Tạp chí toán học và tuổi trẻ số 300 tháng 6 năm 2002

BQ

ur

ctAo DUc vA oAo

rAo

*

n$ roAt nec wgr

Ntnn

odmhoc&
t^t

2

Erg

u6t19s7
u6t2002
NrrA

xuAr nln cdo

D1lC


\rau\

.4 q
.,J
iF-It

ifi

t[*.

]gff

.Ehrl

E ...,r

,.

=

+r/n

&Al l&All eiw Pllwil

r,i*affi

"#

:::qs *'i,

E -i

St =

nr2

vi

S, = .l suy ra x =

thu6c vh compa thi khOng thd dung hinh nhu thd vi
= 0,564i90x. Bang

x 16119
thd giii gdn dLing nhu sau. Til thdi cd Hy L+p da bidt =
1,171r v6i sai sd Ar = 0,006r.
Clci

Ji,

S,, v6i x cho

1,7'72454r vit

x

tru6c.

sai so
Tang phdm dang chd ban ndo c6 ciich dung it thao t6c vdi

o

t 1)r sd siou vi€t'

fffi*dffitnfii;*:;#-;O
. Hinh trdn c6 S, :

il'ili.J 1]

vu618
Ph6p cAu phuong hinh trdn lh dung mQt.hinh
(h'1)'
Tit
cho
dd
trbn
hinh
.o Jigl ri.lr'uang?ien tich

jE;l

ffi+ffi

lJ,tt'ilil'tJ

nho'

I
E

S, = S,

Gitii itdp; cAp crAy o6 rao DA GIA. DfU Hinh l
d
dudng thang d dd A trirng r'6i
M6i thao tdc gtipgidy duoc ki hiOu AB -s A'B', ngfra li gap gia'y theo
A'vd B tring v6i B'.
C6c quy tdc gdp gidy : (1) Brdt AB vi d thi x6c dinh dluoc A'B''
(2) Bidt A vd A'(phan biot) thi x5c dinh duoc d'
(3) Bidt A, bidt D (sE thu6c d) vi dudng thhtg a'(sE chrla A)lhi x6c dinh duoc
e)ui iv,hi a' c6 didm chung A'v6i dudng trdn tAm D b6n kinh DAvb'd L AA'Gap gidy tao luc giric d6u c6 4 dinh thuoc c6c canh hinh vuong ABCD.

Trudng hqp 1. Cdch

G.I
DC-f,r c6GJ;

L

EF

HK

(h. 2). (1) AB -+ DC c6

HP

EF; (2) AB -+ EF

c6

HK

GN

(4) HG -+ HM c6M,P (5)GH -+GQcoQ'N
Tac6HM=HG=ZAH+trt1fu =6GD= 60"=friP =FFG suyra

(3)

IIMI\tPQG

lI

AHEG

luc gi6c ddu.

C(tch 2. (h. 2) Thuc hicn (1) (2) (3) nhu c6ch

(4')AB-+ASc6Se

EF;

NS

l,

Hinh 2
sau d6

:

(5') B-->B'c6N,S,P;

M8
(6') NP -> HG c6M' O

Ta c6 AB = AS = 2AT = AD = SD suy ra HMNPpG ld luc gi6c ddu'
Trulng hop 2. (h. 3)

B

EF
(2) AB -+ DC c6 EF vi O

BD

(1)A+Cc6BD;

QM

G.I

(3) DC -+ EF c6 GJ ;

(4) OF -+ OS c6 QM vdi

Se

G/

QM
BD
A QE G
(6) RN -+ PT c6 P'T
(5) QM -+ run c6 N, R;
Hitfi 3
Tir OMN ld tam gi6c ddu suy ra MNP QKt li.luc giric ddu'
Gdp gidy tao ngii gi6c tldu c6 4 tlinh thudc cic canh hinh vudng ABCD.
F.IC
Trudng hgp 1. G. a)
G.I
HK
EF
(2) AD -+ BC c6 HK vd O (3) DC -> EF c6 G.l
(1) AB -+ DC c6 EF :
EF
(4)GO -+GM c6M e AD.(5) U -+ W c6N e AD; (6)MN -->MS c6S e AB
(7) NMS -+ NPQ c6 P e CD, Q e EF
MS=MN = 2ME = ct..B_t)lz: 0,61804
Dirt AB=AD=a th\GM=GO= on1 A

Hinh 4

=

AM = AE - ME = a(3-'li

=

lt+.

1km riiP bia

J

)


a-

T.IBONACCI
Odnh eha

UA SO OOWG

e&n

OT

HOANG CHUNG

TRUI\IG H8G CO SO
Phi-b6-na-xi (Fibonacci) (1180 - 1250) li
nry tgrirl.hqc 'i. Nam 22 tudi,6ng cho c6ng bd
ryAt lei H0u, rrong d6 c6 "bdi todn vd dAn lhd,,,
dua ddn diy sd 1,I,2,3, 5, 8, t3,2I,.... d6 h

Diophante (thd ki rhrl 3) dd nhAn x6r rang,

trong m6t tam gi6c vuOng c6 canh huydn c vi
hai canh g6c wOng a, b; ta c6 :

[g)'* au -(o*a\2
\2)-2 [ z )

d6y sd Fibonacci rdt ndi tidng.
Nam 27 tudi, Fibonacci tham

gia m6r cu6c
thi Ai vd giii ro6n, do wa ph6-d0-ric II
(Frede_rlc [I) t6 chrfc, vi d6 rhing m6t c6ch d6
dang. M6t rrong c6c bii to6n dfi [ li':
Tim s6' hfru

ti sao cho I t

S ddu ld binh

phtong cila mbt sb'hfru ti.
Fibonacci dd cho

4l
dirpsooungr*=Tr.

=

H,

non (z+v)(

il-v) =

80.18

giii

nhu tr0n rhi rh4r

gian ng6n dE nghi

ri

li

*22,o mbt sd ddng du. Hon nfro, n€u * = ay,

thi d ld sd ddng du.

#.Ndu

li Ai rinh, vi trong thdi

duo.c

MQt tarn gi6c vuOng c6 ba canh a, b, c ddu ld
sd nguyOn (c
canh huy6n) duo. c ggi Ih m6t
tam gi6c Pi-ta-go vh b6 ba sd (a, b, c) duo. c goi
la TOI b0 ba Pi-ta-go. Chri f r6ng, trong hai sd

a, (94nh g6c w6ng) luOn c6 ir nhdt m6r sd
chdn. Ta c6 :
Dinh Ii 2. Ne:a c6 bO ba Pi
t22

.18 Y=
Do d6, c6 thd cho.n z *, = * vaut2
t2
-.
9vi
Suy ra u =
c6 x =
qui Fibonacci
de

ndion tich cta ram gi6c vuong).

!

.Solg 16 Fibonacci dE tim ra sd d6 bang
c6ch nho, p6 tguoi doi{n lbi gi6i nhu sau. Gii s[
t' + 5 = u", I - 5 = vz.Th6thi u, - 12= 10. Mi
10

f!

ring

10

-

80'18

|

cfing.thQt kh6 hec qp dugc gi ql# ror
giii d6, ngodi suth6n phuc ! ''
thd m& r6ng bii tren nhd m6r ding thric
-C6
cfia Di-6-phang (Diophanre) rr) gdn t00b nam
trudc Fibonacci.
Ta goi sd nguy€n'duong d t4 mQt sd ddng du,
n€u c6 sd hiu ti x sao cho
d6u ld binh

Sng

,BA

U*

b

sd ddng du.

* * af

e (;)':

=
'=(+)'
( B\'
vitt-dyz=E *(;)' -, =[tJ
Thrlc vQy,

puo. c5{c

dinh boi a =

-

c-a

q
q li hai sd nguyQn rd ctng nhau vd ch[n 16 kh6c
nhau (a, b c6 thd, ddi ch6-cho nhau). Ndu (a, b,
c) li m6t b6 ba Pi-ta-go , rhi (ka, kb, kc) cfing

bu Pi4a-go, vdi & lh sd nguyOn
ki.
(p, q) mQt sd gi6 tri, ching han (2, 1),
_-Clr.o
(4,
2),
L), (4, 3), (5, 2), (5, 4) ... ta c6 m6t sd
\3,
E
!q 4,ta-go (3, 4,5), (5, L2,13), (8, 15,'17),
(7,24,25), (2I,20,29), (9, 40, 4l), ...

B

tnqt-

9Q_

ducrng bat

Tt

bO (3,

I td

phtong cfia mbt sd hfru ti.
Vi du : 5 li m6t sd d6ng du.
Dinh Ii l. Neu d vd y2 ld cdc sd nguy€n
dugng, thi d ld sd d6ng d* khi vd chi khi dy2 h

Pi-ta-go (a, b, c)

p' - q' ; b =2pq vh c = p" + q, (*) (ban doc tu
giii khi d4t 9! ! - b bing Zy trong d6 p,

4, 5), c6 3.2 = 6

li

sd ddng du

:

[t)'*

Tt

b0

u-(qxz\2
z)
lz):"-(
(5,12, t3),

c6 5.6 = 30 ld sd d6ng du.

TiI (8, 15, 17) c6
60 ld so ddng du.
^15.4 =
Ngoii ru 15.4 = 15 .22 , n6n I 5 Ii sd dong du :

(Y)'*15=[rsts)2

A2

Tt

l+)=^"-( + )

bQ

(9, 40, 4l) c6 9.20 =

ddng du, do d6 5

li

32.22.5 = 5.62 la sd
sd ddng du


Sit{lH I'{}AN {i

+t\' *r_[+otq)'?
[2.6 )
lL6)

(

CO ttrd chrlng

ilinh li

1

minh

duo.

(theo chuong trlnh Todn THCS n*m 1fi02)

c dinh

li dio cria

TNAN PHUONG DI.ING

:

d* thi tdn tqi mit tam gidc
Pi
SGK To6n 6 tflp t
gdm 3 chuong: So tu
nhiOn ; 56 nguY€n ;

Neu d ld mdt d6ng

d6.
nguy€n
"Nhu ndo
v6v. ta c6 didu ki6n c,in vd drt dd milt sd
ld ttdns du.iuy nhien, vi6c sir dqlrq
a
tieu chudn nhy n61 chung:li rdt kh6 khan' D€
iira" ai-e, ,av, ban chi cdi x6t xem trong cdc s6
i, i, 3, 4, 14,'2i, sd nio sd d6ng du, s6 nio

Doan th&ng.

SGK To6n 6 tlp 2
gdm2chuong:PhAn
sd; G6c.
So vdi chuong trinh
vi SGK To6n 6 tit

,ii ilb,

li

khOng.

Fibonacci dd chrlng minh rang

n6m 2001 tr6 vd
trudc, SGK To6n

:

Moi s6'c6 dang
mn(m+n)(m-d nAlu m+n chdn,
4mn(m+n)(m-*) nAu m+n ld,
ddu ld sd d6ng du
Thuc ra, kdt qui niy c6 thd suy d€ dhng tt c6c
dfrng thrlc (*) x6c dfnh cdc bQ ba Pi-ta-go vi
c6c dinh li l, 2 o tron :

6. nim

20Oz

c6 nhtng n6t

mdi
nhu sau:
1) Chuong Sd nguy6n tru6c d6y dAy 6 lop 7
nay dua vdo lop 6.

2)Xdtci

phan sd duong

vi

Ph0n s6 am.

m4t nQi dung, SGK To6n 6 ndm

Nhu v6y, vd
Irtqi ,d cd dqng mn(m+n)(mtt) vdi m, n 2OAZ c6 nhidu hcrn s6ch c[. Tuy nhiOn, cdc
nguy€n duong vA m *n, ddu ld sd ddng du'
chuons dai s6 duo.c vi6t nhg nhing hon. Ci{c
Ding thtlc cira Diophante sC duoc vi6t dudi ph6p t66n'vd s6 tu nhiOn vi phin sd ducmg duo.c
dang :
Oat uan dd nhu bdi tdng k€t cdc quy tic ph6p

lr*)'
Nim

+ mn(mz

-,, r =l*'

-

")*'*'l

1983, nhi to6n hqc ,t' B. Tunnell dd
.OiAiCu kign idn kh6c dd mQt sd d
"t,ing-rrintr
lI ddng du:
Neu d ld m\t sd d6ng du ld thi sd cdch vidl d
duoi dang z? + y2 +-8r2 trong U' x, !: z ld sd
nguy€n va z td sd ld, bdng sd cdch vi4lt d dudi
dang dd vdi z ld sd chdn.
Vi dU : X6t sd 19. Co thd vidt 19 du6i d4ng
trcn theo 8 c6ch khdc nhau va zLb

to6n duoc ph6t bidu thinh ldi, tr6nh dua ra cdc
cong thrlc idng qu6t kh6 hidu ; phgp to6n vdi
phai sd n6i chung duo.c x6t trcn cd sd ph6p todn
Ltra sd nguyon, ihua xet ddn sd hfru ti. Hai
chuong hintr hqc kh6ng gidi thicu bang phucrng
ph6p ticn dd mi duo.c ti6p c6n bang-cdc ho4t
hqng trinh hgc nhu vE hinh, g{p hinh, do d4c... ;

cdclhai niQm duo.c gi6i thiEu thOng qua m6-ti,
hinh v6, tr6nh dua ia nhfrng dinlr- nghia tdng

bli t4p trong-SGK To6n 6

iamZooz li nhtng bii t4p co bin, nhfng bli

qu6t.

Dic

biOt, cdc

mp mang nQi dung thuc

tdvi

nhfrng

bii

d6 vui.

Ve hinh thrlc thd hiQn, trong tung bai ddu c6
nhtng c6u h6i "gita chimg" dd Eng cuitng ho4t
c6ch
AQng"tq hoc ciri hgc sinh trong viQc nam b6t
kh6c nhau :
tlgn tnrfc. ${ch c6 nhidu hinh v€, c6 ci nhtng
2e3)2+(t1)2+8(0)2,
hinh vE vui mit, ng6 nghlnh girip cho viOc hoc
do do 19 khOng Phii li sd ddng du.
c[ra hoc sinh duo. c truc gi6c vd d6i khi dugc thu
Dinh li dio cfia dinh li Tunnell chua dugc si6n nsav trong gid hoc. Sach cbn c6 muc "C6
ct-,:ng minf, hay b6c 6a (tqq+).

itd .r,i"tuu ui&; ntrim gi6i thiqu lich str To6n,
c6c ki6n thrlc liOn quan dd m6 r6ng kidn thfc
Nhd cdc bq.n gidi tidp cdc bdi todn sau ddy:
mOt c6ch nhg nhlng.
1) Chrtn7 minh dinh lt &io cila dinh li 2'
n6i SGK To6n 6 mP 1, aP 2 h nhfrng
Z)froni cdc sd ll, 14, 17,21, sd ndo ld sd C6 thd
cu6n SGK hdp ddn ddi vdi hgc sinh buoc ddu
ddng du, sd ndo khbng ?
vio cdp TI{CS, ddng thhi cfing girip cdc gi6o
3) ChItnS minh rdng m1t sd chinh phuong vi6n td-chrlc hoat d6ng hoc tdp tdt hon.
khang thd h sd d6ng du.l.

2$D2+(t3)2+8(t1)2,
cbn vdi z chdn thi chi vidt duo.c theo 4

2


,Jt

a?

SH T[*I TtrY&r* SErqFf LSP IS

TRffi]IG THPT NfiUA KNIEU DIIOG TP. Hd CHi MI]rrI

nf,il 2001
VONG 1 : MON TOAN AB
(Thdi gian ldmbdi : 150 phtu)

Bai 1. a) GiAi bdt phuong trinh

Jr.l >2x-I
b) Giei hd phuong trinh

I 17
lx+-=lYz
I

VONG 2 : MON TOAN CHUYEN
(Thdi gian ldm bdi : 150 phrtt)
BAi 6. a) Tim sd nguy6n duong a nh6 nhdt sao
cho a chia hdt cho 6 vd 2000a li s6 chinh

phuong.

b) Tim s6 nguyOn duong b nh6 nhdt sao cho
(b-1) kt6ng la bQi ctra 9, b h boi cira bdn s6
nguy€n t6 liOn ti6p vi 2002b ld sd chfnh
phuong.

117
lv*-=-

Bei 7. Cho.r, y lit

t'x3
Bni

2.

Cho a, b, c ld cdc sd thuc phAn bi€t sao

vd y

BAi 3. a) TrOn c6c canh AB vd CD ctra hinh
w6ng ABCD ldn lugt ldy cdc didm M, N sao

+.5

Goi K lh giao didm cira

AN vi DM. C),ttng minh ring truc tam cira tam
gi6c ADKndm trOn canh BC.

b) Ctro hinh w6ng ABCD vdi giao didm hai
duong ch6o ld O. MOr duong thhng d w6ng g6c
vdi mit phing (ABCD) tai O. Ltiy mQt didm S
ftAn d. Chung minh rang @C) L (SBD) vn
(sAC)

I

(SBD).

Bei 4. Cho trl gi6c l6i ABCD c6 AB vu6ng g6c

va CD vd AB = 2, BC = 13, CD = 8, DA = 5.
a) Duong (BA) cit duong (DC) tai E. Hty
tinhAE.

b) Tinh diOn tfch cira tr1 gi6c tlBCD.

Bni 5. Trong mQt gi6i cd Vua c6 8 ki thir tham
gia, thi ddu vdng trbn m6t luot, thing dugc 1
didm, hda duoc U2 didm, rhua duo.c 0 didm.
Bidt rang sau khi tdt ce cdc tr6n ddu kdt thric thi
ci 8 ki thtr nhan duo. c cdc sd didm kh6c nhau vh
ki tht xdp thrl hai c6 sd didm bing tdng didm
cira 4 ki thir xdp xudi ctng. H6i v6n ddu gita ki
thir xdp thrl tu vh ki thir xdp thrl nam dd kdt thric
vdi kdt qui nhu thdnio?

+

:

x

ddu ld cric sd nguy€n.

a) Chung minh*y2

trinhx2 + x + a= 0 vi * + rx + b = }cfrng c6
nghiOm chung. Hdy tim tdng a + b + c.

AM = CN =

*!

*

v

cho cdc phuong trinh x2 + ax + 1 = 0 vi xz + bx
* c = 0 c6 nghiOm chung, d6ng thdi c6c phuong

cho

cdc s6 thgc sao cho

.+
x" y'

ld s6 nguyen

b) Tim tdt ch cdc sd nguy€n duong n sao cho

x nnl'.
"y

I^

la so nguyen.

*nyn

Bni 8. a) Cho a, b ld, cdc sd duong th6a
ab = l. Tim gi6 tri nh6 nhdt cia bidu rhrlc
tr=(a+b+I)1a2+b21*

4

a+b
Id cdc sd nguyOn th6a
_

b) Cho ffi, n
i * 11^ T'irl trilOn nhdt cfia B
;Ai6
=
= m.n.
lmn5- -.

Bei 9. Arc 2 duong trdn C1 (Or, Rr) vd. C2
R) tidp xric ngoii v6i nhau tai didm A. Hai
didm B, C l{n lugt di d6ng tr€n C1, C2 sao cho
(Oz,

g6c BAC = 90o.

a) Ctuing minh trung didm cira M
thu6c mOt duong trdn c6 dinh.

ciaBC

luOn

\) Hq AH vuOng g6c vdi BC, rim tap hqp c6c
didm H. Ctrune minh rang dQ dai doai AH

kh6ne lon

"

hon

2\R2
Rr+Rz

c) Ph{.t bidu vd chrmg minh c6c kdt qui tuong
tu nhu cdu a) vi c6u b) trong trudng hgp C1 vh
C2ridp xfc trong vdi nhau tai didm A.
Bni 10. Giei hc phucmg trinh :
tI

Jr+ t * Jr*a + Jr+5 =S_t

[*+y+

*2 +y2 =gg

+

Jyj*,[-s


ffi

udl xtluc crin nd m unm B[o udu
r?/

VAO GIAI TOAN
NGUYEN QUANG MINH - o6 nA cnu
fdt f ttyf Ddng Hung Hd,Ilang Hd,Thdi Binh)

LTS : Tr€n tap chi TIITT sd 250 (411998) dd
ddns bdi "Mat'ihaong phdp gidi milt sd phuong
trinil bac b6'ni' crta fic gid Trdn Xudn Bang

(Oudns' Binh). Gdn ddv chting tdi nhdn duoc bdi
iet ndi dung phong phil hon. Xin gidi thi4u
cing bqn doc.
bf,c b6n 1I
Tim trgc dtii xtmg cira dd lt t t

,ii

Nhdn xit. a) Tir (**) ia thay do thi hhm bdc
b6n y = flx) nhAn duong thing x = cr lim ffqc
ddi xung khi vh chi khi :

,

b) C6 thd chrlne minh MDl nhd Bd dd : Didu
kicir cdn vi dir ddhnm y = flx) (1) x6c dinh tren
R c6 trgc ddi xtmg x = o thflx) = fl2a - x)
c) Ndu c6 didu kicn ({'*x) thi hhm sd bac bdn
dua vd duo. c hdm sd'trilng phuong bang c6ch dit

f*

h6i itrudng g4f trong c6c bii to6n vd hlm
b0c bdn. NeoIi r'a st dlng tfnh chdt nly cbn
giei duoc n-hidu Uai to6n Ii thf kh6c vd him

cAu

bAc bdn.

dn phu t =

I. X6t him bAc bdn
y=ax4 +bx3 + ci + dx+ e@*a) (1)
Gqi/(x) Ii vd phii cira (1). D0 thdy rang : dd
thi (C) c[ra hdm sd bAc bdn y = /(x) nhAn truc
tung lbm truc ddi xung khi vh chi khiflx)-ld mOt
hdri chin (b = d = 0). Ta tim didu ki6n d6 hbm
sd bac bdn tr6n nhAn dudng thing r = o lirm
truc ddi xfng.
M€nh di l: Dd thi (C) cfia hdm sd bac bdn
y = fl*) nhAn dudng thing .r = cr lhm truc ddi
I f'(cr) = 0
(*)
xung khi vd chi khi l'^ :--'.
.____o

lx=X+a
i
IY=r

Ldi giii. D4tflx) = *4 x6t he

.

Y = a'(X+ a)a + b(X+ cr)3 + c(X+ cr,)2 +
+ d(X+ cr) + e
+ (6aa2 + 3bu + c)Xz +
= oI * (4ao. +
+ (4aa3'+ 3bu2 + Zca + AX+ @d + bu3

Df

+ro2+da+e)
TiI d6 suy ra dd thi hdm Y = 8(X) nhfln ducrng
thing x = o ldm truc ddi xrlrg khi vi chi khi
+3bu2 +Zca + d =0

4aa+b=0

[
=0 (dp.*)
o {/'t"l
"'(o) 0
[.f

4

-

Za

gx3

+

(2)

t6* - l-

o {or'-24x2 +3zx=o
[.f"'(r)=o lz+x-48=o

{,f'{*)=o

Him y =flx) tro thbnh

[+oo'

bx

*4-B*3+16*-1=o

Chim7 minh: Tinh tidn hQ trgc Ory sang h€
truc TW theo vecto d , vdi I (cr, 0). Cong

lh

't
x'+

Kdt qui lz Tt m€nh dd I ta suy ra cdch gidi

mbt dans phaons trinh (PT) bdc bdn f(x) = 0
th'6a mdi ai\, u4n (xx) nhil sau :Tim nghiam
cila h€ (*), n€u h€ ndy c6 nghi€m x= d, ddt x =
X + a t:hi vi€c gitii PT bdc b6'n f(x) = 0 chuydn
vd vi€c gidi PT tring phaong
aXa + (6aa2+ 3bo + i)X2 + f(cr) = Q
Tn d6 tinh dugc nghi€m x = X + a cua
PT f(x) = Q
Vi du L. Giii phucrng trinh

l,f,,,(cr) = 0

thrlc d6i truc

({<* {<)

8a2d-4abc+b3=O

(*x)

R6 ring x -- 2ld nghiOm cira h0, nghla lI dd
thi hirm # y =/(x) c6 truc doi xtlng x = 2. D4t

x=X+2,W

(2) tr&thdnh

(X+ 2)4 - g(x+ 2)3 + l6(X+ 2)2 - | = o
eXa -8X2 + 15 =0
=-rE ,b.=-Jl,

=&

&= Jj ,&= Js
Vay IrI (2) c6 4 ngEQm Ld x1 = 2 - 16'
x2=2-.6, x3=2*.6, x+=2+ Ji
2..Giii PT nghiOm thuc
Vi du'(x+a)a+(x+b)o=,
Dd

eiii

n€u c6ch

blLi

giii

(3)

to6n nly nhidu s6ch tham khlo dd
quYdt nhu sau

:


Ddt x =

-- ((.2)
+!),

2t +3@-D2*

rhay vho (3) duoc

pr

Ldi giii. Gii sir IrI (6) c6 4 nghiem ph6n bi6t
l6p thdnh mdt cdp s(i c6ng, thi dd thi him s6y =
flx) c6 truc d6i xr1ng. X6t he PI

* = -(o!^ul, tt

[2

ddn viec giAi PT (4).

)

Ao a6n

Ta x6t th6m m6t tinh chdt dac bier cira dd
thi hhm bAc bdn c6 truc ddi xrmg

ili

2. Ndu PT
ax4 + bx3 +
+ dx+

r*
e=0(a*O) (5)
c6 bdn nghiOm phAn biOt l0p thdnh mOr cdp sd
cOng thi dd thi hdm sd y = f(x) c6 truc ddi xrmg
trong d5l("r) lh vd tr6i cira (5)
Chung minh: Gii stl W f(x) = 0 c6 4 nghiOm
ro, ro + T , rn + 2y, xo + 3y (y + 0) lap thlnh m6t
cdp sd c6ng. Khi d6

f(x) = a(x-x )(x-x o-y)(x-x o-2y)(x-xo-3y)

rlx**".+)=
/l

?v
tadu-o.c

ld hbm s6 chin ddi vdi X nOn c6 trqc dOi xung
ld X= 0. Suy ra dd thi hdm sd y = f(x) c6 rruc

*

f

fan"-1.

Kdt qui 2. Ndu Pl b,ic boh c6 4 nghi€m
phdn bi€t ldp thdnh mdt cdp sd cing thi d6 thi
hdm s6'bdc bdn c6 truc dA'i xtng. NAlu fim duoc
truc d6i xftng x = a cila d6 thi hdm sd

y = f(x) thi bdng cdch ddt x = X + d ta s€ dua
PT bdc b6'n vd PT trilng phuong o1f + t1* +
ct = 0. Hdn nfra, PT ndy c6 4 nghi€m&,,4 + y,
\ + 2y, \ + 3y (y * 0) ldp thdnh m1t cdp sd
c6ng o PT f(x) = 0 c6 4 nghi€m phdn bi€t x1 =
xo+d, x2 = Xo+a+y, x3 = Xo+a+2y, x4 = Xo + d
+ 3y cfing ldp thdnh mdt cdp sd c1ng.Til d6 ta
cd thd gidi duoc bdi todn tim di€'u kiQn dd PT
bdc bdn c6 4 nghi€m phdn bi€t ldp thdnh mdt
cdp s6'cbng.

L,f

"'(*) = o

tz{3x2 +z(m+ l)x
c>{l+*' -

+

tuJi =o

-z4Ji =o

lz+*

<)Suy ra dd thi

litx=

hlm

s6 y

Ji

=f(x) c6 truc ddi xirng

Ydi m = 1, PT (6) c6 dang xa - qJ-Zx3 + Zi
+ tzJ-2x - 7 = 0. Dat x = x* Ji, pT rrcn 116
thdnh

1 -lof +9=0e& =-3,X2=*1,

& = l, & = 3. B6n nghiOm nly

ldp thinh m6t
cdp sd c6ng (c6ng sai v = 2). Do d6 4 nghi€m
cria PT (6) cfing lAp th]nh mOt cdp sd cOng

xt = 4+Ji, xz = -t+ Jl, x3 = I + J2,
xq = 3 + 0. Vay m = I ld gi6 tri cdn tim.
Yi dU 4. Yot gi6 tri nho cira rz thi duong
thdng Y =2x* 3 cit dd thi hhm so
y= x4 + 4x3 + (I
-ZGm+2)x+ *2 tai 4 didm-phAn biet sao cho c6c khoing c6ch
gita hai di6m gdn nhAt ddu bang nhau
giii. YOu cdu bii to6n tucrng duong vdi

*

-3di

.l* -(+)'][" (;)']

d6i xrlrrg ld ducrng thhng x= ro

lf 'e)=o

lm=l

II.

J,)

- 8= o (6)

\4./

{/'{rl=o o[or*+a)3 +4(x+b)3 =s
L"f"'(r)=0 lZ+1* + a)+24(x+b)=0

Ddtx=X+xo+

+ *3

c6 4 nghi6.m phan biQt lap thlnh m6t cdp sdc6ng.

Vi sao chon duoc dn phu nhu thd'? chring ta

c6 thd giii thich nhu sau
'
Datflx) - (x + a)a + (x + b)4 - c. X6t h0

MQnh

t{u

L"4

-rlryJ -"=0 (4)

H6 ndy c5 nghiom

Vi du 3. Tim m ddYt
*o - a,f* + (m + Dt +

Lli

vi6c tim m dd,Yf
x4 + 4x3 + (1 - 3m)x2

- 6(m+ 1) x +

+*2-m-3=0
(7)
c6 4 nghiOm ph6n bi0t lQp thnnh mOt cdp s0
cOng. Goi ve trdi cfra (7) ld/(x).
x6rhc'

f/'trl=o
.- I*=-t
"'(r) o

Dat x =

X-

|.,f

=

1, PT

[moiz

eR

(7) trd thdnh

t-G*+il*+(m+1)2=0
Dtrt t = * Q r-0) PT (8) tr6 thinh
t2 -13*+5)t+ (m+ I)2 =g

(8)
(9)

PT (8) c6 4 nghiem phan biQt lap thhnh m6t

3{

(theo yOu cdu cfta
cdp sd cOng -3{, -4, Xn,
dd bai) khi vI chi khi PT (9) c6 2 nghiem duong
phAn biQt
duong

tr 1 tz vd, t, = 911. Di6u ndy tuong

v6i

Q{em tidp

trang

13)


BAN vC su nEp

xub cuA HAI od

THI

NGI-IYEN VTET HAI
Gdn day, Toa soan THTT nh{n duo. c rdt nhidu
thu ctra d6c giA, trong d6 c6 cdc giing viOn DH,
cdn b6 chi dao m6n To6n 6 c6c S& GD-DT, c6c
gi6o viOn todn TItrPT vi c6c ban hoc sinh, sinh
vi0n... vdi n6i dung li6n quan ddn bli to6n vd su

tidp xric cira hai dd thi ffen mat ph&ng mQt
vdn dd quan trgng thudng g{p khi khio s6t hdm
sd vd trong c6c dti thi tdt nghiOp THPT, thi
tuydn sinh vao DH, Cao ding.
Bli niy nhim phin dnh c6c thac mic, cdc giii
ph6p vh kidn dd xudt ctra ban doc, ddng thdi
ngudi vidt cflng xin trao ddi thOm vii di6u.

-

f

- f(x) c6 do thi (C) vd him sd
y = S(x) c5 dd thl (O tron mat phing toa dO
Oxy. Dd, tim su tidp xric gita hai dd thi (O vn
(E) ta thuong sir dung hai c6ch giii :
Cho

him

sd y

Cdch gidi 1. Giai h0 phuong trinh

(t)
tf G)= sG)
(2)
U't*l = 8'(x)
duoc nghiOm x.li hoinh dO ctra ti€p didm.
Cdch gidi 2. Tim didu kiQn d€,fl,x) - s(x) horc

trt thrlc cira n6 c6 nghiQm xo b6i k > 2 (nghiQm

bQi 2 gqi th nghi0m k6p), nghia li c6 dang
1x - xo)t QQ) vdi QG)ldda thrlc.
C6 ban h6i : Tai sao trong ciic s6ch b6o c6 cdc
dinh nghia kh6c nhau vd sq tidp xfc ? Dinh
nghia nho chinh xdc ?
Dinh nghla ti€p xric theo quan didm hinh hgc
coi tiOp tuydn tai Mold vi trf gi6i h4n cira c6t
tuy€n MM, crta dudng cong tJ;ri M ddn t1i Mo
(tX hai phia).

Dinh nghia dua vio vi tri tuong d6i ctra c6c
hinh Id didu b6t bu6c ndu mudn gif dugc tinh
chdt hinh hoc c:ira kh6i niOm tidp tuydn, ddng
thdi do mO ti truc quan n6n hoc sinh d6 nh6n
thr1c. Dinh nghla dring cho cA tidp tuydn cfra
duitrg cong c6 phuong trinh (PT) dang tdng
qtdt F{x, }) = 0, mang tinh chdt dinh ttnh, duoc
sit dung trong khi chrlng minh, dtmg hinh.
Kh6ng it hgc sinh dd ngQ nhAn ring dinh
nghla ndy ld co s0 cira phuong ph6p nghiQm k6p
(giao didm M ddntrirng ddn tidp didm M.).
6

(i{c

ban L0

Son Tay,

Hfu

TAm (GV TI{PT

Ting Thi0n,

Hi Tay), Hd Van Tidp (GV THPT

Dfc Trong, LAm Ddng), L0 Anh Tudn (GV
THPT chuyOn VInh Phric), Doln Nhu Tri0u
(GV THPT Hdng Quang, HAi Duong), Pham
Qudc Phong (GV THPT H6ng Linh, Hh Tinh),
Nguy6n Dohi (GV TIIPT Thdng Nhdt A, Ddng
Nai) vd nhi6u ban khr{c ddu cho rang : LAu nay
trong cdc tii li0u 6n thi tdt nghiOp TFIPT vd
tuydn sinh DH dA st dung phuong ph6p nghiOm
k6p mb chua c6 co s0 li lu&n chinh x6c, do d6
c6 hic lam dung phuong ph6p niy ddn ddn sai
ldm trong lAp luAn, c6 hic l6n l6n kh6i niOm tiSp
xric vdi didu kien c6 nghiOm k6p...
Trong cdc bdi b6o cira tdc grit Nguy€n Anh
Dflng (THTT s6 297 thdng312002), Ding Hing

Th6ng (THTI s6 298 th6ng 4120A2) da ffinh
bdy co s0 lf luln ctra phuong phdp nghiQm bQi
dd giii.bni to6n vd su.tidp xric ctra 2 dd thi,
trong d6luu f rang :
o Chi x6t kh6i ni6m.ro ld nghiOm bQi f > 2 dfii
v6i da th*c F(x) = (x - x")kQG).

oD6thlhdmsd

y= 4P v),y= !('),
v(x)'
QG)

trong d6 P(x), Q@), U(x), V(x\ lh c6c da thfc,

tidp xric nhau tai didm c6 holnh dQ x =

xo

e P(.r)V(.r) Q@)U(x) = 0 c6 nghiOm boi x - xo vdi
(thuOc mp xidc dinh cira chring)

*

V("r) + 0.
Ci{c ban : My Duy Thq (GV He NOi), Hd Van
Tidp, L0 Anh Tudn cfing chirng minh c6c mOnh
Q(x")

0,

dd tuong tu.

Dlnh nghia ti€p xrtc theo quan didm dai sd,
gitii tich c6 didu kiOn nhu cdch gi6i 1, trong d6
PT (1) c6 nghiQm x= xo nghia lh (O vn (D c6

didm chung Mo(xo, yn), cbn W (2) c6 nghi0m
x = xo bidu thi tidp tuydn cira (C) vi (E) tai
Mn(xo,yo) tring nhau.
Dfnh nghla nly mang tfnh chdt dinh laong,
cho ph6p gihi cdc bdi to6n tinh todn toa d0 tidp
didm, tim phuorg trihh tidp tuydn...
Ndu xudt ph6t tt dinh nghTa theo quan didm
hinh hqc thi cdch giii 1 coi 1I m0nh dd.


Dinh nghia theo quan didm giii tich chi x6t h€
sd g6c k = tgo" xiic dinh, do d6 b6 qua trudng
hqp tidp tuydn ruOng g6c vdi truc hoinh, ching
han hhm sdy

=

1,1;-

c6 tidp tuydn x =

"r.

ta c6
duong thing
d ln tidp
tuydn ctia dd
thi (C) gdm

(cr),

(Cz\

AB,
hay
khOng, c6
coi hai dd

thi (E) (gdm (81), AB, (E)) vi (C) li tidp xric
v6i nhau hay khOng, m5c di he IrL (1), (2) d
c6ch

giii

nghi€m trong doan lxA,xBl?
Do v6y 6 c6ch gi&i I cdn chri f th0m : ffong
l6n can cria ti6p didm thi 2 dd thi (O va (E)
phii c6 it nhdt 1 duong cong, nghla li kh6ng thd
1 c6

li d6 thi him sd bac nhdt.
giii 1 dtng cho c6c him sd bat ki (trir

2 duong ddu

Cich

nio d6:cua PI (1) thi

srl dung phucrng

ph6p nghiOm boi rdr thuen loi.

C6 -b4n

h6i

: Phdp bi6n ddi tuong duong

(*-t)2 = 0 <> x -

l.

tsan Nguy6n Minh Anh, 12A1, TFIPT Tinh
Gia I, Thanh H6a dua ra giii phr{p : X6t tidp
tuydn dang ! = b (b h hing sO) cira hlm sd
nguo.c*= i6J <; y = x3 + 1 rdi ldy dd thi
ddi xrlng qua dudng y =
Xin h6i:
O hinh oo,
coi

nghi6m

toln tinh chdt
? Xin tri ldi : Hai

L = 0 c6 bAo

nghiOm b6i c[ra phuong trinh

li tuong duong, nhung ph6p
bidn ddi kh6ng li tuong duong vl ptrifnhan,
chia vdi h(x) =; - I mh x = I thuQc tap x6c
dfnh vd ft(l) = g.
Rdt nhi6u ban : H6 COng Dfrng (GV TIIPT
Trdn Hmg Dao, Binh Thu6n), Nguy0n Danh
Cdm (Ing H6a, Hi TAy), TS Nguy0n Cam
(khoa To6n, DHSP Tp. Hd Chi Minh), Lc Hfru
TAm, Hd Van Tidp, L€ Anh Tudn, Pham Qudc
Phgng, Nguy6n Anh D[ng, PGS Dang HDng
Thing... dE nOu ra co s&-li lunn cira ptruong
ph6p nghiOm b6i, ph6rt tich uu thd cira c6ch giAi
niy trong nhidu bii toi{n vi di dd nghi cho sr}
dung phuong ph6p nghiOm boi khi giAi to6n vd
su tidp xric gifra 2 d6 thi him sd vi nhidu dinh lf
d THPT cfing chi cOng nhAn, kh0ng chrlng minh
; vdn dd ld cdn chi 16 didu kiOn str dung phuong
ph6p ndy sao cho chinh x6c, m6r kh6c rrinh bly
phuong-trinh tr€n

ng6n gon dd khong qu6 nang nd.
C[ng xin dd nghf : do th6i quen cdc gir{o viOn
TI{PT v6n day cho hoc sinh phuong phdp

nghiCm b6i nOn c5c bli ldm dring cira hoc sinh
theo phuong ph6p ndy n€n c6 didm s6 th6a d6ng
trong c6c ki thi tdt nghiOp TI{PT v} rhi tuydn
sinh DH sip tdi.

trudng hqp qa hai ddu ld hdm sd bAc nhdt tai lAn

cfn lidp didm), cdn c6ch 2 chi 6p dung

cho

trudng hqp,(x) vh g(x) ddu ld phAn rhrlc hfru ti
(n6i rieng lh da thfc)
Dirng c6ch giii l.ta tim ngay duo.c hoinh dO
tidp di6m, sau d6 mdi tfnh h€ s6 g6c ridp ruydn,
cdn ci{ch

giii

2 cho phdp nhanh ch6ng tim h0 s6

gdc d6 suy ra PT tidp tuydn, tim didu ki€n ctra
tham s0 dd c5 tidp tuydn.

Khi phii tim tidp tuydn dang g@) = tnx + n
cira dd thi hlm y = f(x) dang a* + bx + c,

ax+b ax2 +bx+, thi
/(x) - s@) ho4c tt
cx+a
cx+cl
thrlc cira n6 ddu li hlm sd b6c 2 nOn srl
dung
duo. c ci 2 cdch gi6i. Nhidu b4n dgc h6i ring sf,
dung phuorg phdp nghiOm k6p trong trudng hqp

nhy c6 vi pham yOu cdu cira 86 kh6ng khi hg
thrlc didu ki€n vin li hhm sd bec 2 ?

Khi ph6i tim tidp tuydn cfra dd thi hem
f(x) mi/(x) c6 dang hhm sd bAc 3, bflc 4 thi
d6n ddn giii phuong trinh bflc 3, bAc 4 hic d6 st

y

=

dung c6ch giAi

I

rdt kh6 khan. Ndu bidt mQt

DON DgC THTT SO 301 (7-2A02)
* Ap dung dinh li Viet dAo dd giii hQ
phuorg trinh

.i' Mqr

sd hc thrlc lien h€ gita duong thing
vh dudng trbn

*

Khdi ni6m hdm sd duoc hinh thlnh nhu
the ndo

?

*

Tim hidu vd mr{y tinh song song
.lt MOt bhi todn clra Anh-xtanh.

Mdi cric ban tiep tuc tham du Cu0c thi
Vui HE 2002 ctla THTT vdi dd thi dot 2 o
sd bio 301.
C6c ban se biet ldi giAi diip cric bli : Xep
cdc lon nudc quA, Thidt ke ao h'inh vuong,
COng trinh cira ai ? ...
C6c ban

nhddlt mua THTT quf Itr nh6 !
THTT


A

A

.4.

LIEN rrE
firUC
TI
t

srfra DUo!{GrsAxc vh DUONIGTRON
SO HEg
MOT
t

a

a

LE HAO

(N
Trong hinh hoc phing khi x6t cdcvt tri tuong
ddi giili duong thing va auong trdn ta tim thdy
ntrid-u t

nhidu ung dung trong khi giii to6n. MQt s6 hQ
thrlc nhulhd da duo. c trinh bdy trong bli "Mot
s6 dang kh6c cira bii tohn con bu6m" ctra ban
Phan Nam Hing tron THTT s6 216 (611995).
Trons bhi ndv xin neu ta c6rch chfng minh kh6c
vOi riQt crich nhin tting qu6t hcrn vd c6c bhi toSn
con budm.

I. HQ thrlc gita 4 tlidm trdn tludng thing
1) Ildng didm didu hda
B6n didm A, B, C, D :u:an duong thing c6
hu6ng 6 goi

CA
CB

li

hdng didm didu hda (ABCD)

DA

$u

(1)

DB

3) HA thirc

trudng CDSP PhilY€n)

Niu-ton

M c:&;a
cd = 0

Ndu chon g6c O trirng v6i trung didm

AB thi a =

o

cd = o'

o

-b

ndn (2)

tutc .MD =

o -a' +

MAz

(4)

4. Chilnr didu hda

Bdn duong th&ng ddng quy c6t rnOt duirng
th&ng tai + didm liOn hqp didu hda duo. c goi th
chtm @uns thdnil didu hda
Tinh chdt: Chim didu hda ddng quy 0 didm S
c6t dudng thing y nio d6 tai cdc didm A, B,
C, D th\ A,-8, C,-D lAp thhnh hing didm didu

vI

hba.

Chfing minh. Theo dinh nghia c6c dudng
th&ng SA, SB, SC, SD cit i duong thing 6 t4i A',
B', C', D'l4p thinh hing didm didu hda. Qua B

vi,B'k6 c6c dudng thing song song v6i SA duoc
c6c giao didm nhu 6 hinh 2.

Khi d6 ta goi
doan AB

,u,u:;l:

(h. 1). Ta thdy

c,

D chiadidu hda

(l) <+

li

AC E'
fr=- BD

lric d6 cric didm A, B cfing chia didu
h6a doan CD. Chn f rang ndu ba didm A, B, C
xdc dinh vh C kh6ng li trung didm cira AB thi
tdn tai duy nhl't didm D th6a mdn (1).
Chon didm g6c O 89d6 trOn dudng thl|1rg A,
ddt c6c sd do dai s6 OA = a, OB = b, OC = c,
nghia

a-d

b-d

OD =d thi (1) dugc vidt thnnh 9_9 =
b-c

e

(2)

(a+b)(c+A =2@b + cA

2) He thitc Da'-cric
Ndu chon gdc O trirng v6i didm A thi a = 0

(2)e bk+A=Zcdel-+ =?, o

cdb

,A

112
AC AD

-J--:-

(3)
AB

vi

Hinh 2
Theo (1) ve DL Ta-let c6

CY DY WY N'Y
C'A' D'A' SA' SA'

"1:-:=:--

W y * ru a' --jl nghia ln B'1I trung didm
ctraMN'.SuYruMB+NB=O

=

MB NB CB DB
:'=:-:=:-:
SA SA CA DA
Til d6 A, B, C,D

li

hdng didm didu hda'
(Ky sau ddng ti€P)


l^

vE ofl

Hfi roan Hoc rni GKr 2ooz

Tir cudi th! ki 19, Dai h6i torin hoc thd gidi _
Intqrnational Congress of MathematiZians
(.IC'M) ld su ki6n l6n nhr{t cira gidi todrr hoc.
Cri
4 ndm dai h6i lai di0n ra m6r-ldn. fai Oai tr-Oi
ciic kdt qui ngiien ;,i,
nlpg, nhd toiin hoc.s6 dem"e;oilil";i#;i;
ddn th6ng tin chinh
y9 nyng di vi.thinh tuu toi{n hoc ctri thdi gian
{6. Mqi ngudi ddu quan tAm dbn viec ai d"uoc
doc bi{o ci{o mdi toin thd, ai duoc mdi'b6o ci{o'6

ti6u ban. Dac bi€t, viec trao giii thuone Fields
hoic Nevanlinna lh su ki€n chfnh cua ir6i Aai
!Qi. \am nay ICM s6 di€n'ra vlo ciic neav u)ti
ddn 28.8.2002taiBic Kinh lTrung eudcl. Oantr
sdch cdc bdo cr{o mdi ldn
sdm hon moi nim. Ngu-o.c lai,'rhdi'han
"tt;uil;c dd dinh
ti6i
thiQu cric
cq vicn"iho A; ;iei ituoiie ron
lms
neu.tren dugc
.gi6i han li 1.5.21002. Nam nay
ci{c bi{o c6o mdi toin.thd du kien se mang n6'i
dung dai chring vi mdi brio iao aai OO ptuitl ffii
Tggl nl3 roiin hoc du_o. c mdi doc b6o'ci{o ro}n
the la : Noga. Alon, Douglas N. Arnold, Alberto
Luis^ Angtes. Carffareli, Sun_yung

Pf::*Chan_g,
Alrce
David Leigh Donoho, Ludwig DI
Faddeev, Shafi Goldwisser, Uffe Haalerup,
Michael Jerome Hopkins, Victor C. Xai,-ffarry
Kesten, Frances Clare Kirwan, Laurent
Lafforgue, pavrd Mumford, Uirufu' Nutr:i.*,
Siu,
I.:*-Tlg
.Richard Lawrence faYfoi,'bang
Iran, Edward Witten.
Trong sd d6, hon lb
ngudi dang ldm vi6c tai Hoa (y, cbn l4i li Nga,

Italia, Dan l4?"h, Anh, phiip, Nh0t, Trung
eudc
nhi toiln hob duoc rndi d?c-b,ao
c6o b cic tidu ban. Xr{c suat ve ihdn;-k;; I{inh
h.oc vi phan, Phuong trinh vi phan, fat d to6n,
LIng dr,rng-c[ra. ro6n hoc, L1i t-truydt ,d'... le .6.
linh wc c6 nhidu ngu& b6oci{o. "
thuong trinh nghiOn crfu co bAn DaHiTO
$dQt sd-vdn dd chon loc cira Dai sd - lfinh hoc _
Jgqol:!'r-Yie_t Nam s6 x6t vi trfch rn6t phdn
kinh^phi
hg" tro tidn di tai cho cr6c c6i bo
ngtu€n -dd gittg day trong 3 chuyen ngdnh
tt{r
tr6n dU Dai h6i ICNI Z00Z.
Hd so xin tii rro cdn gti vd dia chi sau rrudc

'
ngiry 30.6.2002:
GS.TSKH N96Vi€tTrung
VidnTodn hoc
Box 631, Buu di€n Bd H6, HA N6i
Tel : (04) 7563474, Fax : (01) iSoqSOS
email :
dunghd so xin li6n h0 theo dia chi trcn
.,(N.Q-i
vh Ixraen. 169

dC bidt cu th6)

Cfrng c5 thd coi H6i nghi qudc td vd

qgu_ tPqng vh ung dung 2002

Giii

tich

ICAAA di6n ra tir
13 ddn 17.8.2002 tai Hd Noi ta Hoi nehi v6 rinh
ctra D4i hQi tor{n hoc thd eioitclii{oi. Nnie,
nhd to6n hoc tt hon 20 nudc'd6 ding kf tham du
Hoi nghi niy.

eheoThaffiTodn

hoc)

TIENG ANH eUA cAC gAt TnAN - BN sd st

Problem. Without using long multiplication,

computer or a pocket calculator, verify ihat
24oa + 34oa + 43oa + sgga =
.

a

aif

Solution. The equation suggests that we should
try to puII a facror lOa out ofthe difference 6514 l
599-. So we calculate

6st4

-

599a =

(65f _ sgg\16st2 + 5992\

52..1250[(625 +26)2 + (625
-26)2]

=
= to4.t3(6252 +2621,

Now we mav write
65f - 5gg4 - 4304 -3404 -240a =t04A with
A = t3(25a + 262) - @34 + 34a + 24a\
= l2x25a + 4x133 - (342 +2q4\ - (434 -2s41
= l2x25a + 4x133 - l6x(172 + L24) - l8x68x2(342+92)

- 41t72+1241easy to check that A

= 4[3x254+133

It is

tsx36}*+q\].
is d-ivisible by the

lr,

numbers 64 = 26, 27 = 33, 25 = 52, 7 ,
13, 17
and hence by 64x27x25x7x11x13x17. Note that
64x27 x25x7x1 1x 13x 1 i > 1600x27x l00l x 17 >

> rc6$)zlr17 > to6x4ox17 > 6xto8.
On the other hand, we have
lAl < max{ t3(z\o + 2627,43a

+ 34a + 24a
l), 3x5041 <
< l3x50a = l3x625x\Oa < l3xl07 < 6x108.
. Th-uq, A is divisible by a number exceeding

< maxl L3x26'(26" +

1

itself. Hence A must be zero.
Deduce 2404 +340a

+

43Oa

Trlmdi:
multiplication
computer
pocket calculator

+ 599a =651a

= phdp nhAn
= mily tinh

=mdy tinh bo trii

= gq y (dQng tir)
try
= thu, thfr xem (dOns tu)
pull (out)
= k6o ra, loi ra iOons tL)
calculate
= 1in!, tinh torin ldOig til)

check
= kijm tra, kiOm soitildong ti)
divisible
= chia het cho (tinh rn)
onthe otherhand = m4t kh6c (thinh ne[)
thus
= do d5, theo d6, vi iay
'" lpnO nl;
exceed
= vuot quri (d6ng tt)
suggest

NGO VIET TRUNG


,l

/

/

A-

,.

'l

"

THI GIAI TOAN TREI{ MAY TINH BO TTII

--

(TiAp theo ki truob)

L

BAi 7. Tim phAn ,O

sao.rr. iA

nhien c6 hai chr s6

li

(tric

m vd n lI

vor

n-

- JZI

TA DUY PHUONG
(Vi€nTodn hoc)
nhau", trlc ld chring bSng nhau holc hon k6m
nhau i don vi. Suy ra m vd n phii thoa mdn
phuong trlnh Pen (Pell): *2 -2r2 =tl '

sd tu

,u nh6 nha't

X6t phuong tr\nh mz -2n2 =1. Nhin x6t
rang: ndu (mn;n) ld nghiOm thi

ln ,-l
xi so v0 ti ./2 tdt nhdO'

tim phan sd xdp

m=3mo + 4n()i n=Zmo

Gidi: Ydi n cd dinh t6n t4i s6 mo sao cho
J, nam gifra hai PhAn sd canh nhau:

!3-..{i.^.+'.
nn

nguyen crta

Suy

nJi),

'

vay tir s6 t6r nhdt

I

mot trong hai sci [n.'

rl

nhocho

lY-Jrl

ln

ta ns=[r"'E]

D)'u [,O]+1

I

*dp

le

*i tr,a,A J1 Gnft tran mdY)-Thi du:

114243

(chon sd

m(n)

daY ta c6 c6c PhAn s6:

,'O' zo'?os' 2378' 13860' 80782

rnnan

m(n)

Tt

3 t7 99 577 3363 19601 114243

-f

80782
phuong trinh

^,

vdi

rn

2

tion phAn s6: J2

n

(vi (tinh ffan mdY)

nen vdi n> 71 thi m(n)>100

=10,11,...,69,70

ItJi

o100.4

phan sd xdp

*i

Ji

tdt nhat ,u

X,

sai sd

Ji

ot)

!1 _ Jz
7A

= a.000072t52...

Ldi binh 1: PhAn ,d

duo.c

tOt

bang c6ch chon so thAp ph6n vO han tudn hoin

i,(41) va chuydn n6 vd dang Phdn

1.(4i)=
-'\ '-l

t40.

gg

Phrn

,6 119

xap

sO

sd

99

Ji tW - J, --o,oooo72148 ), tuy nhi€n n6
.o ttr 39rO c6 ba chfi s6. Nhan x6t rang, ndu

))
x = J2 thi x * -, tfc ld ! cttng xdP xi Jl
L

.

xx
140 .. 2 99
thi -=-.
Chon x
Ldi binh 2: Ndu
m'
10

-2n'.

!!*Jz

cut"

JI

Bdi

:

xi khd tot

fr fnav

hav

Hai sdtunhien m2 vd 2n2 "xdPxi

vir srt

).

8.

Hdy tim Phin's6

L,
n

tortg d6

m

vir

ldcdc sd tu nhicn c6 ba chfr s6 sao cho phan
355'
DdP ,a''
nhdt. uql'
a Lrtr4t'
,A L gdn v6i sd to
""'
'
1

n

355
113

-n=0,000000267

a

lt3

tot nhat

))
li 'aI

sd xdp

xi

sd

xi

@" Ac-si-met tim ra)' 56

*do xi sd zr tdt hon.

PhAn s6 xdp

13

.

Ldi,binh L: MOt thdi gian ddi, phin

355

;x2

t

r*

n

sd

th\

. B6ng c6ch
1

= | +t t(2 + | t(2 +t t(2 + t t(2 +t / 2))ll =

'.'c6t

c6 thd nhAn

r''

dung phim W4 tinh t en
mdy dd, duo.c c6c phdn sd ditng kh6c xAp xi

.

?70

*

sd J7 uang

"cit ctrt" liOn phin ,o tr#'la-cfrng duo.c
nhidu phan sd xap xi tdt ..D. Thi du:

lh sd c6 ba chf sd) vd so s6nh chring v6lnhau ta
cluo.'c

0.000000000 . Tuong tu cho

-2n2 = -1 .
Ldi binh 3: C6 thd xaP xi

nhohon).

Tinh ffan mdy cdc phan sd
n

nghi6m.

+3na cfing l]

x t6t hon tidp theo th:


g

(g-n^,-o,ooooooool) vi
33t02 33102

Gidi: Gii sil sd r thoi
bii. Do l=4x121-'7 x 69

104348.104348

,l _ n+,121-tx6s hay u = no::: =f:'lil!
n,^_, {ns9 ,t

:(3321s'332t5
-_J - n = 0,000000000 ).
Ldi binh 2: "Cdt cut" liOn phAn sd
n = 3 + UG +L lG5+Ue+Ue92+1/(1+...)))))
ta cfrng du-o. c nhidu phAn sd x6p xi t6t s6 n
155.
thi du: 3+llQ+ l/(15 + l)) =

"

3

+ I l(7 +

Theo ddu
,

113'

3+ ll(7+1/(15+ 1/(l + Ilzg\)))=

I lQ5 + r l0 + t legz+

1,03993

lI

sd

htu

ti

R=

Y,
n

v?r n

chf sd, sao

cho hinh vuOng nOi tidp hinh trdn bdn kinh
ci{c s6 tU nhi6n c6 khOng qud hai

R=

L
n

c5 diOn tich gdn vdi di€n tich hinh trdn

I nhat.
Bdi 10. Yot'gi6 tri tu

n-l

quy t6c "gAp d6i

nhiOn nho cira

z

thi

I MF

od

tirrr

1,987'

=(3333.)4 !3340= .ro-

(lgg6..)',

.

l,gg6',

''"'- .rn' = 1,009151070.10^ < n.lO*

theo

1,987'

2); E

ltraJ]

3'3'o]

.

1,986'

@(1,01512=163.1< 512);

E |vt+]

.ro

W n li
m

=2 vi

^ =t,ot392g2gg.to^

.

sd c6 ba chf sd n6n chi c6

n.= 101

thd

.

Ddpsd: n =101.

Nhu vay 5I2chta n bing "phuong ph6p chia d6i":
1aA+s!2

1, 0 1768

duo. c hai phdn sd phii vf rrl,i ta vidt bdt ding
thrlc tr0n dudi dang sau (z = M -9):

lu! thta":

Chia doi doan [512;1024)

nen de so siinh

19867

n nnn4

1,01'

l, 0 lro2a =26612.5>1024).

=

l987t

Tinh trdn mdy:

1.018 lM+l (1,012 = t.o2<
(1,014 xl.O4<4):...;

1

Vrtinh ffAn mdy ta du-o.c:
^^^ ,4
rJJr_
=0.000000001=tttu=

.

nnnn4

vd 1,01" > n .

Gidi: Dung

E

-,-,,-_-------_---;_
rovt .n=G333")1 .G3Y)1- .rcM
(1987)', (1986..)', (1986)',

huygffif-

U4.ro.

bdn kinh bang

1,01'-1 <

(33330...0)4 Gfi3...)4 (33340...0)4
(r9870...0)? (1986..)7 (19860...0)7

332t5

d6 m

n6e bitddu bang 1986, cdn

(r"t )o (3333....14
'= ,rur, =(l9,36J ' ruyra:

t)))) -104348

rrong

bii, sd

c6:

\01

2

= 2083.603436 >7 68.

Yey 5L2d6i" nhu trcn ta di ddn: 650 < n < 652. Cudi
ctng ta c6: 1,01651 =650.4506207 <651 vi
1,016s2 = 656.9551269 > 652
Ddpsd: n=651.
.

Bdi 11. Tim tdt ch cdc sd tu nhiOn n c6 ba
chfi s6 sao cho sd n6e bit ddu bang sd 1986,
cbn sd nt2t biitddu b6i sd 3333.

.

nt21 betddu bang 3333 n6n

ue

33102

Bdi 9 (Bhi torin cdu phuong gdn dring hinh
trdn): Tim

m6n di6u kiOn ddu

nOn

K6t lufln: (}4c bhi thi'tr0n (mor sd duoc chon
tt c6c dd thi cira nudc ngobi) rhd hi6n:
1) Tinh s6ng tao vd d6c d6o (tiOu chudn ctra
dd thi hoc sinh gioi To6n).
2) Tu duy thuAt toiin vI phr{t huy hiOu ning
cira m6y tfnh diOn tt (tiOu chudn ctra thi hoc
sinh gi6i Tin hoc).
3) Kgt hqp hflu co v]L h6 tro l6n nhau gifa tu
duy to6n hoc vh cOng cu m6y tinh: thi6u mOt

trong hai ydu t6 d6 sE kh6ng giii duo. c hoac giii
rdt kh6 khan (tiOu chudn c[ra thi hoc sinh gioi
Giii to6n tr0n m6y tfnh).
Hy vgng rIng c6c bli torin th6a mdn ba y6u
cdu tren cia cdc dd thi hoc sinh gi6i sE ngdy

cing c6 mat nhidu hon trong cdc cuOc thi "Giei
to6n tr€n m6y tfnh", thay cho c6c bdi chi 6p
dung tfnh to6n thudn triy.
11


cAc I6P THPT
Bni T61300. Tim tdt cL cdc ciP sd nguyOn
n _(m2 -nz)nl* -1

duong m, n sao cho

r,C BR HI ]IRV

m .(*' -^!r' f l^ * |
HTfi'}TH TAI{ CHAU

(GV THPT L*ongVdn Chdnh, PhilY€n)

Bni T7l300. Giii phuong trinh

cAc t oP

2'+6* =3'+5r

THCS

rnANruAr'raNn

Bhi T1/300. Gii st phuong trinh x2003 + o? +

bx + c = 0 v6i c6c hO sd nguy0n a, b,'c c6 3
nghiOm nguyOn xy, x2,.r3. Chrlng minh rSng :
(a + b + c + 1)(x1 - xz)(xzchia hdt cho 2003.

4)@3-

x)

*o*?ilift)ro*

DHKIIT N - DHQG T p. Hd C hi Minh)
Bni T8/300. Tim gi6 tri l&r nhdt cira bidu thrlc
(SY Todn

+-+-b J"b,
a+bc b+ ca c + ab
trong d6 a, b, c li c6c s6 thqc duong th6a mdn
a+b+c=l
DAC

o

(6/

Bni T2l300.Chring rninh bi't ding thrlc
ar+a,z+ch.+a4+as
> trfqrrorooo, +
5

tJq

-Gt'

+r,{,2 -J%t'z n
-Jif *(Ji -JA)'

trong d6 a1, a2, a3, a4t a5 ld cdc sd kh6ng 6m.
D&ng thrlc xiy ra khi nio ?
VO GIANG GIAI, MANHTU
(Tp. H6 Chi Minh)

Bni T3/300. Tim gi6 tri 16n nhdt, gi6 tri nh6
nhdt ctra bidu thrlc *Jt+y * ilt*r, trong d6
-r, y

li

hai sd thuc khOng 6m th6a mdn di6u kiOn

x*!=l'

vurRiDuc

"

NGUYENXUANTTUNC
THPT l,am Son,Thanh H6a)

Bni T5/300. Ctro ducrng trbn tdm O

duong

kinh AB. Gqi M th didm deii xtmg cioa O qua 4.
MOt c6t tuydn qua M (khong di qua O)^ c6t
ducrng trbn i4i C ie O.Tim quf t(ch giao didm P
cira c7c duong thhng AC vd, BD khi c5t tuydn
chuydn d6ng nhtmg lu6n di qrta M.
LUONG ANHVAN

(WTp.Hd Chi Minh)

t2

Bni T9/300. Chung minh rang"tam gi6c ABC
li ddu khi vd chi khi 108Rr =7p" +27r',$ong
d6 p, R, r tuong tmg lh nrla chu vi, bdn kinh
dudng trbn ngoai tidp vd.n6i tidp cfia +ABCHOANG THI TUYET
(Hodng H6a,Thanh H6a)

Bei T10/300. Cho hinh tf dicn SABC c6 cdc
canh SA, SB, SC wOng g6c ''rdi nhau timg d6i

niot. coi (P), (Q), (7) ldn luE lI mrt phang
ph'an giic cira g6c nhi diQn canh BC, CA' AB.
ipl "a7SA tai Ar, (Q) c6t SB tai 81, Q) catSC tai
C1. Goi A2,82, C2ldn lug ln-hinh chidu vu6ng
gdc cira er, nr, C, tcn mat ph8ng (ABC). (h cat
SA, tai M, (Q) cat SB, tai N, (\ cat SC2 tai R.
Chiing minh rang hai mat phing (MNR) vd

(ABC) song song v6i nhau.

(Ninh Binh)

Bni T4l300. Cho tam gi6c ABC vi m6t didm
P nam bOn trong n6. Dudng trbn n6i ti€p MBC
tidp xric vor cdc cqnh BC, CA, AB ldn luot tai D,
E, F. Dudng trbn nOi ti6p MBC tidp xric vdicdc
canh BC, CP, PB ldn lugt tai K^, M, N, Gqi Q li
giao didm crta cdc duqqg thhng EM" vi FN.
Ln,ing minh rlng ba didm A, P, Q th&ng hing
khi vd chi khi K tring vdi D.

(6/

TRLONG NGSC
THPT U Qui'DAn, Binh Dinh)

(

sv r o,i n"1Yat#*iffi i N guv € n )

cAc oC vAr r,i
Bni L1l300. MQt tr6i d4n ph6o dang bay v6i
vAn tdc v = 500m/s thi nd thinh ba minh c6
cing khdi luong sao cho d6ng ndng cira hc ting
gdp n = 1,5 ldn. Hdy tim vAn t6c cuc dai v,nu*
ilri mOt trong c6c minh c-6- th€ c6 duoc.
NGUYEN QUANG i{AU
(Hd Ndi) st

Bei L2l300. MQt qui b6ng bin roi xudng m6t
mdng nghiOng 4i", itturfi tnTre d'dn chan mang U 15m vh van tdc
cfii U6ng hic roi xudng m6ng ld 0,5m/s. Hdy tinh
sd ldn va cham tidp theo khe di lctn nhdt cfa qua
b6ng vdi m5irg. Coli va cham li tuyQt ddi dhq hdi'
Ua q:r, srlc cii ctra khongihi. Ltiy'g=I}mlsz '
NGUYENTHANHNHAN
Y€n I'ac,Vlnh Philc)

(C/ THPT


ffiWffitrffiTtrtr
FOR LOWER SECONDAR.Y SCHOOLS

;2oo?+a*+bx*.c=o
with integer-coefficients a, b, c, has three
integer roots x1, x2, x3. Prove that
1)("r1

locus of the point of intersection of the lines AC

and BD when the secant varies but

T1/300. Suppose that the equation

(a+b +c+

5

FOR UPPER SECONDARY SCHOOLS
T61300. Find all couples of positive integers
m, n satisfying the condition

-x)(xz-4)@3-x)
>

-n2)nl'-r
m (*2 -nz)n/^ +l
n._(mz

?tarararaoat

*

(l", -J",,' *(J", -JA)' *(JA -JA)' *
T7l300. Solve the equation

-Jq'

20

whefe a1, a2, a3, a4, a5. &le nOn
numbers. When does equality occur

passes

through M.

is divisible by 2003.
T21300. Prove the inequality

at+a2+a3+q|+a5

ffi

2*

a

b +-'l-rb"
a+bc b+ca c+ab

negatiVe

?

T31300. Find the greatest value, the least
value of the expression

,.v[+y + yJi+x
y are non negative numbers satistying
thecondition-r*y= l.
T4l300. Let P be a point inside the triangle
ABC'. The incircle of LABC touches the sides
where x,

BC, CA, ABrespectively at D, E, F. The incircle
of MBC touches the sides BC, CP, PB
respectively at K, M, N. Let Q be the point of
intersection of the lines EM and FN. Prove that
the points A, P, Q are collinear when and only
when K coincides with D.
T5/300. Let AB be a diameter of a circle with
center O and M be"the point symmetric to O with
respect to A. A sbcant passing through M (but not
through O) cuts the circle at C and D. Find the

+6' =3' +5'

T8/300, Find the greatest value of the expression

where a, b, c are positive numbers satisfying the

conditiona+b+c=l
T9l300. Prove that a tringle ABC is equalateral
when and only when 108Rr = 7pz +27r2,wherep,

& r

are respectively the semi-perimeter, the

circumradius, the inradiuS of A.A-BC.
T10/300. I-et SABC be a tetrahedron such that
the lines SA, SB, .SC are at right angles each to

others. Let (P), (0), (R) be respectively the

angled-bisector-planes of the dihedra with
sides BC, CA, AB. (P) cuts SA at Ar, (Q) cuts SB
at By (R) cuts SC at Cy Let A2, 82, C2 be
respectively the orthogonal projections of 41,
By Cr on the plane (ABC). (P) cuts SA2 at M,
(O) cuts SB2 at N, (7) cuts SC2 at R. Prove that
the planes (MNR) and (ABC) are parallel each
to other.

tltfC DUNG TiNH DOI XtfNC.... (ri€p theo trang 5)
a) 9xa - 36x3 + l7* + 38x - 24 = 0
It, ={l*+rz -4(m+t)2 >o lm=5
b)(.r-3)a+(x-5)a=96
jS=Ir +t2=lQlr=3m+5>0 <+ I
ZS
I
lm=-V6i gii{ tri nio crta m thi d6 thi hlm s6
^
^
L 19 y =Bei 2.+ 4mxT
lP=t1t2=gtf :(m+l)t>0
- i + 3g - 4m)x + 4 cirparabol
t
+
xt
+
3x
4
tai 4 di€m A, B, C, D sao cho AB
=

!
Thfr lai ra thAy m = 5 vd. m = -? ,Uhai gi6
=BC=CD
1.9

tri cfia m cdn tim.
Moi cr{c ban hdy sir dung c6c kdt qui 1 v} 2
0,0 gi4i ci{c b}i tAp sau nham muc d(ch cirng cO
'ptrdn ti thuyOt :
Bei 1. Gi6i c6c PT

Bni 3. Tim m dd dd ihi hhm sd
! = x4 + 2(m - l)*3 - (m + l)x2 + 2mx + 2m - 5
cit dd thi hdm sd y = 2mx3 - ** + 2m - 5 tai 4
didm phan bi€t c6 hohnh dO tuong rlng l6p thdnh
mOt cdp sd c6ng.
13


trong dd bidu thrtc clu?a cdn cd n dilu cdn d tri
s6'vd

Bii T1/296. Cho s6' rtguy€n a lon hon 32. Hdi
tdn tai lruy khdng s6'w nhi€n k thda *dn a10 <
k < aat mr) cd it nhai 6l chrt s6'0 d tdn cilng ?
Ldi giii. Ta thay c6 m = u4r - a40 - 1 sd tu
nhien f nam gifla uoo ua aal. Vi sd tu nhi6n a >
32 nan a> 33, tir d6 m+l > a40(u - 1) > 3340.32
3340.30

= (z*)'n.ro = 11089;20.30

>

= 3.r061
Trong n sd tu nhien lien ti6p thi c6 n sd du
khi{c nhau nOn t6n tai sd chia hdt cho n, do d6
trons 3.1061 sd tu nhien li0n ti6p c6 it nhdt 3 sd
crriJtrgt cho 1061, ba so nly nam gira a40 v), a4l
mI c6 itnhdt 61 chfr sd 0 o tan cDng.
Nh4n x6t. 1) Bii to6n v6n dfng khi sd a khOng
nguyOn, cdn ndu a nguyOn thi chi cdn a > 31 li dt. C6c
ban: Pham Kim Hilng,9A2, THCS LC Quf DOn, f
Ycn, Nguydn Dtc Tdm,9A7, THCS Trdn Dang Ninh,
Nam Dinh dd xic dinh ei6 ti m chdt ch6 hon. Tt
1060.30

<

<

irso]o' c6 140.104 < 338 <
334
1185.103
14r.1010, d6n ddn 17.1061 <3340.32 < 18.1061 vi kdt
luan drro. c : glta a40 vd a4r c6 itnhttt l7 sd chia hdt cho
1061, c6 it nhat mot s;6 c6 62ch[ sd 0 & t{n cirng.
2) C6cban sau cfrng c6ldi giAi dring, ggn

:

Phri Tho: Nguydn Bd Gia,gA, THCS Nguy6n Quang
Bich, Tam NOng ; Vinh Phric t Truong Ngoc Cuong,
6B., Bni Hfru Drlc,gA, THCS Vinh Y€n ; Son La:
Nguydn ViAl Hodng,9, THCS V6 Thi Siu, Phi Y€n ;
Bdc Ninh z Nguydn Dtc Duy, THCS Y€n Phong ; Hii
Dumrg: Trdn Qudc Hodn, 9Al, THCS Chu Van An,
Thanh Hi, Hodng Dinh Phuong, 7/3, THCS Le Quf
D6n ; Hii Phbng: NguydnVfr Ldn, Dinh Khang, Bii
Ngoc Khbi,8A, Pham Anh Minh,Vuong Anh Quydn,
9A, TH NK Trdn Phri ; Nam Dinh : Phqm Duy Hidn,
Vfr Khdc Ki,9A7, THCS Trdn Dang Ninh, PhqmVdn
Hdi,9C, THCS Dio Srr Tich, Truc Ninh ; Thanh H6a :
Trdn Mqnh Tudn, 9C, THCS Tri0u Thi Trinh, TriQu
Son, NguydnTudn Nam,8A, THCS Nh[ 86 S!, Hoing
FI6a ; Dlr NrtngzTruong MinhTi€h,9/3, THCS Nguy6n
Khuydn ; Ddng Nai: Vd Sj Bdc, S/4 THCS Nguy6n
Binh Khicm, Bi6n Hda.
VTET

BdiT21296. Chrmg minh rdng

t4

3-{6+V6+...+V6

6

g-Jo*G;-*G

:

HAI

DdtA=

3-

:-Jo+Jo+...*G
Gidu thrlc chrfa c6n c6
ddu cdn 6 mdu)
o

n ddu cf,n tr6n tit vi n-l

= ,16 * ./o + ... + J6

(c6 n ddu can)

Tac6A= 3ro =l5+ q
3-(a"
-6)

(l)

Mat kh6c ta c6

2,4< J6ddu can) hay 2,4 <

6+

(c6 n

a<3

TU d6 vd (1) suy ra

111
3+3 3+a
.115
' 6 3+a

3+2,4

27

Ta c6 bdt ding thttc k6p cdn chrmg minh.
NhAn x6t. 1. Cl{c ban c6ldi giAi t& : H} Tdyz Cdn
Vdn Bdng,9,A,, THCS Thach ThAt ; Phri Tho: Nguyin
Trudng Thp, 9A, THCS Giay, Phong ChAu, Phi Ninh,
TrdnThdnh Dtc,9A, THCS Lam Thao, h. Lxm Thao;
Vinh Phric z Nguydn Ph,i Cudng,98, THCS Y€n L?c ;

Hii Phbng t Bili Tudn Anh, Phqm Anh Minh,9A,
TIIPT NK Trdn Phf ; Nam Dinh: Pham Tadng Linh,
Nguydn Hdi Long,9A2, TIICS Le Quf Don, Y Ycn ;
Thanh H6a t Bii Khd.c KiAn,8A,'THCS Nh[ 86 Si,
Hoiing H6a; Ngh€ An z HodngXudnTrung,9A, THCS

Huy,9D, THCS NguyEn Tr6i, MQ Dric.
2.84n NguydnTha'TiAn,9A2, THCS Le Quy Don, f
YOn, Nam Dinh dd ph6t bidu vi chtlng minh bi'i to6n
tdng qu6t sau

V6i

:

sd tu nhiOn

/c

>

1 ta c6 bat

ding thrlc k6p

r-m-ktk-3)

:

2k-l

trong d6 bidu thrlc chrla can c6 n ddu can & tir sfi

vi r-1

ddu can & m6u sd.

Cho /c = 3 ta c6 BDT cdn chrlng minh.
3. C6c ban BniHfiu Dtlc,9A, THCS Vinh Yen, Vinh
Dinh Huy, 9A, THCS Nguy6n Trii, Nam
Phfc ;
Mch, Hii Duong cflng dE dua ra hai bii to6n tdng qu6t,
nhrmg trong a6 da pnni sir dung t6i hai bidn tdng qu6t.

k

II

1

dd'u cdn d mdu s0'.

Ldi giii. cha Nguydn Vi€it Hodng, 9, THCS
V6 Thi Si{u, Pht YOn, Sun La.

glnr nnr ri TBuoc

:> m >

n-l

27

rd NcwEN


BdiT3l296.Gid srt phwng trinh rs -f + * 2 = 0 cd nghi€m th{c x,,. Chtmg minh rdng

95.ro.\l+.

xf

+

1

ci

hai vd cira

=r(*".

cho nOn

*).

(*) vdi

vi

d6

bang phucrng ph6p quy n4p toiin hoc

+ ! thu duo. c
.xo

xo

Vdi n = 2,ddt2x = b + c - a> 0,2y = a - b +
c > 0,22 = a* b- c > 0
- a - y + z,b = z * x,
c = x + y, BDT cdn chrfurg minh tro thhnh :
ry3

vd rr6i luon duong,

tt vdphii c6 xo> 0. Ap dung bdt ding

*,* ' I
\ xol

x! *t=z(

thrlc C6-si c6

- b) + bn c(b - c) + c" a(c - a) > O
Ldi giii. Ctrfng ta chrlng minh BDT & ddu bli
b(a

an

Ldi giii. Vi xo ld nghi0m cfra phucrng trinh
cho nen xo* O,J * 1 ve *f *3. + xo=) 1*1
=
Nhdn

Bdi T41296. Gid sft a, b, c lA d6 ddi ba canh
cfia mdt tam gidc. Chtng minh rdng voi bdlt ki
s6'nguy€nn> I thi

> 4. Luu

f

*

yr3 + r*3

f 't

i

-

xyz(x + y + z) >

)

O

I

lx- v- z' (x+y+z)l >0
a xyzl-+J
I

LY z x

(*)

-l

Ap dqng BDT C6-si cho cdc s6 duong ta c6

* 1 nOn c6 bdt ding thrlc thxt su. Do d6
>
": 3vI vixo > 0, ta c6 xo> lll

rang.ro

.

Chia

)^1

a*t
x;
T6m

ci

hai vdctra

=*i

+l

xi

lai lll .

(*) cho

rj

a5n a0n

rZvldod6 *o<

*..

:

FlT

Ql+

li

NhAn x6t: Day
mQt phudng rrinh bdc cao, kh6ng
c6 cOng thrlc nghi€m tdng qudt. Tuy nhi€n phuong trinh
cu thd trong bni c6 thd phan tich thinh thtisd

ha-bgc cria phucrng trinh duo.c, nhu ban phamVdn
-_yn
Hdr, TI{CS Ddo Su Tich, Truc Ninh, Nani Dinir nhan
x6l Ban Nguy€n Chi Linh, 8A, THCS Nh0 Bd Si,

Holing. H6a, Thanh H6a phdt hi€n bdi to6n duoc gi6Li
tro-tg ldi giii bdi T31246 fkhing dinh (2) vn (3)) irei'sd
THTT 25! thdng 4/1998. Ban-NguyinTrung kien cho
bie-t bai T3i296 niy thuc chat H mOt UX trone d€ rhi
hoc sinh S16i lvlOndaviq duo. q
trong cudn ,iChuyOn
l! laiGi6o
dc todn l9lg_ tqp'l cira NXB
duc. C[ng 6an
Nguy€n Chi Linh cho bidt bli torin niy'duo.c in trong

cudn

"Bii

rep todn chon loc cdp

t992-t993.

II"'lfXb Ha Sac]

c6c. ban kd tr€n, ciic ban sau dAy c6 ldi giii
. Ngoai
chinh xric, trinh b)ry s6ng sfra depvi sach :'
_Narn_Dinh : NgryydnThi Hdng Hanh,9A7, (8,{6 cfr),

THCS Luong Thd Vinh, Tp. Nam Dinh, Truong Vdn
Hie:u,
THCS I€ Quf Don, .i ycn ; Son TAy:
-9A2,
Nguy,inVfr Hdi,9E, THCS Son Try ; Di N6ng: fft7i
Thanh-Hdi.9/2, Nguycn Khuydn, HAi Ch6u ; Son La:
Nquy|n vi€.t Hodng,9, THCS V6 Thi Sdu, phri yen ;
Tho: A€uyln Minh Lutin,8A, THCS Nguy6n Viei
9_dn
H6ng, Tp. Cdn Tho ; Tri Vinh : Nguydn enhfudn,cjt
THCS Cdu Ngang ; Vinh Phric : Nguydn Ngoc Tudn,
98, TIICS Vinh Tuong, Dd Hodng Tnng, 6E, Bili Httu

Dilrc'9A' THCS vinh

Yon'

vo oilur Hde

tuong

Tt

)2

I 1- > 2z; z + 2_ 2 2t
.xz
tur

^!

d6 BDT (*) duo.c chfng minh, hav BDT
o2b1o - b) + bzc(b- c) + r2i(, -a; > b duoc
chrlng minh.
Gii srl BDT dring tdi r. KhOng m{t tinh tdng
qudt gih st c < b < a. Theo gii thidt quy nap
ta c6

- c)> -anb(a - b) - cn a(c - a)
+ b"+r c(b- c) > -an b) (a - b) - cn ab(c bn

c(b

a)

.

Do d6
an*rb1a

- b) + b'*r c(b - c) + cn*r a(c - a) >
> ao*rb1a - b) - a"b2 7a - b) - c,, ab(c - a)
+ cn+r a(c - a)
= anb(a-b)2 +c'a(c-a)(c-b) >0
Vf,y BDT dring v6i n + I. Theo nguy€n lf quy
nap BDT dd cho dring vdi moi n > 1. Ding thri,c
xiy ra khi vi chi khi e = b = c hay AA.BC ddu.
NhAn x6t. MOt sii ban nhan x6t rang BDT 6 bii ra
dugc khdi qudt h6a tU mOt bAi thi Tor{n queic td nam
1?8,3 t?i Phdp (r = 2). Trit cA c6c bii giii ddu dring vi

tdnh biy tuong tu nhu trcn. C6c ban sau c6 lli giai i6t :
Hi NOi z NguydnTrung KiAn, NguyAn HodngVi€t,
qH, THqS Lc Quf Don, Quan Cdu Giay; Hii phdng :
Bii Tudn Anh, 9A, Trdn Huong Giang, Ddng Ngoc
Chieh,8B, THPT NK Trdn Phri ; Son La : NguiAniiat
Hodng,g THCS V6 Thi S{u, Phti Y€n ; Bdc Ninh :
Vdn Chi,8A, THCS Y€n Plong, Phan Dinh C6ng,9A,
THCS Le Van Thinh, Gia Binh ; Vinh Phric t BiliThi
Thu Hidn,98, THCS YOn Lac, Nguy€n Manh Hilng,

li

15


9C, Nguy€n Hfru Hodn,98, THCS Vrnh Trtong ; Phti
Tho l Nguydn Bd Gia,9A, THCS Nguy6n Quang Bich,
Tam NOng, NguydnTradngThg, Nguydn Quang Huy,
9A, THCS GiAy, Phong Chdu, Phir Ninh ; Trdn Thdnh
D*c,9Al, THCS Lam Thao, Nguydn Quang Huy,8G,

THCS Vict

Tri ;

Hi

Tdy ; Cdn Vdn Bdng,

Nguydn

Khdc Dilng,9A, THCS Th4ch Thdt ; Hurg YCn z Dodn
Thi Kim Hua', 8C, THCS Ph4m Huy Thong, An Thi ;
Hi Nam ':, Trdn Phan Binh,9B, THCS Trdn Phri, Tx.
Pht Ly ; Nam Dinh z Vfr Khdc K!, Nguydn Drtc Tdm,
9A7, THCS Trdn Dang Ninh, Traong Vdn HiAu,
Nguydn Pham Tuy€h, 9A2, Dinh Xudn Tuy€n, 8A2,
THCS Lc QuI Don, f yen ; Thanh IJ6a :, Hodng
Qu6'c Hoan,98, THCS Trdn Mai Ninh ; Quing Tri:
Phan Hodng Phtong Trang,6/2, TTICS Nguy6n Tr6i,
D6ng Hd ; Di Ning: Thdi Thanh Hdi, 912, Truong

Minh Ti€h, 9/3, TIICS Nguy6n Khuydn, Quan HAi
Chdu; Binh Dinh : Nguydn PhilcTho,9D, THCS NgO
May, Phi C6t ; Kh6nh Hda: Vr1 Thdng Thiti, 93,
Truong ctip 2-3 Ngo Gia Ty, Cam Nghia, Cam Ranh.

gd QueNc vnrrs

Bei TSi296. Cho ram gidc ABC vdi M ld
trung didm cfia BC. Dadng phdn gidc ngodi
ctia g6c A cdt dudng thiing BC tai didm D.
Duong trdn ngoai ti|p AADM ctit cdc dudng
thdng AB vd AC ldn ltqt t E vd F. Gqi N ld
trung didm cila EF. Chatrg minh rdng
MNIIAD,
Ldi giAi. Ggi K ld didm ddi xung vdi F qua
M. Do M ld trung didm BC non suy ra FBKC
id hinh binh hlnh. TiI d6 FC//BK. Suy ra
MFA- AEM . NOn

. Do d6 BMKE li trl gi6c nQi
tidp. Suy ra BEK = FMD - FAD = DAE .
BKM = MFA
68fu =Fm

. Me

NhAn x6t. Gi6i t6t bdi ndy c6 c6c ban :
Phri Tho : Nguydn Trung KiAn A, 8A, THCS Giay
Phong Chdu ; Hh TAy : Khudlt Vdn Son,gA, THCS
Thach Thdt ; Hii Phdng: Biti Tudn Anh, Vaong Anh
Quydn,gA, PTNK Trdn Phri ; Nam Dinh zVil Khdc Ki,
Phqm Duy Hidn, 9A7, THCS Trdn Dang Ninh, Pftam
Vdn Hdi,gc, THCS Ddo Su Tich, Tryc Ninh ; Quing
Ngai: B?l L€TrongThanl,, 901, THCS Nguy6n Nghiem.

VUKIM THUY
BAi T6/296 " Ki hiau pp lii s6'nguy|n td th* k.
Chrtng minh rdng vdi m6i s6' ngtt,tt|n dttong tt
plnldng trinh
Znxz

+9n(pu*t

*

pn+z)x+

1945rr3 =ZAOlx?
nghi€nt duong khdng rld hon n,

ci
Ldi giii : (cira ban Nguydn Vdn Tidn, IOA2,
THPT chuy0n Vinh Phfc). Trudc hdt ta chfng
minh pr*, * pn*2 > 6n .
ThAt viy vdr n = 1,2, ta kidm tra thdy dring.
Gii srl pk+r * p**z 26 vot k >2.Taphii chung
minh p1,*2 * p*+z > 6(k + l) = 6k + 6. Mudn
v6y chi cdn chung minh p1*, - p**r 2 6.
Thit vAy ndu pr*, - p*+r < 6 md pr*t, p**r
16 nOn pk+3 = p**t * 4+ p**t, pk+2,pt*r li
3 sd 16 li6n tidp <) mot trong ba sd phii chia hdt
cho 3. Di6u niy v0 lf vi pt*t ) 3 (k>2)
Dlt flx) = 2OOrx3 - 2n* - 9n(p n*t + p n+z\ x
lu6n

*

1945n3.

= 54n3 - 9n2 (po*, + Pn+z)< 0 do
pn+t*pn*2)6n.
Mh lim f (x) = +@ --t 3xo e [n, oo) sao cho
Ta c6 fln)

.r-+6

flxJ=o
Nh{n x6t. Gic ban sau diy giii tdt : Pham l* Thinh,
Pham Tdn Dd, TIIPT If Khidt, Quing Ngai, Dafng

Trong Nam,11T2, THPT Lam Son, Thanh H6a1 Bni
Quang Hdo, ll, TIIPT Th6i Nguy€n, Nguy,in Nggc
Sdng, llC, THPT NK Hi finh : Hodng Ngoc Minh,
I7Al, TIIPT Hing Vuong, Phri Tho ; PhingVdn Binh,
10A1, THPT chuyen Vinh Phric : Trdn NhQt Thu,

11A1, TFIPT Phan BQi ChAu, Vinh, NghG An ; Phan
Thdnh Nam,11T2, THPT Luong Van Ch6nh, Phti Y6n.

DANGHI.JNGTHANG

Biti

T71296. Giti

srl0 < a

(u,)(n = 1,2,3,...)
Yay BEK = DAE hay ADIIEK
Do FN = NE ; FM = MK nen MN ll EK
Tt (1) vi (2) ta c6 MN ll AD.

t6

(1)
(2)

2,3,....

. Xr, ddy s6'

duoc xdc dinh bdi : u1>

,

vd un*1 = lt,rsin"

. !
2

2ffi2.cos2

"+

un"

a

A

vdi n = l,


Chftns minh rdng ddy sd'

n

(u,,) cd gidi han khi

-) + avd tiru gioi han d6.

Ldi giii. X6t hlm s6li6n tuc

Ninh : Ngrzyen Qudc Khdnh,10T, THpT Hin ThuyOn ;
Nry_Otrll z Vil Khdc K!, Nguy€n Drtc Tam,-9A7,
THCS'Trdq p,ang Ninh ; Ngh6 An : Ddo )tudn Hodng,
Phan_Ddng Hpc,Pl-Cl, DH Vinh ; Bdn Tre : Nguyin

Ti€h Dfrng,10T, TI{PT B6n Tre ; ...

_
.NGUYENMINH

\
) 2oO2.cosza voia>0.
J\u)= r/sln-CI,+ ;
ls.'d.
tu

r)
r-

2oo2

I

Tacdf'(u)=sin2.rl
')

Q t1 =

zo}zcos-

-+t

Do d6 ta c6 bing bidn thi6n sau

n"
( -------;
k^(n-k+ll"-r

Ldi giii. Ap dqng bdt ding rhrlc C6si cho ru s6

:

1+lvilsdltaduoc:
n

>

Nhu vay f@)
u * 2002"o'2'

2gg2"o"2o

vdi moi u

)

O,

)

-

,gg2"n,2

o

Ttlc ld phuong trinh gidi han flu) = u chi c6
mot nghiem u = ZOOZ"o"o. L?i chri f ndu

>

a

2oo2"o'2o thi

u-f(u)=

co.2

(\

ol r-'oP l
[ ,'e'o )

ro

u. url2o

(z)

= 2OO2"o'2 o
u1* 2OO2"o"o
n;1:l

u2) u3) u4)

=

zOOZcos-

_.Nh& x6t.

:

(xtxz ... x1r-1)(xp2... xn-r) < x1x2...xn_1
(n

- 1, * 11n-k+t nn
(n- k +L)l
n!

n!

- k) ko(n-k+l)n-k

(1)

Tuong tu, 6p dung BDT C6si cho n s6

u3

=

...

un

'lil' . vay v6i k> t ta cd

.r1...r1

ZOO2

2002"o"2' :+ ,rl = 42 =

=

m6r d6y tang va

+ 1)!

kt(n

= u.oJ o >

Do d6 ta c6 u1

(t

_- kk
kl

I

vi

,( ,* 1)'
= [,\ * --l-)'*'
n+1/ \-. r) +'"*rDdv
*,=(t*1]'
" \ n) li
(t*

(1) vn

f ( zooz"",'o')
"\

...

o'

li

1) Day
Toa soan nhAn duoc

bdi torin dang co ban, c6 f hay.
ldi giii cira'hon 70 ban, n6u h3t

c6c ban c6 c6ch giii theo n6i dung nhu trOn. MOt sef ft
b?n.chrlng minh c6c BDT (l) vi (2) bing bat ding thric
C0si suy r6ng. M6t vii ban, do kh6ng xJm x6t cdi than
a

o
2gg2cos'a , u2) u1) u4) ...
2gg2"o'2
\_9ec ban sau c6 ldi gi6i tdr: Ph[ Tho: Nguydn
Th4'Ting, 10A1, TI{PT chuy0n Hnng Vdong ; Bdc

-

n

A;Dq;7;r);;''

l=o

,cos'a)

\

a'

DT1C

Bni T8/296. Cko cdc s6' rtguvirr duons k,
sao cho k < n. Chrtng minh rd.r,tg(r * [)"nl
nl

r't, - rt,

1-1

vilsdltaduoc:

,[, -*) + t > (n.,) .iF-*)i
.(n+l)"*l (,, )'
=[.;J'[;.i

li
' * 1')'-'
= Diy sd ,, =[t
n)
lr...lr=
Suy ra

(fr +

-

ddv eiim vr

l)t*l

fr!

(ylz...yr)(yttz...tn+) > !J2...!,

t7

I
I

(ft +

+l)/r-t+t (n + l)n*r
(n - 111t.
nt.

l)(*l
kt.

(n _ k

(n + 1)'*l

+--..--> nl
kt(n - k)t

Tt

(1)

vi

(k+1)k+l

(2)

0t-k+l)z-t+t

(2) ta c6 dPcm'

chrlng minh uOn ta thdy (1) dring v6i /e
> 1 Vn >,t. Trudng hqrp k = I' n = Zthi (1) khOng dfng'
2) C6c ban ddu giii dring theo phuong ph6p quy n4p
vh dua vdo hai nhAn x6t vC t(nh don di€u cia ddy lxnl

Nh4n x6t.

Tt

vh {yr,}.
C6c ban sau dAy c6 ldi giAi tdt :
Hda Binh: Ngb Thdt Son, l0T, Hd Hfiu Cao Trinh,
Vfi Hfru Phaong 1lT, Nguydn IfrmTuyin,l2T, THPI
chuyOn Hodng Vin Thu ; Phri Tho z Hod.ng-Ngoc
viin, tlA., TIIPT chuy€n Hing Vuong ; Bdc Ninh :
PhamTtuli Son,l0T, tityt Uan Thuy€n ; Ddng Thdp:
Ngiuydn Vd VTnh Lac, 11T, THPT Sa D6c.; Ddng Nai:

f n,rong,1 lT, THPT chuyen Luong Thd Vinh ; YInh
Phric : Od enn Ddng, L}Al, TIiPT ch. Vinh Phtic ;
Phri Ydn ; PhanThdnh Nom,llT2, TIIPT Luong Van
Ch6nh ; Hn Nqi : Vfr DinhThe', LIAT,TrdnThdi Son,
Trdn Hdi Sar, tOg Tin, Ic Hing Viil Bdo, fiAT,
DHKHTN - DHQG, Nguydn Chi Hiep,11A1, DHSP

ti

Hii Duong ; Pham Huy Hodng-. PhamThdnh
guy€n
T haih N om, I lT, N guyd n T i €h T hdnh'
N
rung,
T
llL,-THPT Nguy6n Trdi ; Binh Dinh z Nguvin Hodi
Phuong,10T, TIlm chuy€n Le Quy DOn ; Bdn Tre :
Npuy€n Tidh Dfine, 10T, THPI chuyOn Bdn Tre ; Tp.

D4t

9A7,

THCS Trdn Dang Ninh

llT,

; QuingTrit Li Nhdt

Trdn Ti€h Hodng, 12T, THPT chuy€n l-€
Quf Don, Phan Qudc Hung, l2B, TfiPT Hai Lang ;
Ngi,e ,q,n : Trdn Nhdt Thu, llAl, Nguy€n Tli Qulnh
Tiang,10A2, THPT Phan BQi Chau, Thanh H6a: U
MqnhTrung,lll, Mai QuangThdnh, B'ni Hdng Qudn,
11T1, Nguydn Minh Cdng, 12T, THPT chuy€n

Tdn,

Lam

Son'

Ta thdy

=

clttdrtg thdng OA, OB, OC cdc gdc nhon a, B, y'
ChLlrig minh rcirtg gid tri cia bidu thitc
sin4A.sinz

a + sinlB"sirtz B +

td ktfittg ddi khi drrdng

sin4C.sin2

rhclng

d quay

y
quanh

:2(z -

l8

(mod 2n)

:

:2(x - z) (mod2x)
4C:-2(y -,r) (mod 2n)

48

Tt

d6, ta c6

:

sin4A.sin2cr + sin4B.sin2B + sin4C.sin2y

= sin2(z

y)sin2x

+

sin2(-r

-

z)sinzy +

sin2(y-r)sin2z

I
I

=

6in2(z-yX 1-cos2x) + sin2(x-z)( 1 -cos2v)

+ sin2(y - rX1- cos1))
= !1rinZ1r- y) + sin2(x

2'

-

z) + sin 2(y

- x))

- 1(rin 2(z - icos 2x + sin 2(x - z) cos}y

2'

+sin2(y

-

x)cos2z)

(1)

Tac6:
sin2(z

+ sin2(y

- y)cos2x + sin2(x - x)cosZz

= 1(rin(,

2'

z+

y)+ sin(x - z - Y) +

+ sin(y -x+z)+sin(Y
=

z)cosZy +

- y + r) + sin(z - y - x) +

+ sin(,r -

Ldi giii. (Dua theo ldi giii cira ban Dd Hodng
Huy, lOTl, TI{PT chuY0n Luong Vin TuY,
Ninh Binh)

*0).

(mod 2x)

Y) (mod 2tt)

Tuong tu nhu vAY

didm O.

Lt{y M thuoc d (M

:4A=2(OB,OC)

2(ett ,oe)-@14,oil)

NGUYEN vAN Mau

Bni T9/296. Cho tam gidc ABC cd ba gdc
nhan nii tiip dudng trdn tdm O. Mdt dudng
thfrng d di clua didm O ldn luot tao vhi citc

1@fr,o81=,

Itofi,oe)=,

Ha Noi ;

H6 Chi Minh : Trdn Vd Huy, llT, DHKHTN Tp.
HCM ; Nam Dinh: Nguydn Drtc Tdm, Vfi Khdc Ki,

fr* ,oA)= *

-*-4)

|{Ointr-y+r)+sin(v

-x-z))

+

+


+ (sin(z - y - x)+ sin(x - z + y)) +
+ (sin(,r -z- y)+sin(y -x+z)) =

Cht?ng minh rdng n€u M ld rnot itidm tily y ndnt
trortg t[t di€n tli :

MA+MB+MC+MD>
> 3(MA'+ MB'+ MC'+ MD')

1

i(0+0+0)
=Q
2

=

(z)

TiI (1), (2) suy ra :
sin4Asin2o + sin4Bsin2B + sin4Csin2y

: (sin 2(z - y) + sin 2(x - z) + sin}(y - r))
/.'
I

=

I

= :(sin 44 + sin 48 + sin4C)

2'

= 2sinAsinBsinC (khong ddi)
NhAn x6t. l) Day ld bii to6n hay. Tuy nhi€n n6

44

dang:
s6

hay hon ndu duo. c ph6t bidu & dang tdng qu6t hon.
Cho tam gi6c ABC vi didm O kh6c A, B, C. Vdi moi
didm M khdc O, chfng minh ring gi6 tri bidu thrlc sau
khOng phu thu6c vio vi trt crta M :
srl.zlc,E,

oe)

snz @A,ofr)

+ sn2(oc, oA)

sinz @

B,oW

+

sinz@4, oilsinz 1od,ofr)
i : Tam gi6c ABC kh6ng buQc phAi nhon. Didm
O khOng bu6c phii le tam dudng trdn ngoai tidp tam
ChLi

gr6c ABC.

2) Nhd bdi to6n tdng qu6t tr€n, c6 thd giii ngay dugc
bii toiin chung sau d6y.
Bdi todn 1. Cho tam giiic ddu AAC, Am O. Duong
thing d di qua O. Ggi A;, B', C,tdhinh chidu cia A, B,
C l;en d. Chrlng minh ring gi6 tri cira bidu thrlc

Zue, >3\tttA',
i=l

(YM e midn tr1 diQn[A1A2\Aa])

Vi trl di6n ArAz&A.41) gdn ddu nOn ci{c rnit lh
nhftng. tam gii{c bang nhau, vI do d6 c6 diOn
tich blng nhau. Sir dung cOng thrlc thd t{ch, d6
dlng chung minh duo. c :
- Cdc duong cao A;H; bang nhau, ta dat :
AiHi= h (i = 1,2,3,4)
(1)
- Tdng c6c khoing ci{ch tir moi didm M nirn
rrong rf di€n d6n c6c mil lh khong ddi :
4

c6c

M2+BB2+Cc2
ld khong ddi khi

d

quay quanh didm O.

Bdi todn 2. Cho tam gii{c nhon ,A-gC, rruc mm H.
Dudng thhng d di qua H. Goi Ai B', C, h hinh chidu
cia A, B, C trcn d. Chrlng minh ring :
,1,,1,2

tgl

+.B82

tgB +

CCZ

igC

=25(ABC)

.3) Nhidu b4n rham gia giii bni to6n niy. Tuy nhi€n,
vi kh6ng bidt st dung g6c dinh hrrong gifia hai v6cto

nen da sd c6c lbi

giii

(*)

trong dd A', B' , C', D' ldn luo't ld hinh c'hi€ir cila
M" tr€n cdc rndt BCD, CDA, DAB, ABC.
Ldi giii. -Tru6c hdt, dd cho ti6n trinh bdy ta
hdy thay ddi kf hieu : A, B, C, D ldn luor duoc
thay b0i A1, A2, A3, A4, cfrng v6y, A' , B', C', D'
ldn luot duo.c thay b6i A'1, A'2, A'3, A'0. Thd thi
BDT (*) cdp chrlng minh b0y gid duoc vidr dudi

IMA',

Quf DOn ; Hda Binh : Vil Hfiu Phuong,llT, THPT
Hoing Van ThU ; UAi Duong z li euang Dao, l0T,
THPT chuyOn Nguy6n Tr6i ; Hi Nam: Irdn Qu6ic Son,
THPT chuy6n Hi Nam ; Quing Tri: Phan eudc Hung,

12B, THPT Hii Lang ; Hi TAy : Nghi€m DuyTrung,
12A1, THPT Ddng Quan ; Hn NOi : Nguy€n phuoig
Nhung,l lAT, DHKHTN - DHQG Ha Noi.
NGUYENMINHHA

1'

(2)

M4t kh6c, slr dung BDT tam gi6c vi BDT vd
duong ru6ng g5c vi duong xi€n, ta lai c6 :
A;M + MA'i > ArA'i 2.A,H, =
1.1

(i =
Tir d6 ra duoc

1,

2,3,4)

(3)

:

4

Z(MAi
vI do d6

'

MA'i)

>-4h

'='

l=1
Cudi cirng,
cdn tim.

+

44

Zu+

ddu phg thuOc vdo hinh v6.

4) Ngoii ban D6 Hodng Huy, cdc ban sau dAy c6 lbi
gi6i tuong ddi t6t : LAm Ddng : pham Hodng Duy
Thdi,12T, THPT Thang Long; Ngh0 An: Nguy,inThi
Qu), nh T r an g, l0 A2, Le T hily N gdn, I 0A t, THpT phan
BOi ChAu ; Di Ning z Phan AnhTudh, t}At, THPT Lc

=

i=1

tt

> 4h

-

Zue,i

G)

l=1

(2) vh (4) ta thu duoc BDT (*)

Ding thrlc 6 (*) xiy ra khi vi chi khi M e
AiHi (Vi), nghia lh cdc dudrng cao cira trl di6n
ddng quy vd didm M tring vdi didm ddng quy
d6, ld trgc t6m H cira rr1diOn. N6i kh6c di, ding
thrlc xiy ra khi vi chi khi vi trl diOn gdn ddu
AAzhA cf,ng li m6r trl di€n truc r6m, vh do
d6, ta phii c6 th6m cr{c didu ki0n sau:
4A? * AtAz = AlAl + 4,4 = ere,|. + e1,$

Bii T10/296. Cho tr Jiitt gdtt cldtLt ABCD
TU d6 suy ra : A2A3 AtA+
AtAr A2,44
h BC = DA, CA = DB, AB = DC). ArAz= AlAavit trl di0n=lh ddu, =vh didm=M tring=

(nghTa

T9


i
v6i trong titm G, truc mm H vi.t6m O cdc mat
I

I

Nhfn x6t. 1) BDT (*) cdn tim c6 thd dugc.suy ra rdt
nhanh ch6ng nhd sir dung ba kdt qu6.(tinh chdt) sau d6y
vd tinh chaicira tf di€n li0n quan dOn "didm !Uc^ti€u""
(didm Fermat) v) BDT gifra Cdc brin kinh r, R cira c6c
mit cdu nQi, ngo4i tidP mQt ttl Ci6n.
Trong mQt tri dion gdn ddu ABCD, ta c6 BDT sau :

-

I

T

cdu ndi, ngo4i tidp cira trl di6n d6u.

MA + MB + MC + MD > OA + OB + OC + OD = 4R
(YMl vir tam O cfia m4t cdu ngo4i tidp li "didm cuc
iidu" hay didm Fermat ctra tf dien gdn dOu.
- V6i mgi tf di0n, ta c6 BDT : R > 3r
D&ng thrlc

xiy ra (R = 3r) khi vi chi khi

trl dic.n

h

ddu'

tt

mpt dipry lT! trolq.mQi
trl di€n Een AA, ddric6c m4t cira n6 ld khong d6i' vd
bang chidu cao cira tf diOn, hay 4 ldn biin kinh.r mit

-

Tdng c6c kho6ng c6ch

cdu-ngi tidp. B4n nio chua bidt cdc kOi qui ndy' hiy tg
chrtng minh ldy.
2) Ldi gini cira bii to6n nhu dd n€u ra & trcn chi ttbi
kieh thrtc tdi thidu, don gian nhdt. MQt sd
h6;;t;,lil;"phuons
ph6p thd tich cfing cbn su dung
bun. nnoii
ohuons" oh6i v6ct6 tiicli vo hudng) vi ldi giii dua ra
nhidu
[;;-;ffi ;t; khong !qn. Rdt deing-tiec. cung cbn
ban chua ciri ra dudCddu dang thrlc xiy ra khi ndo hoic
itri nOi duoc khi bdn dudng iao ddng quy vi-M trilng

voi didm ddne quv d6. C6 ban v6i vhng kh[ng ditth
ra vi tf dien dd cho li gdn dOu,
rtins thfc khoie-the xiv-truc
tam ! C6c ban niy kh6ng
kho"ne phai la If dicn
iiidl.iioiii* itrat ran dd ld tim didu kiQn-cdn vd dir vd tri
Jien auoc xet vh vi tri hinh hoc cira didm M d6ivutA
dien d6 dd ding thric xiY ra.
3) C{c ban sau dAy c6ldi giii tdt, nrong d6i go:t g}mg :
Hi N6i : Dd Xtdn Diin' llAl, Vdn Anh Tudn,

liA2, PICTT-DHSP, Le Hdns ViQt Bdo, 10A CI
DHKHTN ; Hii Duong : Phan Tudn Thdnh, l0T,
Nsurin Th.€' Iic. Phail Thdnh Trung, l7T, T'I7P[
Ngry6, T.ai ; pt,i Thg z I'e Jaiet HL Nguvan Th€'

rini. rcel, TriPT chuven HDng Vuong ;

A1 chl

Ldi giii.
. Khi Kr, K2 ddu d6ng, ta c6
gifrn dd vectd nhu hinh vd

r

sing =

Binh:

=U=Un

R2

JtF Hqng
Minhz Trdn Vd Huv,

Dao,'Phan"Thidt; Tp. Hd Chi
rir, rnvr DHQG ip. HCM ; Bh Ria

- V[ng Thu:

NGUYENDANGPHI\T

Bii Lll296. Cho mach di€n nhu hinh ud'
uAB = 25OJrsinl\\nt (V), C = 20ltF' cu1n
ddy thudn cdm. Khi Kt,Kz ddu ddng, ampe ke'

;

(lc= IZs:

U

=(lc +Un

.(z)'

UR

U

R

- rl

or('-s-

lzr

UR

bii Ip =

Theo dd

*.61,

(1)
12

Iil{udn

Thuan: Nguy€n Phudc Tai, 1lA2, THPT

20

fl

3=i{

I I.R
?i

Le Vdn

128, TIfff Hii Lang; Binh Dinh : Ng-uv|-n Hodi
Phiong,10T, THPT chuycn Le Quf-D6n. Quy l{hdn ;
Phri 'iien z Phan Thdnh Nam, 11T2, THPT chuy€n
Luong Vin Ch6nh ; LAm Ddng z Phyy lodng luy
Thdi," l2T, TIIPT Thang Long, Tp. De I4t. ; Binh

;

)(J=

Th6i

Qudn,-11T, THPT chuy€n l-uong Vdn
T{y, Va Vdn Phin, 12C1, THPT B Yc-n Mq ; Thanh
Hna z Mai XtdnTrdng,l lA, THPT l,c Lgi"Thq Xuan ;
Thft, ll{l,
NghO An z Phan Cbng Manh. I'e
-Duy
Hung,
Phan-Qudc
z
Tri
1
Lqc
ugni

ffrff
; Quing
NghTa,

uls : ulP vd'

,o=*,U=Z;cosc=+=Z

: Nguy€-n Anh Thdi, llT,

TftFT Jhuven I-r Hdng Phong : Ninh

,57A. Khi K t, K2 ddu md ampe k€' A2 cht

t,SZe. Bd qua diin trd cila ddy ndi, cdc kh6a
Kt, Kz vd cdc ampe k€'.Tim R, L-ViAl bidu thfic
ddng di€n mach chinh khi Kt, K2 ddu d6ng'

Nsu"ven z Npuvdn Minh Gidne. i1A2. THPT Luong

N[oc Ouvdn": Nam Dinh

1

!-=

250

500i

ZC

R

+ZZ

=

1,57A. Ta c6 nhtn

= 1,57A=
7r

+

nen

tir

(1)

tQtnl )L=0,51H.
suy ra i ZL= Zc= tn
o

Khi

Kr, K2 ddu rn6, v6i

sdchil=

I
R

Zy= Zs, ampe kd 42

=1.,57A3R =

#"159(O)

. Khi Kr, K2 ddu d6ng, theo ttEt26 = ZL= R
vi / = tpJz ; ngoii ra :
3/p=12-d,=
TE

4

F
F

c

f

F
o


Ua

InR = 250 (Y)

=

; Uc = IZc = J2.ln4 =
zsoJ, (y), mi U = 250 (V), nghia li
uZ =uk +(J2 > u tuo,hay 0 tro pha gdc
1

o=

I4

i=

so

(1,

vdi

7 . Oo d6 bidu thrlc cfra

sTJi)Ji

i

li

-

vi

gon

:

To6n,

Vinh ; Bdc Ninh : Nguydn Thi Thu
NK Hin Thuy6n ; Ttraitr H6a z L€

Ddng Hdi,l2G, THPT L€ Van Huu, Thi€u H6a ; Vinh

Phric

:

Hodng Vinh Hung, Ddo Ddng Hda,

llA3,

,u ,

=-4.1 =
-'99
AB

h= AB+BC
PP9^=

AB

=

vu6ng g6c

t'di truc chinh cfia h€.

25cm : dz = BC

Thdu kinh L ddt tai A c6 thd thay the'he (Lt,
L2) sao cho voi bd?'ki vi tri ndo cila MN ddt
trtdc L ddu cho dO ph6ng dai nhu cila h€.
Dat MN tai A:
+ L1, L2 vdn d B, C ; sau d6 dtio vi trl cho nhau
ta duoc rinh qua h€ sau khi drio bdng 4 ldn tinh
qua h€ khi chua drio..Hai dnh ngao. c chiA"u nhau.

+ Ch; dt)ng L2 dqt tqi B thi cho tinh crta MN
tai C.

=L
- 105,

k=k') + ,= ,f, , . -f, ,
f-d fi_dt fz_dz

"uy

320

-5
64d-320 -5-d 'r

NhAn x6t. C6c em c6 ldi giii ddng vh gon :
YGn B6i : PhanThi Kim Hoa,ltAt, TFIPT chuyOn
Nguy6n Tdt Thnnh ; Quing Ngii t Trdn Ouy Xnibm,
11 Li, THPT chuy€n Lc Khict ;.Hir finh : L€ Hfiu Hd,
U Dtc Dqt, ll L(, TIIPT NK Ha Tinh ; Vinh phric :

Trdn Bd Bdch, 12A2, Chu Anh Dfing, ttA3, T-tlpt
chuy€n Vinh Phdc, Nguydn Duy Binh, 12A2, TIryl
NgO Gia TU, L{p Thach ; Hii Duong :Vfi XudnThdne,
11 Li, THPT Nguy6n Tr6i ; Ngh6 An :Trdn QuangVfi,
I lA7, THPT chuyOn Phan BOi Chiu, Vinh ; Bdc Ninh:
Nguydn Vdn HOi,llAl, THPT ThuAn Thinh sd 1 ;
Thanh H6az NguydnVdnThi1n,llK, THPTHAu LOc
2 ; Bic Giang : lc Nho Thity, l0B, THPT NK NgO Si
Li€n ; Quing Binh : Duong Dfic Anh, 12 Li, TIIPT
MAI ANH

inh

:

,LL,L.
MN __-> M,N, vd MN .-:+M N thMzN
d d'
dt d't '- 'd2 d'2''-'

z

Khi d4r MN t4i Ata c6 d = d = O M'N' =
=
MN = L
= k = 1 (1). Theo dd bii, chi ding 12
o

, k'= .--?0f',.,

chuy€n Quing Binh, Ddng Hdi.

AB.

Ta c6 cdc so dd rao

.

c:iua

f
f-d

Ldi giii.

du-o.c

_., 100 - 3n- d'1=
25_ft.

) ft = 16cm. Ngodi ra, vdi vi tri bdt k\ MN
tru6c Lthi : d1 = d + 25
= dz= BC - d't =
84d + 500
.^ . Nhrme theo dd bli, vdi d b6t ki ta
d+9
lu6nc6

Timf,f 1,f2vd

k= I vI kr= -4k',suy

2000

Bdi L21296. Cho h€ hai thdu ktnh ddng truc

Lt, Lz vdi BC = lrn, Vdt sdng MN

:

=Zocm.Honnta :d1=fl+AB=

Thay vio (3) ta

MAI ANH

ld

AB =Zfucm,vi do d6

TIIPT chuyOn V-rnh Phric ; YOn B6i : Ngoc Anh,lLA2,
THPT chyen Nguy6n Tdt Thanh.

L)

fi-dr fz-dz

Theo dd bdi ta c6 : k' =

Bdc Giang z Ii Nho Thily, t}B, TT{PT NK Ng6 S
Licn ; Ngh€ Ln ; Nguydn Manh Hilng,12A3, TT{PT

ll

BC' 100
,ABAB\&

Mat kh6c dO ph6ng dai qua hq (tr,
ft
fz
t-,

sinfroor,, * 1')

NhAn x6t. C6c em c6 ldi gi6i dring

Ph.an BQi Ch8u,

.

:

(4/
/
,r\
= 3,l4sin I toorr +: I (A)
4)'
\

Hdng,

dat mi B cho inh o C, ta thdy inh niy cfrng
chfnh ld inh cira MN quahQ (4,L2) sau khi dio
vi trf cria L1, L2 cho nhau. D6 ph6ng dai khi d6

Nhdn tin :
Cdc ban dtoc nhdn tdng phdm tr€n bdoTHTT
tit thdng 5 ndm 2002 hdy g*i dia chi mdi nhdt
vdTda soan dd ti€n li€n

hA.

THTT

2t


2) Tt sd thdne U2001 ddn sd thdng612002 dd
c6 bao nhicu bII vidt cho chuyOn muc Todn hoc

vii ddi

sdng

vI

cho chuy6n muc Lich

sr?

todn hoc ?
3) Ban c6 bi6t c6c chuy6n muc cho 6 bAng
auOi dav xudt hi6n ldn ddu tiOn tit th6ng, nam
,a" K,olrg ? Hdy'didn vdo cdc 6 trdng ctra bing
cho dring nh6 !.
Thdng, ndm
xudt hi€n

Chuy€n muc

'tuy clni Nat : 4/g/ffu/l//i{/lloll?
,,*tt^e

VinL l4tQtl
cHi ToAN HQc vA rudr rnE
Ddn nav To6n hoc vi Tudi tre dA ra m6t b4n
TAP

doc duoc"300 so. Nhan dip nny CLB thir
cdc b4n thOng qua c6c cau h6i sau

tii

cfia

:

THTI ddu tien xudt bin vio th6n-g,
nam nio ? 86o TH&TT Adi *renn TaP chi
TH&TT tilth6ng, n6m nio ?
1) Si b6o

Kat qud;

uor

sd

DInh cho c6c ban THCS
Di6n din day hoc To6n
Oarn cho c6c b4n chudn
bi thi D4i hgc
To6n hgc mu6n

miu

Nhin ra thdgidi
T4ng phdm dang chd cdc b4n
nhanh !

giii

ddp dfng vd

THANH HOA

pulr

MINrI T*AN Hqc

qua bing
CLB xin gi6i thi6u mQr sd cong trinh tieu bidu cbaicic nhI to6n hoc thdi xua th6ng
thdng k0 chinh x6c sau :
Narn sinh, mdt
Cing tinh
huong
Nha todn hoc
Qu0

Trung IJoa

Luu I{uy

A-ri-ap-ha-ta

1

(Aryabhata 1)

6-ma Khai-am

(Omar Khayam)

E-ra-to-xten

ft6*10-.m0 C.

An DQ

Iran

Hv Lap

(Ptoldm6e C.)

cdc hQ phuong trinh, tim ra

x x 3.141
BAng tinh sin ciic g6c nhcn

Hv Lap

(Eratosthdne)

Giii

+.o.2o = 1 ; n * 3,1416
Phin loai, giii phumg trinh bic Z,bec 3

Tl^,d

ki

iii

kho&ng 550

vdi y nghia hinh hgc

khoing
i048 - 1131

(PP sing

E'-ra-t0-xten)

I

--!silcft---l
+io-

sin2cr

Phuong ph6p tirn sd nguven t6

i

l

Klioing 276-194
(TrurJc CNi

Bing t(nh sin. COng thfc c6ng cung.
3lt

Khoing
r00-178

,L +

120

Td Xung Chi
(Zu Chong Zhi)

22

Trung lloa

-

7tN--.Tt:c
'1

355

it3

-.

430

- 501

h{a phucrng bQc 3

vd c6c nhi to6n hoc
Nhfing b4n sau dd didu.chinh dring bing thdng k0, ddng thdi trinh bhy th€m
d6, vh dluqi nhan tang Phdm ki niY :
Ninh ; Pham
Daong'Hdn! Hu,Q,-ffinhi 48, d 5t, khu 6, phu-ong Hbl pai, rp 4l!gg:*a"*g
Phri'
X.rrdn
r,rtng Duong, e7 van cao, quq Ng9 Quvdnl
4yi,t-oni,99,]!tT
THPT
Nga,10C,
phdng ; Cao phuon;r;y,'ffiS NEti oi.i", NeniP.:,^\ehQ An ;VdThiThuy
Thenh'
Binh
P13,
Quan
i,(ei:ird;fiont cioni,46it72 No rrang Long,
b; r",

IIii

liyir

i; i;:

Tp Hd Chi

22

a;a"'s

Minh.

cLB

,t


qinr ildp lfii
a
Ua

7vr-n
2'-n-2s
.4;7
s+

:

NGUY BIEN
"Nhi torin hoc nhi"
.'^a
,ot^ r ^
d"a 4p lu{n-sai
6ch6:Gidsftpld

p phdi chia h€i cho m1t sd

nguy€n td ndo db thubc ddy pt, p2, ..., p* Thi du
sau dAy cho thAy p = prpz...pn + I c6 thd chia hdt
cho mOt sd nguy0n tO ldn hon p, :

p = 2.3.7.11.13.17 + 1 = 510511 = 19 x 26869
D6 chinh li phin thi du b6c b6 "phr{t minh" tr6n.
^cria
gi6 tta
56t fr
di c6 nguyticn
nguy
tr0n vi tdc
t6c gii
"dinh li" d6
da
ph6ng theo ph6p chfng minh
ninh cria
ctra O-ctit
O-clit vd rinh
tinh v0
vo

(1nQt kdt oui
lpn cua
han
gua_44y
ddv s0

so nguygn
qui h,
nsuvOn td (m6t
het sri'c dep
dE c[ra So fiqc) mT fiin cdi ban cbn irhd t
Ciic ban thdv dAv, ph6p chfrre minh cria O-clit
"nhh toiin
"nhh
vd
vhr "nhd
toi{n hoc
tioc nhi"
nhi'' chi
c kh6c
kh6c -nhau
nhau drine
drins mOt
m6r t&
til
"h[u han". Tiei thav ! tt ndy lai l]r m6u Etrdt'ctra
van dd ilan den ch6 sai cria "dinh lf' ay ddy !
NhAn x6t : Nhfrng b4n sau day d5 chi 16 sai ldm
trong ph6p chfing minh "dinh lf", ddne rhdi dua duoc
cric ph1n thi du drine dd b6c b6 "phr{tinintr" cira "ntri
to6nhoc nhfl' duo. c nf,An t4ng ptrdni ta :
Hn Nqi : Vdn A;h Tudn, ltA2, khdi PTCT-Tin
DHSP-Hi_NQ| ; T6.Xain Nam, 6F, THCS Giang V6,

Qqan A" Dinh

; Bic Ninh : LuuThi ThuTran{,8A,

THCS YOn Phons ; Nam Dinh : PhamThi Linh Ouy1n.

T2lC, truc,ng CDSP Nam'Dinh ; HAi Duons T ir,ii
Quat Hodn,gAl, THCS Chu Van An, Thanh IIi.
Khen them bdc ban .' hldng Hai long, 8B THFI NK

Trdn Phi, Hii
N-inh, Thanh

Phdn! ; BnjVariNshta, S,{ THCS frdn I\,Iai
H6a t N4uy,in U Minh Hodng, vir Nguydn

Qlr"lg Huy,l0 Ctruy0n To{n DH Vinh, Nghd-An ; Ng;ydn
Hd Phrong,10 Tod& TIIPT chuy€n Thdi-Niuv6n :Pn"Thni Son,lOI, TIIPT NK Han Thirycn, Bdc Ninh.

NGQC HIdN

cHo m.iy prdu'l
Trong mQt dd thi tuydn sinh DH nam 1997 c6 bii:
Hdy xnc dinh a dd diong cong y = ? (t-a+4 +
-3a ti€p xuc voi duon! congy = ? + L
MOt thi sinh dd hm bei nhu sau :
Hai dudne cons n6i trone dd bii ti€p xfic nhau
khi vichi kfu PIsau c6 nsfiemk6o ^

i

:

3'(3'- a+Z)+oz -'3o= j.* I
(l)
D4t dn phu- r = 3* (t > 0) thi pT (l) c6 nghiQm

k6p khi

vi chi khi PT s^au c5 nghi€m k6p :

t'+(l-a)t+a'-3a-l=0

Didu niv xiy ra <+ Bi€t thric
6 = 1l-a)2 - i1r2-3o-lj = 0 <+
vul

)

5

vi, = 3*,

t

3a2

-

IOa-

5=0

a

th6a mdn dd bei

.

c6 siltri a-

Girii ttdp: BAO NHIEU HQC SINH?
Goi s0 hoc sinh khOng dat

5+

Ndu ban li gi6m khio thi ban cho
may didm ? Visao ?

2fi0
3

bli lim niy

HoANG NcuyEN
(Thdi Nsuy€n)

giii li

"brd
(0 < a, b, c, d < 9 (1)). Til gii thi0t bii ra ta c6
:

1000a + 100b + l}c + d + a + b + c + d = 2392
(2)
hay 1001a + 101b + llc +

2d=2392

Tt(2)suyra0
Tt (1), (2) tac6

(3)

= 1366, suy ra a22 (4)
(4)taduo.ca=2
Tt(3),
:+ 101b + l1c + 2d = 390
(5)
Tt (1), (5) c6 101b > 390-9-18 =273
b23
=
Ttd6vi(3)taduqc b=3 +llc+M=87 (6)
Tt (1), (6) c6 1lc > 87 - 18 = 69 + c 2'1
Ttd6 vi (6) chi xiy ra c = 7, do d6 d = 5.
Sd hgc sinh kh6ng det giii ld 2375 vh
2375+(2+3+7 +5)=2392
V0y sd hoc sinh dat eiei oudc te li 2. dat siii
quoc gia li 3, dat giii'tiih li 5, dat giii cta tiu8ng
lit'l .
1001a > 2392-9@-99-18

Nh0n x6t. Hon 100 bii

siii eui vd tda soan ddu
drine. G{c ban sau c6'ldi"eiii telt vn siri bni
"
ildn sdrntron duoc nhAn tang phdm
Hn NOi z Bii Vdn Dai, Ll To6n, DHKHTN
giii

:

H4ryQi i Bdc Ninh : Trdn BtnhTing,

UlaG

-

L0

To6n,,THPT NK Hin Thuy€n ; Nam Dinh :
N_Syydry Minh Dtc,1141, THPT XuAn Trudng B;
Ngh6 An z NguydnThA'Anh,12Q, THPT Lc Vidr
Thu4t, Tp. Vinh ; Thanh H6a z Trdn Hodi Thu,
9A, THCS Trdn Mai Ninh, Tp. Thanh H6a.
ssu_cqng c6 ldi giii t6t : Nguydn Hodng
_ _Clic -b_qn
Vi€t, 8H, THCS^Le quf- Don, W{uyen euynih
Hoa, l4C, X3, td 72 Dii,h Vong, edu Gid,, Hn
\Qi :_@ l6ng L4a, qg, THCS-TT Tc Tici;My
Dri'c, Hi TAry ; Hodng Drtc Quang, 11A, THPT

Dio Duy Tt, Ddng Hdi, Quing Binh ; Nguydn
Th! Cdm Nhung,ldp 8', THCS Nguy€n Cu Trinh,
Huong So, Tp. Hue
NGOC HIdN

sfiu TflIt DUilG THUOc

2Ji-o

o ien chi

I

Hidn do

UA GtlMPA

\1d. : Cho 3 didm A, B, C khOns thdng

p cd:c! Qdu ba didm d6. lrtuO'n dVql
tdm dudng trdn n1i ti€p tam gidc ABC bgn can
pfuii mAy ldn ding thudc kd vd mdy l,in d,ing
hdng vd.didm

compa

?

Hd : Tbi chi cdn ding thudc kd vd compa cd
thdy 6 ldn thai.

Ban Hd dd ldm nht th€'ndo nhf

?

NGUYfiN DUC T{N
(tp. Hd Chi Minh)

23