Tạp chí toán học và tuổi trẻ số 302 tháng 8 năm 2002

Qlld*SWslqr

#

$**&i*ffi & kffi*Tg4*MtrK&:1fi#€

ffiffiMffi€ffiffi@wffiffiffi€ffi@€€


\rr{r\'\r

Ki 'rHl 'fc.lArrl auOc

'ri uAt t 'tilu 43

VU DINH TIOA - NGUYEN TUAC MINH
khdng ra dd), thuOc c6c linh vuc dai s6, h)nh
hoc, td hqp vir so hoc. Cdc bli thi dr-toc phAn
loai lhm 3 loai: d0, kh6 vir trung binh. Bdi ra
cho m6i ngiry gdrr mot bdi d0 (bz\i so I vi bli
sd 4), rnOt bli trung binh (bhi so 2 vh b)ri so ,5)
vh mot bni kh6 (bi'i so 3 vh bdi so 6). M6i bhi
duoc toi da 7 didm.
Theo clidu 1€ cua IMO, khdng qu6 ll2 tong
so thi sinh cluoc trao huy chuong, tinh tir didm
cao xudng thfp theo ti Ie 1:2:3 II huy chuolrg
vhng, ba- v}-cldng. Narn nay Ban td chfc
IMO cia trao
39 hLry cl-tuong vhng (HCV) cho czic th( sinh
dat tir 29 diOrn tro-. Ien,
73 hLry chuong bac (HCB) cho ciic thf sinh

dat tir 23 diern tro len.
120 huy chLtong d6ng (HCD) cho cdc thi
sinh dat tu I4 diern tro len.
Bing khen cho m6i thf sinh giAi cluoc tron
ven mdt biri todn.

Ki thi todn quoc te ldn thf 43 (IMO2002)
duoc td chfc tai thlnh ph6 cAng Glasgow cua

Scotland, Vuong quoc Anh til l8l7 tdi
301112002 vdi 419 thi sinh tir 84 nudc vh khu
vuc (c5 th6m I nudc viL 6 thf sinh tham gia so
vdi ki thi to6n quoc te lZin tht 42 td chfic tai
Hoa Kj'). DQi tuydn Viet Nam gOm 6 hoc
sinh: Pham H6ng Yiet Q2 ilCTT DH
KHTN-DHQG Hd Ngi), NguYOn XuAn
Trudng (Fr|CT Vinh Phric), Vlt Ngoc Minh vir
Ph4m Gia Vinh Anh (l2T FTCTT DHSP Ha
NOi), Mai Thanh Hohng lop 12 vd Pham Th6i
Kh6nh Hiep lcrp 11 (PTCTT DH Vinh). Dodn
c5 mOt quan s6t viOn lb thdy gi6o Mai Vin
Tu, chri nhiOm khdi THFrf chuyOn To6n Tin
DI{ Vinh.
IMO2002 chinh thrlc duoc khai mac vho
ngay 23.07 tai h6i trudng Barony cua Trudng
dai hoc Strathclyde 6 Glasgow. Trudc d6,
trong ba ngiry lidn tt20.7 tdi22.7, c6c truong
ctoin cilng Ban td chr1c ki thi soan tliAo dd thi.
Bdi thi bao gdm 6 bdi trOn co sb c6c biri todn
duoc ciic nudc dd nghi (Anh I) nudc cht nhir

:

Sau dAy lh bang diOm cua ciic

Nam.

Bai I Bhi 2 Bei 3 Bei 4 Bei
Ho va tcn
b
1
0
7
7
1 Mai Thanh Hoans
7
7
7
7
2 Neuycn Xuhn Trudng
7
7
7
.,) Pham Gia Vinh Anh
7
7
2
b
7
b

2
4 Pham Hdne Vi0t
tr Pham Th6i Kh6nh Hi0p
7
7
7
U
1
7
7
7
7
7
V[ Nsoc Minh

TT

1

'fheo thOng le IMO kh6ng c6 giii ddng d6i,
nhmg neu tir-rh theo tdng sd didm, doin Vi0t
Nam cting thrl nam (v6i tdng sd 166 didm).
Mudi nudc c6 tdng sd clidm cao nhdt lh Trung
qudc (212 didm), Nga (204 didm), Hoa Kj'
(171 didm), Bungari (167 didm), Vi0t Nam
(166 didm), Hhn Qudc (163 didm), Dhi Loan
(i61 didm), Rumani (157 didm), A, Oo 1t-SO
didm) vd Drlc (144 didm). Trong budi le bd
mac c6ng chfa Anne da den du vh trao huy
chucrng vhng cho ciic thi sinh.

L-

thi sinh Vi0t

5

Giei
HCD
FICV
HCV
HCB
HCD
HCV

Bei 6 TOne so
0

21

0

29

0

35

U

24

n

22

0

35

Tdng so 39 huy chuong vlng duoc trao cho
thi sinh c6c nu6c sau:
6 HCV: Trung Qudc, Ngu ;
4

HCV: Hoa K!

;

3 HCV: Vi6t Nam, Bungari

;

2HCY:Dr1c, Rumani ;
I HCV: B0larut, Canada, Hungari, Hdng
KOng, An OQ, Nhat, HAn Quoc, Na UY, Dli
Loan, Thd Nhi Ki, Ukraina, Oxtralia, Niu
DilAn.

Uf;lt OUN$ TINffi CffifrT
OUON6 Pltf;N fitAE Tffi&NG Tf,III SIAO

q-)Afti,L eflo*

sr fit$f,T&firu

e&-e,

(W

Tmuru& ffioG G{r se}
Tap chi To6n hoc vd TuIdi gi6i bdi T51293 v6i hai cdch glhi, trong d6
c6ch thf nhdt s[t dung dd.n tfnh chat dudng ph0n
gii{c trong tam gi6c dd chtmg minh AB + AC >
ZBC. Ta thdy di6u kiOn 1G I .41 trong gii thidt
ld dd cho N > 2DI vi tam gi6,c ABC khOng cin
tai A. Ndu tam gi6,,c ABC c6 thOm di6u kiOn A^B
< AC thi mudn c5 N > zDI ta chi cdn cho rdng
buOc : IG cit tia MB ld dir. Trudc hdt c6 nhAn
x6t sau

:

NhAn x6t 1. Cho LABC v6i AB < AC. Goi AD
lh duong phAn gi6c trong, Aivl lil dudng trung
tuydn cria tam gi6c d6 thi M ndm gifa C vd D
(h. 1) ThAt vay ta co
> AB BD -)

AC CD
---

!9r!9

.BM
CM
--l

CM CD =

CD > CM ->

hI

niam gifra

C

PHAN THE HAI
THCS Hd htdn Httong,
Qu)nh Lau, Nghd An)

TU (1), (2) suy ra AB + AC > 2BC.

Tit kdt qui bli to6n 1 ddt ra cho chring ta cAu
hoi : Khi ndo thi AB + AC < LBC ? Kdt qui sau
dAy sd trA ldi cAu h6i d5.

Bii toin 2: Cho AABC (AB < AC) Soi I lit

tdm dudng trdn ndi tidp vd G ld trong tdm tam
gidc vd GI cdt tia DC tai K. Cht:tng minh rcing
AB+AC<28C.
Ldi giii : Goi giao didm ctra AI vd AGv1i BC
ldn luot li D vh M. Qua G k6 duong thing song
song vdi DM, c6t ID tai "/ thi.r ndrn gifa I vd' D
theo Nhdn

xil I

nan

A

vh

D.
Goi

1li

tAm dudng trdn nOi tiep AABC

thi

:

AB+AC_
_ AB+AC _
-ttt1r,,

tD BD CD BD+A
6a
Bii to6n | : Cho" AABC (AB < AC). Goi G,l
A_l

_AB _AC

ldn luot ld trong tdm, tdm dudng trbn ndi tidp
tam gidc vd GI cdt tia MB tai K. Chung minh
rang AB + ,4C > 2BC.
Ldi giii : Goi D, M ld cdc giao didm tuong
img cta AI vd AG v6'i BC. TU / ke duong th&ng
song song vdi BC, cit GM tai "I, khi d6 theo
Nhdn

xit l, J nim

giira Gvi M nen

{=
-{-,
ID ]M

llDlvIKC
Hinh 2

AINAG
ID JD- --/
GM

(3)

TiI (1), (3) suy ra AB + AC <28C.
Ta x6t xem khi nio AB + AC = ZBC.

Bii to6n 3 : Cho AABC (AB < AC). I, G ldn
luot ld tdm dudng trdn ndi tidp, trong tdm tam
gidc d6. Khi dd IG ll BC n€h vd chi n€u
AB+AC = 2BC
Ldi

giii

z

IGIIBC <>

Theo (1) didu ndy

N

=z
GM
ID=AG

xiy

(h.3).

ra <> AB + AC =28C.

Phdn tidp theo s6 ta sd khai th6c bhi to6n 3 dd
c6 th6m cdckdt qui kh6c.


NhAn x6t 2.D4t BC = a, AC = b, AB = c
"
c < b, c + b = 2a. Khi d6 : IG ll BC,theo Nhdtt

bcb+c
xitI th\: =*=;='
=

MC

b-a.

2 -'

Suy ra

H1t THt TtN HoC lRE tU1Nq (jilU]yEN
\
^/

vd,DM=DC-

rc = ?oPt
3

b-a
J

BDM
Hinh 3
Ndu ldy a = 3, b = 4, c = 2 th\ IP= 1/3 va day
chinh li nQi dung bdi todnT4l227 trong THTT

,iitqgo Ndu ld! a = 6, b = 7, c =- 5 thi
16 = tl3 vd d6 li nOi dung bdi T41229 trong

THTT IUI996.
Bay gid gqi N 1)r trung didm cira AC' P lb'
t.un! aldtn dua AB. Khi d6 NDC = NNC (lC
chung, NCI =DCI

uitDC=bz

=NC)'Dod6

tn AIP=ACB (4)'
Gia st K ld giao didm cira CI vd AB thi K nim
gifra B vh C (Yi theo Nhdn xit I vd a =
b+c b+b b).Y[Y frp oD frn
_<_
=
=
a1
CID (theo (4)),

phAn girlc dua

Eei

). Tuong

.

.

Lb

(Afi =AD). ril

d6
:
quA
kdt
c6
chring ta

dc. 6iir eqpo vi

Bhi to6n 4. Cho AABC (AB < AC)' Gqi G' I
ldn laot ld trong tdm vd tdm drtdng trdn ndi tidp
to* gid, d6,' i h trung didm AC' ChlnS minh
rdng n€u AB + AC = 2BC thi GICNhAn x6t 3 : Ndu MBC c6
vit

b = 90", AB = 6

Bt = 8 thi hic d6 AC = l0' Tam gi6c nhy
l4i phdn

tta" *a, didu

kiOn b

+ c = 2a. Ttb

ta c6
piran ,i.t dd d6n ogn uai todr. 4 (hinh 3),
frN=Ga = 90o. Day chinh ld nQi dung cfia
bdi to6n T4l2O5 trong THTT 711994'

Hi vong rang c6c b4n c6 ttrf 1in dugc nhidu
tgique th',n vi irong c6c bii to6n kh6c ntra' l'
2

^\

Jt

-

T|,/'N QU0O L/lN THV|VIII
VO KIM THUY
Khai mac nghy LL.IOOL.HQi thi Tin hoc tr6
khong chuyen-tohn quOc ldn thf VIII, td chfc

iai HX NQi; da bd m4c ngdv 3.8'2002' Tham du
hOi thi td 154 hoc sinh tt 49 tinh, thlnh vI
nsanh duoc tuvdn chon ttt c6c cuOc thi co s&' TP
ftb Cni trnint,, Cdn Tho, Tiing COng ty Buu
int Vi6, thOng lh c6c don vi td chfc tdt cuOc
"t
tni Queng Tri li m6t dia phuong c6 nhidu kh6
Hrar-ntuig d6 tien hdnh hoi thi c6 kdlouA tdt'
Ng"ai cdc"co quan Trung uong Dodn TNCS Hd
Cfri Minh, no. fnoa hoc Cong ngho vh Moi
;;drn. n6 ciao duc vi Dro t4o, Dhi Truydn
tinf, Vie,'Nam, Tdng COng ty Buu ch(nh.Vi6n
i-tong, rlQirin io. flet Nim, nam nay ddng td
chrlibn'c6 th0m BQ Tdi chinh' Ngodi c6c phdn
iti nfru moi nim (Bing A cho Hoc sinh Tidu
hoc,bingB cho hoc sinhTrung hoc co sd,bLng
i ctro hoc sinh Trung hoc Phd th1ng, bing D
cho c6c phdn mdm sdng tao khdi kh1ttg c.huy€n
vi bing'E ctro phdn mdm sdng tao khdi chuy€n)
na. ,Iy cOn iO them phdn tti f in hoc vui dd'
tang phhn HOi cho Hai +i vI c6.cuQc Chung
khlo Tin hoc rui' 98 giAi thuong dd^dugc trao'
trone d6 c5 9 eiii Nhdi, l l giii Nhi, 32 giii Ba'
Ooai fdng Cong ty Buu chinh" Vien thong
eihnh eiii Nnet aong d6i vdi didm binh quAn
"os,llt1O. GiAi nhdt bing A ld Nguydn Anh
Turdn,}Jriti Phbng, nhAt bing Bld Nguy€rt Ddng
Vi€t Anh,Tgt Buu chinh Vi0n th6ng, nhat bing
C'litThdn Qudc Ldm, Quing Ninh, Huynh Ly
Thanh Truig, Dd Ndng, Trdn Ngoc Dfic' Tp'

Hd Chi Minh, nhAt bing D Ld Huynlt NguyAt
Thanh, Cdn Tho, Nguydn DdngViil Anh'TCl
Buu chinh Vi6n thOng, Nguydn Ngoc Qu)nh'
Hd NOi, nhdt bing E ld Phan Vinh Long'-Hd
' Noi. Hulnh Duong Quy, An Giang, 10 tudi le
thi sinh ti6 tudi nhdt dat giii.
NEm nay dd thi kh6 hon nhmg kdt qui cua
tang ddu kha, chdt luong hqlsinh khd ddng
"a"
;d;. i# ddu ti6n c6 3 thi Jinh rtung hoc Phd
thOng dat didm tuY0t d6i 100.
D6ng tidc li nhidu tinh midn nfi Bac B0 dA
khOne"tham gia duoc HQi thi' NOn chang' c6
;'rrn"gitt khr"vgn khi,ttt kh6c cho ciic tinh mi6n
n,iira"rirg tto ttan vd didu kien d4y vd hoc
iir frq. ad" aong vion vd khoi day phong trlLo
cho nhffng n6m sau'


MOT SO DAC DIErul
tt I

CUA SACH HINH HOC LOP

6 MOI

PHAM GIA DUC
1. Dac

diim chung cira hinh hoc lop 6

Ffinh hoc 6 (HH6) lh phdn chuydn ti6p

ti

giai

doan hinh hoc bang quan s6t, thuc nghiOm 6 bAc
tidu hoc sang giai doan tiOp thu kidn thrlc bang

suy di6n 0 cdp trung hoc co s0. O tidu hoc, m6i
hinh le m6t chinh thd, bay gid m6i hinh duoc

tao thinh tt m6t sd "b0 ph6n" c6 liOn h0 vdi
nhau vI ngay gita c6c hinh cfing c6 mdi quan
h€ ndo d6.
Hinh duoc hidu theo nghia khdi qu6t vd thdng

nhdt : Hinh ld m6t tdp hop didm. M5i hinh
phing ld m6t tap hqp con cita mit phing vi m6t
ptring lA tap hqp didm cho trudc. Dudng thing
lb mot tap hqp vO han didm, li mot hinh, n6 le
m}t b6 phdn c:iia mit ph&ng. Tt d6 quan h0
thuQc, ki hiou e gita phdn ttl vd tip ho. p, dd bidt
trong li thuyOt qp hqp tr6 thlnh quan h0 duo. c
thta nhAn trong hinh hoc. M0nh dd thOng
thudng didm A ld m6t phdn tft cfia tdp hop a,Y't
hi€u A e a vd doc li didm A thudc dadng thdng
a. Tit cdc didm, ta xAy dung c6c hinh don giin
vi tt hinh don giin ddn ciic hinh phric t4p hon,

d6 lh l6gic phdt tridn cira hinh hoc phang.
Ching han: Doan thdng AB ld hinh gdm didm
A, didm B vd cdc didm ndm gifra A, B.Tam gidc
ABC ld hinh g6m ba doan thdng AB, BC, CA khi
ba didm A, B, C khAng iltdng hdng, v.v...
Tuy nhi6n,,quan niOm Hinh ld tdp hop didm

ngdm hinh thlnh cho hoc sinh. Cfing viy,
quan hO e duoc hoc sinh quen ddn, cbn quan h€
bao hhm (c) thi khOng sir dung mOt cr{ch tudng
minh.
Day hoc hinh hoc 6 l6p 6 khdc vdi day hinh
hoc trong c6c lop tidp theo & ch6 hoc sinh nh6n
thtc cdc hinh vd c6c mdi quan hd n6i tr6n bang
mO ti truc quan vdi su h6 tro cira truc gi6c, cira
tuong tuong. Tuy nhien, tir mO ti truc quan, hoc
sinh phii di d6n khdi ni6m trtru tuong vd hinh
hinh hoc. qua quan s6t, thuc nghiOm, do doan
thing, do g6c, ... hqc sinh hidu duoc ciic quan
du-o. c

he triru tudng trong hinh hoc. Hoc sinh phii
phan biet dugc cili thu6c, c6i gdy vdi doan
thang, cdi mic 6o vdi hinh tam gi6c, hinh vE vdi
hinh hinh hoc. Su phdn bi6t niy n6i l€n tinh triru
tuong ctra hinh hoc lop 6 so vOi hinh hgc 0 Tidu
hgc.

llinh hoc lop 6 dugc xAy dmg theo duong ldi
quy n?p, cung cdp nhftng bi€iu tuong ban ddu,

cdn thidt 0d tridu thdu m6t s6 kh6i ni6m m& ddu
hinh hoc phing, chudn bi co so vfrng chac cho
vi6c chung minh suy di6n o c6,cl6p tidp theo.
NOi dung cira llinh hoc 6 gdm :

- MQt sd hinh don giin vb quan h0 gifia
chfng.
- So do doan thing, sd do g6c vd c6c dung cu
do vd

-

MQt sd tinh chAt li€n h0 gifra cric hinh dd nOu.

2. Circh trinh biy cfra hinh hoc l6p 6
- ttrnh hoc lop 6 duoc trinh bdy theo kidu tidp
cAn quy nap, tU quan s6t, thtt nghiOm, do, v6,
n€u nhAn xdt, di ddn ddn kidn thrlc mdi.
Chdng han : Tir vi6c do doan thing di ddn
kh6i ni6m dO ddi doan iltdng.
- Trong trudng hop c6 thd, SGK ncu tinh

hudng dd hoc sinh khdm.ph6.
Chdng han : C}ro ba didm A, M, B thing hlng.
Do AM, MB, AB r6i so sdnh AM + MB vdi AB.
Cho hoc sinh v6, do, so s6nh, nOu nhAn x6t

trong hai trudng hgp : M nam gita A, B vh. M
kh6ng nim gifa A, B, tU d6 di ddn kdt luAn :

M nim gita A, B e AM + MB = AB.
- SGK cfrng chri trong n0u cdc khi{i ni€m phlr
dinh nhau vi v6 hinh minh hoa d0 rdn luyOn tu
duy thuAn, nghich, ching han :
Ba didm thing hdng vI ba didm khOng thing
hlng.
Didm nam gifta hai didm vh didm kh6ng nam
gifia hai di0m.
Hai tia ddi nhau vI hai tia khOng ddi nhau.
- SGK coi trong vi6c sir dung thhnh thao ciic
cOng cu do, vE, n5i r5 t6c dung cfia m5i loai
c6ng cu d6. Chlng han :.
Vdi thu6c thing, ta v6 duo. c vach thing bidu
di6n m6t duong th&ng. Dd bidu diSn m6t doan
th&ng phAi vE 16 hai mrit.
Vdi compa ta v6 du-o. c duong trdn, chuydn
dugc mQt dQ dei tir vi tri niy ddn vi tri kh6c.
- SGK coi trong vi0c sfi dung c6c c6ng cu
khdc nhau dd giAi quydt cing m6t vdn d6.
Ch&ng han

:

D0 v6 doan thing cho bidt dQ dai thi c6 thd
dtng thudc do d0 dhi hoac ding compa vh
thudc thang.
3


Dd xric dinh trung didm cira doan thing thi c6

thd dilng thu6c do d0 dei hoac gdp gidy.
Dd r,6 tia phAn gi6c cira g6c thi c6 thd dirng
thudc do goc hoac gdp gidy rdi t0 lai ndp gap
bang brit vh thu6c thang.
Chf y rdng viOc sit dung m6i c6ng cu c6 co s&
kh6c nhau. Vi0c dilng thudc do dO ddi, thudc do
g6c dua vlo tinh chh x5c dinh didm trOn tia,
iac dirir tia tron nria mat phing, cdn vi6c gap
giAy,dua vlo ph6p d6i ximg truc.
- SGK rait cliti trong viOc trinh bly dinh nghta
kh6i niOm.

Cho hoc sinh tidp xric vdi kh6i ni6m duo. c
dinh nghia trudc khi dinh nghra n6, tidp xric
bang cdch quan siit qur{ trinh hinh thinh kh6i
niOm d5 hoac bang c6ch quan s6t h)nh vE ddi
tuong d6.

Ching han quan s6t hinh vE trung didmM cia
doan thing AB ta thdy didm M c6 hai tinh chdt :
M nam gitta A, B vd MA = MB, ti d6 YOu cdu
hoc sinh tri ldi cau h6i : Trung didm c[ra doan
th&ng AB Id gi

?

ti, don
vio hinh v6, c_o mrlc d0 triru tuong,

XAy drmg ciic dinh nghia theo kidu mO

giAn, dua

[frai quat thich h-o. p. Chfng han ta dinh nghia
doan thing AB chfi kh6ng dinh nghia doan
thing, ta dinh nghia tam gi6,c ABC chrl kh0ng
dinh nghia tam gi6c.

C5 trubng h-o.p, ta ph6t bidu dinh nghia vd
cing mQt ddi tuong bang nhidu c6ch kh6c nhau.
Ch[ng han

:

Dinh nghia thrl nhdt : Didm O chia duong
thing;y thlnh hai phdn rieng bigt. M5i phdn d6
cirng v6i didm O li mQt tia g6c O.
Dinh nghra thrl hai : Tia AB li hinh gdm didm
A vd tdt ch c6,.c didm nim cilng phia vdi B ddi
vdi A.
Dinh nghra thf ba : Ilinh tao thhnh bdi didm
O vi phdn duong thing chrla tdt ca c5c didm
nam c[ng phia ddi vdi O l] mOt tia gdc O.
Tuy nhiOn SGK khOng vidt ti6u d6 "dinh
nghia", chua yOu cdu hoc sinh xAy drrng moi
dinh nghra phAi thoa m6n tdt cil cdc yOu cdu
lOgic. Trong cdc kh6i ni0m ctra Ffinh hoc 6, mOt
sd kh6i ni6m ban ddu duoc nOu ra 6 dang mO ti,
dua vho hinh v6, cbn mOt sd kh6i duo. c dinh
nghia bang ldi dua tren c6c khrii niem dd bidt.

- Trong hinh hoc 6, c6.c tinh chdt duoc di6n
dat chinh x6c, nhung don gi6n, 16 ring, it ding
nhfng thuAt ngfi todn hoc kh6 nhu tdn tai, duy
nhai, ba't ki, xdc dinh...
SGK quy udc : Khi n6i "cho hai didm (dqong
thing, tiai' md khOng n6i gi thOm thi ta hidu ld
4

"hai didm (ducrng thtng, tia) phAn bict".

Sir

dung quy udc d6. SGK kh6ng vidt : "C6 mOt v)
chi mOt dudng thang di qua hai didm phAn biOt
cho tru6c" mi vidt : "Cd rnOt vi chi mOt dudng
thing di qua hai didm A, B" . Qtty udc ndy chi stlr
dung d ldp 6.
* SGK coi trong hinh v6, xem kOnh hinh c6
t6c dung gay bidu tuong, tri tu0ng ttrong kh6ng
gian dd thuAn loi trong viOc nhln thrlc kh6i niOm
hinh hoc triru tuong. Do d6 trong s6ch c6 nhidu
hinh v6, c6 hinh v6 kdm theo chri thich.
- SGK Ffinh hoc 6 cdu tao bhi giAng theo $,
m6i $ mert tidt (cA li thuydt vh thuc hnnh)' Ri0ng
"$5. Tia" (chuong I) vh "$6. Tia phAn gi6c cira
g6c" (chuong II) c6 thOm mdi $ mQt tidt luyOn
tap. Sd bii tap o m6i $ chir ydu nham cirng c6
kidn thric, rbn luyOn c6ch ph6t bidu chinh x6c,
nhdn dang vd thd hiOn khdr niOm. C5 chri trong
loai bhi tap ren luyOn ki nang do, v6. Cbn ciic

cau h6i, bli t4p li6n h0 v6i thuc td, rlng dung
vdo thuc td, cflng nhu c6c bhi tap tdng hop c6
chrit ft dlo sAn, tirn tdi, khai th6c duoc trinh bhy
nhidu hon trong siich bii tap.
S6ch gi6o khoa, s6ch bhi tap, siich gi6o viOn lh
mOt bo s6ch hoin chinh. Bli tap trong si{ch bdi
tap khOng tring lip bdi tap rong SGK nhtnrg
duoc xay drmg tuong tu, ld su bci sung hhOng
thd thidu cho SCX vd mat s6 luong bai tap, vd

dang bli tAp, vd loai hinh tu duy. S6ch bli tap
thuc hiOn yOu cdu phAn h6a hoc sinh, nAng trinh
dO hoc sinh til trung binh l€n kh6. gi6i. M6i bli
tAp ddu kdm theo huong d6n, hoic d6p so, ho4c
frintr vC minh hoa, holc ldi giii t6m tat.
3. Nhirng chfi

f

khi day llinh hoc ldtp 6

SGK vidt theo kidu quy n?p, dting trinh tu lOn
lop. N6i chung, gi6o vi0n c6 thd hudng d6n hoc

sinh stt dung SGK, girip hoc sinh tu hoc, nOu
thSc m6c, ph6t bidu, tranh luan. Gi6o vi€n ldm
trong thi, goi y, chdt kidn thfc.
Gi6o viOn ra xen bhi tap dd cttng cd ttmg
phdn. Hdt m5i tidt hoc, giii quydt xong khoing
75qo s6 bhi mp trong SGK. Hoc sinh vd nhh lhm

tidp sd bei tap cdn lai ctra SGK vd nhu vAy li dat
yOu cdu d4y hgc doi v6i hoc sinh trung binh.
Gir{o vi6n c6 thd hudng d6n hoc sinh lim thOm
bhi tAp trong sdch bdi tap nh[m nang cao ki
nang giii to6n, cilo sAu, khai thi{c vd mat li
thuydt, v1n dung ki0n thrlc vdo thuc td.
Gi6o viOn tu chgn phuong ph6p d4y hoc thich
hqp vdi moi ddi tuong hoc sinh, dAm bAo mgc
dcu bAi giing, kich thich duo. c himg thri vI hoat
dOng hgc tAp c[ra hoc sinh.


so tw'i lfi{ GIAI
iln^\it*/Y
TrrI TUYIN SINrrn$HQC KrrUANlUn 2A02
mCIr

(Tidp theo ki tru'dc)
DANG THANH HAI _ NGI"TYEN ANH DIING

Cdu IV. 1) Cho hinh ch6p tarn gidc ddu
SABC dinh S, c6 dQ dai canh dciy bdng a. Goi M
vd N ldn lum ld cdc trung didm cila cdc canh
SB vd -(C. Tinh theo a di€n tich tam gidc AMN,
bi€t rdng rndt phdng @l4N) vudng g6c vdi mdt
p!fing (SBC).
Hudng dAn giii. Vi SABC li hinh ch6p tam
gidc ddu nen ASAB = ASAC * AM = AN. Goi F
l-e tru,rg didm cira BC. SF c1t ivIN t+i E (h. 1).
Ta c6 MN li duong trung binh cira ASBC cdn

vi E li trung didm cia MN. Vi AAMN cAn vdi
ddy MN nOn AE L A,IN. Lai do mp(AMN) I
mp(SBC). Suy ra AE I SF (1).
s

Cdch 1. Tinh theo cOng thrlc

MN.AE
^tvtt\222416

I A 'JTO

St = aho:2
O, JIO

Cdch 2. Tinh theo cOng thrlc hinh chidu S'=
Scoscr. Gqi H lh tam LABC ddu vA My, Nlldn
lugt lh trung didm cta HB,IIC.Kht d6 MMrNl
li hinh chidu cira LAMN lOn mp(,48C). Qua A
k6 ducrng thhne dllBC * d ll MN th\ d li giao
tuydn c[ra hai mat phing (ABO-vit (.AMN), mit

AE L d vh AF J- d ndn g6c EAF lir g6c phing
nhi diQn tao bdi 2 mat ph&ng @BC) vit (AMN).
Theo cOng thrlc tinh diOn tich hinh chi6u ta c6 :
seaz,,*1

=StMN..ordAF =

AE

Mflt ^,,E :_2J1 o a5Jl

=
ntYtt\
AF 2
-.-.ru-r

Jtttxt

-<

=

Soyt,.#
.ito

4

12

o'{to (dvdt)
.-..
16

Ciu IV. 2) Trong

khdng gian vdi h€ tqa

Dicac vuilng gdc Oxyz clrc hai dudng

i

.. ------_ i
----H\ri
rl,

\

r
['r'=l+
lx.-zv+z-4=o
-r
vh A-r: 1t,=2+t
At:
' 1'
lx +2v_22+4=0

l'r=t+Zt

M,9

a) Vi6t phuong trinh mdt phdng (P)
dudrtg thdng Ai
B

'IU

(I)

LE-=SM

=!
sFsBz

I

=

cd dd ddi nhd nlfi't.

t;
a:1
.

2

Xdt ASBF vudng c6 Srt = SB2 * BF2 = SA2

o

2

BF2-

-

a-

L )SE=

2 -SF= Jz
zJz

X6t ASAE vuOng cd AE2 =
3i oz 5ar
uJto
.."..............._--=-

4884

anp

=

dudrig

b) Cha didm M (2; I ;'4). Tim toa d9 di€'m 11
thuac dudng thdng A2 sao cho doan thdng MH

Ql

(2) suy ra SA = AF

vd song song vdi

chil'a

thdrug A2.

Hinh

Mit kh6c

dQ

tfuing

SA2

- Sf

Huong d6n giii: a) Cdch 1. Slr dung cich
thiei lQp PT chilm rndt phdng. Vi mp(P) di qua
A1 nOn n6 thuOc chirn rnlt ph&ng chila hai mit
phing c6 giao tuydn lh Ar. il mat phing (P)
"Q
d4ng 7"(x-2v+z-4) + lt(x+21t-)7a4) = O (t-2 + pP
* 0) hay (},+p)x + (-2L+2,p")y + ()"-2yt)z -

$'+4p'=g
=

i = (l'+p,
A2
c6
vecto chi
thing
Duong
-2tu+2trt,-i,,-2p).
phuong u = (1,1, 2). Theo gii thidt Lzllmp(P)

M4t phing (P) c6 vecto ph6p tuy€h

-r
5


n6n z .u = 0,suy ra I + p - 2?u + Zyt + 2)" 4yt
= 0 hay ),. = tL (* 0). Thay vio PT mflt phing (P)

-

taduo.c2x-z=0(1)
Luu j, 1. Chring ta chua thd kdt luan ngay
mp(P) c6 PT 2x - z = 0 vi chua bidt A2 c6 thuOc
mp(P) khOng ? Thay x = I + t, ! = 2+t, z = I +
2t vdo vd trdi [rf (l) ta duo. c 2(L + t) | - 2t =
1 + 0 (Vt), didu niy chrlng tb A,2 e mp(P) hay
Lz ll mp(P). Vay mp(P) c6
2x - z = 0.
Cdch 2. Xdc dinh tqa dQ vecto phdp tuy€h cfia

-

PT

mp(P).

tqi A, phuong trinh

dudng thdng BC ld

{Z*-y-J1 = 0, cdc dinh A vd B thuxc truc
hodnh vd bdn kinh dudng trbn ndi ti?p bdng 2.
Tim toa dd trong tdm G cila tam gidc ABC.
Hudng d6n giii t Cdch 1. Vi B lh giao didm
cliua BC v6i truc holnh n6n B (1, 0). Mat kh6c do
duong thing BC c6 PT y =
vdi h€ s6

Jix-Ji
.6 ruy ra trong MBC thi I = 60",
e = 30o, i =90". Gia srl A(a, 0) 3

g6c bang

c @,,r3a-J1 $.2)
C

(*, r,
[3' :)

Duong thing A1 di qua

chi phuong v

vh c6 vecto

,l I ,
=(l-,
lll)=
I I z -4'l-z

,l

1l

(2,3, 4).

(I, 2, 1) c6 v6cto chi
1, 2). Khi d6;=
[;,;] =

Duong thing A2 di qua B

phuong

i

= (r,

*= -t * o - ;, ;,

(-2, 0,1) . Ta co
li,;)

aE

W,ong ddng phing,

nhau. Mp(P)

[;, ;]

=

tt

d6 A1

vi L2 ch€o

di qua o r1,0,:'l vh nh6n

[3

3,/

e2,0,1) llm vecto ph6p

tuydn,

il =

tt

(P).li 2x - z =0.
b) Cdch l. MH (H e Lz) nh6 nhdt e MH L
Az. DWrg mp(Q) di qua M (2, 1, 4) vi vuOng
g6c v6i A2 + mp(Q) nh4n u=(I,I,Z) llm vecto
phdp tuydn. Do d6.mp(Q) c6 PT

l(y- l)+2(z -4)=0hay

x+y+22-ll=0.
H li giao didm cira A2 vi mp(:Q) n0n toa dO
+

I

H

lI

x=!+t

1.,:r*,

nghiem cira h0

m {' .

-

lz=l+?t
lx+y+22-LI=O
Giai hc ndy tim duo. c roa d0 H (2,3, 3). Khi
d6MinMH = 16.
Cdch 2. Do H e A2 non H(l + t,2.+t, 1, + 2t),

MII=e-D2 +(r+ t)2+(2t3)2 = 6tz - llt +11 > 5 ) MH > Ji
Ding thrlc xiy ra e, = 1, hic d6 H (2,3,3).

re R.Tt

(tB)

Tac6AB=

r[cote-:+c*s;)

hay

dAy

suy ra dugc PT mp

l(x-Z)

Hinh 2

d6

CAu V. t ) Trong m\t phdng voi hA Qa dQ
Dicac vu1ng g6c Ory, xit tam gidc ABC vudng

la-ll=2(l+,,1t
Y6i a

- | = ze*Jj) thi A( 2J1$, o),

B (1, o), c
thrlcG

[33)

o+2.,6)

3

3 )

-tt-----;-t-----:-|.

Ydi a
B (1,

Ap dung cong

(xo+xa+xc. )a+)a+)c) .6

^ (t*+Ji

'[

QJi$,a+z$).

- | = -2(l + .,6) tr,i AezJl -l,o),

o), c ezJ,-r,-6-zJt

rric d6

o"(t-+11

"'[ 3 _ryr[)
3 )
Vay bdi to6n niy c6 hai nghiOm.
Cdch 2. L4p PT duong phAn gi6c trong g6c A
crta MBC
lr = -x + a L4P PT UH,

:

,#,

gi6c trong g6c B cia
Ttr day x6c dinh
ti€o L,ABC

,u

duo. c

MBC, y =
;r-T.

toa d0 mm duong trdn nQi

a-I s,i ouns
,l.g-.
sia
'I t*.6 ' t+J3
).

-


thi6t d (1, AB) = d(l, Ox) = 2, ta tim du-o. c a, tir
d6 tinh du-o.c toa dO c6c didm A, B, C vd toa d6
trong tAm MBC.
CAu V. 2) Cho khai tridn nhi furtc

( r-r

-*

lz, *z'

t

( *-, )"-' ( ., \
)'
+Qlz,
lrt
| =c,olr, |
(

)'

*-r

,l (.,
(.,r)(

r)'?-r

I
l, ][,,

l

D0 ra

tim n vd x.

ki

nity

din giii. TI
(n-l)(n-Z) _J

di6u kien Cj =5Ct,

)

- tt = 7. TU didu ki0n

6

/ ,-r)a( -,)3
r-'lr
II
w7 lL : iI ir:

lL

t ,1,

l

=
-

140

<> Z' 2

-4e"=

4.Y4yn=7,x=4.
Xin gi6i thi€u m6t s6 bii

BT1. Cho hinh ch6p tam gi6c ddu SABC dinh
S c6 ci4c g6c phing tai dinh S ddu bang o (0 < o
< 90o), canh bdn hinh ch6p bAng b. Goi M, N
ldn lucrt li trung didm cua.tB, .tC. Tinh di6n tich
tam gi6c AMN theo b vb. a.
( 9-7 coscrXl

"Ji
- vdb = :':
32

hoc

:7'j.%o.

(h.I)

ta c6 kdt

qui ciu IV

1).

Seu1w1=SAMN.cosB. Theo

SE_SASE

BC SF Sasr

AE.sin(cr-9)
AF.sincr

sinA.FE sin(cr-P) _ sin(o-P)

--------:-

slncr

it ngudi thich hon :

tic

gih, Gi6i thiOu cdc trudng,
Tin hoc, Vui mOt ti, CAU lac b0,

Guong mat ciic

2. Ffinh thric trinh bhy cira tap chi :
587o cho ll dep, 33% cho lh trung b\nh,ll7o
cho li xau.

3. Ban doc thudng c6 tap chi vho cuoi thdng
mi tap chi phrit h.irnh : J9o/o, c6 5Va trh ldi lh ddn

sin(cr+F)'
Q{em

ileb tang 9)

4. Con duong brio den vdi ban doc dd rdt da
dang. Ti l0 dat b6o 0 Buu di€n lI 57Vo, mua o
cdc cka hhng si{ch ld 23Vo, sd ban doc mua il
ciic dai li, cdc sap b6o via hd ld 20Vc.
5. Trong sd phi6u giti vd c6 75Vo lh nam

vi

s0

hoc sinh phd thong trung hoc gui phieu vd
chi6m 507o sd phidu nhAn duoc.
vdo

BT2. Cho hinh ch6p tam giiic ddu SABC, canh
dr{y bang a, edc canh b6n nghi€ng ddu trdn dr{y
m6t g6c cr. M6t mat phing qua A, song song vdi
BC, tao v1i mp(ABC) mOt g6c B (0 < F < o) cit
SB, SC tai M, N. Tinh di6n tich tam gi6c AMN
theo a, a, B.

Gqi i, Sidi.
DL Ta-16t

:88Va,

gi6i :847o, Chudn bi thi vlo
dai hoc :79Vo, Ddnh cho Trung hoc co sd :
76Va,Didn dln day hoc toiin :727o, Giii tri todn

6. Cdc y kidn dd xudt ctra doc gii tap trung

-coscr)

8
1

:

thi:lug sau mdi co biio.

tAp li€n quan ddn

c6c dd thi tren dd c6c ban luyen tdp.

b2
.c
, JAMN _

:

:90ai, Giai bli ki trudc

Cdc chuyOn muc duoc

Hudng

sinAEF

Thring 4-2002. tap chi Todn hoc vh Turji tr6 da
td chric tnmg cdu f kidn ban doc. Ngay trong
tudn ddu tiOn dd c5 phidu tham dd g&i vd. Kdt

Circ dd thi hoc sinh

(n ld s6' rtguy€rt duong). Bi€i rdng trong khai
tridn d6 C?, = 5C1, vd s6'lrung thrt u bdng 2An,

MrNr _

i KIEN BAI.I IItlG

1. Cdc muc duoc ban doc y6u thich nhat lh

I

=

THIM Do

quA cira cdc phidu tham dd cho thay

(-.)'

+"'* r,,tl.>1
llr. | +c;lzr
u,,
t" li." .,l ',,[- )

coscr,

TTET QUA

:

- Tang bli cho chuy€n mucTodn hoc vd Ddi
sdhg, Lich sfrTodn,Tidu s* cdc nhd todn hoc.
- ThCm c6c bli vd phuong phdp vh kinh
nghiOm hoc toiin.
- Gi6i thieu thOm vd toiin hoc hi0n dai
- NOn c6 cdc cAu d0 vui, thong minh
- Gidi thiOu cilc guong mat hoc sinh gi6i.

- Duy tri h6p thu giao luu gifra tda soan
vidt, ban doc

vi

ban

- C6 bai cho hoc sinh lop 6lop 7
- Dd ra ki nhy c6 mOt sO bhi cbn d€
- Tang them cdc dd thi to6n vd c6 hu6ng d6n
gihi cdc dd thi vlo ci{c trudng chuy6n, dd thi cria
ci{c dia phuong.
- Brim sdt chuong trinh phri thOng mdi, thOm
c6c dd thi trSc nghi6m.
- Gidi thieu s6ch hay vd toiin.


7" CAu hoi dd nghi nCIu tdn c6c titc gih cdc

bli

bdo trong Tlim mi ban thfch d6 bi hidu nhdm
li cdc tric giA sr{.ch to6n khdc nOn thdng tin nhAn
duoc cira cAu hdi ndy chua that chinh xdc. C6
50 tdc giA duoc neu ten. Cdc tdc gii duoc nhac
ddn nhitiu ld Nguy€n Minh Hd. Nguydn Ccinlt
Todn, Dartg tling Thdng,ltil Dinh Hda, Ng('
Y iit Trung, N gr.rydn Vdn MdtL, V' r'i Thanh Khi0r'
Hadng Chting, Ngu1,6n Drlrc Tdn, Nguyin Dutia
Phd't, L0 Tho'ng Nh.i't, Hd Quang Vinh, Vii Kim

Thtly, Ngtty1n Viit Hiii, NgO Dat Ta. Diittg
T lrcrrlr Hai, Nguviu Mitth Dttc,...

c!;Qc THAM Do
I'lhin chung ban doc hli ldng vdi tap chi v)
nhi€t tinh c6ng tdc. Cdc ch,*y€n muc cfta tap chi
v€ co bin da ddp ring duoc yeu cdu ctra dOc gii.
CA nOi ciung vi hinh thfc cira tap chi lh tuong
doi phn hop vdi mot tap chi phd bidn kidn thrtrc
r'rH[NG rHr ruArrr

sc

BQ $AU

toiin hoc so cap.
Ciic chuy0n muc n6n thr"rong xuy0n xudt hi0n
siuh giai, Ddnh cln Trung
ld : Cdc tli' thi hoa
,!
lLoc Co so,Tim hiiu sdu ll$nz todn koc so cd'p,
ia, aA ili hoc sirh gidi ctia cdc dia pfutttng,
Chudn bi thi vdo Dai hoc, Gidi tri todn hoc va
tdt nhiOn 2 muc thubng xuyen ld D€ ra ki nd-v,
Gi(;i bdi ki truisc.
Cdn tang thOm c6c bli vd Lich st? todn hoc,
Tidu sti cdc nhd todn hac, tng dung todn hoc,

Trong ki thi Olympic to6n quoc td ldn thrl42
nam 2001 c6 bii toiin chtrng minh bat dang thtlc

(BDT) do HIn Qudc dd nghi nhu sau :
Bii to6n l. Chung minh rdng :

ab
J o2

,

+8bc

(l)
J b2

+gro

,lrz +8ob

cho moi s6'tltilc duong a, b, c.

Ldi
-r

-*-S>r
giii bii niy
dd dang tren THTT s6 296

(212002) tuy nhiOn c6ch ddnh gi6 bat dang thfc
c6 ve khOng tu nhi0n. Tba soan nhAn duo.c c6c
ldi giAi khi{c vi cr{c bii toi{n md rOng cira ci{c
tdc giir: Trdn Xuin Dring (GV THPT chuy€n L€

Hdng Phong, Nam Dinh) vd Pham Thinh (Xi
nghiOp sira chfia cdc c6ng trinh Ddu khi - Vfrng
Tiu). Xin gidi thi6u cirng ban doc.

I. (cira ban TrdnXudn Ddng)
Cdch girii I BTl. Dit vd tr6i c[ra BDT (1) le
T, rip dung b6t ding thrlc Bu-nhi-a-cdp-xki hai
PhAn

ldn ta duo.c

:

(a + b +

c)2.

<

Tin tttc lioar dAtrg tadn hoc. Diin ddn da;;
hoc to(tn.

Cdu lac'bd c6'n vui, di dtlm hon vh phii hoo
v6i hoc sinh ham thich to6n. Muc Sai ldm d duu
cdn chon bdi c6 nhftng sai ldm phei bicn hoac
tinh vi m) ngudi doc kh6 phrit hiOn hon.
C6 thOm bdi cho hoc sinh ldp 6, lop 10. Tang
cdc dd thi hoc sinh gi6i kdm rheo cdc ldi giii,
dac biOt ld c6c dd thi vdo cric lop chuyen THPT.
Tirem'c6c dd thi tric nghiOm Od froc sinh llm

quen v6i c6ch ra dd thi trong tuong iai.

-J;'t7.srbr)
< TJ@+b+c)(a+b+c)3 =T(a+ b + c)2
Ttr d6 suy ra T > l.
Ding thtlc xiy ra khi vh chi khi a = b = c'
Cdch gidi 2

O Uia 4 sE dang xen kE vi0c gi6i thieu cdc
tnldng vdi cdc bii khdc. Vi0c nOu guong cfc
don vi gi6o duc ti0n tidn li didu cdn thiei, clong
vi6n ci{c trudng vh lim cho hoc sinh chim hoc

,=!

toiin hdn.

todn?:

Ctin ltru y viOc grii tap chf di midn T'rung r"i
mi6n Nam dfrng ngAy.
Tidp tuc t}n tbi dd cii tidn ciich trinh bdy bia
vh hinh minh hoa cho deP hon.
THTI xin chAn thlnh c&m on ciic ban doc dd
nhi€t tinh cOng tlic vdi Tap ch( v) hi vqng tidp
tuc cluo.c su ung hO cira ciic Lran trong ci nudc.

THTT

BT1. Dat x =

bc
^

at

ac
,J

1

t

b"

thix, y, z > 0 vh *y, = l.

C

Ta chuydn viOc chtrng minh BDT

(l)

vd bdi

Biri to6n 2. Chtmg minh rdng

--l---!--!=t
Jt+8x t+sy ./t+82

(z\

.i

trong d6 x,

!, 2 ld cdc sd thuc duong thda mdn

xyz = 1.

Loi gitii. Ap dqng BDT Co-si ta c6

x+y+z>31[ry2=3

:


MWrMsMrereffiwffinnTqffiffi&T$
Til d6 suy ra :
(1+8x)(1+8y)(1+82) =

|

+ 5!2ry2 + 8(x+yt'z)

+64(xy+yz+zx)>729=36
Ap dUng BDT CO-si ta c6

Jt+8r.+Jl+8y

rnAN xUAN OANC - PHAM TI{INH

31p;* =l

xy+yz+zx)

*Ji*9,

(3)

Phdn II. (ctra ban PhamTliinh)
C6 thd khdi qudt h6a BT1 rhlnh hai bIi todn
dudi day vi sir dung ciich chfug rninh tuong trr
nhu 6 BT2 tren.

:

>

Bdi to6n 4. Chitng minh bdt ddng thirc

>:@>o

(4)

Bidn ddi BDT (2)

...*-=3"1 ,12 -l
q+--

:

e

(Jl+8x Jl+8y+f+8y.d+&+
+ .G8; Jl+8x;2 > 1t+sr;11+8y)(1+82)
(2)

<+8(x+y+z)+z@,
. (Jr*r'*++8y+Ji+&)

>

sro

(5)

Thay thd (3) (4) vho v6 tr6i cira (5) ta thdy (5)
dfng
= (2) dring.
x = y - z = l.
Ding thrlc 6 (2) xity ra
Tdng qu6t h6a bii todn2ld bhi to6n sau:

e

Biri to6n 3. ){et bidu thitc

oz

l', ,2-I
' ,lai,+--la,

zI

(6)

trong d6 ae a7, ..., an ld cdc sd thuc duong th6a

mdnar.ar...a, = l.
Bdt d&ng thric (6) tuong ducmg vdi

E_*l_L_*.*i4
>1(7)
'4;7" lo,-u

rl4.rr-r.1

trong d6 bp b2, ..., bnld c6c sd thuc duong th6a
mdn brbr... bn

= l.

Biri to6n 5. Ctlnrg minh rang :

"-111

'
v{.tr tlt.-k,
'lt*k*

trong dd x,
z ld cdc s6'thuc duong thda mdn
p

_1-a-

-

!,

xyz = 1, cdn k ld tham sd daong.Vdi gid tri ndo
crta k thi bidu thtc E c6 gi6 tri nhd nhdt vd tim
gid tr! nh6 nhdt d6.

Ldi gitii. o V6i /< > 8 thi chrlng minh tucrng tu
nhu & BT2 ta c6 gi6 tri nh6 nhdt cira E li
-+,3 gii{.. tri n}y dat duo.c khi I = y = s = l.
./1+

trong d6 a1, a2, ar lit citc s6 thuc duong th6a
mdn ararar= l.
C6 thd md rQng BDT (7) vI (8) ddi vdi n sO
thuc duong ap a2, ..., a, thba mdn arar...an = L,
cf,n bdc rn vi mdu s6 cfia ph6n sd trong cin lh

ai*nm-1.

/c

o V6i 0 < k < 8 thi cfing b6ng c6ch chfng

minh tuong tq nhu bBfz ta ddn d€n E > 1 (v6i
moi x, y, z duong th6a min ryz = l). M4t khdc
cho y = z = n

vd.

*=

in

vdi

n e M th6a mdn

di€u kiQn dd bAi thi

_111
"n- l--{ ' ,lr+k, ' .'Pa
L

,/I*
Y'I ,

Ta

l-

!

--

ru0t s0

[m i

-

c6 limEn = 1. VAy vdi 0 < /c < 8 thi E
tl-rc,

kh0ng c6 gi6 tri nh6 nhdt.

cliloi

T$1... 6iep ffarg7)
BT 3. Cho hinh ch6p trl gidc ddu SABCD,
canh d6y bang a. C{c mat b6n nghi€ng ddu tren
d6y mOt g6c cr. MOt m[t phing (P) qua AB vir
tao vdi mp(ABCD) m6t g6c B (0 < B < u) c6t
SC, SD ldn lugt tai M vd N.
a) Hdy x6c dinh dang ctra tir gi6c ABMN.
b) Tim diOn tich ttt gi6c ABMN theo a, cr, B.
DS : a) ABMN li hinh thang c0n,
b) Smuu =

KHt

a2 sinz cr.cosp
sin2

(o+B)


@@ LIff{ sC wwt wnw wnt

${tq,frt

ylw sd

LE THI HOAI CHAU
(KhoaTodn trudng DHSP Tp. H6 Chi Minh)

"Him sd" (tidng Ph6p : fonction, tidng Anh

:

function) c6 gOc la tinh ld functio (ldy tit fungi,
nghia lh "thuc hi6n"). Ngudi ddu tien ding thudt
ngtt fonctior ld Lai-bo-nit (Leibniz), vho nim
1673. NIm 1734 O-le (Euler) dua vdo ki hiQu/
vh cic ddu ngo6c. Nam 1797 La-grlng
(Lagrange) ding kf hiQu/x dd chi hhm sd ctra
bidn ,r, nhung stl dung ci{c ngoac khi vidt flf)
hay fla + bi dd chi hdm sd cira I hay cira

a+bx.

Gi6ng nhu c6c kh6i niOm to6n hoc kh6c, khi{i
ni6m him sd kh6ng duo. c hinh thdnh trlc thdi mh
ld ket qui ctra m6lqu6 trinh dhi ph6t tridn. BIi

vidt niy didm qua nhfrng mdc quan trong cta
qui4 trinh hinh thirnh khdi niOm hlm sd.

l.

Thdi Cd dai:

d

thdi ki niy ngudi ta

d6

nghicn crlu mOt sd trudng hqp v6 su phu thu6c
lin nhau gifra hai d4i lugng. SU phA thuilc duoc
thd hian *€n cdc bdng s6', cho bi€i ma'i lian h€
girta hai t,ip hqp hfru han, rdi rqc. Ching han

ngudi Ba-bilon (Babylone) di lAp du-o. c bing
binh phuong, bing can bAc hai, bing lAp
phuong, bAng c[n b6c ba, bing cr{c bO 3 sd
Pi-ta-go ... ; ngudi Hy Lap thi c6 bing Sin.
2. Thdi Trung dai : D6y le thdi ki mi ngudi
ta tim cdch dinh luong mOt sd hiOn tuong nhu
nhiOt d6, mAt d6, vAn tdc. .. Cdc ludt ctra tu
nhiOn b6t ddu duo. c nghiOn crlu nhu mOt dai
luong nIy phu thuQc dai lqgng kia. Vdi su ph6t
tridn cira khoa hoc chiu Au 6 thc ki 14, kh6i
ni0m dai luong bidn thiOn vi kh6i ni6m hhm sd
ldn ddu ti0n duoc di6n dat m6t c6ch 16 rdng

(dudi dang hinh hoc vd co hoc). Nhmg, cfing
gi6'ng nhu thdi cd dai, m6i trudng hW phrl
thudc cu thd gifra hai dqi luqng d€u duqc dinh
nghia qua mil sa md td bdng ldi hay bdng so dd
chtl chta phdi bdng m1t cbng thrtc.
Ching han, liOn quan dOn vAn tdc, O-ret-smo
(N. Oresme, 1323 - 1382) dI dua ra mOt ph6p
chrmg minh bang hinh hoc cho kdt qui sau :
trong mOt thdi gian x6c dinh, m6t vAt chuydn
dOng nhanh ddn ddu sE di du-o. c m6t qudng
duong b[ng qu6ng duong ctta v6t thrl hai
chuydn dOng ddu vdi vin tdc bing trung binh
c6ns c5c vAn tdc 6 hai ddu mrit ctra vAt thrl nhdt.
Od Jhring minh, Oresme bidu di6n thdi gian trcn
m6t dubng nam ngang AB vd v6n tdc trlc thdi
tr0n du&rg vu6ng g6c AC. Ndu D ld trung didm
10

ciua AB, F li trung didm cira AC, thi sd do di6n
tich hinh cht nh6t AFGB bang sd do quSng
duong mi vAt thrl hai di duo. c, vi AB x DE lit
tich ctra thdi gian vi vin'tdc. 56 do diOn tich cira

tam gi6c ABC tuong rfuig v6i qudng dudng vAt
thrl nhdt di duo. c, mi diOn tich niy bang dien
tich hinh chfr nhat AFGB.
3. Th6 ki 16 vir
L7 z Giai doan nhy

c6 nhidu su kiOn

quan treng anh
hu&ng ddn qu6 trinh

hinh thdnh

!

kh6i

ni0m him trong

E

to6n hoc. Tru6c hdt
li su s6ng lAp ra nghnh Dai sd v6i Vi-et (Vidte,
1540 - 1603)dA cho ph6p ghi m6t bidu thtlc bao
gdm ci cdc dai luong d6 bidt vd chua bidt. Torin
[oc duoc nh5n thrlc nhu m6t ngOn ngfr 0d Oldu
di6n thuc ti6n vdt chdt cfia thd gi6i tu nhiOn.
Vdo cudi the'ki 16 vd ddc bi€t trong the'ki 17
cdch bidu di6n hdm sd bdng cdng thrtc vd bdng
bidu thftc gidi tich bdt ddu thdng th€'. Lop cdc
him s6 giAi tfch duoc bidu diOn mOt cdch tdng
qu6t qua tdng c6c chu6i vO han nhanh ch6ng tr&
thhnh lop hdm duoc sttdgng rOng rdi.
D6-c6c (Descartes, 1596-1650) vi Phec-ma
(Fermat, 1601-1655), hobn todn dOc lAp vdi
nhau, dd 6p dung phucrng ph6p dai sd vio
nghi€n cuu hinh hoc, xAy drrng n6n ltrnh hoc
giii tfch, vi bang c6c phuong trinh dd hinh dung

duo. c su phg thuOc l6n nhau ctra hai dai luong

bidn thi6n. N[m 1670 nhi b6c hoc Ba-rdu
(Barrow), dudi dang hinh hoc, dd xdc l6p tinh
d6o nguo. c qua lai gitta cdc ph6p tinh vi phAn vi
tfch phAn (dl nhiOn khOng sit dung cdc thu0t ngf
nhu vdy). Nhtng di6u trOn chung to 1I thdi d6
c6c nhb bdc hoc d6 nhAn thrc rdt 16 ring kh6i
ni6m hhm.
Kh6i ni€m him cdn duo.c o Niu-ton (Newton,
1642-1727) n€u ra dudi dang co hoc vh hinh
hoc trong cuOn "Ph6p tinh vi ph6n vI c6c chu6i
v0 han".
Newton chon thdi gian li mOt khdi
'

vi giii thich nhtng bidn phu
nhftng dai luong sinh ra til d6 theo
mOt c6ch thrlc 1i6n tuc.
Song song vdi Newton, Leibniz (1646-1716)
dd phdt tridn ph6p tinh vi phAn vd tich phAn
t :
t./
ni6m phd bidn

thu6c nhu

li

tt

hinh hoc c6c dubng cong. Chinh 6ng ld ngudi
ddu ti6n dirng thuAt ngfr "him sd" (fonction),
v6i 1f nghia : trong bhi to6n x6c dinh toa d0 cfia
mot diem th6a mdn m6t tinh chdt nio d6, hlm
sd ld nhfrng doan kh6c nhau, c6li€n h0 vdi m6t
duong cong nlo d5, ching han nhu tung d6 c6c
didm ctra n6. Cfrng chfnh li Leibniz ld ngudi dd
dua ra ci{c thu6t ngt "hdng sd", "bi6n sd", "tham
sd", "toa dO".

Dinh nghia ddu ti6n vd hlm sd gdn gfii v6i
quan didm hi0n dai lh cua Bec-nu-li (Bernoulli
Johann I, 1667-1148) : "To goi hdm s6'cr)a nfit
dai laong bieh tli€n ld mdt dai ltong clildc tao
n€n theo mot cdch ndo dd tt dai ltong bi€it
tlti€n ndy t,d tit cdc hdng sd'. Trong chidu sdu

cira dinh nghla chua thit hodn chinh dy lI f
tu0ng bidu di6n hdm sd bang mOt cOng thrlc gi6i
tich. Nhrmg duhng nhu khong phii Bernoulli d6
hidu rang hhm sd cbn li mOt c6i gi ddy khdc vdi
nhftng bidu thrlc giii tfch duo. c bidt ddn 6 thdi
d6. Nhu th€, d giai doan ndy, khdi ni€m hdm sd
mdt d,in di cdc ddc tinh cd hoc vd hinh hoc.
4. Thd ki L8 : Tru6c kia, tinh todn tr6n c6c
bidu thrlc v0 han dd xudt hiQn khi bii to6n g6n
li6n vdi b6i cinh hinh hoc. Euler (1707-1783)
dd ph6t tridn tinh to6n tron nhftng bidu thrlc nhu

vay bing cr{ch chdp nhAn mOt quan didm hinh
thrlc chir nghia. Theo 6ng, tinh to6n tren cdc
bidu thrlc vO han phii ld m6t cOng vi0c hinh
thrlc c6 bin chat dai s6 mi thoat ddu kh6ng c6
li6n h0 gi vdi hinh hoc, vd do d6 viOc nghiOn
crlu n6 cdn phii c6 tru6c vi€c nghi€n crlu tinh
to6n vi phAn. Ddn ddn, trong nhfrng cOng trinh
cta minh, Euler n6i 16 vd c6c kh6i niOm "dai
luong kh6ng ddi", "dai lugng bidn thi6n", "hlm
s6" : "MQt hdm sd cfia m1t dai luong bi1h thi€n
ld m\t bidu tlulc gitii tich duoc tao thdnh, theo
milt cdch thftc ndo d6, ttl chinh dqi luqng bidn
thi€n ndy vd til cdc sd hay cdc dai luong khdng
ddi... MQt hdm s6'cfia mil bieh cfrng ld m1t dai
luong bi€h thi€n". Luu y ring Euler cflng dd
phAn bi€t hhm don tri vh him da tri, hhm "liOn
iuc" (theo 6ng, li duo. c x6c dfnh bdi mQt bidu
thrlc duy nhdt) vI "him gi6n doan" (duoc x6c
dinh bdi hai hay nhidu bidu thrlc giii t(ch).
Tidp theo Euler, kh6i ni6m him sd duoc holn
thiQn ddn trong c6ng trinh ctra nhidu nhh to6n

hgc nhu Da-llm-be (D' Alembert, 1717
C0ng-d6oc-x6 (Condorcet,
- 1783),

17 43-1794), La-grLng (Lagrange, 1736-1 8 1 3),
... vh ngudi ta dd n6i ddn him nhi6u bi6n. C{c
nhi to6n hoc thdi ki niy ddu xem hdrn sd ld mQt
bidu tlu.?c gitii tich.

ki 19 ngudi ta nor
khdng
nhdc gi tdi cdclt
sd
md
vd kh6i ni}m hdm
gidi
n6.
tich
cila
bidu di6n
5. Thd k1i 19 : TX ddu thd

:

"Ndi
chung, hdm s6'f(x) bidu di6n mdt ddy cdc gid tri
duqc sdp md mdi phdn t* dd dutoc ldy tily y".
Di-rich-I0 (Dirichlet, 1805-1859) thi dinh
nghia : "Gqi a, b ld hai gid tri cd dlnh, x ld mdt
dqi lud. ng bieh thien gifra a vd b. N€'u tuong trng
voi m6i x ddu cd mdt gid tri xdc dinh y = f(x),
vd y biafu thi€n mil idch li€n tuc khi chinh x
bi€h thi€n mdr cdch li€n tuc ir a d€h b, thi ta
n6i y ld m1t hdm s6'li€n tuc tr€n khodng ndy.
Theo quan didm hinh hoc, rt€it xem x vd y nhu
ld hodnh db vd tung dO cila mil didrn, sao cho
m6i gid tri crta x ilruac khodng dtqc xit ddu
ttong tng voi m1t vd ch[ mdt gid tri cila y, thi

sil li€n tuc cila hdm s6' sd xtiy ra ddng thdi vdi
Phu-ri-e (Fourier, 1821) ph6t bidu

vi€c dadng cong lidn mQt khodng".
Cbn theo L6-ba-sep-xkt (1792-1856) "... hdm
s6'cila x ld mQt sd dtqc cho vdi mdi x vd bidn
thi€n d,in ddn cing voi x. Gid tri cila hdm sa' c6
thd duqc cho bdng m1t bidu thtc gidi tich, hodc

bdng mdt didu ki€n ldm phuong ti€n dd thft tdlt
cd cdc s6'vd chon mQt trong chilng, hodc cu6'i
cilng, sr1 plru thu6c c6 thd t6n tgi nhung cdn
chua duoc biAi". Nhu v6y, dinh nghia hi0n dai
cira him sd (thodt ly moi c6ch cho bang bidu

thrlc giii tfch cira chring) duoc coi ld
Dirichlet vi L6-ba-sep-xki.

cira

Dgc sAcu
TAp san The gidi to6n - tin hoc cria
khoa To6n Tin hoc, trudng Dai hoc Su
ph4m Tp. Hd Chf Minh dI ra mit ban doc
so ddu tiOn ngdy 25.04.2002 nham nAng
cao chAt lugng dio tao sinh viOn, gitip sinh
vi6n c6 thOm co h6i ph6t huy khi ndng hoc
tAp vh nghiOn cuu. TQp san cdn ld di0n ddn
cfia thdy vh trb vd phuong phrip gi6ng d4y hoc tAp - nghiOn ctlu khoa hoc.
Todn hoc vi Tudi tr6 xin gi6i thiou

cilng ban doc hai bhi b6o rft tt tap san :
MOt bei to6n ctra A. Einstein (cria GS.
Ilohng Qu!) (de in ffong sd 301 (7.2002),
Lich sir hinh thdnh khdi niQm hhm sd (cfra
TS Lc Thi Honi Chau).
THTT

ll


2) VOi m5i s6 nguy€n duong ri thi t6n rai duy
nhdt mOt s6 nguyOn duong n sao chof(n) = lr

finhf(Zooz)'

,,aY

DAo rHr r-oAN
(S/ Bl K46 khoaYdt Li DHKHTN

T'C NR HI ilMry

-

DHQG Hd NAi )

CAC IOP THCS
Bni T1/302. Chrmg minh rang n0u a, b, c lit
cdc sd nguyCIn (kh6c 0) th6a mdn di6u kiOn
= 3 thi tich sd abc litlAp phuong cta

!*!-*SD(A
ntot so ngu)'en'
rHi.N Neu DtrNG
(6/ DHKI{IN - DHQGTp. HCM)
Bni T2i302. GiAi phuong trinh
(2001 - x)a + (2003 -,r)a = 2000

BiiT7l302.

xv" +:-+vz z.\
z" ,xn y"

rm+,I

^

,m+n

4o

y"-+) z*

trong d6 r, y, z

x-

)

.xn

+

\,,,

+

2,,

li caic sd thuc duong vd, m, n ld
coNG

vu

(5,' K47 kln'LTodn,
DHSP Hd N6i)
'HAM

Bii

T41302. Goi AD, BE, CF ld ci{c duong
phAn gi6c trong cira tam gi6c ABC vu6ng & A.
AD cit EF tar K. Duong thing qua K song song
v6iBC chtAB viAClAnluot taiMvdN. Chrmg

t
:' (AB+AC)
> ' --E
L

NGUYEN ruTAug }*GUYEN

(W THCS Hdng Bdng, Hdi Phdng)

Bni T5/302. Cho tam gi6c ABC vh didm M
nim trong tam -ei6c. Cdc didm A1,81, C, theo
thrl tu thu6c cdc canh BC, CA, AB vb th6a m6n

VO TRi DUC
(Ninh Binh)

Bni T8/302. Chung minh rang

{3.
\_:-lr_I

r
^
ft-l
,ti ci
trong d5

Cf

trong d6

S

= lStOAai
5

chi di€n tich tam gii{c.

NGLYENMINHHA
DHSP Hd Nli)

(6/' YI'CTT

cAc IOp THPT
Bni T6i302. X6t him
man cac oleu Kren

sd

f :7

-+ Z+ thba

:

l) V6i m6i s6 nguy0n duong n thifln+1) hoac
bang/(n) - 1, hodc bilng4f(n) - l.
t2

cj_i
c!,-l.c!,

ta tei hgrp chAp k cira i vd n

2 k22.
t

Bdi T91302. Cho tam gil.c ABC cir ba g6c

nhon. C'4c ducrng cao AA1, BBb CCt cit duong
trdn ngoai ti6p tam gi6c ldn nfra tai Az, 82, C2
tuong rfurg. Chung minh rang :
46ro

P +t,.r
+crAr)
B1L', 1 tt

<

3_-@rBt
a'
=213

Mt

+ BB1 + CC.t

tan*BC+cA\

'

*cwEN r-Ar
(6/ |HPT LtongVdn Chi)nh, Phri Yin)
Bni T10/302. Goi V, r vd, R idn luo't la tnd
tich, bdn kinh mlt cdu n6i tidp vI ngoai tidp tf
rlicn ABCD. Chring minh rang

,.,

24V2

R AB.AC.AD,BC,BD,CD
Ding thrlc x[y ra khi nio

?

NGUYEN VAN HAO

didu ki0n A$rllAM, BPrllBM, ClAtllCM.
C)rung minh rang S(ArBrCr)

:"

PHAM DiNTI TRUONJG
(Trung tdm CNTT, PS\ . Hi Not

c6c sd nguY€n duong'

minh rang MN

> -r" + \r' +

!, z, m, n lb. cdc s0 thuc duong thoa
mdn didu ki6n-lr 2y > z vh, m> n.

Bni T31302. Chung minh rang
vm+n

r[ng

il

trong d6 x,

I.IGOHAN
{Bdc Ninh)

Chrrng minh

mnmnfiI

(S/ 00C.lC DHKI Dn Ndng)

cAc oC v4r

li

B iIlBAz. Tr0n m5t ban nam ngang c6 mOt
chidc n0m c6 khdi luong M, c6 mdt cdt li hinh
tam gi6c ABC vu6ng o B. G6c gita hai canh AB
vd AC li 0, chidu cao tt B ddn mat sin ld lr.
TrOn ,mat phing
nghiOng AB tai ,4r
dar mor vat c6 kh6i

luong

ld m

(hinh

vE). Lric ddu vAt vh
nOm ddu dtlng yOn.

Sau d6 cho

rdt rr


chuy6n dong theo hucrng AB vdi vf,n tdc ddu vu. 86
qua ma s6t gifra nOm vi rnit sfur vh ma s6t giln vAt

vlm[tAB.
H6i r',, phii lon hon gi6 tri
thd r.uot clua duoc dinh B

bao nhi0u dd vAt c6

?

(Thi Olympic Tr!!!_g:d, ldn thu 2)
(TO GiAt\lG sr)

BiiLZl30z.llai

thanh ray d6ng chdt, tidt dien
ddu, c6 ciurg chidu dei le 2L, ddtt c6 di0n tr6
sudt p dat tren mat phing ngang , song song v6i

nhau, c6ch nhau m6t khoing l. Cdc thanh ray
duoc ndi vdi nhau qua c6c ngudn di0n nhu hinh

r,6. Suat di6n d6ng m6i ngudn
trong rit nh6.

MOt thanh kim lo4i khdi lucrng ni, chi€-u dhi /,
ti lOn 2 thanh ray vI cri thd truot
khOng ma sdt trcn chdng.
H0 th6ng dugc
d4t trong tt trudng
d6u B c6 phuong

diOn trd R

thing drrng. Tir vi
trf cAn bang cria

thanh, dich chuydn
thanh mOt doan rdt
nh6 theo phuong song song v6i 2 thanh ray.
Chung t6 iang thanh dao dOng di6u hda. Tim
chu ki dao dong.

lh E, diOn trd

NGTIYEN XUAN QUANG

$Hff

chuY(nVinh Phic)

ffiSSffiITffiffi
FOR LOWER SECONDARY SCHOOLS

if a, b, c are integers
!
(distinct from 0) satrsfying ! * *t = 3 then
Tll302. Prove that

hccl

TZi302. Solve the equation
(2.001 -,r)a + (2003 - r)a = 2000
T3/302. Prove that
ym+

n -ln'ln

---+:-_.+:--..
y* z'n x*

>

.trn

t!-*l!:-*'"' *" ) r" + \"' + z"
milmn

zn

n are

T4l302.Let AB,.BE, CF be the inner angled
bisectors of a triangle ABC right at A. AD cuts
EF at K. The line passing through K, parallel to
BC cut AB and AC respectively at M and N.
Prove that
,>

_.6

MN > :_ _::1eA+lC1
2

TSl302. Let ABC be a triangle, M be a point
inside it. Ar, Br, C, be the points respectively on

the sides BC, CA, AB suclt that A,BrllAhI,
B rC ,

ll

BIlt , ('

tAtllCM. Prove that
I

5(A,B,C)< :S(ABC)

J
rvhere S clenotes the area of triangles.

f : Z* -+ Z*
:

y"

n

T81302. Prove that

T I =,*-l' ' k-r -'';i
c1-? cf,
?r ci

where

C!

are binomial coefficients and

n

)

k>_2.

T9BA2. Let ABC be an acute triangle. Its
altitudes AAy BB1, CC1 cut the circumcircle

again at A2,82, C2 respectiveiy. Prove that

^G

fta,a,

+BrC1+CrA1) < AAz+ BB2+ cc2

='{rou*BC+cA)
T101302. Let V, r and R be respectively the
volume, the radii of the inscribed and the
circumscribed spheres of a tetrahedron ABCO.
hove that

FOR UPPER SECONDARY SCHOOLS
T61302. Consider a function
satisfying the following conditions

xn

for arbitrary positive numbers x, y, z, tn,
satisfying the conditions x ) y ) z and m 2- n.

1-y'+2"

wlrere ,y, y, z are positive numbers and tn,
positive integers.

integer n, the numtier

f(n+l) is equal tof(n) - 1 or to 4f(n) - | ;
ii) for every positive integer rn, thete exists a
unique positive integer ru such lhatf{rt) = 177.
Calculate l(2002).
T71302. hove that

the product ubc is a perfect cube.

xrnl.tl

i) for every positive

24V2
R

AB.AC.AD.BC.BD.CD

When does equality occur

?

13


KhO ; Binh Dinh

I

: Nguydn PhilcTho,9D, THCS Ngo

May, Pht C6t; Phf YAn: Hu)nhThiThily Lam,9Cl.
THCS Phd LAm. Tuy Hba ; Ddng Nai : Laong Minh
Thdng, g/3, THCS Nguy6n Binh Khicm, Bi0n Hda.

.
VIETHAI
Bdi T21298. Tim gid tri ldn nhdl cila bi€'u
thirc i + *y +y2, trong 116 x, y ld cdc sA' th*'

otnt Bnr xi Tfiuoc

thda mdn 2 di€'u ki€n

I

Bni Tl/291J. ('hmg ntinh rdng sA'cdc chfi s6'
t'i€lt trong lft thip phdn c'iu lrui sa'200f001 vc)
2002200t + 22oo! ld bdng nlruu.
Ldi giii. GiA srl s6 2OO22oor c6 m chfr sd thi
10002001 <2002200r < lo* (1)
= m> 6003.Ta
sd chfng minh bang phin chring : Gii srl sd
2}O22oot + 22001 c6 nhidu hor- m cht s6, trlc li it
nhdt c6 m+I cha sd thi 20022001 + 22001 , 1gm
(2).

Tt (I),

zooz2oor

(2) c6


<2ooz2oor +zzool
< z*.5* < 22001110012001 + 1;

= 22001.10012001
= 10012001 .2m-2oot.5m < 10012001 +
yi 2m*2001.5- lI sd nguyen non chi c6 thd
1

2m-2oor.5m = 10012001 + 1. Didu niy khong xiy
ra vi vC tr6i chia hOt cho 5 nhmg vd phAi luOn

c6 t6n cirng

li 2. Yqy hai sd 2OA22oo1

2OO220ar +22001 ddu

vit

c6 dfingmchtsd.

Nhqn x6t. 1) Nhidu ban dd giai bai to6n tdng qu6t
hon vdi chfng minh tudng tu nhu tr€n
C6c sd sau c6 cDng mQt sd

cht

sd

:

lx - 3yl < I
Ldi giii. Tir c6c bdt ding thfc d5 cho, blng
c6ch nhAn lOn theo hQ sd 3 vd.Z, ta c6 cdc bdt
ding thrlc tudng duong
l2x

-yl

-<3

vd

l6x-3yls9vhlx-3yl<1,
llx - yl < 3 vh lZx - 6yt <2

(1)

(2)
<
lAl + Bl (ding
Str dung bdt ding thrlc lA + Bl
thrlc chi xhy ra khi A vi B cing ddu), til (1) vI
(2) ta thu duo. c :
l5xl = l(6x - 3y) + (3y - x)l < l6x - 3yl +

lx -3yl <9+1=10=lxl<2,

- y) + (6y -.2x) I < lZx - yl + l2x 6yl<3+2--5=lyl<1.
Nhu vAy, ? * ry + y2 < x2 + lxyl + y2 . 4 + 2
+ I =7, vdi ding thrlc x6y ra chi khi lxl =2vd
l1yl = l(2x

d6.r vi y cUng ddu, hay (x, y) bang
(2, l) ; (-2, -!),hic d6 * * *y + yz = 7 .
Nh4n x6t. Kh6 d6ng ban gti ldi gi6i vh ddu c6 kdt
lyl

=

1 trong

qui dring.

:

VU DiNH HoA

:

Bni T3/298. Chtng^minh rang n€u phuong
a) % 2OO2n vit s6 2O02n + 8.2't vot n e N*
trinh axl + bx3 + r*2 -2b, + iu = 0 (a *0)
b) Sd a' vd sd a" -2" , s6 a" +2" v6i n e N* trong
f) c6 2 nghi€mx1,x2thda ntdn x1.x2 = I thi
d6a=l0t+2vdit>3.

5a2=2b2+ac.
2)Cdcban sau c6ldi giii tdt:
Ldi giii. Ndu phucrng trinh 6 dd bhi c6 2
Phti Tho t TrdnThdnh Dftc,9A1, THCS LAm Thao,
Nguydn Dftc Thdng,gAl, THCS TT. Thanh Ba 2, Vinh
nghi€m x1, x2 thi da thrlc bAc 4 b v€ trdi cia
Phtic : Nghi€m Thi Ngdn, 9A, Nguy,Sn Phil Cudng,
phuong trinh phAn tich duoc thlnh
Ngu_,tdn Huv Tirng,98, THCS YCn Lac ; Bdc Ninh :
ax4+bx3+r*2-2bx+4a=
Lau Thi Thu Trang,8A, THCS YCn Phong ; Hi TAy :
Nguy€n Khat Dfing,9A, THCS Th4ch ThAt, PhqmVdn
= (.r - rrXx - x)(axz + mx + n)
Khi€m,88, THCS Nguy6n Truc, Thanh Oai ; Hh NQi :
= (x2 - px + l)(ax2 + mx + n)
Pham Hodng Le, 9B, Trdn Nam San, 7T, THCS
Nguy6n Trudng TQ, Ddng Da ; Hii Duong : Ngut€n
= o*1 + (m - ap)x3 + (a -. mp + n)xz +
Hdng T hatil . 9/3, THCS lr QuI D6n, Tp. HAi Duong ;
+ (m - pn)x+ n, trong d6 p - x1 I x2
HAi Phing z Drdng Hdi Long,8B, THCS NK Trdn Phri
Vi hai da thrlc ddu vh cufii 6 cdc ding thrlc
.' Nam Dinh : Phqm Duy Hidn,gA7, THCS Trdn Dang
Ninh, Pham Kim H,ing, Dodn Duy Thuy€i,9A2, THPT
Le Qui DOn, Y YOn ; Thanh H6a: Nguy,in Ngoc Tri,
98, TIICS Le Dinh Ki6n, Ycn Dinh,Trdn ManhTudn,
9C, THCS Tri0u Thi Trinh, TriQu Son, Nguydn Ti€h
Trung, 9A, THCS Quang Trung, Nga Son, Nguydn
QudcThdng, L€ThiThu, BiliVi€t Anh, Bii Khdc Ki€n,
Nguydn Chi Linh,8A, THCS Nht 86 Si, Hoihg H6a ;

Nghd An : Manh Phric Tho, 98, THCS Trung D6,
Vinh; Hi Tinh : l,d AnhVfi Hd,9A, THCS TT. Huong

t4

trOn ld d6ng nhdt nOn ta c6

:

(l) ; m-pn=-2b
n=4a
a-mp+n=c (3); m-aP=b
Thay (1) vio (2), (3)

duo. c

:

m-4ap=-2b (5) ; 5a-mp=c
Tt
Tt

(2);
(4)

(6)
(8)
(7)
(4) (5) c6 b = ap
vh m =2b

(6) (7) (8) c6 5az = map + ac = 2b2 + ac.


NhAn x6t. l) Ban Trdn VTnh Trung. 9/2, TIICS Vo
Van Kj, Nha Trarrg, Khrlnh Hda dua ra lbi giii t6m tat
nhu sau :

Vi
c6

x1 ;e 0

:

vi

ax{ +bxl
a + bxy

I
xl

-

12

li

nghiOm cira PT

(*)

nOn

^
HOANGNGUYEN

+r*l -zb*r*4a=O

* t*l -ZU*l +4axf

Tri hai ding thfc niy

o*f

ddu

dAn ddn

=g

:

-u"l +bxr-a=o

<> (-r, + tXx, - lX (ax?

X6t 3 trudng hqp

U*, + a) = g

-

Dttdng Hrii Long,gB, TI-IPT NK TrAn Phri ; Thanh
tl,6a ; Trinh Thiinh Ki|tt,8A1. aA Anh Linh,88, THCS
Trdn Mai Ninh, Irrrrl Thdnh Ddttg,Vfi Nguy?nThilng.
9A, THCS TAy DO, Vinh L6c ; Ngh€ An : Nsal'ar
Hodi An, 8D, Ddng Bdo Dftc,8E, THCS D4ng Thai
Mai. Vinh.
\

:

-l = x2= .tl = -1 thi da thfc 6 vd tr6i cira (*)
phAi chia het cho 1.r + 1)2 = ? + 2x+ I nOn da thrlc du
phii ddng nhat bang 0, nghia ld. -4a + b - 2c = 0 vi
a + 2b - c = 0. Tii d6 c6 b = -2a, c= -3a, suy ra2b2 +
-2
ac=Jo
b) xr - |
= x2 = ,t = 1. Lhm tuong tu trudng hqp a)
dAn d6n b = 2a, c = -3a
= 2b- + ac = 5a'.
a) x1 =

Bei T4l29S. Goi P k) tt'ttttg didrn canlt BC c'iltt
tam gidc ABC t't) BE, CF ltl lrui cltk)Itg c'tttt.
Dti,ttg rltdttg clttu A, r'trotrg, goc' r'rri PF, ctit
cludng thdng CF tai M. Du'o'trg thiing qrta A,
vudng gdc v6'i PE, cttt dtotrg tlrung BE tai N.

Goi k ,a G lrin lrot lti trung didm cila BM t'it
CN. Gqi H td giuo cliiitt ctia du'dng thdng KF t't)
GE. ChLtttg minh rdng AH vudng g6c voi E F.

Ldi giii. Xdt LABC

nhon. Goi 1 lh truc tAm
MBC, D ld giao didm cua Al v>a BC . Ta c6
Ar=Ct (g6c c6 canh tuong rlng w6ng g6c)

crta

A

c) lxll * 1 thi x1, x2 l?r 2 nghiem cia-oi - bx + a = 0.
Chia da thfc & vd tr6i cta (*) cho o? - A, + d thi da
thrlc du ph6i ddng nhat bing 0, nghia lir (o - bx)(5a2 2b2

-

ac)

=0

=

Vi x * 0 kh6ng

= 2b2 + ac.
giii nhu sau :

5a2

2) Da s6 c6c ban

li

nghiOm cira PT (*) nen

s1o o( r'.4)* u( --?)+c
=o
x'l \ x)
\ )tA^
Dat y = r - :
= *' * -n= y'+4 thi PT tren tr&
thlrnh ay2

* ay | +o*,

=o

l-;)

))
D4t yr = xt - a (9) vd y2 = v2- J- (10). Ap dung
x1
x2
c6ng rhrlc

y1.y2=

vi-6r

dcti v6i PT

(**)

c6

tt + lz= -la

ua

ao)' .fU^, (9) (10) vho c6c ding thrlc niy vi

bidn ddi d6n ddn 5a2 =2b2 + ac.Chff rang ta 6p dung
duoc cdng thrlc Viit chi khi yr tzld 2 nghi€m cria PT
(**). Nhung ndu y1 - 12 a xt = x2 thi lt = !2 chi mdi
li I nghi€m ctra PT (**). Nhu vxy cbn phAi x6t 2 trulng
hqrp a). b) nhu o nhAn x6t I neu trOn.
C[ng tir su nhdrp lAn trOn n€n c6 ban^cho rang PT (l)
c6 4 nghi6m. Di€u nhy kh6ng dring, chang han

, x'-l-i+5x+4 = 0 e (x+1)(x+1)(l-3x+4) = 0
2) Ngoii ban Trung c6c ban sau c6 ldi giii dring :
Vinh Phrfc z Vfi Vdn Quang, 8A3, THCS Vinh
Tulng ; Hda Binh z Nguy6n Hodng Anft, 9A1, THCS

Hii Duong l Le Dinh
Huy,-9A, THCS Nguy6n Trdi, Nam S6ch, Trlnh Qudc

Bdo,9Al, THCS Chu Van An, Thanh Hi ; Hii Phdng
Trang,8A, THCS Y€n Phong ;

BDPC
Mit

khr{c tam gi6c vuOng BFC c6 BP = PC
nen Cr =Fi . MA Fr=Fz . Lai c6 F2=A2 (g6c

q=tr.

c6 canh tuong fng wong g6c). Do d6
Cing vdi CF L AB suy ra LAMI c6n. Do d6 MF
= FI ttrc lI KF lh duong trung binh c:iua A'MBI.

ll BE vd suy ti6p ra KH L AE.
Tuong rV EH L AF. Nghia le H li ffuc mm
LFAE.Y4y AH L EF. Trudng hqp MBC ti bdi
Suy ra KH

to6n v6n dring.
NhAn x6t._l) C6 thd chrlng minh theo c6ch thrl hai

h frF+dFH

= eon (Do

trFH=trcF

) dd dan t6i

FH LAC.
2. Bii to6n niy duoc rdt nhi6\ b4n gti ldi giii. Giii
tdt hon cA li : Phf Tho zVtrong ManhTilng,9G, THCS
Vict tri; Vinh Phric z Nguydn Thi Hd,7A, THCS Ycn
Lac ; Hii Phdng: Pham Huy Hodng,SB, THPT NK
Trdn Phri; Hii Duong: Trdn Qudc Hodn,9Al,THCS

Van An, Thach Hi ; Nam |inh : Daong Dd
NhuQn,9A2 THCS Le Quf D6n, Y Y€n, Pham Duy
Hidn,9A7, THCS Trdn Dang Ninh, Nam Dinh ; Ngh0
lrn z Nguydn Bd Giang,88, THCS Lang Thdnh, Ycn
Chu

15


Ttranh ; Binh Einh : Nguydn PhaongAzft, 8A1, THCS
Lo Lqi, Quy Nhon; Phri Y€n z HuinhThiThily Lam,
8Cl, THCS Ph'i Lam, Tuy Hda ; Kh6nh Hda : Le Thi
Hdng Nhung,Sla lt1CS Th6i Nguy€n, Nha Trang ; Hir
NQi : Phqm Hodtig Li.9B. THCS Nguy€n Trudng TQ.

VU KIM THUY

Bni T51298. C.lro litrh t'uonB ABCD. Liy didm
E tren cunh AB vti cliein F tr?rt canh CD suo
clrc AE = CF. Ctic cltdtrg rluing BF vii CE L'tit
du'ottg tldttg AD litt ltrn tui N vir M.Tint qtti
tic'lt iiao cliirn P cia crjc rlu'dng ildng Bl'4'vd

CN khi E, F tli chut,dn tr1n cdc carth AB
'"c)

C

D.

Ldi giii. Phdn thudn.
Ap dung dinh li Ta-l6t ta duoc:

CF CB ( 1).
FD ND
-=AE _AM ,.\
\L ).

EB

D

A

dudng trdn duong kinh BC. nlim trong hinh vuOng

ABCD loqi b6 c6c didm B, C
2) Cdc'banc6 ldi giii t6t :

vi

tam hinh vu6ng ABCD.

Hn NOi : Phan Xudn Dfing,9A, THCS Nguyen
Trudng TQ, QuAn Ddng Da, Nguy€n Trung KiEn, Vfi
Nhdr Minh,SH, THCS Lc Quf Don, Q. Cdu Gidy ; Hii
Phbng : Ddng Ngoc Chidn, Pham Huy Hodng, Dudng
Hdi Lone, Trinh Tud'n Linh,8B, THPT NK Trdn Phri ;
Bdc Ninh : LtuTltiThuTrang.8A, THCS Yen Phong ;
Phri Tho t Nguy,5nTrung Ki€tt,8A, THCS Giay, Phong
ChAu ; Hlr Tiry ; Pham Anh.Tuy€\,gA, THCS To Hi€u,
Thrtdng Tin ;'Nam Dinh : Ddo Manh Chiih,.Ngutin

Quf Don. Y Yen ;
Thanh H6a:TrinhThdnh Ki€n,8A1, THCS Trdn Mai
Ninh, Tp. Thanh H6a, Ii Phil Khdnh,8A, THCS Le
Hfru LQp, HAu LOc ; NChe An : Hd Ngpc KiAn, Vtr
Trpng Quj,, Bii DuyTodn,SA, TIICS Hd XuAn Huong,
Qulnh Luu, DQng Bdo Drtc,8E, THCS Dang Thai Mai,
Tp. Vinh ; Khfnh Hita t l,e Thi Hdng Nhung. Phclttt
Kidu Antt,8la, THcS Th6i Nguyen, Tp. Nha Trang.
Trdn Vinh Trung, 912, THCS V6 Van Kf. Tp. Nha
Trang ; Tp. Hd Chi Minh ; Ngttyln k Hodtry, 81,

Thdnh Ludn B, 9A2, THCS Lc

---

,^i

dring. C6 thd nhdn thiy ndu E thuQc tia BA (BE > BA) ;
F thuQc tia DC (DF > DC) thi tap hap didm li n[ra

BC

VrAE=CF
MnAn

THCS CLC Hdng Bnng, Q.5.

sd eueNc vtNH

cF

TU (1), (2), (3) vh

vuong ta suy t&

FD=49,,1
EB

.

_
ND AB

r ---1--

, vay hal tam gmc

vi BMA ddngdang v6i nhau. Tir d6
frEA=1Nb = Fufu*Ffrlt = frEA*FntN
= 90o hay BPC = 90o. Do d6 P thuoc ntta

du-tng trdn dudng kfnh BC, nam ngoli hinh
vuOng ABCD rir di hai didm B vd C.
Phdn ildo. Ldy didm P bdt ki tron n[ra dudng
trdn dudng kinh BC nam ngoii hinh vudng

vu6ng NCD

A,BCD (P

*

B, P

Bhi T'6/298. Tim s6'du cila phdp chtct

theo giA thidt ABCD lh hinh
CD AM

2(X)3

fli=l tt o i21 cho 2oo3

Ldi giii. (cira ban Hodng Thi

10NK Trdn Phri,IIii Phbng)
Nhfln thdy 2003 h mOt sd nguyen t6. Ta

giii
bdi todn tdng qudt sau : Tim sd du cta
p-

fl(t+;')
j=1

V] i2 = (p

=

*

,u,

*=*
BC ND
AB ND- *=9L

lE
=cF )
=9L
BE DF AB CD

AE

AE=

cF (dpcm)

K{t luQn. Qu! tfch didm P ld ntra duhng trbn
duong kinh BC, nlm ngoli hinh wOng ABCD,
b6 di hai didm B, C.

NhAn x6t. l) Tru m6t sd ban chua loai b6 hai didm B
vd C trong tlp hqp didm cdn tim, sd cbn lai ddu giii

16

-

D2

(modp) nOn ndu d[t
p-l

p-2^

didm cfra PB, PC vdi AD, E lI giao didm cira
MC va AB, F li giao didm cira NB vdi CD.Ta
phii chun! minh AE = CF.ThAt vay, ao frFN

LMAB(e.e).Suy

khi chia chop.

* C). Goi M, N ldn luot ld giao

= 90o n€n PMN + PNM = 90", PMN +
LCDN cn
frEA = 9oo = frEA=FfrM

Tuy€it Mai.

s=

fllt+iz
i=l

)

,T =

:t

p-l

s=

fli=l tt+i') ttri

fl{t+;')
i=l

=T' (modp)

p-1

X6t da thrlc P(x)

=

z

(*-i';-l
IIi=r\/

Ta c6 b6c cira da thrlc P
P(iz)

=0

(mod

D6 thdy i' F

p)

(p-t

vot i

,t

-,

)
I

p-I
li degP < 'va
2

= 1,2, ...,'2

+

i'(mod p) v6i 1 < i, i

<

p-I

Theo dinh li'La-grdng (TTITT 298,412002).

2


thiP(,r):0(modDvotmgixeZ.N6iri0ngr"=0,i{=.llahainshi€mduvnhAtciiaphrrorrgtrilrh
'-'^u bihg c6ch su dung uit aing tt"tt B"*o'ili hoal tinh
p-t

:

P(-1)

0 (mod

f

il of

f
I
-t

-l(-1)i'
=
l{mod r)
)
L' '

12

n-,
-t' |, =
I
I
[0 mod n) ndu o = 4k+l
f4 (mod p) ndu P = 4k+3

Vnv .S
"!r o -= Il\f-f '/t; '

d6 thi c6c him s6 3x v}r I + 2x'
2) Clctan sau c6 ldi grai tot : H'da Binh: Hri Hfru Cao

ctrft

1t.a

ffiiii1ffiu:'J"T,'^effiN##ilFfr',il:+#.
llT, TlIPTLuons

Van Tuy :

Hii

Duong :vtrVtinThiin,

1B3, Tt{pT Mi-Chidu ; Bdc Ninh :TrdnXtr)n Quydi.
11,{6, THPT Thuan Thinh . Nguvin Duv Twih, llA9.
'i}tl'f Qrg Vo l; 'Vinh Phnc: iliang Vin Congi:, 11A1,

f

Ndup =2003?s=4(mod

2oo3)

NhAn x6t. Ci{c ban sau day c6 ldi

giii

tdt.

Phong ; Thanh \l6a' :

k

Cbns Huy, 11A4. THPI' Tneu

tr"'i#J: i\;,::il:;;Xz"'t$i1ffiffif;'i;

Long ; TAy Ninh : Nguvin Anh Tuah, 1 IT, TIIPT Hoang
L€ Kha' "'

;

Hii Phdng: Trtin Hdng Trang, Pham Khoa Thu
NGUYENMINHD6C
Huong,lo THPT Trdn Phri ; Hi NOi : Nguyin Kim
Td'n,t2At, THpr_Nguy6nTa'tThanh, Ilsuy€n.l""uif Bni rg/29g. Day s6'(x,,) (n = 0, 1,2, ...)
Tlranh. 11A1. DHSP Ha NEi, Vfi Qudc Mi, ll
DHKHTN I Binh Dinh : PhamThdnh Phrt.nl-cQuf r --,- .-/,i:.-t- L:: . ..
.. - |
Dot :x"=
-, t' \'t=
Dorr : Dong \1i .. Le Plutong. ll Luong The Vinh '. dtld( Yuc dtttlt
2'
Vinh Phric : \/fi Nhar Hto,. Phan Bd L€ Bi€n : Pbi
Y€n : Ilrrj'nlt Lanr Litth, Phatr Tldnlt Nant, 11T2,
n-rl"xn
THPTLuongVanCh6nh;Tp.HiiClriMinh..Trdn\,,d.fl,,]=
I
4't 'vv"\tt -"" " 'tt+l"-tI
Hrr.r'. llT -\K. Ngrrl,"rr Lanr Hntg: Ninh Binh zTrinlt
M! Chdu.11, dft Luong Va"n Tuy ; Hda Binh : v(t'i moi rt = 0,1,2""
Nguyd'n LimTuydn, Hd Hrtu Cao Trinh; Hii Duong ,
Hriy titrt cortg thtlrc dng qudt ctia x,, th.eo n.
Nguyin Anh Ngoc, Nguydn Thdnh Nam, Luu Vdn
Hiin, IlAl,THPTN_guy6nTrdi;ThanhH6a:Trdn Ldi giii. (Ldi giii ctra ban Hd Hiru Cao
chuv.n Hohng van rhs'

.#"["::::,ffi:ffi,,,

ffinu,lt l"in'

DANGHUNGTHANG
BniTT/298. Gidi phuong

ril

rrinh;

l

3*=1+x+logj(l+2x)

Ldigiii.Di6uki€n:l+2x>0ex>-r'I

rtut

gia thidt suy ra
20OL

vdi moi n vi

+2OOZ +2000 v6i

xn+z xn+t
DAt

xn*o

n=0,1,2,...

xn

1
1000ts cO
- +-tt =
vv /(r
!, = x,,
' 2ool ta

3001

2ool'

phuong
" lt = 5002 Yd !,*z = 2001y,,nt +20A21', vdi
trinlitrcn c6 d4ng
ffi
g(3') = q(l +Zx),trong d6 g(r) = t +log3t. n=0,I,2'...
Chf i0lehdmsdddngbidn,dor Suyra i !,,.2+!n+r = 2002(y,,r *)n)
yi log3r ld c6c hhm sd ddng bidn tr€n (0, +co), - 20022(1,,, +),-r) =... = 2002"" (y, n.rn)
COng hai vd cira phuong

trinh vdi x thi

T::::;=I; :^X|.:',.i: =ffi zooz'*'
t

Ddta= Vr*Vn=-

8oo3

vaO=2OO2.tac6

200I
bien.
Q !n= a.b"-t -yr-, = a. b"-t - a. b"-z *!n-z = ...
=/(1) = alb,-, +(_t)b,,-z +...+(_t),,-rb+1_1;,, ,] +
VAy phuong trinh d6 cho chi c6 hai nghiOm
,(b' -(-l)')
+(-l)")',
x=Qvhx=l.
+(-1)'!o= *
b+|

-

Ta c6l'(x) = 3'.1n3 zlb, hdm sd ddng
Do d6, theo dinh li ROn, phuong trinhflx) =
c6 klrong qu6 hai nghiOrn. R6 rdngflO)

NhAn x6t. Tba soan nhan dugc ldi giAi cira gdn 500
ban hgc sinh. hdu hgt.i;-b* ceu fia theo'phuong Suv ra
--r ---

phiip tren - d6 Ie phuong ph6p ung dung cira dao hl'rn
irong viec- tinh sd-nghiQm cfia phuong trinh MQt so it v6i n = O,
ban dua vho dac thi ri0ng cira bii todn dd chrlng minh

3000,-l)"

,- =

ril
2003'

+ *oo'-,roorr'
2001.2003

'

1,2, ...

t'l


Tt
r=

"

d6, cong

thfc tdng qu6t cira x,, lh :

/cosB+l
cosC+l\2
l-r-l

A

2001.2003
2002".8003 + (-1)" 3000.2001

-

\ sinB sinC /
(cosB+1 .)'(cosC+l .)'
=t-+lt+l
sinC
sinB

1000.2003

I

NhAn x6t. Nhidu ban de bict sit dung thdnh thao viOc
Uidu di6n nghiOm cira phuong trinh saiphan tuydn tinh

nhd da thrlc dac trung nen c6 ldi giii ggn hon vidc s&
dung h0 thrlc truy h6i nhu de ffinh biy 6 trcn.
Day Ii mOt bii to6n co bAn, do vAy c6 ddn gdn 200
bdi

giii

giii

giri ddn tda soan c6 ldi

dring.

o

NGUYENVAN MAU

Bei T9/298. Goi AH vd r ldn ltror ld dtdng
cao vd bdn kinh dtdng tritn n1i ilip ctia tant
gidc ABC. Gqi pt ld ntlra chu vi A,{BH vd 11 ld
bdn kinh clthrg tt'dn ndi ilAp cila 16. Goi p2ld
tu\a clrtt vi "MCH vd 12 lc) bdn klnh dLrhrg trdrr
rtt.ii riep cilcr

vd

clti khi t'

=

4!!!!!
ta

Ldi giii. Khi

s6c A

H thuoc doan BC.
Dat h = AH, p lI ntra chu
vi MBC.
90o dd

BHC
Goi S, 51, 52 ldn lugt

A,ACH

li

diOn

tich MBC,

. Lttc d5 ,- = Pt\*Pzrz
12 2
tth+Pz

S S,+S, at)
.>
^ l.> +pi
Y
lpi

)

'

-

p?

*pi

( a

[cotsr+*rs, ) [*,rr*,J
BCBC
cots-cots-2 "2
"2 "2 = cots-+cots-+l

BCBC
ts-+ts-+ts-.ts"2 "2 "2 "2

=I

BC
ts-+ts-) -)

(a+c)
'
- =t<=lSi
B C
\z ^ )i=l
I-ts-ts
-) "')
A=90o (dpcm).

Nhin x6t. l) Bii ndy c6 rdt nhi6u ban tham gia giii.
Tuy nhi€n sd c6c ban nh6n thdy su cdn thidt cira clidu

H thuOc doan
BC. Dd chrlng minh chi6u
nguoc lai ta phei bd sung
vuong thi

o

o

ki6n max{B, C} < 90o khOng nhidu. Ban Hd Hrtu Cao
llT, THPT chuy€n Hohng Van Thu. Hda Binh
da dua ra phAn vf du tot chrlng t6 khi thieu dieu kien bd

Trinh,

sung thi c6 th€ g6c A kh6ng vuong.
2) Nhidu ban c6 nhAn x61 ring biri to6n ndy (o dang
h€ thrlc ('t)) l)r CAu III3, dd l2l cta Bd dd thi ruydn sinh
Dui ltoc c6 bd sung them didu ki€n cdn.
3) Cric ban sau c6 ldi gi6i tuong doi tot :
Phti Tho z Hodng Ngoc Minh,11A1, TI{PT chuydn

Hing Vuong ; Hii Phbng l Luong Th€'Hing, l0T,
THPT NK Trdn Phf ; Vinh Phric : Ddo Quang Son,
10A1, THPT chuyOn VInh Phfc r Bdc Ninh l Nguyitr
VdnThdo,llT, THPT NK Hdn Thuy€n: Hrmg Ydn:
CaoThi Hu?,llA, TI{PT Duong Qudng H}un ; Thanh
H6a : Nguy€n Danh Quang, SAl. THCS NhS 86 Si,
Holing H6a ; Quing Tri t Hd ,Si Sdng, 10T, THPT
chuy€n Le Khidt ; Bi Ria - V[ng Tdu : Tntong Cdng
Dinh, | 1A4, THPI Vflng Tiu.

NGUYEN MINH HA

Dc thdy

I +cotnc* I )
'o=L(rotrg+
"
sinB " sinC/
2[
h(
I _*r')
Pr =
;[cotso*rinr )
t *t) nen
," = L(rorrc+
"
2[
sinC )
(t)e [.o,ntr* I *.o,n6'*-l)'
\. " sinB " sinC )
1 *t')'*[.o,nc* I *t)'
= [.o,nB*
\. " sinB ,/ \ " sinC )
,

l8

o

, rrir a

'JPt + Pz

klfiug rti
.1

MBH,

<>

nit. ChLhtg rrtinh rdng tani gidc ABC

tttong rcti A klri
g6c B, C

<-)

J I
)
( c -)'
c)2 (-+lln -\2+[cotsr+t,
=

Bli Tt0/298. C|rc., ttr diirt gtitr diu ABCD (AB
= CD, AC = BD. AD = IIC) Cit.ti R vd r ldn lucrt
lir bht kitrlr cdc mdt ciu ttgoui tiep vti rrdi ti1p
tricr lt| diirt ABCD. Goi R,, t'd r,,ldn laot ld bcirt
kitir t'tic' drdry trdn ngoai tiip vd ndi tiip ct)a
totrt gitit'ABC. Chlrng minh riing
;

Ddng ilu'rc xdy, ra khi ndo

Ldi giii.

?

(Nguydn Ldm Tuydn, 12 Todn,
TIIPT chuyen Hohng Van Thu, Hda Binh)
Goi O lh tam mat cdu ngoai tidp trl di6n
ABCD.Y\ ABCD ld mot ttl di0n gdn ddu nan O


LCDA = MAD = MBC. Ggi do,d6 v2r d" ldn lucn
ld c6c khoring c6ch til tam O'dudng trdn ngoai tidp tam
gt6c ABC ddn c6c canh BC, CA vi AB. Ta sir dung h€
thrlc sau dAy (Dinh li Carnot) :
do+ du + d, = Ro + 5 (do ABC Lit tam gi6c nhqn) (5)

=

(Ndu tam gi6c ABC c6 g6c A.

ti

thi ta c6:

4+4+4=Ro+ro)
[C6c ban c6 thd chfng minh h€ thrlc tr€n bing c6ch
li ft01€m0 ho6c thay da = RocosA. v.v...
rdi sft dung cdc he thrlc luong gi6c quen thu6c trong
srt dung dinh

tam

B

trDng vdi trong tAm G

vi

tdm

/ mit

cdu nQi tidp

cira tf di6n vb do d6, O e DD'ra ..|!=1 ,1ru,
DD' 4

= ]n
loo
JJ

tit: oD' =

(Eong d6

D' lh trong

li

tAm cira tam giiic ABC). Ggi O'

hinh chidu
(wOng g5c) cira O trOn mat ph&ng ABC th\ O'lit
mm dudng trbn ngoai tidp tam gi6c (ABC, vi do
d6: O'A-- O'B = O'C = Ro, vi OO'= r.

Nhmg lai c6 OD' > OO' = r, nghia lh
R > 3r (1) vn OAz = OOa + O'A2, suy ra

4=n2 -r2

-

z{fr
Mit

-

:

3rz

-# r,t;A -4=RoJi

(3)

khi{c, trong moi tam gi6c ABC ta c6 BDT

quen thuoc Ro2}ro,hay

Cu6i ctrng,

(*) cdn tim

li

,"=

tt c6c BDT (3) vi

+

@)

(4) ta dugc BDT

:

R.. +r^
-?-:-e-

tlR2

:

(2)

Tir (1) ra duo. c : 4fr l2rz > 3P
Kdt hqrp v6i (2) suy ra :

-3r'

s

3R

--+<.,5.
zrln2

dp.-.

-lr2

Ding thrlc xiy ra 0 (*) khi vi chi khi ding
thrlc xiy ra ddng thdi 6 (4) v) (1) nghia le khi

vi

chi khi tam gi6c ABC th ddu vi trl di0n gdn
ddu ABCD cfing li trl di6n truc tam (O = I = Q =
I0, vi do d6 ABCD li m6t tft di€n d€'u. (Tuy
nhiOn, cfrng d6 thdy rang d&ng thrlc xiy ra 6 (1)
k6o theo ding thrlc xiy ra 6 (4), vd nguo. c lai vi
chi cdn xiy ra m6t trong hai ding thrlc d6 thi trl
diOn gdn ddu ABCD sE tr6 thinh mOt trl diOn
ddu).
NhAn x6t.
cdc

mit

li

l) Y\ ABCD li

giiic: f

mQt trt diQn gdn ddu n€n

nhtng tam gi6c nhon vd bdng nhau : LDCB

,o =

cosA

= qrn4r lr

ll,

+Ro.injr^f,t*ll

Sau d6. thidt ldp BDT:

(Ro + ro)2
<
u(.8)

/ z ,2-!--2
\
!-ll:.t
I

!

a,AC = BD=b,AD= BC =c),
Sau

Ti

(AB = CD =
(6)

ctng, chrlng minh h0 thrlc (c6 thd sir dung v6cto):

o2 + b2 + c2 =8fr
d6 suy ra BDT (*) cdn tim.

(7)

2) Nhidu ban cho ldi giAi qu6 ddi dbng hoac sit dung
qu6 nhidu d6ng thric hoac BDT trong tam gi6c nen ldi
giii cdng kdntr. Dring tidc. cdn c6 mOt ban do quii cdu
thi n6n di v6i ving kdt luAn dd to6n lir sai I Ngoii ban
Ldm Tuydn. cdc ban sau d6y c6 ldi giii tdt vd tuong d6i
ngin gon hdn ca.
HA NOi z D6 Xudn Diin, Ddng Dinh Khdnh, Nguydn
Hodng Thanh, llAl, Vdn Anh Tud'n, 11A2, IrICT-T,
DHSP He NQi, y, Qudc M!,Trdn AnhTudiu LlA., Le
Hilng Vi€t Bdo, l0A To6n, DHKHTN - DHQG He
NQi; Bdc Ninh : Ngr_vdrtVdnThdo,llCI, THm NK
Han ThuyOn; Vinh Phric :\'ii Nhtir Huy, lIAl.Trdn
Bd Bdch,l1Al, THPT chuyOn Phti Thq ; Hda Binh :

Hd Hfru Cao Trinh, 11 To6n, TIIPT chuyen Holng Van
Thg, Hba Binh ; Hii Duong z Khdng Minh Thdo, l0
Todn, Nguy€n Ti€h Thdnh, ll Ly. THPT chuyen
Nguy6n Tr6i ; Nam Dinh : Drtrttg Dd Nhudn.9A2,
THCS Lc Quf Don, 'i yen ; Ninh Binh : Ngtq,[l1
Tuy€i Mai, llT,TrinhThiy Nhung.12T, THPT Luong

Van Tqy

; Thanh H6a t Mai Quang Thdnh, 11T1,
l2CI, THPT Lam Son, Le Khdc

Nguy,in )fudn Hda,

Huy€n,l1Bl, THPT L0 Van Huu, Thi€u H6a ; Quing

Tri z Bach Ngoc Bdo Phudc. Dodn Quang Tri, ll
To6n, THPT chuy€n Ir Quy DOn, Q.T ; Binh Dinh :
Pham Thdnh Phil, ll Todn, THPT chuyOn tr Quy
D0n,; Tp. Hd Chi Minh :.TrdnV6 Ha'v, llT, THPT
NK DHQG Tp.HCM ; D6ng Nai : Ngaydn Hodng

Dfrng, l0 To6n, Nguydn Cdnh Lanr, ll Tin,
THPT chuyOn Luong Thd Vinh, Bi€n Hda ; Bh Ria V[ng Tiu ; Bii HodngGiang, llAl, THPTVfrng Tiu
; TAy Ninh z Nguydn AnhTudn,ll, TIIPT Hoing L€
Kha ; Ddng ThSp : Nguy€nVdVinh LOc,LLT,TtIIrt
Sa D6c ; DuonsTi€h Cudng,l1A2; THT Ng6 Gia Tg .

Minh

NGUYENDANGPHAT
19


Br\i L11298. MOt qud cdu kh6'i luong m = l ,2
kg nA'i bdng 2 soi ddy khAng kh6'i lu'ortg vdo mdt

tkanh thdng dimg
dang qua1, vcti vdn t6't:
gdc at. llai ddy bubc

v,dt bi cdng vd tart
thdnh mbt tam giitc

cdn ABC

t,oi

AB = AC = I = l,Swt,
BC = 1,8m. Luc ciing
ctia ddy ,48 ld 7'1 =
35N. Clto g = l()ntls-

llrlt' .t'ir.

,:lirtlr

Ltc cdng T 2 cira cldt AC
b)\'an to'c g6c a.
Ldi giii. a) VAt m chiu tiic dung cira 3 luc :

trong luc F=mi, luc ciing { cna tlAy AB vi luc
a)

.

cing i

cta
day AC. l,uc
tdng hqp F =

Ninh : Cau: Xtdn Binh, i0 Li, 'I-HPT NK Hdu ThuyCn .
V*oneThdnh Nam,l1Al, THPT ThuAn Thdnh sd I ;
Nam Dinh z Trtin llodng Anh, lL Li, THPT 1, ltglg

Phong,'ilp. Narn Dinh, Ngri T'i€n Dat,l0A5, U{Pl'Hai
Hau A : Phf Tho z Bili Ngqc Khang,l0A2,Vti\/tin
Tiung, 1181, THPT ch.uyen HDng Vrrong, Vi0t Tri ;

: L€ Drlc Phuong, 10A8, THPT Ilhrt
R,irre: Trd,? Ati'r Son, 1043, TFIPT Nguy6n Trdi, Tp.
Thanh H6a

llna*r

HOa

Ngoc Hrng,l0{l0' 1llPI i-€ tdj'
Li, Bili Hodi
z L€ Dirc Dat'

Nguy'in Xudn Drtc, 11 f i, TIIPT NK He

.Trdn

Hi finh

Il

Thach; NghQ An : Pltam Cung Sort,

llA3,

Tho Xu&n r

l0 Li,
Tinh ;\tnh Phfc : Pham Quang Chie?i. l)AL, Trdn
Vlnh
Dtc Phti, Trdn Dinh Cung, 10A3. THPI ch.
'fq. L4p
Phric, Hd \rdn LA. l0Al, THPT Ngo Gia
Nam,

Phant.T rttrtg.

ifim pnun Bgi Chau, Vinh. Ngrn'r'rr v'rli
Dilng, 10A, II{flI B6c YOn ltrinh. L.i Tiri Thu Hd.
Trudng,

THfrI

10A,

Nam Dhn 1.
N,IAI ANH

Bii

L2l298. Ba qtrci ctiu nlrc) gi6hg h€t nhau
ktfii ltttrng trt, di€n tich q) ndm y€n
d r:d.c dtnh cila tam gidc ABC vtrdrtg tai A nhd
cd.c srri ddy ndi khdng gidn, kh(tng d,in di€n,
k!rui tuong kh1ng ddng kd vd chi€u ddi cdc
sqi ddy hdng db ddi cdc canh trtortg itrtg ct)a
(tnni quti cit

tam gidc.

r nri+4

Hi ilfi'rtg dtit trin mdt phdng ngang,

dring vai trd

nhdn vd

luc hudng tAm
.(
,.
t'
F = nun'R (l). o

'I'r0n hinh cri t,E luc tdng hop Frthco cltu vi
drtdng da gi6c, truc Ox hu6ng theo Ai1). i-il
hinh vE ta c6 :
F = Itcosc{, * Ircosc (2) v} Tlsincr = P +

cich diin. Goi trFC = a, BC = t. Hay xdc dinlt
lir; r(ic ctia r:dc cluci cdu ngay sau khi ddy BC bi

I2sincr (3) vdi sino = 9H

luc

=0,9 = 0,6 vd
1,5

BA

1-0.62 = 0,8.
Til (2) vI (3) tim duoc : Tz= l5N ; F = 40N.

coscl =

b)R=Afl=

I,r'

4,*

(D

=

.l

'

*

tuctng

t{c gita
qui cdu B
vdi cdc
quA cdu A

vh

:

Hba Binh t NgtiydnTu(in Anh.ll Li, THPI chuyOn
Hohng Van Thu ; Ninh Binh : l/fi Thi Ngoc Anh,
l0A3; THPT Yen Kh6nh A ; Di Ning r 1l Tdn Huv,
10A1. [{PT chuy€n Le Quf D6n ; Quing Ngai :

Nguyin Vdn Thdng. llT, THPT I,e Khidt ; Quing
Ninh : Le Quy Anh, 10A4, TIIPT Hoing Qu6c Vi6t'
M4o KhO, Dong Tridu ; Hii Duong : Trdn Thanh
Cidng, 10 Tin, TIIPT Nguyen Trdi. Do-dn Manh Hd,

10A1; THPI Thanh HI ; Hn NOi : N guvin Hodng Duv,

10L2, THPT HN - Amsterdam ; Ti6n Giang : Traong
Gia Thuan, 11 Li, THPT chuy0n Tidn Giang ; Bdc

20

Chon h0 truc xOy nhu hinh v6. Ki

C. Xdt
h€ gdm 2
qui cdu A

5.3(rad/s).

V rnR
NhAn x6t. C6c em c6 ldi giii gon vh d6p sd dring

ttg,t)t.

Lcri giii.
hi€u 4,
Fl n"a"

vi

= 1,2(m).

Itr

TU (1) tim duo.c :

tlirt dr.)t

C,
tt-

ngoai luc

tdc

dqng

len

I

hQ

c

theo
phuong

Ox

le

F1 sincx, do d6 : Flsincr .= Zmax

o,= lLsino

^ 2m

*

aA, = acr, =

,2

lL;sin.-.
= 2mlz

a.


flruFf QU$ CA# g&r rnA,el *BAr sd sa
Problem. Show that it is possible to cut a diametrically opposite to S, and by T and V the
hole in a cube, through which another cube of points on 6 such that SfilV is a square. Since
"{t€,Nq

the diagonal of this square has length 1B , its
sidi:s are of length 1. Now construct 4 planes
perpendicular to the plane cr through the edges
of the square STUV. The region enclosed by the
4 planes creates a hole inside the cube
ABCDEFGH, through which one can push
through a cube of the sar'ne size.

equal size can pass.

Solution. Let ABCDEFGH be a cube of edge

length 1. Construct a plane cr through the center

TrI mdi:

= c6t (dQng tt)
= 15 thirng
= hinh lQp phuong
= kich thudc, d0 lon

cut
hole
cube
size

DPC

Tuong tu xdt h0 gdm 2
phuong

qui

cdu A vd B'

theo

Oytac6:

F,

oAy= oBy= -J.-cosct..

DO ldn gia tdc cira

= c?nh

bdge

= c?nh, m6p
= xay dung, v6 hinh (dong tit)
= vuong g6c (tfnh tit)
= dudng ch6o
= trung didm
= da gidc
= nhu nhau, tuong ddng (tinh tI)

construct

O of the cube, orthogonal to the diagonal HB.
The plane o meets the edges AE, EF, FG, GC,
CD, DA in their midpoints K, L, M, N, P and Q,
respectively. The polygon KLMNPQ has
congruent edges of length equal to tlz D. fAe
distance of the center O from the vertices of the
cube is also 1B /2. Thus the polygon is a regular
hexagon. Denote by V the circle inscribed the
hexagon. The circle 6 touches KL at R.
Choose an arbitrary point S on'€, rrear R but
different from it. Denote by U the point on 6

f,

kq2

orthogonal
diagonal

midpoint
polygon
congruent

regular
hexagon

diametrical
region
enclosed by
create

push

SuY

qui cdu A

li

{
tg| = b=
Vdcto

2mlz

g6c B x6c dinh bdi
vdi
Ax
hqp

;

tt)

or=

,@,*t, = 2mlz

kq'

t

1+3cos

td aB , ac hqp vdi trgc Ox cdc g6c y

6 tucmg rlng md

:

:

X6t qrfr cdu B theo phuong Ox vd qui cdu C
theo phuong OY ta c6 :
ks2

P,
aB*= - asing = --+-SmO
mmt
kt-roro
4
.or..
vl =- m "oro= - ml"

.[*:tir'"
=E2mlz

,@*ir,

vi

cotgct.

aA,

vilng

= bi ch6n.b0i
= t?o nOn, gaty ra (dQng
- ddy, xO (dOng tt)

ou =

C6c v6c

o21r+o?1,

-

ra:

= ---cosa"
:

= ddu, chinh quy (tinh tt)
= lqc gi6c
= thuQc vd duong kinh (tinh tit)

NGO VIET TRUNG

kq?

aA=

.

side

E^t=X ].orso rtB6= ff='".,ro
NhAn x6t. C6c em c6 ldi

giii

ddng

:

Ninh Binh t Li Hdng Linh,IOLi,THPT Lrrong V-an

ruv'i'iiar mni z Hoing

Nguvdn

vi€t Cudng' l0 Li'

llBo
rrirr xr< Hd Tinh ; Hn NQi t Bili Gia Khdnh, Cung
t
Phqm
An
NghQ
NOi
HAi
;
DHQG
DHKHTN Son,|LL3,THPIPhan Boi ChAu, Vinh'
MAI ANH
2t


3. Thdng, ndm xudt hian cdc chuy€rt muc duoc
cho b&i bing thdng k0 dring sau :

Dinh cho c6c ban THCS
r/1998

Di0n dhn day hoc Toiin

Vuic tlmi voat :

,+tttfWEU//01/?
aI - rs6 NAo r

Tt thd ki thrl II m6t nhi khoa hoc Hy L4p d5
xAv dtme mO hinh v[ tru dia t6m (coi Tr6i ddt

nain ycn" d trung tam vd tril). Ocn thd ki XVI
m6t nhl khoa hoc Ba Lan d5 dd ra HC nhat mm
(coi M4t trdi & trung t6m vfi tru). Nim hdnh tinh
sdn sfii cira H0 Mat trdi mi con neudi bidt ddn
ftr rai sOm n Kim, Moc, Thiry, H6a, Thd. Mot
nhd khoa hoc Ao dd xAy dung n6n ba dinh ludt
ndi tidng, tiong d6 mqt dinh tu4t U : Cdc hdnh
tinh chuydn dbng quanh Mdt trdi theo qui dqo
hinh elip md Mdt trdi ndm tai mdt trong hai ti€u
didm cfia elip d6. Nam 1964 HQi Thi0n vin
qudc td dd x6c nhAn b6n kfnh Tr6i ddt vdo
khoing tt 6356,78 km d6n 6378,16 km.
CAu h6i dinh cho c6c ban.
1. Hdy nOu tOn c6c nhi khoa hoc Hy Lap, Ba
Lan vi Ao n6i tron. Ban bidt gi thom vd ho ?
2. Hdy xdp thrl tu 5 hinh tinh tr6n ctng vdi
Tr6i ddi tin[ theo thf tq tt gdn Mat trdi nhdt
ddn xa nhdt.

Ngoii c6c hdnh tinh tr6n ban cdn bi6t c6c

hinh tinh nlo thuQ.c H0 M4t trdi ?
3. Hai s6 do b6n kfnh Tr6i ddt n6i tren rfurg vdi
c6c khoing c6ch nio ?
VKT

Knl qili.. 'filA i{l{U )'TA? Cili
TOi\Nl

il?C

V+\ TUOI Ti(i

Dlnh cho ciic ban chuAn
bithi Dai hoc
Toiin hoc mu6n mlu
Nhin ra thd gidi
C4c ban sau c6 ddp r{n dring duoc nhAn tdng
phdm : N guydn Chi HiAp, 1 1A1, chuy0n Tor{n Tin, DHQG He Noi, He Ngi ; Phan Thi Kim
Hoa, 11A1, THPT chuyOn Nguy6n Tdt Thhnh,
YOn B6i ; Li Qudc Hi€p, 10T2, THPT Lam
Scm, T. Thanh H6a ; Dodn Thi Thiry, Ddi 2,
DOng Tho, Di6n Tho, Di6n Chiu, NghQ An '
HOAI THUONG

Gidi ddp bdi :
CHO MAY DIEM

*r

Ban thi sinh dd

sai ngay tir ddu
do dS lAp luAn :
Hai dadng cong d d€ bdi ti€p xtic

-a + 2) + a2 -3a
cb nghiom kip, bbi vi ta chi n6i ddn
kh6i niem nghiCm k6p (nghi6m bQi 2) doi v6i
hdm da thrlc me th6i. Nhidu ban d6 cho thi sinh
no didm "2010" ! Sau d6y ld ldi giAi dring cia
bii to6n: Hai du&rg cong 6 dC bei tidp xfc nhau

nhau khi vd chi khi PT ; 3'(3'

= 3' +

I

+l
[2.:2' trl+12-a)3'ln3=3' ln3

<>ilY G' -a+2)+a2

HQi To6n hgc ViQt Nam) cirng v6i co quan chir
tiy ban Khoa hpc Nhd nudc (tu 1966 h
quan
Uy ban Khoa hoc vd Ki thuflt Nhd nuoc). Ra m6i

vi

b6o

li

th6ng m6.t sd, 16 trang, in 2 mdu. TUt$ng-lll993
B{o tr6 thinhTqp ch{Todn hgc vdTudi trd.
2. Tir sd thdng U2001 ddn s6 thdng612002 dd

c6 7 bdi vidt cho chuy0n mucTodn hoc vd ddi
s6'ng vd,2 bii vidt cho chuy€n muc Lich srt
todn hoc.
22

(1)

(2)

o=l

2J1,0
a=-5 + -"-

SO

-3a=3*

(a> l)thayvho(1)ta
TU(2)suyra 3'=
duoc 3az - tOa - 5 =20 mh a > 1 n€n chi c6

TII&TT ddu tiOn xudt ban vbo th6ng

10 nlm 1964 theo s6ng kidn ciua BanVqn ding
thdnh lqp HQi Todn hoc ViQt Nam (tit 111966ln
1.

?

thoamdnddbei.

3

Nhfln x6t : Q{c b4n sau dd ph&n tich ch6 sai
nOu ra ldi giii dring duo. c nh{n tang phdm :
Bach Hodng Ydn, l0 Arrh, Nguy€n Kltoa Dfing,
!2 todn, THPT NK Hdn ThuY0n, Bdc Ninh ;
Vfi NhQt Huy,l1A1, THPT chuy6n Vinh Phtic
; Hodng Minh Dtc,12E, THPT Kim Li6n, Hh
Ngi ; Irrin Tudn Anh,l2{l, TIIPT Qudc Oai,
Hn raY'
NGoc HrdN


Trong m6t cudn s6ch tham khio c6
Gidi PT :

logl! +l)

3(x-2)

bli

to6n

- 2(4x-6)logr(x+l)+4x = 0 ( i )

vdildi gfuii nhusau:
Di6u ki€n x > -1. De thdy x = 2 khong lh
nghiOm c[ra PI (1). Vdi x *2, datr = logz(x + 1)

(l)

rhi FrI

c5 dang

3G-2ll-214x-6)t+4x=0
Ta c6 A,

= (2x- 6)2 suy rt tr=Z, tr= 33(x-Z)

Ydit,-2

x = 3 th6a m6n di6u ki6n bii

to6n.

Vdi r,' =

- ''' - ta c6 logr(x + l) - =4
- et
3(x-2)

3(x-2)

NhAn thdy

him

sd

y = log2(r + 1) ddng bidn

+oo). Hdm sd

rrcn khoeng

y=

#

nghich bi0n trOn t4p (-1, +oo\{21. Vi vay PT(2)
nOu c6 nghiOm thi nghi0m d6 duy nhdt. D6 thay
x = 0 lh nghiOm ctra PT (2). Do d6 IrI (l) c6 2
nghiom r = 0, x = 3. Ldi giii tron c6 van dd gi

f

khOng

f

ndn ctra ban thd ndo ?

(W

,
VU VAN BANG

THPT Nguydn Binh Khi€m,

\:inh Bdo,Tp. Hdi Phbng)

DOND0O.
-;:

:,' :

THTT So 303 (gilefiLi)
^'

o Sri dung dn phu dd gini phuolg

irmt

hai ph6p to6n nguoc nhau.

.Tim c\rc tri

mnLE

udru mffirue Tffi[$sff $e

Theo ban Ngd Phudc Nguy€n Ngqr,
ffadng Qudc hoc,ThitaThi€n Hu€' :
* D\mg duong trbn tAm
Dbdnkinh DA.

|.
,,,

,.;.,-);

mQt bidu thrlc bing ffii€x.-.

.

6$,fffii'tffin:einh D4i hoc

khdi B, D nam 2002.
Kdt qui cuqc thi hoc sinh gi6i TTIPI toan
qudc nam 2002.

,

iodnrv i; Cau t4c U9....
i,
sE li ban hoc tdt trong nam hqc t6i.
Hdy c6
trcn gi6 sdch cria ban !
Vh, cdc bdi

T"l

t..
2

4-

4*
.i-

_

6-

TIITT

Todn,

* Dirng hai ldn compa vh

A) "ung
tai E.
* DUng duong thhng AE.
* DUng dudng trbn tAm E b6,n kinh EB c6t
doan AE tai 1 thi 1 lh tam dudng trbn n6i tidp
tam gi6,c ABC.
Chfing minh : Theo ci{ch dumg thi AE l} phan
chtra

Mit kh6c Al2 + IBA =
= 6in = frr = fdi*ddi = Atz*iEZ

Tt d6 suy ra lEa=lEA hay BI ld phan gi6c

giiic cta g6c BAC.

.

cia

{ic.

v4y

/ li

tam duong trdn noi ti6p

NhAn x6t. Cdc ban sau cfrng c6 c6ch dmg
dring, trinh bhy gon, gui bli ddn TS sdm: L€
Nguydn Di€u Hdng,9A, THCS Nguy6n Trudng
T6, D6ng Da, Hh NQi ; Phan Thdnh Vi€t,8D,
THCS Luong Thd Vinh, Tuy Hda, Phti Y€n ;

Trdn Qudc Hodn,9A1, THCS Chu Van An,
Thanh He, HAi Duong ; Nguydn Thi Huydn
Trang,10C5, THPT Phan BOi Chdu, Ngh€ An.
HONG QUANG

thi tuydn sinh vIo lop 10 THPT trudrng

DHSP HE NOi 2002.

o Nhfrrrg cr{ch gifli

ll

1 ldn ding thudc kd dung
duong trung truc cta doan
63 (khong
BC cit

crich
o Dd

*effipA

MBC.

bnrla
:;,:',

{iirii d&p:

A{ Tilqi-{ hIF{AI\H i-{#N

?

Ban h6y cho bidt quy luAt ctra bing s6 dang
tam gi6c dudi day vi h6y tim c6ch tinh nhanh
sd duy nhAt 6 hine cudi ctng cfia bing :
vr
r i 2 3 4 ".. 2Ai)1 2AA2 ._

r
i

,1 -5 i
s t:

40{l

i

4ili13

800.i

;j,-

i.',i-

(t*

o,