho 6Ei =90" G ld trgngt6m tam gi6c ABC, KG cit OT taiZ C6c di6m

P, Q thuQc

"doqn BC sao cho LP ll OB, LQ ll

OC Cdc clii5m E, F l6n luqt thuQc do4n CA, AB

sao cho QE, PF cirng vu6ng g6c v6i BC Gqi

(O ld dudng trdn t6m 7 di qua B, C Chimgminh ring ttuong trdn ngo4i ti6p tam gi6c AEF

ti6P xirc v6i (f'

TRAN euANG Hr)NG

(GV THPT chuyAn KHTN, DHQG Hd NA,

.TOAN HOC

t O ' 6l'udiua s6 455 (5-2015)