TOÁN HỌC TUỔI TRẺ SỐ 455 THÁNG 5 NĂM 2015

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2015
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Tru s6: 187B Gi6ng V6, Ha NOi.
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HÃY ĐẶT MUA TC TH&TT TẠI CƠ SỞ BƯU ĐIỆN GẦN NHẤT !


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xuAt ban Giao duc Viet Nam vua cho ra
m6t phien brin tieng Viet cOa cu6n Tu dien
Bach khoa Britannica c0a M!.

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Ttr dien Bach khoa Britannica gOln$OO0 rnuc
tLr, 2.500 hinh minh hga
beni:&.|51 lTnh vuc
khoa hoc vEr dcvi s6ng, gan 300$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.

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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$j.d&,891
soan - bi6n d!ch, do ong Ng0,TradAirlite,h$
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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.
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.
Theo
bdn
l,9JQ,l

1Nhu Loi nha xu*t
.kho khin, c6c dich
tap vren da lam vrec
rthe cr:ng 'kho tr6rnh

HÃY ĐẶT MUA TC TH&TT TẠI CƠ SỞ BƯU ĐIỆN GẦN NHẤT !

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r

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

K6 ttuong phdn giSc BD' ciru goc ABC,
tqi I. Ta c6

ABC+ACB=180o -BAC =180o-60" =120o

ncn fr

*U,:*P=60,

suy ra

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

BIC =120" vd Ir=lr=9Q".
Ggi 1r(ld duong phdn gi6c cila L,BIC thi

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

I,=Io=60o, LBIE=LBIK

nhau

sao cho CED

=30'.

T[nh s6 do g6c BDE.

A

cF.t CE

(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,
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.

ABC

Hinh I
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
Tim hrdng

=40".

=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.
ABC + ACB

Ldi gidl (h.2)
Hinh

3

Tim habng gidi. (h.3)

Bing c6ch do tryc ti6p, ta thiy fEE =25o, tftc
liL ABE = 65o. Do tI6 ta
sao cho

trung

fri'=65"

,. -.},
vor dremz.

6y di6m E, tr€n tia Ax

r(ii chimg minh di6mE'

sdnrr6-*ro

T?8I#EE

1

HÃY ĐẶT MUA TC TH&TT TẠI CƠ SỞ BƯU ĐIỆN GẦN NHẤT !


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
cdn,

AI: AB,vd NEl=180o -65o=115'=BAC.

Tac6 LNE'=
l4i c6 BC

:

Do 116 frE

ABAC (g.c.g)

AE

=

n€n AE

65"

, frE

:

nAnAE': BC.Ta

AE' , suy ra E'

= 9oo

-

tting E.

65o =25o

Iiinh

Ldi gidi. Ta c6 NBC cdntqi I n€n IB
-nee

.a

,

J
1'
trong tam giac do sao
I ndm
choiili=ieE=30". T{nh s6 do cdc g6c IAB

vd g6c IAC.

Tim hrdng giai. (h.4)
A

:

IC

(l)

=180" -(70" +50"):60o,

6;={Ea -i,=70'-3oo = 4oo,
^
Cz : ACB - Ct :50o -30o = 20o.

1'hf ctg 3. Cho tam giac ABC cd ABC =70",
ACB =50o, di€m

5

C6 the chimg minh bing phin chimg theo hai
c6ch:

o

Cdch 1. GiA sir A,

NAB,

do

Ar>40"

>40" thi

=8,

ndn

4<20'.

Xet

IB > IA. Xet

NAC, do Ar<20o =Cz ndnIC <11. Suyra
IB> IA> IC,triLi voi (1).
Gi6 sri At<40o thi 4120'. Chimg minh
tuong t.u ta dugc IB < IC, tr6i voi (1).
Ilinh 4
Bing cSch do tr.uc ti6p, ta thdy iiE= 40" tuc ld
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

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
tr+Tr.60',

^
trdivot BAC :60". Gia sir 18 : IC > IA. Chung
ir*$>

60o, trhi vu

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

minh tuong t.u ta dugc

IAB citn c6 IBA = 70o - 30o = 40o ndn
iiE =40". Suy ,u itri = 600 -40o =20o.A

Bni 1. Cho tam gi6c ABC, Gi =75",
frE:60", iti6m l nim trong tam gi6c d6 sao

2. Phuurng phtip phiin cltil'ng
Tr0 lai thi du 3 n6i tr6n. Ngodi c6ch gi6i bing

BAC = 60". Vay IB =

IC: IA, suy ra

4--A=4oo'

i:20"'J

Cdc b4n thr? dtrng phuong ph6p chli'ng minh gi6n
ti6p iI6 giii c6c bii tip sau:

cho

tni=tCn=45'.

Tinh sti tlo c6c g6c IAB

ctra tludng trdn ngo4i titip tam
c6ch vE tdm
gi6c ABC tOi chtmg minh 1' trung lnhu tr6n,

vd goc IAC.

cdn c6 th6 chimg minh bing phuong ph6p phan
chung (h.5).

ACB = 60o, ACD =70'. Gqi O ld
giao
I'

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

=30',

^

^

HOC
A TONN
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HÃY ĐẶT MUA TC TH&TT TẠI CƠ SỞ BƯU ĐIỆN GẦN NHẤT !


Hrlilng df;n gifii uE THI {UYEr* s:rqrt unu

lffr tu nnmru ronnl

THFT cHuYEr*r ruAc hrrt*xruAn't Ht){"3t114 -:t}}5
Ciu I.

1) Ta c6:

*r+@Grf

r =(Ji-r)(t+J,
=

(.tr-

2) Ta c6

r)(r

+

:=r=
(t+Jx)

J*)'

)_

J, -r.

ld udc cria 2 g6m:

i.reG-l

+1,L2. Tt d6 tim rtugc x

Ciu

2. 1) DK: ryzabc

.

{O;+;e}.

* 0. Ta c6:

, >l+J3
o lm-tl,
.t- Ji
"" el*
-

l*.t_Js.
1*1t -!-,v*,y
'xyx+y

2) Ta chimg minh tlugc

iting thirc x6y ra khi vd chi khi x = J . Tt gi6
thidt a + b - c > 0,b + c - a > 0,c + a - b > O.Ta c6

r \-z t*0
+t(_t
y6+a-g- q+6-g)'7--b a

ayz+bxz+cry=0'

xvz +- l3
-+'
abc -

e

x2 y2
1+Br+

22

c*'

>0,

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

a b c = U^..ayz+bxz+cD_n
-+-+xyzryz €

e

3. 1) Tn PT thri nh6t ta c6 x=3m-W,
thay vdo PT tht hai ta c6 ! =2ax =,?r. Suy ra
* - zx * ! : m2 - 2m -2 : (m* 1)2 - 3 > o
CAu

Mit 2c+b

(:.#',1'='

= abc

o?u*l= a

ncn

S>za+9>4J1. Eing thirc xiy

(#.#.#)=,

a

khi a=

ra khi vd chi

b=c:J5. VayminS:4J1.

C6.u,l.

\2 zz cry+bxz+ayz _,
*fr*F*'
o xz
o,
abc
\2
o xzrr* w*7=''
22

A,: 4a2 +164-151. OC pf
nguyOn thi A: n2 v\in e N. Khi d6
2) Ta c6

e
e

4az

+l6a-l5l=n2

(4a2

+16a+16)-

(2a + 4 + n)(2a + 4

n2

=16'l

-

n) = 167.

c6 nghiQm

Do 167 li s6 nguydn td vd 2a+4+n>2a+4-n
n6n ta c6 c6c trudng hqp:

o

l2o+4+n=167
{-'''"'=4a+8=168)a=40.

l2n+4-n=l
l2r+4+n=-l
"'" ^

{*'

=) 4a+8=-168= a--44.
l2a+4-n=-I67
Yu a = 40 thi PT c6 hai nghiQm nguy6n ld
x=0,x=83.
Voi a - -44 thi PT c6 hai nghiQm nguydn ld
o

x

=-l,x

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 c6

111111
16 OB2 ' NC2 OB2 ', NBz BKz

-1-I

=BKz:16= BK=4=BC=8.
2) Ta c6 LNBP c, LNMB fe.g
-a-=-

=#:ffi,

(e.g)

=

^NCp@ ^NMC
Mn NC = NB n6n suy ra

= -84.

#=ffi

#=#
T?s#?E,

"e

"ru,u--ro

(1)

3

HÃY ĐẶT MUA TC TH&TT TẠI CƠ SỞ BƯU ĐIỆN GẦN NHẤT !


i
Thdi gian ldm bdi: 150 phrtt

C6u 1 (2 diem). Gi6i c6c phucrng trinh sau:
$xaa-JTx.+t =1.

{ ,
b)

.)

(x+ Jx + l)- =2x2 -3Ox +2.
Ciu2 (2 dia@
a) C6 hay khdng sd nguyOn t0p th6a m6n
8p - | vd 8p + I ctng ld c5c s6 nguyCn t0? Giei

thich.
b) Tim tdtcitchc sd nguy6n duongr,y sao cho

3* Y:1.

Cflu 3 (2 diery)
a) Cho hai s6. thyc a, b th6a mdn a + b

Chtmg minh ring: a2 +b2 l aa +ba.

b)

:

2.

Cho c5c sd ducrng x, !, z th6a mdn cli6u ki€n
zx: 1. Chtmg minh ring:

x.v + yz +

y

,{

z

-3

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

(}- uat khdc, do AM llBC ndn AMCB ld hinh
(2)
thang cdn

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

3) Gqi

Qld

AC ,AB.

grao di6m cira

AP vit BC.Ta chimg

minh BQ=QC.

Y\

LBeP

tr€n.

NGUYEN DUC TAN gP. HA Chi Minh)
duong thdng BC tai K. Kd AH vudng g6c voi
BC tai H.Ta co AH < AK < AO < l.
AH.BCa 2,1 (mauthuanvcn
; ,.
Suyra 5ou=

2

r=l

gia thirit). Suy ra dpcm.

a,b chinthl a2 +b2 li hqp s6. Do d6
ndu tdp con X cia A c6 hai phdn tu ph6n biQt
a,b md a2 +b2 ld m6t s6 nguy6n tO thi X
2) N€u

c..,

(e

^Aec

@

#
= ffi=#

e)

(e.e)
^cQP
^AQB
Tir (3), (4), (5) suy ra

'

>Q=/(.

voi BC, (dr)ldduong thing qua E vd vudng g6c
voi CA, (&)ld duong thing qua Fvd vu6ng g6c
vot AB. Chimg minh r[ng (dr), (dr) vd (d3) ddng
quy 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 tr6n
cung nh6 AB. Goi H, K, P, Q li,n fuo.t ld hinh
chi€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.r6rg
cl6ng thoi bing nhau. Hiy tinh t,Ong ctra t6t ci
c6c tich ba sd trOn 3 dinh li€n ti€p cira da gi6c

= ffi=

(4)
(s)

ry=ry=BQ=gg
AQAQ

Vay BC,ON,AP cl6ngquytaiK.

i^

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

Cflu 5.

:/N--)
" n\

so nguyen t6 nhu sau:

(t;+),(z;3),(s;s),(0;tt),

L

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

1) Gie str O

(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

(Gl/ THPT

Y€n Phong SA Z,

Sdc Ninh)

Swu

xA

im

HOC
,+ TOnN
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HÃY ĐẶT MUA TC TH&TT TẠI CƠ SỞ BƯU ĐIỆN GẦN NHẤT !

cHUfu Bt
CHO

M

Eno DUNE TiN+t eflfir

THI

K+rdr

TRUilG H0C

pud

e[l

rfl ol$x yu6xo

Yilg sil rgnN TiN.fI I[+IgiNE

rn0ro

ou6c on

EflE+I

NGUTEN NGQC XUAN
(GY THPT chuyAn Hodng Ydn Th4 , Hda Einh)

bdi vi6t ndy, chring tdi xin gioi thidu
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.

.6/rong
v

(ee' :(enc)) = ( ee' : tc1 =

A'G=AGtan6o"
A'

-Zalj
1
3

lQ!

= 66"

;

:ot

. Dod6 v,...uvw
vA|AH------'AAtfl:
3'

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

D0 dai OH :

ring

#= oorl*#.#

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

MBC

li

c6sin cho

co:

AC2 =AB2 +BCz -2AB.BCcos60"

-2.a.2a.| =3a2 = AC = ali.
Suy ra ACz + ABz = Bt, dfin d6n LABC
vu6ng taiA.Do A'G L(LSC) n6ntath6y
= az +4a2

Hinh 1
d(G,(A',BC))=h.
Ke GH ll AB, GK /l AC suy ra

.Dlt

GH

vir

rGK.

GH:|*:i

Ta c6

GK:Io,

A

B

=+

Hinh2

Do GA' L(CUX) n6n G.A'HK li tu dign
vu6ng, suy ra

111139351

h2 A,G

"

Suv ra h

GHz

'

GK2

4a2 ' a2 '

a2-

4a2.

J
=ZJsl

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)

.

(e'nc).

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

B

Stnco=az. Gqi

G li

trgng t6m LABD,

= (so,(ancD))=.fr = 60..
Ggilldtrungdi6ml Btac6 aC=lU =of
33
a'll5
-': -r-i
vd sG = DGtan6o,
sG

L(ABCD)

- 3

,vltne
1 -_ a3JB
Vr.*ro:iSG.S-",

=t.

ta,

nru,u-rorr,

t?El#ff

5

HÃY ĐẶT MUA TC TH&TT TẠI CƠ SỞ BƯU ĐIỆN GẦN NHẤT !


ofi

vr.*o=]sa.s*r, =l32

.2Jia, =a,J5.

s

tlinh

3

Hinh

.

(A.(

)a
# 1=
Ke GJ song song v6i BD
(t e aC)+ GJ LGC mit
d

=

sBC)) =

1c,1snc)) = |n.

ovi cD ll
a

5

(sAB), dSndcn

(c o; sn) = a (c o;(set

1) = a

(c;

(sar))

+a(n.;(sen)) = +t(a > o).
Gqi E ld trung di6m cria AB, tac6
=

tam diQn r,u6ng dinh G.

Do A GJC vudtg
n€n GJ: GC.

cdn tqi G

1-

HE=ltB
Isuyra AH LHE
,2

llll

..........._

.-....................._!-L-

h2 \G2 GC2

Hinh

GJL

I

diQn vudng

_g-1
-15a2'lo

/c

-\2

lt"n

)
9

-\2

l;'r2

)

9 9
_ 57 _lr=2oS
_
- l5a2 _' gaz ,' gaz 20a2
JSI
Suy ra

an=1rc:+,

a(a;(sac))

=f,r.

.

a

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)
Tam gi6c SAB c6n tai A suy ra SA = AB =2a'
Ta c6 BD = {3AC
= BI = J1,q = Jixn€nv6i
x= N(x > o) . Ma N2 +BI2 = AB2
o

x2

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

Tu d6 so-ac, =
SHz =SA2

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

-A1p

=4a2

nuy st=$.
"' 2'
-+--E*
Z- 4 ""t

roHN !-loc
-b';q"3it@

vit

ddn d6n H-SAE ld kh6i

t'i

dinh 11. Ta c6

I
ll: l'
HEz
HAz
h2 sH2
I *l*-1
: l5a2
' a2 3d =-28-,-IJTL
sor=tt- A '

444

vfly c6

a(co;sn):'f o.o

Ldi binh.

DC giai bdi to6n tr6n ta dd

st

dlmg
ld hai ttudng

tinh chSt quen thuQc: N6;u d*d,
thing ch6o ntrau, (r)U m[t phing chira d, vd

d, khi d6 moi di,5m M e d, ta
c6 :d(d,;d,)=d(M;(P)) . Nhu v6Y bdi to6n
n?ry sau khi xdy dpg m{t phing (r) ta l4i quy
song song vor

vCUaito6ndqng 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
'linh tk1
G ct.)q tunt gidt: BCD, bi1t SiG :Ztt.
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

HÃY ĐẶT MUA TC TH&TT TẠI CƠ SỞ BƯU ĐIỆN GẦN NHẤT !


ABCD

Vr.*ro=o{

ldhinhchtnhat.

NLSC=ASOC-LNC)

o,.

#=#-sc=aJ6=so

^S

Vr.*ro=o'f
oKO IM llSB (u enC)> SB ll(NM),

Vithe

I -::l--"-

a

K6 cluirng thing qua B

va AC clt oC vi,

DA tai I vd J.
Qua G ke GH ttto(ru ent),

I

AD(K e B,I) suy ra
GH LGK din d6n G.SKH
GK ll

AD

a(ec;(xt))

tlinlt

7

HE

=

= a (c;(snx))

aJT. nr

GH = AB =2a,GK = BC =

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

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

ld tu diQn ru6ng.
Do AC ll(SIJ)vdsB e (Sr,r), nen
=

:

aO

1i111111

h2=-a-I-GS2 ' GH2 ' GK2--I-l-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

LHF

H.IEF

li

Hinh 9

llDC (2,,r . AM)

llAD,HF

HE

n€n

h.

-

md IH L(HEF)

tu diQn vudng tpi 1L

1 I
r
taco. I
*= 1112+ 11pz+ *z
,u -l on oJS
ttt-lru3

..
vot

.

,

gp =1ruc

"-" =1.! nc

r8
6'3""=5'8
un =luN =1tr* =So. Suy ra
,

6

1ll12975a

r

J--_

h2- IH2' HE2' HF2- 25d-"-

---l-l-

gr",,, r'r:/l ilt;'itrr.q thirtg ,1i
''ilt',n!: r10i i'ii'l .t{.i.

Loi sirtL G.8,

ri

,5tt hiOt t'ting

,41

h.9)

,rI)..

/

,1,."t ''

-,:. B ft.,.

|

t,,
..

,\

\.

-.)o

' i 7"t'

.o\ :.;

AC =

JAB\ BC =2a )

OC =

a vit

d(es;sa)=+

#=ffi.

dinh A,B,D . M liL trung th6 tich mroi tr6p ABCD.A'B'c'D' vd khoang
cSch tu di6m M cltin mflt ph5ng (e'nO) .

-----.,:'-fl-].:V
---*- -E--:.V^

ML
llinlt 8
o Gsi O ld tdm hinh chfi nhet ABCD suy
So L(ABCD). ra c6

vpy c6

3$3'
a

BAI TAP
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

,s

/,./r" i\;\\

suy ra

Ke rH

song song

a(ec:sn)

.

llso (tt eoc) A
= IH L(eaco) ve
HC IC 1-laco
(f SC 3
s
-=-=-.
a(a,(uu))=u(c;(utt))

D
ftinh 6
o

=aJ|.

ra

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 trung
di6m c4nh CA', I li giao di6m cria hai duong

q"*'-,u*,', tsEI#E! 7
HÃY ĐẶT MUA TC TH&TT TẠI CƠ SỞ BƯU ĐIỆN GẦN NHẤT !


thing AM vir A,C. Tinh theo a th,s tich kh6i . DUng
ch6p I.ABC vd khoing c6ch tu A d6nm[t phing Ta c6
(rBC).

3. Cho hinh ch6p S.ABCD c6

ddY

la

) 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

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

rudng tqi

HI=AC=a

2.V,or, =

t

o'

=
;a

3.vr*ro='f

+

;

aQw:(t' no)l

(e;Qnc))

=

=*

zJ-sa

o,; a(ec;sa)=

*Lo.

: fr.

e

rc)

vd

HJ = AB = ali n)n

I - | * |
I
h2 HS2 HI2 HJ2 -h

o

r,

AH tl(SBI).

tu diQn vu6ng. Ta c6

5.

o, r,

>

mdACLAB=HILHJ vit I
SH L(HIJ) d6n dtin H.SIJ tiL

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

r.v *ro

AH ll BI

HI ll AC.Ke HJ lt AB Q

BC. Tinh th6 tich kh6i ch6p S.ABC vd
kho6ng c6ch gifra hai cluong thdng AH vd SB.

60". Tinh theo a thc tich ttroi ctrop S.ABCD vit
khoing cilch gliraABvdSD. ,.'
HI,ONG NAX CTAT

>

O6 ttr6y AHBIliLtlinh binh hdnh
ndn H ld trung tliOm BC suy ra

doqn

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

= HE

(en ; sn) = a (t n ;(sm)) = a(a; (sa))

a

hinh

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

fr

Ggi 11 ld trung tli6m

AB, this[ t(esco).
HK LCD, SKH:60".
Xdt hinh thang ABCD,
gi6 str CD cht AB tai E B
l...........

Tam
thl BC=1,4D.

'z
gi6c ADE vudng cdntai A

+AC LED+HKllAC

=l.ir"A =Uf
sH = HKtanUr. =tJ--fo . vr.*ro ='{ or.
Trong (rcco) aung DF = E sry ra
AB l (sDF)= a(ea;so)= a(a;(soF))= h.
= HK =1o,

o

4.

Ke IN llAC,HrJ

llcD (u,v eor)

>HU=ED=2a$=W.

vd Sll t(ttW)

Ta
n}n H.SUV lir

c6

HU LHV
U

tu diQn vu6ng suy ra

I_ I _ I _

I-4"'
L6y

>

r

sr( ncn

SH = HKtan 60.

.I

A

h2- HS2' HU2' W2

K ld trung di6m AC suy raACr(Sar)

AC

I

((sac);(arc))=

=*

,r*,

=

+

ffi

= oo".

_16,l,l_59
'
'
54a2

8a2

-Yav h =ol-]

t

59

t

o.

8a2

H
108a2'

B

J

U tcruagw
TONN HOC

HÃY ĐẶT MUA TC TH&TT TẠI CƠ SỞ BƯU ĐIỆN GẦN NHẤT !


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:
lro-2)-3 --3e.roe {t;a}.
'(*'-?)'
C6 hai ti6p tuy€n cdn tim v6i PT l6n luot ld:
:

-3x +13.
Cdu2. DK: x e m.f {O;1}. Phucrng trinh ttd cho
J = -3x +1,

tuong tlucrng
<+ log,

l"' -

Y

voi log, lxl+ log, lx tl: logs 2 olxz -

tl

: 1sg, 2

x2

x

Hai tam gi6c ABM vd BB' Kbingnhau n6n

=dFE'-AM LBK = AM r (ac'r)
:
= AM a (NsR) = a(eu,tttr) a(n,lrs).
Tac6 il : SB - !- gr =K[/ =1 BA
B'U SB' 3
4

6trfu

=

sd c6ch chsn th6a m6n

X6c su6t cin tim
CAu 5. Ta cb

I'

h

C?.C?.C|.

iE =(z;-z;-z);Ai :(+;o;o);

A,B,C,D ld b6n dinh mdt kh6i tu di€n vd

v

MLD -llTan
6IL

*)

^rl=

f

ra*r.

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
BK=.[B'Ie+l*'z

=+

= rH, uz
a(n,rus)=

+

=ff* =f =cx=+.

I
a2JB
, ^, I aJB
Vay S.,r,u, :-C'K.A'B'=
Zl\' .o =?.

Ke

=+,
l

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

$a1r,ns

)=+

CAu 7. Gqi d ld tluong thing qua M, song song
vot AB; N : d I BD,I1ld trung di()m MN, I lit
t6m hinh cht nhdt ABCD vd P
d0 di6m B thda min hC

[x-v-1:0

-;,

lo'

*s
PT MN : x-3y+

.
Cur, -20
77'

M = (-rt -s;\, lai,Ml= (rz;-z+;t)
=[/B,AC] A5:t4o*0.
YQay

=)nr

111
=

1;

cz'c?'ct'z

=i.r'*

rH NS, ta c6
TP IIC'T
=TP T(INE'N')>TP LST

=

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.
Si5 c6ch

PT

M TA TA 3 SR TR 8
-5,7-5'
RU_U-B'I+UB'

C0u 4.

!1-r
: -]rn;r -,ll] -i*lr *,1|] = -]nz*]r,:

-

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'.

rs

lt-t'

-l-

V*r.o,r,r,

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

=t=2:,x=93 1y=J.
1 ='1J:4 6t = --1i-q-f i-qr
2
211+r

Vfy

tl6

"l=2.

- x =2<]x €{-t:z}.
THZ. x2 - x = -2 (PT ndy vd nghiQm).
CAu 3. D{t r=lnx+ 6y -!,
16 =p,

THl.

Do

:

IH

I

AB.Toa

lx=4
B(ar3)' ra co
t; =, =

= 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).
15 = O

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

e(-z;t) suy ra c(s;o).
Cflu 8.DK -r+y>0, )>0.
THl. y = 0: kh6ng th6a m6n hC di cho.
TH2 y > 0: bi6n dOi phuong trinh diu thdnh
x2 + xy -2yz +,{i + y -,!4 : 0

e (x-y)(x +2y)+,[x.+y -J4
(,\

=o

o(r-r)l
' -'[ x+Zt'

l=o+-*=-r.
Jr+y +,lZy )

Thay vdo phuong trinh thir hai ta

ts"uu,r-*ru,

duo.

c

T?8I#EE

G

9

HÃY ĐẶT MUA TC TH&TT TẠI CƠ SỞ BƯU ĐIỆN GẦN NHẤT !


THusucrpuocxlrru

aEsds
(Thdi gian ldm bdi: 180 philt)
bing 60". Tinh th6 tich kh6i 15ng try ABC.A'B'C vit
Ciu 1 (2 dia@. Chohdm siiy = 4
khoAng circh gl1ahai'x+/a) Kh6o s6t s1r bi6n thi6n vd vC dO thi (Q cira hdm sd
CAu 7 Q diA@. Trong mflt phing vdi hQ tn4c Oyy,
dd cho.
hinh tu6ng ABCD c6 M(-3;1) ld trung cho
b) Viet phuopg trinh ti,5p tuy6n cria eO tni 1q, Ui6t
ci.r. AB, di6m. E. thuQc do4n thing BC sao cho
rang ti6p tuy€n d6 w6ng g6c voi tlucrng thang
EC = 5EB. Bi6t ring DE:23x + 9y - 10 = 0 vi dinh
d:x+7Y=Q'
D c6 hodnh dQ duong. Tim tga d0 dinh D.
C6u 2 (l di€m). a) Gi6i phuong hinh
/_ \
CAu 8 (l did@. Trong kh6ng gian v6i h0 tqa d0
sinx+J3sinl !-*l=2.
Oxyz, cho mflt phdng (P): 2x + 2y + z - 5 = 0 vir
\2 I
r-1
--',
b) Cho s6 phirc z:3 - 2i. tinh -6 dro cua so phric
a_=a=,
dudns
_' =""-'\_" . Tim toa d0
------o L:
---"o th6ns

--

*=

Ciu
Ciu

I

22

ur'

3 (0,5d1A@. Gibiphuong

trhh 2"-t +2'-'

=1.
2'

4 (1 die@. Gieib6t phuong trinh

2(l

Ciu

5 (1

-

x1'[x)

+/r -

1

< x2

-2x - l.
4.

di€m). Titthtich phAn
' t=

l--!L--:
x2+x,l x

i

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

ff=

x3 _5x2

e

dugc chgn c6 t6ng c6c cht s6 blng 10.
Cffu 10 (l di.Afi. X6t c6c s6 thUc khdng 6m a, br 9
nh6
th6a mdn di6u kiQn a+b+c =3. Tim gi5

fi

nhStcriabi6uthric

+l4x_4=6{V, _x+l

f ({&t':3x.3)
+x+l=VBF:8x+8=,r=1.

HQ itd cho c6 nghiQm

ciu

e.

yi b=#*,"c"

= (t;

.2,

sO/(r),

suy

ramaxP = max/(r)

r;

2'D=iJ'

Ngodi cach gidi trAn hoc sinh c6 thi
sir dqtng phucrng phdp tpa de nhu sau

c2

o(,,i,,)

=

.,y,u -E

=l

gidi cdu 6 theo cdch

toa dO

(* t' 4

ru (aj;g),

(K;KC',KA',rr)

tfri

r($,a,), r'(*
* (S,at)',

nn,,

(*-i")

*,-+,"), *=[0,-",-;),
** =(+;,ol
.,/ vay
lr-

(11

1_P-'

l-d =1+qb=c=T'q=
b-a J, lr,

1-rqt=3'c= 2n

_21*--3!-

=-2t+3t (r-tz)=-Ztz+t =f (t).
Kh6o s6t nam

I

c

a, (o; - g;o), a

r (r-o.)' ]-zclc(r-a'\+2"f
-,[
-'laz+t- (a,+t)(cz+t).]
(a,+tXc,+l)

=.,=#

t=

dusc khi

vot AG. Chon h€ truc

fi1-fi

(ocrct). ruri d6 P<

4+)=3'dat

t).

fi$*+o, =Loatr=*J=tJ
- .
=-(a,+l)F/c:+l
/c:+l t2

-aa

YE tia Kt ru6ng g6c vu A'B', voi chi6u cua tia cr)ng chidu

=

(x;y)

+16'

PHAN VAN"THAI

=

1)3

Pr (*) c6 dqlg.f (x + t)

l==!b4 +16
:+,b,=*
' ca +16

(GV THPT chuy€n Phan B6i Chdu, Nghe An)

+3(x+ r) =(ax' -8x+8)
+:vsx, -sr+8 (x)
tac6
+3t,telR.,
oit f (t)=f

f '(r)=3t2 +3>0,vrelR.
(x+

2

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

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

gi6c

I

didm A thuQc.dudng th.ang A sao cho khoing crich tu
,4 den mit phang (P) bdng6.

l au.c,a
AM.NP\=L-- ).ur,tl o$i
-r

-l

NGUYEN VAN THONG
(GY THPT chuy€n LA Qui'D6n, Dd NSng)

I 0'?8I-HELsF .uq(q-?,,)HÃY ĐẶT MUA TC TH&TT TẠI CƠ SỞ BƯU ĐIỆN GẦN NHẤT !

ffisffiml s]affil
{."
lLy'-<
-1'

nAroiuarnut'nffi

,
"

vA uOr vAr Ap,,DUNfi

. -- .- , 'i. n-,'
,_ i:,

...' -, .. i".-;
\'.. .is
.- l..i
-6q$-'-"*-'- -'

tr.

;

6rtroJt7 bai viit

?!

,,

I

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+ ..,+

lir Trong cdc t6ng c6 dgng
(),, dng ndo ld ktn nhiit, t6ng ndo ld nhd
Cdu h6i ttugc d{t ra

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....lit q>q>...>-an vd 4 > br2...2b,).
Hai diy ttugc goi tiip nguq" tha tqr n6u mOt d6y
gi6m (tuc

so thqrr: vd (.r, .,\..,...,-r,,
(b, ,b.,...,b,,)

t4r

+,+,...,1
At AZ An

,u hai ddy

sip

,a2,...,an yd

nguo. c

thri tu, trong

a,ar,...,a, vd af , af ,...,ay lithai
ddy sip ctng thri tg, trong khi a, ,a2,.-.,an yd.
duong thl

+,+,

'atr ld hai ddy s6p ngusc thu
a( af' .+

r.u.

lir ntdt hoon vi

lir.t,

1-

va b.b.,....h, sitp cirng

tht?

thi

+...+{r,).\',t (l)
. lVett ut ,u1,...,(ttt vtr b,.b......b,,sip ngtro'r: thti'
u,b,+a.l\ +...+u,,b,,)a,l,

+zz.-r.

tu thi

+...+a,,-y,, ill)
Dittr ":" :t'ons (l) rd (ll) xcil, ro O q,:q.=...-s,
u,b, +u.h, +...+ {1,b,,(rl,,r, +rr..i.

hoat' b' =ly =...:

11,

hoac (t5x2,...x,,)=(b,,14,...h,,').

Chilmg minh.Xet hai ddy

...,b,, cing tdng vd

f

a1

,a2,...,a,1 vd,

(x,x.,...,x,)

cta(b, ,b2,...,bn). Gi6 str x,
S

:

S' =

b'b2,

ld m6t hoSn

vi

) xr.

atxt + a2x2 + a1x3 + ...+ anx, vd

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 -

= ar(x,

'*" "l '-'2""'-'n '*
en_1*An' ' Az*h, Ar+a,
li hai.ddy sip cr)ng thf t.u.
c) NCu 0(o,. ar<...
)

r'ac

.

'': trlLl-,....d,,
c Neu

DAt

a1

TRil!{'rr0u NAM G'ii Nd,)

BAT DANG THU'C HOAN VI
Cho a,,ct.,...,u,, t,it b,.b.,....b,, lit hui tlti.t,

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
t.u.

,,,,,..

2.

tuy

b) N€u 0
'

:r:r.r.:, i,,.

a1,a2,...,a,vd af , at,...,a!, ld hai ddy sip ctng
thri qr.

tdng vd mQt ddy giim.

hai d6y sip nguoc thri

,

! i

d) NCu ar1ar1...1(1, vd ft 1d t5 te tni

cia

anxn. (*)

-

| .

,,.,

-."

-

xr) +

ar(x,

a2x2

- rr)

=(xz- xr)@r-ar)> 0.
Do cl6 S'> S . Nhu vdy khi d6i vi tri cira x, vd

r, thi gi6 fi

chi c6 thii tlng l6n. Do d6
n6u chring ta cl6i ch6 tbt ch cic c{p (x,;,rr) v6i
cria

,S

,'. t6ng
,:,
x,) x,,i <7. thi
chi c6 th6 tdng l6n. T6ng
ifat gi6 tri lon nfr6t mt (x'xr,...,x,,)=(4,b2,...,b,)

T?31#EE
=,;

"rr,r-rrru,

11

HÃY ĐẶT MUA TC TH&TT TẠI CƠ SỞ BƯU ĐIỆN GẦN NHẤT !

tuc ld khi

S = arb, +

I

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

,l , cos,r
slnJ
nOn theo

HQ qud

vi ta c6 hai hQ qui

hoSn

t,

+

al + ...+ al, ) atx,

(a,ar,...,a,) thi

+ a2xz +

HQ qud 2. N€u a1,a2,...,an

daong

"'+

va (x.xr,...,xn) ld

thi
\a,,a.,....a,)
' I

9L

hoan

vi

xr x2

xit

l) CAu hoi o phAn ddu: Khi ndo t6ng dqng
NhQn
(*) 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

(II) ta c6:

<>1=sin2r+coszxlA.

x=14

thi

sin3x
cos.x
-

A:l.VflyminA=l.D

Thi dU 2. Cho ba sd duong a,b,c. Chang minh
. a8+b8+c8- I 1 I

rdng

ffi.;*

a*i.

Ldi gidl BDT cdn chimg minh tuong duong
voi

ot *bt *., ,f*l*1.
b3c3 a3c3'a3bt-a b

c{'ra

r.
*lz*...*L>
xn

siP ngugc thu tu

sln-r cosx slnr

anxn'

ld cac s6 thac

x

cossx-coslx,
sinjx, _-__=___

Voi

N€u a1,a2,...,a,, ld cdc sd thryc vd

(x, ,xr,...,x,) ld hoan vi cila

al

sau:

sin3 .x, cos3

__-:_T

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

Tt BET

vd

c

BDT d6i xtmg ddi vbi a, b, c n6n c6 th6 giA sir
a> b2c > 0. Khi d6 hai ddY as,bs,cs vd

11t.t
siP ctng thri tu n6n theo (I) ta
;F,;A,;*
bs cs
. a5 bs cs -a5
z

co:
bry
aJb3o*+ otrr* ot1 t)c3+
os b5 cs _a2.b2,c2
(l)
*#.;F++;>+*Y,*

u

Hai ddy a2,b2,c2

t, *.#.*

tU nOn theo (II) ta c6:
a2 *b2 *c2

sip ngusc thri

.o, *L*L

a3'b3 cl-c3 a3
I I 1 ,a2 b2 c2

b3

o!+ O+:
Q)

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

do c6 th6 ldm dugc f(a,ar,...,a-)kh6ng aOi voi mgi ho6n vi cria

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

(a,ar,...,an)

tam giac. Chung minh riing

thitit

.

Sau 6p dpng BDT ho6n vi.
3.

MQr So rHi DU

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

A=sin:'*-ry
COSX SlnX
Ldrt

gidi. Voi 0 .

2
".lthi

voi o<

r.\.L

sinx > 0, cosr > 0 .

BET ddi xtmg d6i v6i sinx vd cosx n6n ta c6

th6 gi6 sir 0
rz'?Eil,HB!

I

Thi dU 3. Choa,b,cld d0 ddi ba cgnh ctia
a2

(b + c - a) +

b2

(c + a

-

b) +

c2

(a + b

-

mQt

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:

**

HÃY ĐẶT MUA TC TH&TT TẠI CƠ SỞ BƯU ĐIỆN GẦN NHẤT !


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

Thi dU 6. Cho A,B,C ld ba goc cila m6t

Lqi 6p ftrng (I) cho hai dAy sip cung thri
vd, a, c ta c6: ba+ cz ) bc + ac

giac (do bdng radian) vdi cac canh ttong img
a,b,c;p ld ruha chu vi. Chilmg minh riing

\r b, c

4.4,+_e>l_r"
p-a p-D p-c p

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

(2) vd (3) suy ra (1)

xhy ra

€ a=b=c. A
NhQn

xit.

1) X6t c6ch gi6i kh6c cho thi du 3. Ta c6:

a2 (b + c

-

e

+c2

a(b

a) +

b2

(c + a - b) + c2 (a + b - c) < 3abc

-az)+b(c

[{.;{)r,)<3abc

(4)

Theo dinh l1f c6sin trong tam gi6c ABC ta c6:
(4)
2abc cos A + 2abc cos B + 2abc cos C < 3abc

o
<>cosA+costr+cosc<]
2

(5)

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

.J-n
f

{ri - y)2

i:l

{\{*, - r,)'

.

I <-+<
Ap-D p-c
1

LAn luqt 6p dpng BDT Chebyshev (thi du 10) vd

BDT Cauchy,tac6'.

I *l,*
t )
vr(l)>A+B-+C.(p-o
p-D
p-c
t
\
)
=

fi

tt -ri,,,y,*it =it -zi*,,,.i+
i=l

i=t
nnnn

,=1

oZ*,y,2\x,2,

(

- o* p-

n

(*)

j=t

i=l

@o

=|41.
Zt
i=l
i=l

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

-3n
--. p

n sii

(II) ta c6:

Thi dg 7 (IMO

r-l -, I
-b, ' trr-t""-r-'
[t"'-A

o,+9+...+&->_h+b*
'
y72

(p-a)(p-b)(p-c)

Cho a,b,c td cac sii

1995).

,,r1 ,*,r,1
,*--l ^-12
a3(b+c)
b37c+aS c31a+b)Ldi

gidl

Ddt x = bc =

Khi d6 x,!, z ) 0;

).,

= ro =

ryr=J_=l
abc

(l)

|., = ob =1.

vd (1) trothdnh:

!->tr

!-*):-+

y+z z+x x+y

2

e)

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
y+z
z+x

x+y

Theo(r)tac6:

x+y y+z z+x
n-2 t,,2 + '-2
VT(2)>
' x+z+y+x
z+y

cria

l>1>...>4
4n2 ndntheo
o!t'

#)

*2
,,2
-2
VT(2)> o - ! + '

a1ta2t...ta, sao cho brthi}t a,ar,...,a, ngly€n duong ph6n biQt n6n
O),Yi=1,2,...,n. Do

\(h. h.

dwongvd abc =7 . Chtng minh rdng

a.all

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

'4n'2n
Ldi gidl Gqi bt,b2,...,bnld m6t hodn vi

-,

-r
u

ngry\n duong phdn bieL Chffng minh rdng

4

p

>fi'z'{n-db-db4'21

j=l

j=t
i-_t
Nhmg theo (I) ta c6 ngay BDT (*). O

h>i

a+

i=1

Ldi gidl BDT cAn chimg minh tucmg rlucrng vcri

(r)

Ldi gidi Kh6ng m6t tinh tdng qu6t, gi6 sri:

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

tam

CQng lung
m2 + nz

.

vti hai BDT tr6n vd su dpng BDT

,@y,

ta dugc:

YT(2)>!( *' + Y' *Y2

+ z2

+ x2)

r\ x+y y+z *22z+x )

o, nuu,u-roru,

T?EilrHS

13

HÃY ĐẶT MUA TC TH&TT TẠI CƠ SỞ BƯU ĐIỆN GẦN NHẤT !


(x+y)2 (y+z)2 (z+x)2'\
- 2[2(x+y\ 2(y+z) 2(z+x) )
\.._l

Vcri cdc sd duong a,

I/

-L-L-t
=aa1],BDrcauchY

a| ...ai"

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

.

St dpng BDT ho6n vi ta c6 th6 chimg mfuh

irW=).a

Nesbit). Chu'ng minh

Thi dU 8 (BD,r

ai'

)az,...,(tn, ta lu6n c6:

nhi6u BDT cO dien kh6c.

ring

nAtt

a,b,t' ld ha sd duong thi

o _ h _ .=1.
b+t'' t lti u+b- 2'

Thi dU l0 (BDT Chebyshev). Cho hai ddy cac s6
thltc ar,(12,...,a, vd b,b2,...,b,,. Chilng minh
a) l,lJu hai ddy sdp cilng thtlr tu'thi
a,b, + arb, + ...+ a,,b,,

riing:

Ldi gidi. BDT d6i ximg theo a, b, c nOn c6 th6

O
gid su

l1
6*r,r+r,

I

. ta hai ddy sdp cirng thri tu nOn theo (I) ta
-:
a+D

.abc-abc
""' h+c c+a' a+b- a+b b+c c+a
ahc-abc
b+c' c+a a+b- a+c b+a' c+b'
-_

:

a

ePt

c

b

t=-

3

b+c- c+a- a+b- 2\a+b' b+c' c+a ) 2'

-!-!

Thi

":"
d1.t

xhy

ra

o a=b- c. D
-!-!-

9 (nOr hoan vi dqng lfiv thira).

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

51t l'rb

;c )

61b

lrc

ru

Ldi gi,rti. Cach 1. BDT khdng ttdi qua
ho6n vi vdng quanh n6n c6 th€ gia

(III) e

sa-csc-b

)

) 6a-cfic-b

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

)

anbn

atbz + azbt + ... + anb,

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

)

arbn + arb, + ...+ anbn-,

(IID

CQng tung v6 c6c BDT tr6n ta dugc:

ph6p

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

(a, + a, + ...+ a^)(b, + b, +

ifrici'

"'+ b),

ta c6 BDT cin chimg minh.

= 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
BAy gio xdt a> c2b> 1. Ta c6:

Ldi gi,fii a) Theo (I) ta c6:
arb, + arb, + ...+ anb, = arb, + arb, + ...+

arb, + arb, + ... + anbn

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

a

D lleu hai ddy tdp ,gu'P', thtt ta thi
t1

trdn ta duoc

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

b,,

nn

:-L-\-f-l-

CQng timg vi5 Z

DAu

b, + b, + ...+

(1)
6b-c

.

(2)

)

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

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 (*)

cAng fttmg binh nhdn). Chlrng minh rdng n1u
,:
a | .a ,.....a,t la cac so khong am thi

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

nhat mQt trong c5c
a1,a2,...,an bing 0 thi BDT hi6n nhi6n dung.

(*).n
NhQn xdt. BDT (IID v6n dung khi a, b, c ducrng vir
thi du 9 c6 th6 dugc md rQng nhu sau:

M'?EI#BLe

Ldi gi,rtL a) NOu it

s6

=1,...,n) tldu duong, tlflt

N6u c6c s6 a,

Q

M=q8,0"-.a.,
r' -

a(to
X'=
*,=!1,
M
M,""'ln=

atoz"'*"

A4,

'

Khi d6 theo he qub2tac6:

455 (5-2015)

HÃY ĐẶT MUA TC TH&TT TẠI CƠ SỞ BƯU ĐIỆN GẦN NHẤT !


xn

{+{r+...+

>n

e 9l!4:l!u>
M

n

xn J1
xn_t
A,*A,*...*A>tt1lA112...An,
' t

€'

DAu"-"

xey ra

a.,

<> Jr =,..= xn e at = ...= ar.A

12 (BDT Bunyukovslg). Chtimg minh ring
,ar,...,a,, va ht ,b),...,b, ltt 2n s6 thryc thi

(arb, + u.h. + ...+
<

"-"

Dtiu

xdy

(ai

+

ct,,b,,)2

a! +...+ al)(bl + b] + ...+ bl).

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

Ldi gidi. N6u a, = a2 = ...=

an =

0 hoic

0 thi BDT hi6n nhi6n dfng.
N6u t6n tqi a, * O vd b, + 0,ta ddt
b, = b, = ...= bn =

N=@+b|+.*fi

M = @+E+*+a?,

;

a,b,.
x, =ft . x,, =fr $ =1,....n).
i

Khi d6 theo hC qui 1 ta c6:
n _ a?

--

+oj+...+a],

- b?

w

+

trung binh

la cac s6 thtrc thi
_

n

, < L+L+...+ x, = N +{*...*{
X2 X3

aj +...

+

,ArA2An
":" xiry rakhi vd chi khi
Xt = X2 =...=X, € At = A2 = .,.= An. A
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:

BAI TAP
1. Chrmg minh ring n6t a> 0, b > 0 thi
a) Z(as +bt)> (a3 +b3)(a2 +b').
b) ae +be > a2bz(as +bt).
c) a' + b" <2n-1(an +b,), z e N*.
2. Chimg minh ring n}u a,b,cld c6c s5 ducrng
th\ a) ab + bc + ca> albc + bJca + cJab .

CQng tung v6 cdc BDT tr6n ta duoc:
n(al + al + ...+ a|) > (a, + a, + ...+ a,)z

€al=Az=...=ar,J

a'+b"+c,
'---'
^J

(IMo tgs:) Biet ring
az

+ a2at + ...+ anln_l.

Suy ra BDT cAn chimg minh. D6u

cta

a, b, c ld dO dei ba cpnh

rlng
- a) > 0 .
MO lgl4) Chrmg minh ring n6u a > 0,

ctra m6t tam gi6c. Chrmg minh

a2,

a] ) ara,

An

D6u

b(a

5. (USA
+

X1 A1 A2

e"@,q-A=.;1}=

4.

) ara, + ara3 + ...+ anal
al + al + ...+ al ) ara, + aza4+ ...+ ana2
al

fi ,x, = #,

a.a^...4
| i- ' . Theo h€ qud 2 ta c6:
....r''nN4=

a)t?,

Ldi gi,rtL Theo hC qui 1 ta c6:
al + a| +...+ af, = a? + aZ +...+ al
+...+

lll
'+ +...+
0t 4.2
d,

hay gi6i c5c bdi tflp sau.

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

-\4

n

BDT kh6 nh.mg cl6 clugc chimg minh kh6 don
gi6n nho BDT ho6n vi. De luy€n tdp, cdc b4n

X2rXn

Suy ra BDT cAn chimg minh. D6u ":" xdy ra
€ r; = Xn*i € a,N = b,M (i=|,...,n) . A

a? +

s6 du'ang thi

BDT ho6n vi trong chimg minh BDT, nhiAu

MN

dt + a1 +...+ atl

c'cic

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

= x? +

-

ld

Ar34S4
N2

Thi du 13. (BDT tntng binh c\ng

tl

Ldi gidi.D6t N = a[apr-.a,:x, =

bi +...+ bi

x! +...x] + xl-, +...+ xi,
) XtXr*l*...* XrX2n* Xna1X1*...*
_2(arbr+ ...+ a,b")

- TB diiu .hia). Chultg

TB nhdn

mlnh rAng neu U t.Q ).....11

'fiP"-(',,>-

Thi du
n€u

Thi dtl 14 (BDr

b) +

":" xhy ra

c(b

b>O,c>0, neZ* th\

Tii
.

b2

c) +

c2

aafu,b6c

a(c

>(abdt?

.

liQu tham khflo

1. Dragos Hrimiuc. The Rearrangement Inequality n in the sky. PISM, issrie 2, page
2312000.

21

2.Phan Huy Khdi vd Tr6n Hiru Nam. Bdt ddng th*c
vd drng @rng. NXBGD Viet Nam, 2009.

Sti ass 6-2015)

TOAN
1

HO(.

cruaga

.d"

5-

HÃY ĐẶT MUA TC TH&TT TẠI CƠ SỞ BƯU ĐIỆN GẦN NHẤT !


BdiT71454. Cho tam giirc ABC. GSlm*ntb,rll,
ld d0 ddi cdc dudng c6c trung tuy0n 16n lugt
img v6i c5c cpnh BC : a, CA : b, AB : c.
Chrmg minh rSng:
(a2 + bz

.,,)(:. ;.:)

=

211 (m. + mo + m,).

Eing thric xiry rakhi ndo?

.:'

DTIONGVAN SOI.{
(GI/ THPT Hd Huy Tqp, TP. Vinh, Ngh€ An)

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
i..
; -,
thi ta dugc mQt sd m6i md
cuoi cung cua so do
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 duvd 6 th6a *in,iici ki€n a20r5 - a 1 : 0 vd
b4o3o - b - 3a: 0. HEy so s6nh a vd^b?
LE XUAN DAI
(GV THPT chuy€n Wnh Phtic)

Bni T3/455. Gi6i phuong trinh:
,,
x+Y+x/l-Y2 +Yll-xz

'6
=+.
z

NGUYEN DUY THAI
(GV THPT Nam Hing, Hing LTnh, Hd Tinh)

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
m6t ducrng trdn cd
NGUYEN QUANG NAM
(GV THPT 8u) Hqp 2, Ngh€ An)

Bni T5/455. Tim c6c nguy6n t6 x,y th6a m6n
:2ya + tt y2 +
1x2 + zy2
if *P
TRAN VAN HANH
(GV DH Pham Vdn Eing, Qudng i{gai)
CAC LOP THPT
BiliT61454. Giai h9 phucrng trinh

["+Y'=4Y2-5Y+3x+4
r.'r r -t - 422 -52+6y +6.

1'r
I

J

l3zr +x3 =4y2

-5x+92+8

-

CAO MINH QUANG

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

.

.TOAN HOC

t O ' 6l'udiua

s6

Bii

T8/455. X6t tam gi6c nhqn ABC c6 c6c g6c
lit A, B, C. Tim gi6 tq lon nh6t ctra bi6u thirc
tan2

A+tafi

B

'" - wo A+tan| B '

C
larf B+tan4 C

+tatf A
4o C +tarf A'
KIEU DINH MINH
Hitng Vuctng, Phu Tho)

tNP B+tan2

(GV THPT chuy€n

tar2 C

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 mdn
di€u kiQn: Vdi 3 .O tt U. a, b, c sao cho
lrl*lal+l'lnghi6m:
l*to +

or'+

bxo

1l*'o- rn+11+

l.llll

+ cx + 15

l*o

(

0

- r +tl
'

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 trgng
t6m tam gi6c ABC, KG cit OT taiZ. C6c di6m
BC sao cho LP ll OB, LQ ll
P, Q thuQc
"doqn
E,
F
l6n luqt thuQc do4n CA, AB
Cdc
clii5m
OC.
sao cho QE, PF cirng vu6ng g6c v6i BC. Gqi
(O ld dudng trdn t6m 7 di qua B, C. Chimg
minh ring ttuong trdn ngo4i ti6p tam gi6c AEF
ti6P xirc v6i (f'
TRAN euANG Hr)NG
(GV THPT chuyAn KHTN, DHQG Hd NA,

455 (5-2015)

HÃY ĐẶT MUA TC TH&TT TẠI CƠ SỞ BƯU ĐIỆN GẦN NHẤT !


t3*i t,!i455" Cho do4n m4ch RZC nhu hinh
du6i; zas : 100\6 cos(100tt) (V).

Bi5t vdn kti c6 diQn tro vd cirng l6n, tu diOn c6
dung khiing lon g6p 3 l6n rli6n tro R.
L

/

v6i z6s. Khi

t-

U=- iU,
SO

L: Lr:zLtthi v6n k6 chi

vd dong diQn trong mqch tr6 pha gz

Bni L2l455. MQt m6y bay bay theo phucrng
',1
^ toc
ve : 320km/h, bAt ngo cl6i
ngang voi v{n
hucrng vd b6t dAu chuy€n dong len tren theo
.:
cung tron ndm tren m4t phdng thang ilimg. Vdn
tdc cira m6y bay liic ndy thay cl6i theo clQ cao
theo qui lu6t v2 : vfi- 2g,,vit tai di€m cao nh6t

+Timgrvdez.
thfc uflt) oiatlidn 6p giira

hai dAu

:

160km,h. Gia t6c cua m6y bay bing
bao nhi6u tai di€m cao nhdt vd tai rli€m vQn t6c
cria n6 hu6ng thing dung l€n tr6n?

bing

V01 zlas.

+ ViCt bi6u

:

:

CRB

/ h,r
\iv

a) Khi 116 tu cAm c6 gi6 tri L : Zl thi v6n ki5 chi
gi6 tr! Ur vd dong cliQn trong mach sdrn pha gr
so

vdn k0 img v6i trudng hgp L: L2.
b) niet R:20 f), cho Z biiin thi€n. Tim gi6 tri

L Lt d€ s6 chi cua v6n t<6 Oat cuc rlai. Vi6t
bi6u thuc u{t) ct-n cliQn 6p giira hai diu vdn k6
khi d6.
c) Vdn giir R 20 Q. Tim gi6 tri cua L de UL
dqt csc dai. Vi6t bi6u thric at(t) khi d6.
vrsT cuoT.{G (Ha N1i)

v1

ttll,rln*a

TRAN KHANH

Thian - Hu€')

p*ou,rilsfitmf Isl,[
FOR SF,CONDARY SCHOOL

Problem Tli155 (For 6th Grade). Find a
natural number with more than 3 digits
knowing that if we delete its the last 3 digits,
we will get a new number whose cube is
exactly equal to that wanted number.

Protrlem T5/455. Find all prime numbsrs x and
y such that
(r' + 2)z : 4f + tt f + xzyz + 9.
FOR HIGHSCHOOL

Prohlem T61454. Solve the following system

ofequations

(,

Prolrlem T21455 (For 7th Grade). Given two
positive real numbers a and b satislring the
following conditions a20r5 -a - 1 : 0 and
b4o3o

- b - 3a: 0. Compa

re

a

and b.

Problem T3/455. Solve the equation

x+y+xrliJ*yJi*
=$..
-t
Problem T11455. On a circle centered at the
point d fix two points B and C. ApointA varies
on the circle such that the triangle ABC is
always acute. On the side lC, choose M so that

Ir'+Y'=4Y2-5Y+3x+4
_
12u,

l-r +zr =422 52+6y+6.
l3zt
t"' + 13 =4y2 -5x+92+8
Protrlem T71454. Given a triangle ABC.
Suppose that the length of the sides are given

by BC : a, CA : b, AB: c and r/t,, rltb,and m"
are the length of the corresponding medians.
Prove that
(az +

'

b2+c'

(r r r\

> zJl(*,+mo+
)l' +|*1
\
b c)

'\a

I

m"l.
tt

When does the equality happen?

MA

3MC. Let H be the perpendicular
projection of M on the side AB.Prove that H

(Xent ti6p trang 27)

always lies on a fixed circle.

.

se ass_rs-rorsr

TOnN
H9C a ,
'TciiiiEE

17

HÃY ĐẶT MUA TC TH&TT TẠI CƠ SỞ BƯU ĐIỆN GẦN NHẤT !


HE vu6ng g6c v6'i CD tai E. Chttng

AH,

VA

minh

ring f,lg =90o.

Ldi gidi.

Bii

T1l45l. C'ho a, b, c td cdc s6 nguyAn
rll
cltro'nc tltou rtttin I + . + - < l. Tint giir tri lo'n
tth."

nhar cit

thi

t'trcr

t - ;'ll'i., :

ohai c6

azb> c> 2. Tu d6

l*l*l<
abcablat):

:1+1+1=

ra

a>

b

*,

.u,

ra

>3. Do

<16

s--s

s

l nen l ,-1.-1-.ru,

!+l*1< I ncn 1<
a
a 3 2

a)7. ru d6 s < +*+*j

nav

ss

,r, ra gi6 tri lon nh6t cira S bdng #.,uu
*,
42
niy tlpt tlugc khi (a; l; c) bdng (7; 3; 2)
f,o6c c6c'ho6n vi cira b0 ba s6 AO. D
Nh$n xit, MQt s6 bpn cho k6t qui thing nhmg lqp
luQn kh6ng chflt ch6. C6c ban sau c6 ldi gi6i t6t:
gi6 tr!

Thg: Nguydn Ch{ C6ng, 6A3, THCS LAm
Thao; Vinh Phtflc: Id Minh Vi€t Anh,-6A, THCS
Wnh Ydn; Nguydn Thi Thily Linh,Nguydn Vdn Nam,
Nguydn Duy Toan, LA Anh.Qudn, Nguy€n Khdc Tu,
OLi, fuCS Y6n Lac; Trin Minh Huy, Ddo Ngpc
Hai Ddng,6A, THCS Ly Tu Tts1g, Binh_Xuy6n;
Hi finh: Trtin Thi Kim Oanh, Nguy€n !{di Ly,
Phrri

inh NguyQt, Phqm Hi€u Ngdn,
Phan LA Vdn Nhi, Trin Nhw Qu)nh, 6A, THCS

Phqm

YAn

Nhi, Phqm

Hodng Xudn Hdn, Dric Thg; Kon Tum: LA ViAt Lmt
TY Trqng, TP. Kon Tum;

Duy, 6A4, THCS
THCS Pham V[n
Ki€t,6A,
Tuiin
Zd
Ngfli:
Quing
D6ng' Nghia

Li

Hdnh'

HAI
'IFT
Bni T2l451. Cho tam giac ABC cdn tai A' 4H
!d dtrt)ng r:uo. Goi D lit tt'ung di1m doqn thdng

HOC
- ^ TOFIN
t cryudigd
s6 ass(s-zors)

IU

:

do d6 AF

=AF=HC,ffi;=5r(:-=90o,

AF - BH e Hq, suy ra LKAF = A'KBH
AK = BK,KF = KH; LBAH -- MHA

=

Ldi gidi. Do vai tro cta a, O, c nhu nhau n6n gi6
sir a > b > c > 1. Tu di6u kiQn cira gi6 thi6t

1* 1*l< I
ahc'

:

HC. TrCn tia dOi ctra tia DC ldy
DF DC. Gqi K lir giao di6m
cho
sao
di6m F
clera AB vd HF. Ta c6 LDAF = LDHC (c.g.c)

Ta c6 BH

ll

BH,

(g.c.g)

(c'g.c)

=AB=HF.
Xet NIEF wdng tqi E (HE LCD), c6 EK lit
ducrng trung tuy6n, n0n Er(

EKld

=+ = fX=ff

.

clucrng trung tuy6n

Xet LEAB

co

(AK:

vit EK =+,do cl6 LEAB vudng

BIQ

t1iE.YA"y AEB=90'. O
NhQn xit. Bdi todn tuong d6i d6,t6t ce c6c b4n tham
gia ddu c6 loi gidi dirng. C6c bqn sau day 96 loi gi6i
ti5t hon c6: Phri Thg LA Na, 7 A, Thi tran II, YCn
LQp; Trigu Quang- Mqnh, 7A3., THCS L6m Thao;
Xgnq An: t'/guydn Dinh Tudn, Dinh Thi Qu)nh
Chdu, Hodng Thi Bdo Ngoc, Thai Bd Bdo, 7C,
THCS Ly Nhit Quang, D6 Lucmg; Quing Ngfli:

Nguydn LA Hodng Dyy1n, D6 Thi Ui _!?,' .7-f,
THCS Phpm Vdn D6ng; Nguy€n Thi Ki€u Mdu,
THCS Nguy6n Kim Yang, Mai fhi lhu Thdo,'7C,
THCS Thi tr6n S6ng V0. Tu Nghia.. Cin Tho:
Nguydn Hoang Nhi,7 A6, THCS ThOt NOt.

NGUYfN XUAN BiNH

Bii

13/451 Giai

phu'ong trinh

hA

rr-l +--.r
,1r-l

v.\

-)

1)

.\'2-vl+11
-q
.I
Ldi gi,fii.DK: xy * 0. Ta c6:
PT(2)e xt -xy'-9x+12=0

,l)
/

(2)

(3)

PT(l)e x'+Yt -x-Y-3ry=0

-

HÃY ĐẶT MUA TC TH&TT TẠI CƠ SỞ BƯU ĐIỆN GẦN NHẤT !


-(x +y) -34r(x +y + 1) : Q
(r+y)[(x +y)'z -1]-3 ryG + y+ 1) = 0
<> (-r+y +l)(xz + yz - ry-r-y) : 0.
oV6i x+y+l=0 thi y=-x-1, thayvdo(3)

e
e

(x + y)3

Suy ra NA LO'A, do d6 N,4, NB ld cic ti6p
tuy6n cria
M

+5x-6:0. Tri cl6 tim dugc nghiQm
(x;y) ctra HPT ld (1; -2) vd (-6; 5).
. Vcri x2 +y2 -ry-x-y =0, k6t hqp v6i (3) ta

ta dugc x2

dugc

(x' - xy' -9x +72) -2(xz * y2 - ry-r-y)
<> (x+y -3)(x' - xy + x-ZY -4) = 0.

=0

-3 = 0 thi kOttrq,p vOl
x2 +y'=ry+x+y ta tim tluqc nghiQm (x, y)

+Ni5u x + y

cria HPT la (1; 2) vd (2; 1).

+ N6u

x2

-

xy + x-2y

-4=O thi 2x -

yz + y + 4,

th6 viro 4(x2+y2-4y-x-Y)=0 suy
yo +9y' -6y+8:0<) yo +(3y-1)2 +7=0,

ra

PT v6 nghiQm.

Vay HPT c6 4 nghi€m (x;y)

(r,-2)

K
Gqi K ld giao
li (l;2),

(2; l),

(-6; 5). D
F Nh$n x6t. Bdi niy cin c6 ki ndng Ui6n dOi aai sO

cria rludng trdn

loi gini dring. Thanh H6a: Dfing Quang Anh, 8A,
THCS Nguy6n Chich, DOng Son; Hir Tinh: Hodng
Khdnh Trung, 9A, THCS Hoa Li6n, Nghi Xudn;
Phri Thg: NguyAn Nhdt Ddng, 9C, THCS Phong
Chiu, TX Phf Thq.

NGUYf,,N ANH QUAN

BAi T4l451. Cho hai dud'ng trdn (O; R) vd (O''. R''1
c'dt nhau tai A va B. TrOn tiu ddi cila tia AB la.v
diAm M bdr ki. Tti M ke tbi dtrdng trdn (O': R')
hui ri€p yyln MC vit MD (C, D td cac ridp cti1nt
t'd D ndm trong (O; R). Citc: dui'ng thang ,-1D
t,d AC tdn laqt cdt (O'. R') tai P vd Q Q, Q khuc'
A). Chu'ng minh ring dw)'ng thdng PQ ltron di
c1u.a mdt di€m cd dinh khi M tha1, d6i tr)n tia
doi c'itcr tia 4ts.

Ldi gidi.
Ggi E, f, lAn luqt ld trung di€m cira CD vit AB;
N1d giao cli6m cira O' F vd CD.
OC ttr6y LO'EN
O'E.O'M: O'F.O'N.
M[tkh5c, O'E.O'M: O'Dz: O'A2.

f,fu = fiB

= frdE

;

QKB =QPB-PBK = BAC'KQB

:6Da -dEN

:dfii

,

suy ra LKQB ca ANCB, n6n

BQ-BK >BK=BQ.BN

BC BN

vd

vd phAn tich da thric thdnh nhdn tu. Chi c6 ba bpn c6

(o). Ta c6

BC

(1)

Do A,KQB cr> LNCB, n}n LDBC !, LPBQ vd LDBC lAn luqt nQi ti6p duong
tron (O; R) vd ( O' ; R'), suy ra

ry=L
BC

e)

R,

Tt

(1) vd (2) ta co BK=

d6

di6mf

*.r,
R'

(kh6ng doil, do

cO ainn.

thingPQ ludn di qua di6mKcd
dinh khi M thaytlOi tr€n tia cl6i cira tia AB. J
NhQn xdt Bdi torin ndy c6 th€ giei blng c6ch sir dpng
ph6p vi t.u quay. Chi c6 hai ban tham gia giii bdi torin
niy ld ban D(ng Quang Anh, 8A, THCS Nguy6n
VQ.y rlu
Chich, D6ng Son, Thanh Hor[ vd b4n Hodng Vdn
Hieu, gE, THCS Vinh YOn, Vinh?hric.

NGUYEN THANH HONG

Bni T5/451. Tim gid rri nhd nhdt cua

2 + l-t3
A: 3ry

!.v+l

dtrctng

tctng d6 x, y ld cdc s6

thu't:

thol mdn x *,y < 3.

Ldi gidi. V6i hai s6 a, b kh6ng 6m ludn c6
a + b -zJab = (JA - JE)' > o, suy ra

su*,u-*t

j$#EErg

HÃY ĐẶT MUA TC TH&TT TẠI CƠ SỞ BƯU ĐIỆN GẦN NHẤT !


(1)
a+b>LJab
:
tling thric xiry rqk}ri a .b.
Ap dung li6n ti6p bAt tl6ng thirc (1) v6i hai s6
ducrng x, y ta c6

Ldi gidi. Gi6 st phucrng trlnh

I
I
co nat ngntem l? - v3 _
n2
(r+

nguy€n duong. Theo clinh

y' 6

O

y(x+1)+4
,
-2
-1-."
6
Theo gi6 thi6t 0 < x
suy

6

t'

**I*l )' .[1)'
\, 2 ) \2)

ra y(x+ r)

'

=4.

Do d6

Tt
suy ra
' * (r)
'* [r-r)'
"7 "-'r
(

4

y+1=3

2=ry
3xy 6
6 4+v--J.y=Z'
[x=l
J

4+y- 6

n+l

r2b

li Vidte, ta c6:

)

+' n(n+t)
,2 , =-bo

- n'(n*l)' n(n+l)
f6t

M[t

2 =__
b+c
,
n(n+l) a

t45a+t44b+t44c=o=T=-ffi (4)
1- 145 / -\ 288
Tt (3), (4) suy ra
=ffi=n(n+t)=

-it)

n\n*t)

Di6u ndy kh6ng th6 x6y ra

288

x+Y=3

(3)

kh6c tu gi6 thiet

A)

)=x+1

a

hqp voi (2),ta dugc

LL

a>2 *)-4+4 =1.A=ikhivdchikhi
"-3'6

1)'

ln'z(n+l)'

_-T----

4,

(

e\
- =g

a
"

l---l

.

* !<3 n6n 1< x * y + l<

,. n ,.
vol
la so

(r)
l4*-l
,,
" =-l
a
ln, fu+t),

2

2 6 -t (2
rry)f
6.a+y)t__Lry+4+y
tlt _L____L
_L_!_
-_r_
-3ry' a+y-(3xy'o]'[++y'
6)

6
ry , t-f-6 4*y y(x+l)+4
f2
-..,
t.......'....,._.-TL
"-\!4+
--L -V:ry'

ax2 + bx + c = 0

t45'

eN.

rt'+

i145.

vl n(n+l) e N

cdn

t

Vfy nghich ddo
l

Vfly minA=] tJ

: 1. y : 2. A

NhQn xdt.
1) D6y ld bdi torin cgc tri hai bi6n c6 chira cdn thirc
mh vai trd cta r, y kh6ng nhu nhau n6n c6 it bpn
tham gia gi6i bdi.
2) Tuy6n duorg c_5c bpn sau c6 ldi giai t6t:

Pht[ Thg: Nguyin Thao Chi, Nguydn Hdi Duong,
Trdn Thi Thu Huy€n, 8A3, Nguy€n Ti€n Long,
9AI,THCS Ldm Thao,' Vinh Phric; NguyAn Minh
Hiiiu, gD, Philng ,Vdn Nam,9E, THCS Vinh Y6n;
Kon Tum: Le ViA Lmt Thanh,9A, THPT chuy6n
Nguy6n r6t rnann.

PHAM THI BACH NGQC

Bii

T61451. Cho cdc sd thrc a. b, c voi a+O
thda mdn 145a+144b+144c=0. Chang minh
:. , , ,.
rdng nghich dao cia hai sd chfnh phanng khdc
0 hAn tiiip kh6ng thi cirng la nghi€m ctia
phaong trinh bdc hai ax2 +bx+c =0.

cria hai sd chinh phuong kh6c 0
li6n ti6p khdng th6 ctrng ld nghiQm cira phucrng
trinh b{c ltai ax2 + bx + c = 0 . Bdi to6n dugc
chimg minh.

NhQn

Tt

xit,

C6 th6 suy ludn nhu sau

:

(1), (2),(4) c6

(r

t( t

) b+c

[a-'][a;r-')=-*t=Di6u ndy kh6ng
I

r

\,,

r

xiy

l

w+

ra vi v6i n nguyCn ducmg thi

')r
ll

i[(,+r)'

)

144

Bqn Hd Xudn Hitng, 11T1, THPT Dd Lucrng I, Di
Scrn, D6 Luong, NghQ An dua ra bdi toan t6ng qu6t :

Cho cdc s6 thtrc a, b, c voi a*0 th6a mdn
ka+b+c=0 (t >t). cntug minh riing phuong

trinh bQc hai

ax2

+bx+c=O kh6ng the c6 hai
I hoqc cimg nhd hon l.

nghiQm citng lqn hon

Crlc ban thir ki6m ha bdi torln nh6!

HÃY ĐẶT MUA TC TH&TT TẠI CƠ SỞ BƯU ĐIỆN GẦN NHẤT !


C6c ban sau ddy c6 bdi gi6i tiit:

Hn NQi: NguyAn Vdn Ti€n, 11A1, THPT D6ng
Quan, Phir Xuy6n; YOn Bii: Vfr Hing Qudn, 1l
To6n, THPT chuy6n Nguy6n f6t fnann; Thanh

B,llT, THPT chuy6n Lam
Trdn Bdo Trung, LA Phan NhQt Duy,
Nguydn Vdn Th€, Nguydn Duy Tuiin, 11T1, THPT
chuy6n Hd finh;NghQ An: Cao Hiru Dg| Phan D*c
Ti€n, Nguydn S! Mqnh,l0Al,THPT chuydn Phan BQi
THPT D6
Chdu, TP Yinh, H6 Xudn Hilng,
Luong I, Dd Son, D6 I-uong; IJung YOnz TriQu
H6a: Nguy€n Nggc Vinh
Scrn;

Hi finh:

Ap dpng c6ng thric cluong trung tuy6n
DB2 +DCz BCz
LBCDtac6DM2 ,-

cho

4.

,

MAt kh6c, do BECD ld ru gi6c nQi ti6p

89' . Suv ra

MD.ME = MB.MC =
Adt

4"

2(DB2+DC2 BC2)
,T-BCz

.AtA2=ol

.,

/\

llTl,

4)

-

6

_ DBz +DCz +BCz

Ninh Ngdn, Duong H6ng Son, Chu V{l NguyAn

(1)

9

Hqnh,70A9, THPT DuChu Minh Huy, l0 To6n l, THPT chuydn Hrmg
YCn; D6ng Nai: Cao Dinh Huy, I I Torin, THPT
chuydn Luong The Vinh, TP Bi6n Hda; Vinh Phtic:
D6 Vdn Quy€t,1041, THPT chuyen Vinh Phric.
NGUYEN ANH DUNG

Tri gi6 thi}t At ld trgng tdm LBCD, theo c6ng
thuc Leibnitz ta thiy

Bdli T7l451. Xet ta di€n ABCD nQi nAp m7tt cdu
(S) cho trtroc. Ggi A1, Bt, Ct, Dt ldn luqt ld

AAt2

trgng tam cac tam giac BCD, ACD, ABD,,ABC.
Cac drdng thdng AA. BBt, CCt, DD1 cdt (S)
ldn lffqt tai cac di€m A, 82, C., D. . Tim gia
nhd nhdt ctia bi€u th[rc

P_

Lal + BB| +CCi

tri

AB2 +AC2
-------------;-\2)

+AD2

BC2 +CD2 +DB2

:9

Mt_

Tt (l)

vd (2) ta thu dugc

-

AB2+AC2+AD2

M,(AAr+ A,4)=

- AAt.AtA2 suy ra

AB2+AC2+AD2

AB2+AC2+AD2
= AAr.AAr:

(3)

Tucmg tu ta cf,ng c6

+ DD"')

BA2+BC2+BD2

BBr.BB, =

AB.CD+AC.BD+AD.BC

(4)

J

CA2 +CB2 +CD2
ccr.ccr= -._--;-

Ldi gidi.

(s)

J

DD,.DDr=

DA2 +DB2 +DCz

(6)

Tri (3), (4), (5), (6) suy ra
AAt. AA2 + 84. BBz + CC r.CC, + DD r
2(ABz +ACz +Aoz +BC, +cDz +

D D2

BDz)

J

(7)

VAn tu c6ng thac Leibnitz tath|y
AArz +BBr2 +CCrz +DDr2

A-

=](ou,

+ AC2 + ADz + BCz

+cD2 +BDr) (8)

St dpng biit dang thuc Bunyakovslq
(AAr.
A2

Gqi E ld giao di6m cira DAl voi m6t cAu (S), M
ld.trung cli6m cira BC. Yl ADA2E li tu gi6c nQi

ti6p, n€n

M,.A,4

= A,D.A,E

=!r*(!oru

)
=;DM (4M

+ un)=

?r*,

+ ME)

*lotr.rue.

ll,

<(M,

ta c6

+ BBt. BB2 + c c t.cc 2 + DD

+BBr2 +cCr2

+DDrr)

"(14,

Tt

t. DD2)2

+B\2 +ccrz +DDz2) (9)

(7), (8) ve (9) suy ra
AAr2 +BBr2 +CCrz +DD22

> AB2 +ACz +AD2 +BC2 +CDz +BDz
>

2(AB.cD

+ AC. BD +

AD. BC).

r. nuu,u-ro,u,

T?3I#t!2

1

HÃY ĐẶT MUA TC TH&TT TẠI CƠ SỞ BƯU ĐIỆN GẦN NHẤT !

Tt d6 P > 2. Ding

thr?c xAy ra

khi vd chi khi

minP:2.

ABCD ld tu diQn gAn d6u. T6m lai
NhQn

E

xit. Ddy ld mQt bdi torin tim

cpc tri hinh hqc
thitc luqng trong ducrng tron, tinh

c6 sri dpng d6n hQ
ch'5t trqng tdm tam giirc, c6ng .th*c

Leibnia vd biit

ddng thvc Bunyakovslg,. RAt ti6c k:h6ng c6 b4n ndo
tham gia giri kri giii cho bdi torin ndy.

HO QUANG VINH

BAi Tft/451. Oho cat: .sit thtrc o, b, c. Clttin,q
1 ,i +nli ll +nc2-:----c') +nul

: I' | -:----ll
ntirrh rirttg
7
'
r (l)
Ll lt C (t

1

ttl)'

lt +tlc'

t" *tltt

trcng cfui ne N, n>2.

Ldi gidi. (Cila nhiiu bsn). Truoc ti0n ta chimg
t
n
a2+nb:

Tucrng t.u, ta dugc:

D+rr2 I (l rz\ -. c2+na2
I (l r\
'
=*$.;.J ,\*,,

ffi

[:+rJ

CQng (6), (7), (8) ta c6

(sl

(l).

2) R6t nhi6u ban tham gia giii bdi ndy vi tl6u c6 loi
giii dimg, circ cfich gidi ld kh5 phong phri- C6c ban
sau c6 lcri gi6i ngdn gon: YGn B6i: Yfi H6ng Qudn,
11To6rn, THPT chuyCn Nguy6n T6t Thdnh. Bic
Giang: Nguy€n Vdn Nghia, 10,{1, THPT Lang
Giang sd 3. Nam Dinh: Trlnh Tudn Giang, 10T2,
THPT chuy6n LE H6ng Phong. ThSi Nguy6n:
K25, THPT chuy6n
Ngy€n Thanh Titng, To6n
Thrli Nguydn. Vinh Phlfrc Nguydn Cdm Dung, Dd
Vdn Qay€t,lOAl, THPT chuy6n Vinh Phirc; Hodng

ll

Vdn HiAu,9E, THCS Wnh YOn. Bic Ninh: LA Huy
Cudng,ll Toiin, THPT chuy6n,Bic Ninh. Hi NQi:
Trdn ThiQn Nam,llAl, THPT Ung Hda 2;.Nguydn

" ln+l)a' ln+l)b- a] + nb3
Vdn Cao,gB, THCS NguySn Thugng Hidn, Ung

b+na
a:+nb2
Hi:a;
Vil Duc Vdn,l0 To6n 1, THPT chuy6n DHSP
laco: (Z)e
@+l)ab> a\nb3
Hd Ngi. Hnng Y6n: Trigu Ninh Ngdn, Nguydn Phic
Hodng, Chu Vfi Nguy€n Hqnh,10A9, THPT Duong
adb+nd +nba +fialiZndb+db+#aF +naF
Quing Hdm, V[n Giang. Thanh IJLia: Nguydn Ti€n
e na4 - na3b + nba - nab3 > 0
Tdi, llT, Vfi Duy Manh, l0T, THPT chuy6n Lam
Son. NghQ An: Truong NhAt Phi, Lop Al-Kl,
(3)
e n(a-b)2(a2 +ab+b2)>0
Trucnrg PTDTNT THPT 36 2 ; Cao Hiru Dst, Phan
BET (3) dirng v6i mgi a, b duong vd n)2, ndn D*c TiAn, Nguy€n Si Mqnh, 10A1, THPT chuydn
(2) dtng. Tuong tu ta c6:
Phan BQi Chdu. Hd Tinh: Mai fhi ry UyAn, Ng6
Vi€t Hodng,,Phan NhQt Duy, Nguy€n Anh Tri€u,
I
n
b2+nc2
_r_
(4)
10T1, Nguy€n Khdnh Htrng,10T2, Trdn Bao Trung,
(n+l)b (n+l)c- b3+nc3

, 1.. *,
',' c3 +na3

'
(r+l)c
ln+l)a-'c|+na|
BDT (2), (4), (5) ta cluqc
Ding thric xiy ra khi a= b=c . J
CQng c6c

(5)

BDT (l)

l) C6 th6 giai bdi ndy bing c6ch dtng
BET BunyakousAT vd BDT Schwarz nhu sau: Ap

11T1, THPT chuy6n Hd finh; Trdn Ngrydn D{rc
Thp, 10A1, THPT Nguy6n Thi Minh Khai, Dtic
ThS. Quing Binh: 11o Anh Tien, 11 Toriin, THPT
chuy6n Vd Nguy€n Girip. Phri YAn: Hi Minh
Hodng, Ng6 L€ Phuctng Trinh,l0Tl, THPT chuyCn
Lucrng Vdn Chrinh. Binh Dinh: Trdn Van Thi€n,10

NhQn xdt.

To6n, THPT chuydn L6 Qulf D6n. Edng Nai: Cao Dinh

dpng BET Bunyakovslq ta c6:

Huy,11 Toan, THPT chuydn Luong ThC Vinh. Binh
Phuric: Ngtry,€n Vd Trung Hi€u, Biti Vdn Binh, llA,

(a2

+nE)2

=(,r; *,ii
I

t

,\r,i)

<(d +il)(a+nb),

r \2

(a+nbl2 =la. l+ n2h.n:

I a+nb nrl
^Suv
ra:a) +nb: S-<--J'-'
-

\i

a1+nb\

+nb2

l\l+n).

a)+nb:- a+nb'

Ap du"g BDT Schwarz ta c6:

1 1 r_ I > (r+l)2A_ n+l < _l I (l_!_ n\
o ! "" b- a+nb- a+nh-a+lla'b)'

_r_r

Tdn Kim DuyAn,l0T, THPT chuy6n Bac Li6u.

I

.-l

Do dir:

THPT chuy6n Quang Trung. Kh{nh Hita: Hd Xudn
Khang,l0 To6n, THPT chuyCn L6 Quli D6n. Tidn
Giang: NguyAn Minh Th6ng, 11 Torin, THPT
chuydn Ti6n Giang. Long An: Phgm Ddng Khoa,
10T2, Phqm Qu6c Thdng, l0Tl, THPT chuydn
Long An. Vinh Long: Nguy€n Minh Thtrc, l0Tl,
THPT chuydn Nguy6n Binh Khi6m. S6c Tring: Id
Long Qudc, l0AlT, Vwtng Hadi Thanh, llA2T,
THPT chuydn Nguy6n Thi Minh Khai. B3c Li6u:

;-r!:.If

1*1')
ut+nb:' n+lIa b)

TRAN HUIJ NAM

Biri
,0,

. Tim c'cic' b6,si ngu.yAn du'o'rtg
p) rrong d(t p lit :;6 ngtryAn td ,sao c'ho
thiu mdn: p' i(7t - l;:' - (2p-l)'

T9/451

(x, .v, z,
c'hung

.

HO(
-a z
- TORN
grrldiuL
!" fslg-2o12
'

HÃY ĐẶT MUA TC TH&TT TẠI CƠ SỞ BƯU ĐIỆN GẦN NHẤT !


Ldi gidi. (Theo bqn Tri:n Quang Huy, llA,

THPT chuy6n DH Vinh, NghQ An vd bqn Vd
Duy Khanh,11T1, THPT chuydn Hi finh).
Trudng hqp l: zl€,,tac6
p. +(p _l)" =1 (modp), (zp -t)' : -1 (modp).

Vay

1

=

-1 (modp)* p =2. Phuong trinh

thdnh

2'+l =3' e2' =z(z' '+...+3+l)

N6u x

> 1 thi

tr
e2x-1 =3'-t +...+3+1.

vC lr:ai ld sd chin, trong khi v6
phii h tdng cria z s6 rc do d6 h s5 le. Uau
thu6n. Do <16
Ydy moi nghiOm ld

y-lsz=1.

(1,k. 1,2)voi keN..
Trudng hqp 2: z chln, z:2t, / e N*

.

Phuong trinh trO thenh

o' =l?n -t)'

-(, -,)' ]lQn -tI

+(r - r)' rrl
]

Dop ld sd nguy6n t0 n6n (l) tucmg aucrng rOi

llrr-t)' -(p-t)'
1.
,

i) NCu a>

Tt

(2)

lQp-r)' *(p-r)' = pu
b>a)0,b+a=x.

(3)

l= b22

'

(2) vd (3) suy ra

l

(4)

lz(o-1)'= pu -po (5)
Tfr (4) suy rap 12> p =2 Thay vdo (5) ta dugc
2=2b -2o =z'(zo-, - t).
Suy ra a=1, b-a=l+ a=1, b =2= x = 3 vh

l:

l. Tir

d6

z:2.yAy

2'*'=

Khido

3b

8 (mod16)

=

y+

l=3= y =2.

=9-b=z={'=l=2 1r-

2.

lx
b0 nghidm thf ba ld, (2,2,2,3)

Y$y ta clugc
Thu lai cA ba bO nghiQm tr6n ili:u th6a mdn.
Ddp s6: (l , k, 7 , 2); (3, k, 2, 2) vd (2, 2, 2, 3).

A

NhQn xdt. Bdi ndy duoc kh6 d6ng c6c bpn tham gia
gi6i vd hdu h6t ciic loi gi6i ld tron vgn (tuc ld tim
dugc tdt ci ba b0 nghiQm n6n tr6n). C5c b4n sau ddy
c6 loi girii t5t:
Hi llinh; Nguydn Vdn Thd, I 1T1, THPT chuy6n Hd
finh; Ding Nai: Cao Dinh Huy, I To6n, THPT
chuy6n Lucmg Th6 Vinh. Hn NQi: Vfi Bd Sang, 1l

To6n, THfT chuyCn Nguy6n HuQ; Vinh Phric: Dd
Vdn Quy€1,10A1, THPT chuy€n VTnh Phfc; Quing

I

Binh: Htj Anh Ti€n, 11 To5n, THpT chuydn V6
Nguydn Gi6p; Binh Dinh: Trdn Vdn Thi€n, 10
To6n,_THPT chuy6n LO Quy D6n, Ti6n Giang:
1l To6n, THPT chuy6n Ti6n

Nguy1n Minh Thing,

= p.

fzQp-l)'= po + pu

YAy

ta c6 nghiCm thri hai

(3, k,,2,2).

Giang'

DaNG HUNG THANG

Blri T10/451 . Tim hing ta K k n nhdt sao cho
bat ddng tht?c ,satr lu6n clung

F.41

.\E.Kl,.-,f *nl-*xl,-ul

v,6i ntoi a > 1 vd a, b, c' td
amthoqmdn a+b+t: =1.

<2.

cac, s6 thuc kh6ng

Ldi gi,rti. Diiu ki€n cin. Gih sir K ld s6 thuc
th6a mdn bdi to6n. I-iy a: l, b : c : 0, a =l
tac6 .J-1+JK+Jr< 2*zJK<1=0 =4'

X=j

ii)NOu a=0,b>1. Tt (4)tac6

Diiu ki€n dir. Ta si! chimg minh

+1, suy ra z(-t)':1 (modp).
Ni5u I chEn >2:-1 (modp), khdng xity ra.
Y $y t ld
=-2:l (nndp) =3 = O (""4p) ) p =3.
Khi d6 (a) vd (5) tro thdnh

mdn bdi to6n. ThAt v6y n6u a, b, c ld c6c sd thgc
khdng dm, a t b + c : I vd a ld s6 thUc, a>1

z(zp-t)'

= pb

lZn +t=2.5'

13'-1=2"*'
(6) ta c6 3b:4(mod5).

(6)
(7)

b:4k *
r e {0, l, 2, 31, ta c6 3' : 4(mod5) ) r =2.

Tt

YAy

b:

4k + 2.Thay
3on*'

DEft

vio (7) ta duoc

*l

= 2,nt .

-1:81r

.g-l:g

ta c6 0.

Fp.

(mod16).

1,

o+4lc -

td

al

.F.T-ur

Do vai trd cria a, b, c trong bdi to6n ld nhu
nhau, khdn g mat tinh t6ng quiit ta gii su
a> b2. c. Chu y: Ybi x, y ld c6c sd thuc khdng
dm, ta c6 birt ding thfc Cauchy cho hai s6
Z"fxy

Xdt ddng du theo modl6 ta c6

3ak+2

r,

thtta

<.r+y.

Tri

t16

a+illb-4+ b++1,-,1+
Sif ass (s-2o1s)

,*jv-ul

5h\HEEzs

HÃY ĐẶT MUA TC TH&TT TẠI CƠ SỞ BƯU ĐIỆN GẦN NHẤT !