Tạp chí toán học và tuổi trẻ số 301 tháng 7 năm 2002

Bo GlAo Duc vA DAO TAO

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HCIc

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NAM

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2002

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Pho Chu tich nurje N-uur,en T'hi Rrnh trao ltruAn chuong Doc ilip hnng Nh:ir
611r \Jhrt Xual hlri Ctic ,.lu. rlttlh lr.'n t.
Tl.ru trr-ro-ng B0 Griro duc va Dao iao L€ Vu Hnng trao Birng khen cua
lhu tuOng Chinl'r phu cho T;ip chi Toan trroc va TuOi tre (anh trdi)'


Gidm doc NXBGD Ngo Trdn Ai Ooc di6n van trong le ki niOn

,Ai W nii1,n 45 nrumilnnufi tap
O(ffi(B qid- fui oi ilin nlnfun

lfiunn ohrm70\o
-_96ng

W

kqos qffinr

4..1.2002, HOi trudng chinh Cung Van h6a

H[u nghi trang hoing ruc id. Hon 100*0 dai bidu
dI quy Lu ve chung vui trudc su truons thanh oua

45 nam cua Nhh xual bdn Gi6o duc. 5.n au Uidi
le trong.thd' ki niem 45 nam thhnh lap NXB Ci{io

dyq ye d6n nh€n HuAn chuong Doc lap hang Nhat
c6 Nguyen.Tcing bi
$g Fu, itra'p nan'n TW"Dring
cOng san Vi0r Nam D6 Mudi, ph6 Chu tich nudi

CHXHCN ViOt Nam Nguyen Thi Binh, ph6
flr.ong ban Tu tuong Van Edi fw Olo-nry quat,
Thri truong E0 Van h6a Thong tin phan rctrlcifii,
Thri truong Bg Ngo4i giao N[uy0n phi Binh, ph6
chtr nhi€m Uy ban V[n h6a Giiio duc rhanh thieu
1i$n ve nhi ddng Qu6c hoi Luong Ngoc Toin,

Vi€n truong Vien rhi dua khen thuEng NhA nudc
Cao Kim Hudng, ciic dai bidu cira Uy ban bio v€
vi cham s6c trd em Vier Nam, Vari phbng Chu
trch nu6c, Van phong Chfnl pfrt, Ban'KhoI girlo
I

rung uong, Ban Tu tuong Van h6a TW. Ban-Var

giri Chfnh phi. Ban Ddi-m6i doanh.nghi€p. Bo
Lao.ddng_vd Thuong binh xI hoi, Tiin! cuc day
nghd, BQ Tii chinh, Ngan hdng cong thiion!. Cuc
xuat bin, Cuc bin quydn,
-dfi fdniConi tv pnai nann
srich Viet Nam vI
aien n[idu fo,"eun nginh
cua Trung uong. Dai bieu cui 4olrire iriiua,
trong ca nddc cfrng v€ du chia vui cirng NXB Girio

1y.. Dai sf Vuong quoc Campuchia rai Ha NOi,
Tham trin van hda Dai srl qudn nudc CHDCND
T io, Bi thu thrl hai Dai sri qu6n Trung
eudc, Dai
bi6n lam thdi Dai srl[u6n Lien Uang:N]a, Girim
'duc
dd'c NXB Gi6o
Lho, Tro ti CiTm'JOc ptrq
tl49l q Kurusap_a'Thrii Lan,'Ph5 Tdng'bien^tAp
NXB Girio duc Li6n bang Nga.dai dien Dai iii

quiin nudc Coirg hba PnripiChri nhi6m du an song

.

Cdn b6 Toa soan Tap chi To6n hoc va Tudi tr6.

ngfi Phrip - Viet la cdc vi khiich nu6c nsoai ttcm
d€n cho buOi.le nhfing tinh cam dat dho c-ua nhung
ngudi b4n gdg xa trong may thAp ki qua da ho!
tric,, gitip dd lAn nhau l(c thuAn l,ci clng nhu ktii
gian kh6. Cric Thf_tru6ng BO Girio duc vh Dio rao
1,0 Vfr Hirng, Nguy0n Van Vong, Dang Huj,nh Mai

(km ti|p

trotr,q 2)


BAI PHAT BIEU CUA
PHO CIIIJ TICH N l-0c

tcuynn THIBINII

(Trich)
Trudc h6t, t6i nhiQt liOt chric mimg todn thd ci{n
bO cOng nhAn viOn chrlc NXB Gi6o dqc
(NXBGD) nhAn dip L0 ki niOm 45.nam ngdy

ina*, mp NXBGD... Hion nay, vd quy m6
NXBGDI} mQt NXB lon nhdt ci nu6c, v6i doi
ngfi d6ng dAo Gi6o su, Tidn si, Th4c si, Ctt nhan

uI tbin Sdch in tr€n gid:y nhu trudc dAy, mI cbn
xudt bin cA s6ch diOn tir, phim gi6o khoa. bang
hinh, bang tidng, dia CD-ROM, bin d6 vh" tranh
inh gir{o i,}", .:bn dim duong nhiep vq td chrlc

phdt henh s6ch gi6o khoa vh xuat bin phdm gir{o
^duc
ddn tAn tay gir{o vi€n, hoc sinh vi ph6i hoo
chi dao cOng t6c Thu vi€n trubng phO th6ng
trong toln qir6". Trach nhiQm d6 tdt n-4ng 1d
nhtn g th4t vlnh quang. Didu d6 chtmg t6 sr,r tin
cay Ctra Ding, Nhh nudc vh nhAn dAn ddi v6i

NXBGD.
TOi d[c bict bidu duong c6c d6ng chf d6 thuc
hiOn vi td chrlc chi dao viOc thuc hi6n chinh
;;h hoi ddi v6i s6ch gi6o khoa, dd td chrlc
^t vi€i truyAn ngdn gido duc dao dftc cho
cu0c I/ri
rhiiit nian, nhi d6ng g6p phdn gii{o duc dao drlc
cho hoc sinh, didu md toln xd h6i dang rdt quan
mm.
D6nh gir{ th}rnh tich vd d6ng g5p cira NXBGD
cho su nghiOp gi6o duc vh dho tao trong 45 nam

qua, Nhi nudc dd tang thudng cr{c ddng chi
nhidu lluan chuong cao quli, vi h6m nay t0i
vinh du thay m6t" Chir tich nudc trao cho
NXBGD Huan chuong DQc lAp hang Nhdt.

... NXBGD cdn phii d6ng g6p tfch cuc vho
vicc ddi mdi, nang cao chdt luong vd rroi dung
vd hinh thrlc cfrng nhu cAu tnic SGK hdn nfra,
g6p phdn vlo muc tiOu nAng cao dAn tri,.dlo tao
ntran tUc, b6i dudng nhAn tli cho ddt nudc'
Chring ta hAy phdn"ddu dd ngly chng c6 nhi6u
bo s6ih gi6o khoa t6t vd noi dung, dep v6 hinh
thric, ddng thdi phii tim moi biOn ph6p dim bAo
gir{ bdn hop li, giim gi6 SGK, g6p phdn giim
[t O tct an cho nhidu gia dinh nghbo, vi hdy tidp
tuc duy tri, phi{t ridn c6c chinh s6ch xd hOi vd
SbK. Ngoaira NXBGD cdn tich cgc g6p phdn
vdo vi0c" xAy dqng thu viOn truitng hqi ve {dy
manh iroat aong itru vi€n truhng hqc. T6i bidt
nhiem vu niy thong d6 ding, nhtmg d6.li ddi
h6i'cira xa 6oi ta hign nay khi ddt.nudc cbn
nhi6u khd khan, trong hic nhu cdu vd gi6o duc
ctra nhAn d0n ta rdt 16n...

orErrr
\

vAN rt
r
A

rurErrl
a/'

45 NAM

1-- ^.-l ^ BAN
GIAO DUC
XUAT
NHA
THANH LAP
DO GIAM OdC NCO TRAN AI TNiNlT NEY

(Trich)
... B6n muoi lam nim qua NXBGD di co
nhftns budc phiit tridn vtng ch6c vd moi m[t vd
mdi ouan h-e nhidu chidu, da phuong khOng
nsuns duoc cirng cd vi m& r6ng dd thuc hi6n 2
"nang
: xudl btn, in, phdt hdnh S6c}- g1ry
cfir1c
duc vi cfii d4o Thu viQn trudng hoc. NXBGD
trung thenh vdi t6n chi kinh doanh ld phuong
ti€n, phuc vu cho sil nghi€p gido duc ld muc
dich crtu cdnh.
Hhng nim, NXBGD d6 td chrlc in vh ph6t
hdnh ticn 140 triOu bin SGK, vdi mong mudn
khip moi midn dat nu6c, m6i hoc sinh c5 dir
m6t b6 SGK dd hoc.
NXBCO cfing thubng dOng viOn khuydn khich
hoc sinh dilng lai s6ch cfi, dd cham lo xAy dlrng
Tir s6ch giriJkhoa dirng chung dd hoc sinh c5
thd muon SGK dd hoc, nham giim b& kh6 khan
cho cdc bAc cha me vh tidt ki6m vAt chAt cho xd
hQi ; cldng thdi cfing rdt chri trong viOc tang.s6ch

cho Thu-viQn wng sAu, vitng xa, cho nhfrng
vtng bi thiOn tai lt* tut, thgc tiign d4o li truydn
fnOig ia hnh dim ld rdclt, n€n d6 duoc nhi6u
dia phuong ghi nhAn vh YOu mdn.
M6i nam, NXBGD cfrng dd Phdt hinh h)ng
triOu phidu uu ti€n cho hoc sinh conem.gia dinh
chinh si{ch, hoc sinh nghEo, hoc sinh gi6i nham
khuydn khfch c6c em vuot kh6 vI c6 vfr tinh
thdn hidu hoc.
... Ngodi ra, dudi su chi dao ctla 86 Gi6o duc
ve DeE tao, NXBGO iile, khai tdi ciic hoat
dong Kd chuy1n ilteo sdch, Thi gido vi€n thu
viei gi6i,Thi vi€lt chfi dep,Thi vi€i truy€n dao
dftc cho thii'u nhi...
... Ai trong chring ta cfing ddu biCt : Srlc song

vd su hmg thinh ctra m6t Nhh xuAt bin 1u0n
g6n lidn v6i tri tuc cira c6c nhh khoa hoc, cdc
ina giao duc, c6c hgc gi6... goi chung ld ngudn
nguy€n khl qudc gia.Y\ thd, NXBGD lu6n trAn
trqng, thidt tha mdi goi quf tr{c gii d6n-vdi mlt
trdn-uan h6a gi6o duc, tham gia vidt s6ch gir{o

khoa, biOn so4n s6ch tham khAo, lim s6ch phuc
vu cho nhidu dAn tQc anh em.
Cing vdi nhfing cong viQc trcn, N,XBGD cfrng
dd m6-rOng mdilicn kdt vdi cdc Nhd xudt bin
ban & nridC ngoii (Li0n barrg Nga, P16p, Trung
Qudc, Th6i Lin, Lho v.v...) dd tidp thu chqn lgc

n-htns thhnh tuu tinh hoa cfia nu6c ban, llm
gia, Eno ndn vin h6a, gido duc nudc nhi, d6ng


thdi cfrng gi6i thi6u vin h6a cta nu6c ra ddn
mOt sd nudc ban thAn thidt tr0n th6 gi6i.

Hing n[m toln NXBGD d6 xudt bin

gdn

2500 dn phdm kh6c nhau. Nhin theo chidu cao
lh m6t nrii s6ch, nhin theo chi6u r6ng li m6t
rfrng s6ch.

Tuy m6i cudn sdch m5i mIu m5i v6, nhung
didm chung ln thd hiQn duoc dnh sdng v-d
nguy€n li giito duc crta Ddng, dQm dd bdn sdc
ddn t1c vd gidu tinh hi€n dai.

HOm nay li ngdy vinh dU nhdt, li ngly 16
trong dai vh ngiy h6i ldn cira rodn NXBGD.
Chring tOi dai diQn 9 don vi thdnh vi6n, 61 COng
ty S6ch - TBTH vi 1300 cr{n b6 c6ng nhAn vi6n
thuOc NXBGD xin hrla v6i cdc ddng chi ldnh
dao DAng, Qudc hOi, Nhh nudc, Chinh phir, vdi
BQ Gi6o dqc vI Dio tao : NXBGD sd phdt huy
truydn th6'ng dodn k€i hodn thdnh t67 cdc nhi€m
vry, g6p phdn tich cilc vdo sa phdt tridn gido
duc, ndng cao ddn trl, tao tidn d€ quan trong

cho cdng cubc hi€n dai h6a ddt nudc. Dac bi€t
ld thuc hi€n thdnh cbng vi€c thay sdch trong

... HOm nay, NXBGD vinh du duoc nhin
Hudn chuong E\c t\p hang Nhat irta Cirt
tich nudc trao tdng, day ld mOr ldi khing dinh nhrtng ndm sdp d€h...
quf gi6 vdi f nghia vO ctrng sAu s6c :
Trons gdn nfta the'ki, lilc thuQn loi cilng nhu tE KY NIEM 4, NAM ... (Tiep bia 2)
nhrtng lilc khd khdn, cdc the' he ctia N)BGD
... vi nhidu c6n b0 c6c VU, ViOn, COng doln

l.ubn n€u cao truy€'n thdng do.dn k€i,,phdn da.u
khdng mQt mdi, q.uydt tdm hodn thdnh chtc
ndng nhi€m vu cfia Bd Gido duc vd Ddo tao
giao ph6, phuc vu tlch cuc cho ngdnh vd g6p
pluin dn dinh xa hili.
1. NXBGD c6 duo.c kdt qui to l6n nhu
ngiy h6m nay
Trudc hdt ld do su quan tdm cta c6c ddng chi
Ldnh dao Ding, quOi troi, Nhh nudc vd Cifnn

:

phir.

i(garn, iii*;;Arg.6.;.rdd-ca.
d, iiq", Girim ddc
ce"c 56 Ciio au" I Deo 13o,
ffr*'tru6ng no
GD-DT d5 vd irghi dudng, dai dien UBND" c6c

tinh thinh pho Dn Ndng, Hii Phdng, Nam Dinh,
Wnh Phfic, cdc Girim ddc Cong ty S6ch Thiei bi

ffe,H,xrn$r:jirlr,ffi;1Tr,ll

"*r'lr6ur.,.J;
n[an'vicnNxE ci6o dui-fuo" ao fiao
su, ridn si,
nhi gi6o dai di€n cdc tdtc gi6 srich d-a c5 mat trong
budilE trong thd ndy. Doig dir crlc thd h0 c6n bQI
c6ng nhAn vi0n NXB Gii{o duc vi c6c don vi thhnh

Dac bier li sq chi dao cira BQ Gi6o duc vi Dho vicn da.vd dlr Ic'
t4o i sr; gr'ip a'o.cira'c6c 86, c6c co quan ooan
,#fl"31,i#rilKaf?*Hier".T:,f#S;ff:f,:
thd trung uong vd dia phuong.
NXB.Gi6o duc.

r"l".:XJiffi,'1T,ffi i#,'frf#fr'fJ;i
, iT^yi:Idn
vi Ntra xludt ta, nudc ngoei ---" "1" jff

Ai, ci6m d6c Nhh xu.t b6n Girro

$kftffi;$3i1lffii'.flH1?#f;,#,'ilf;:
phu Cr,,i ti"t ;;; N;ric;'rr,..i'gi"h da trao Huan
hbng triQu doc giA vi cira nhan chuong Doc lap hing Nt ei;n" Nig ciao dri;i
lyynt, hoc sinh,
sudt
45 nam qua dE gilnh cho hai Hrian inuoirg La"o aong cho Nhi mi{y in's6ch

9$_qgrg
Ld tinh cd,m uu iti tin cay cia^c6c b{c

NXBGD.

2. C6ng lao ndy .
Trudc het, thuoc vd cdc bac tidn bcfi da dny
un o+t nr,rn; ;d;

!!r^,::
sy:
;ad;ta .l::s
S1ir!!,'o[l'lrX!;r;1';i;';i;',m?ni";r'i

i#
1,"*gJB:i

lhr?. ha!,c6ng rao !!uac vp.nhftng thdy

8ir{.o da vd nghi dudng khong tidc c6ng tiec

chat chiu, vun v6n dd dd lai cho chring ta

c0 P{rtlff#}"fi'3bTJ;fT.?ilH$l;$l"6lij

sfc,

nhidu

tdi san v0 gi6, nhfrng kinh nghiem va nfrfmg

truydn th6ng tdt

gi6o khoa Dong Anh, Chi nh6nh NXB Gi6o dirc
tai Tp. Hd Chf Minh vd phdt bidu f kicn.
r.!y+s.
,,r!! Nry G.ido dlrc,

dep.

Thft ba, c6ng lao niy thuoc vd c6c t6c giil,
oht gido duc dp ttmg

Tap chf T66n'hoc vh Tudi ,tr6, Chi nhTnh NXB
Gi6o duc t4i Tp. Di Ndng vi Trung tam bAn dd
tranh 6nh gi6o duc.

Thri' tru6ng

c6c 9ll

*r;il;
i,ri,rofac-"r'"?di

Lc vfi Hing, Gi6m ddc Nhi xu6t
,Ig,l,_$i1T,9d:"p\q_H*. j,

9il *:, fi9:

fyyf-"*:ik?::
Nga l.l6]d+q^lt.*,qb$P9l1:

vi GS rrdn Dinh Sri dd phr{t
;d.
urrqv 6rrvrur
vuu tr6ut4, tul tuUL Ug Ull(J ULrI llIIUllX
z r.o.
r:-!
;hdrilArfui;il
*:*:,:i?LB"ng
"ffi;
trang s6ch quf, q-ua bao th6ng ngiy dd dct
neil '::If:}
PGS TS Vfr Duong Thuy, Ph6 Gi6m dOc, Bi thu
nhtng tdm thuongquf hon ci ngdn vdng ...
Dqng riy, Tdng bi€n tap NXB Giiio duc dd phdt
Thit tu, vinh du ndy cfrng thu6c v6 9 don vi Ui6riOi'.r* o".
thinh viOn vi 1300 c6n bQ-c6ng nhdn vi0n Nidmvuinhucdndonglaitrongttngiinhm6t,
NXBGD, khOng ngmg phdn ddu xdy dmg don nu.uOi.
vi vd NXBGD ngiy cing thOm vftng manh.
VKT
nhn kho.a r,q",Za;

2


s( outta c0ua um

^'i^

tti PttffittcTilintl

oEatRl
Odnh oh+ ohn

rnAN rHI HUoNG
(G/ THCS Bit

IBUNG HtlG GO S

Son, Hodng H6a,Thanh H6a)

Biri todn l. Gidi vd bi€n ludn
theo tham sd a :
Trong chuong trinh to6n THCS c6c ban hoc
sinh thudng hing tdng khi phii gini hQ phu-o-ng
trinh (h0 Pil tto,-ne d6 co cac bidu thrlc lh tdng
troac iictr cdc dn,lio4c tdng cdc lf,y thta c6c dn.
nai ney trinh bdy mOt phuong phdp giii h9 trf
nhu th6 dua vho cOng thtlc Vi-6t.
Ndu PT bAc bon
(1)
Ap3 + 12* + Ag + A+=0
c6 4 nghi0lrl .x1, x2, x3, x+ ff c6 cdng thfrc
Vi--et licn hQ gifra c6c nghiOm vi cdc h0 s6 ctra

/

trl (l)

-

+

nhu sau

A2= x{2* x{3*

x4

x1x4* x2x3* x2x4 1x3x4

- At = x1x2x3 * x{2x4 + x{3x4 * x2x7x4

(2)

A4= xf2x7x4

viy, vi x1, x2, x3, x4ld c6c nghiOm ctra PT
(1) n6n (x-x)(x-x2)(.r-x3)(x-xo) = Q (3)
Khai tridn vdtr6i cira PT (3) vI so s6nh vdi PT
ThAt

(1) ta c6 c6ng thtlc

N6i ri0ng, ndu x3
= 0 c6 2 nghiOm

'

At = x1 *

\,

x2vd'

Vi-6t

(2).

x4= 0 thi PT i + Ap + A2
=
x2v€t c6ng thrlc Vi-6t ld :
A2=

xfi2

@)

N6uxa = 0 thi PT x3 + A1* + A2x +A3 = 0 c6
3 nghiOm x1, x2,"r3 vdi cOng thtlc Vi-dt li :

- At = \ * x2+ 4 ; A2= xLxZ* xfi3 * x2x3

-

(5)

At= xfi2x3

Khi gap

f

hQ phuong

trinh mh vd phii cdc PT 1I

hang sd,-cbn vd trai c6 dang tdng-cr{c lfiy thira
dua-cdc dn, ta c6 thd coi c6c dn d6 ld c6c
nghi0m ciua m1t phuong trinh,rdi sir dung c6ng
thrlc Vi-6t dd thict l4p PT m6i niy. Nhu v4y ta
dd chuydn vi0c giAi hQ n phuong trinh n 6n v0
gi6i mot PT bQcir mOt dn,-ndu PT b6c n mOt dn
nghiOm
nny giii duo. c dd ding thi d6 chinh

hA

phuong trinlt

x+v=2a-l

{ -'
l*' *Y'=a2 +2a-3
Ldi giii. Ta c6 f * y'=

(6)
(-r + Y)>

-

ZxY =

el -Ze, theo cong thrlc Vi-6t (4). Ta tinh
duoc: 41 = | - 2a vd2A2= (1 - 2o)2 - (a2 + 2a
- 3) = 3a2 - 6a + 4. Til d6 x, y ld hai nghiOm
ctra

i

PT i
+

.

+

A6 + A2-

OhaY

0 -2a)x+3a212-3a+2=O

Bi€t thrlc

:

At= x1* x2* x3*

ttFET

ndu a

A,

- 7. Til d5

= -2a2 + 8a

(7)

:

.z - E hodc arz * [th)
22

A<

o

= h0 PT (6) v0 nghi6m.
.neua=z- J, ) x-y= 3-J,

n6n PT (7) vo nghiOm

Z

z

J,

vJi
on€Ua='Z+-'
-'=y=
-T)-')
t;
u't; thiA=
o ndu2- l'2
.12
8a

-

7>0

vI

hQ

(6) c6 2 nghi€m (x, Y) th

(uq-'[i z-r*.'[) (zo-*Ji
2 )\

[ 2

-2a2 +

tl

za-r-Ja)

'>
.

I-

)
t

Khi he PT c6 3 horc 4 dn thi ta chuydn vd gi6i
PT mOt dn b6c 3 hoic bQc 4, ndu PT ndY c6
dqng dac bict thi ta tim duoc nghi0m cta n6.
Biri to6n 2. Gidi ha phuong trinh
(7a)
lx+y*z=3

l'

1*'*r'+22

=21

(7b)

|.r'*y'

=57

(7c)

t'

+23

Ldi giii. Coi x, y, zld

3 nghiOm xy, x2,

xj cta

cira h0 n phuong trinh dd cho.

mQt PT b4c ba, theo c6ng thrlc Vi-6t (5) tt (7a)
d651 = \* x2* x3=3 = - At) A1= - J

Vdi he W 2 dn thi phuong phdp niy rdt hiOu
qui vi ta dua vd m6t PT b4c 2 luon gili duo.c.

-2(xp2* xfi3+ x2\) = e? -ZAz

li

rJ fzul c6 s2 = *? *i+xl

= (x1 + x2 +

xz)2

.J

>

-

fxyzr=l

- 2l = -12 = A2= -6.
Tt (7c) c6 Sj = *l + *l+ xl = (x1 + x2+ xj)x
2Az= A?

Sz= 9

x Ql + x) + *! -rg2

-

xtx?

-

xzxt)

*

= -Ar(Sz - 3A1= -tr3 + 3A1A2 - 3A7
) 3A1= -57 + 27 + 54 = 24 =Ar = 8
Nhu vAy .rr, xz,x3 ld nghi€m c[ra PT
*' - 3*2 - 6x + 8 = 0. D6 thay pT niy c6

nghi0m ,r1 = 1 suy tt x2 = -2, x7 = 4.
VQy-hQ Q) c6 6 nghiem (x, y, z) ld (1, -2, 4) ;
(1,4, -2); (-2, 1,4); (-2,4, l); (4, I, -2);
(4, -2. t).

Bli

-r]**r*r*,=l
l.ryzt

3xp2x1

Az)

14a)

I

1*l*l=5

lxy+yz+zt+tx=4

04b)
tl4c)

Ldi giii. TU (14b) (14c) c6 (x+z) +

(-v+r)

vI

=

5

(x+z)(y+t) = 4. Giei hC PT ndy duoc nghiem
x*z = l, y+t = 4 hoilc x+z = 4, y+t = 1 (15).
Theo c6ng thrlc Vi-6t (2) ta coi x, !, z, t ld cdc
nghi€m cira PT
x4 + Ag3 + A2x2

+.Ag + Aq=O (l)

tt (l4b) c6 41 = Aq, su)
ra ndu x1 lh nghiOm cira (l) thi x2 = I cUng id
Tt

(14a) c6 Ao = 1 vi

xl
n1n.rp2= 1. Tuong ttt x.;l"o= l.
Xdt cdc truamg hqp sau (chri f x, y, z, t ddu
khr{c 0) :
a) xy = 1 vi zt = 1. TU (14c) c6 yz * tx = 2

to6n 3. Gidi h€ phaong trinh

nghiOm cira (1)

(8a)

fx+y+z=o
t+22 =10 (8b)
1*'*y'
t'
[r'*y' +27 =350 (8c)

l^ + xt
+
='2 = \xt - l)2 =O + rt= 1. Suy ra
Ldi giii. Coi x, y, z ld 3 nghiOm cria m6r PT
-xt
x = z, ! = t . TU d6 vd tt (15) c6 ciic nghiOm
bAc 3, theo cong thric Vi-6t (5) tir (8a) c5
Sr = xr + xz + 13 = 0 - -A1. Tuong tu ldi giii
x=z=1,-r= l=2vb,.r=:= 2, 1,=1=l
:

:

bii

2'

todn 2 tD (8b) c5

+rl+"! = 10 = ,+l -Ze,
)2A2= A? -Sz = -10 * A2- -5
(9)
Tfnh53 = r?*l+xl = -a,'* 34A2-3At
> Sr = -3A:

(10)
Sz= x?

D4t

S,,

=*l +xl +"r{. Khai tridn

(x1 + x2 +

+ *';*2) = 0 ta du-o. c
";*2
(11)
-S,r+3 = Sn+t.Az+ Sn.Asvdin> 1
(l
(9),
(8b)
TiI l),
c6 :
SzAz+
(t2)
SlAl = -50
-Sq=
TU (l l), (9), (10) c6 :

4)

Qf*.z

+

.

-Ss = S:Az + S2A3

Tt

=25At

(12), (13) c6 :
-J7=S5A2+S4A3 =l75At
TU d6 vd (8c) suy ra At = -2.
Vay r, y, z ld c6c nghiem cira PT

(

4

C6 thd v0n dung cOng thric Vi-6t dd sdng tao ra
bdi todn moi bing cdchldy m6t phuong trinh bAc
n (n = 2,3, 4) d6 tim nghiOm, tinh n gi6, tri cira c6c
tdng cr{c lfry thta ci:r- cdc nghi6m 51 r6i yOu cdu
giii hQ phuong trinh mi vC trdi li ci{c tdng Sl cbn
vd phii ld gi6 tri tuong fng ctra chring.

Mdi c6c ban vdn dung c6ng thrlc
c6ch€, phuong trinh sau :
Bni 1.

ta

duo.

c

PI

todn 4. Giai

trinh

'

lx+y+z=6
I

Bni 2. 1*2 +v2 +22 =lg
t-

[Vx+ry +'lz=4

:

(8) c6 6 nghiOm.

hQ phuong

t:'.:='+tJ-z
[x'+Y'=6

(1 1),

\thi

Bii

vO nghiOm.

13)

,3 * 5, - 2 = O<>x3 + 8 - 5(x + z) = o
<>(-r+ zlfi -2x-l)=Q
Phuong trinh ndy c6 3 nghi0m xt = -2,
xz= | - J2 ,r3 = 1 * Ji .Thay x, y, zbbi x1,
x2,

2

b) xt = I vd yz = 1. Lap lu0n tuong tU nhu tr€n
cfing suy ra h€ dd cho c6 c6c nghiOm nhu rhd
e) xz = I vd yr = 1. Kdt hqp vdi (15) suy ra h€

lryzt=1

le
lx+v+z+t=t'2

Bni3.

]l*l*1*ll2
lx y z t 2
ll
I

l.

I r'

I

I
t'

r

1 125
I_=_

L*' yz z' t2
1

4

Vi-6t

dd

giii

},tAy

rilt

soNo'soNc tA

'

Hins neav chfng ta phii ddi diQn v6i rdt
uai tban c6 thd
rrri^c, Ea"ial toin. Vie. liai
"a"

thuc hiOn bbi mOt hof,c nhidu ngudi. Tuy
ir" aonn tap thd cira nhidu ngudi duo. c td chfc
khing dinh sric manh-lc'n lao ctra
;3i;;'ry
"a,ig
thd, rn6i ngudi thuc hiQn
dgng-tap
la5
itrohe
nO.
;a# cira miih, c6 trao ddi-ttrong tin v6i
;hd;
'nguOi
mac theo mot cdu truc nhdt dinh'
- "Tu*n
tu. dd eiii c6c bli todn, m6y tinh c6 th6

.ot"t oi" nt[du uo xt 1l M6y tinh c6 mQt 4
chi
"o
it U, iai *oi thoi didm chixrl-li duc.c-m6t gie!
;hi,;rd" goi ln md.y tinh ndi.tiep;vir-thuft
tuiing ime"ggild thidt sidi ndiliab.rvl6y tinh c6
uo-*I'li, thr"rc nie, nrridu chi thi tai m5i
"rriafi
tt Oi aidm, ggil.il niny finh song song. Thuit to6n
tuons fnsloi ld thult to6n song song. C'{c bO
* ffpt al d'tioc td chrlc theo mOt cdu trric nhdt
dinh.
Nsudi ta mO hinh h6a td chrlc cfra c6c bO xtr li
.orr[ tottg blng m6t dd thi Graph) G = (V, D,
trong d5 :
. Tap dinh Y li qP htu han c6c b0 xrl li'
o Hai dinh a, b thu}c V duo. c ndi v6i nhau (goi
.ld naOJ canh) khi vI chi khi hai bQ xir li a, b trao
Adl tirong tin trgc tidp duo.c v6i nhau. Tip c6c
canh cta V ki hiQu lir EDi nhiOn, G tdi thidu phai lir m6t d6 thi lic1
th6ng, nghia 1I v6i hai dinh bdt k\ a, b thuQc Y
thi # finhdt mot dudng di qua c6c canh ndi

oi

q

NcurENvANNGQC
(DH Md Dia chdt)

A

B

D

(_

vqY

avdb.

Nhu v4y, hai dinh kd nhau trao dtii tholg-tin
truc ti6p'vdi nhau, cdn hai dinh kh6ng kd nhau
r*rai truo oitintr Urac. N6i chung m6t mo hinh xft li song
sons nOn c6 chu trinh lla-min-tcnr (Hamilton),.
di tir m6t dinh qua tAt ci
,,ntfru te tdn tai drrdne
-dinh
dring m6t ldn, vi d6n
cic dinh ciaV, m6i
(ducmg
di kn6p kin).
ph6t
xudt
dugc dinh
d5 duo.c nghi€n
song
MOt mo hinh xir li song

N6 c6 cdu
phuong.
crlu nhidu li n - khdi l4p
:
sau
tnic duo.c x6c dinh nhu
. Tap dinh y e6 t difir, duo.c di{nh sd tt 0
ddn 2n - 1 theo h0 ddm nhi phan. Trlc li m6i
dinh 1I mQt ddy nhi phan vdi d6 ddi n.
. T0p c+nh E gdm nhfing canh n6i-.hai gay ohi
phan t'hi tYdi n =Zttfi'Z - kh6i lQp phuong gdm 4 ttinh
A, B, C, D, 4 catth AB, BC, CD, DA vI 1 chu
trinh Hamilton g6m 4 canh d6 (h'1)'

ll

I
Vdi n= 3ttti

I

Hinh

3 -khdi lflp

Hinh2

Phucng

2 - ktdi l4p phuong ABCD vi'
A,B,C,D,. n5 edm 8 dinh, 12 canh (h"2)' Dudng
ai encob'c'BaA'l,ld, chu uinh Hamilton cua n6'
Ttmg tg ta xAy dr;ng Q*l)-khdi lip phuctng tt)
hai n -lfidi lQpphucnrg bang cdch:
M6i dey nhi phan @9i ld m6.t tir nhi phan)'
dinh cria rkhdi lQp plucrng thf nhdt dugc th€m
vlo kf tu 0 bcn tn{f (tld dugc day ki tU nhi phan
do dni n + 1), cdn dinh ct;a n - khrti lQp phuong
thrl hai thi rluoc th€m vdo ki tu I b€n tn{i.
Ndi m6i dinh cfra htrdi lQp phuong thrl nhdt
v6i dinh tucrng rfurg (c6 cDngddy kf tg dQ dli rr
ben phii) cira kh6i lQp phuong thrlhai'-Gii sft dd xdc dinh duoc chu trinh Hamilton
ciua n - kh6i l4p phucmg. Khi d6 mOt chu trinh
Hamilton ctra (n + l) khdi lQp phuong duo.c
xdc dinh theo hai n - kh6i lQp phuong tao thdnh
ld
n6 nhu sau : Gii sir vf , .-.,r,1 ,r,t*, , ...," ,,
duoc xdy aring

ttrr

-

,'!

ri- kh6i lap phucrng thrl
nhdt, vi ,! , ..., t , fur, .--,
fn, ,f le "ho
trinh Hamilton cfia n - khdi lQp phuctng thri hai'

Khi d6 16 fang ,f , ..., ,! , ;? , frr, ..., t , tn ,

chu trinh Hamilton cira

..., ,?*r,

,,!,,

...,

i", ,f lI chu trinh Hamilton

khdi l4p phucnrg. (C6 thd tim duo.c
nhidu chu trinh Hamilton).
C{ch x0y drrng chu trinh Hamilton cfia n kh6i 16o pliuonelrcn dav ld noi dung co bin c[ra
ptiuooii 6rap xEy drmg-md Gray dd tim ra ddy
ph6n do dii n sao cho hai tt liOn tidp,
2'i tir nhi ^tt
ctns nhri Aau ticn vi til cu6i cirng chi khdc
nhal dring mQt kf tu nhi PhanDOi didu tr0n d6y vd m6y tffi song song vh
*tr lf torg song hi vqng gi-rip clc bll hinh duqg
;td vi6c xaY 4rn'g c4u
fid;;a; finh"vuc md
qp
bu6c di ta't ydu cira
mot
phiic
tint
tr,ii*ai
khoa hoc tinh toi{n.

cira (n+1)

-


nrOr so tutt'i nm uAl or'THt
$IN}I DAI HOC IffiOI A NAM 2OO2
t,.f

TTJYTN

'

Qua viOc giAi dd thi tuydn sinh dai hoc kh6i A
nAm nay, chfng t6i trinh biy m6r s6 luu vd
phuong ph6p giii c6c loai todn c6 trong dd thi.
Ngodi ra, xin nOu th€m m6t sd bhi dlnh cho cdc
b4n c6 nhu cdu luy6n t6p th0m.

i

cAu

t: cho hlm sd

y = -x3 + 3mx2 + 3(1 - m2;x + *3 - rn2 (1)
(m l) tham s0).
1/ Kh6o s5t su bidn thiOn vd v6 d6 thi him sd

(1)khim=1,
2! Tm k dd phuong trinh: -*3 +
3k'= 0 (2) c6 ba nghi€m phAn bi€t.

Yidt phugng trinh duong thing di qua hai
didm cuc tri ctra dd thi him sd (l).
Hudng dtin gtdi:
1l Ydi m = 1, ta duo.c hhm sd y = -x3 + 3x2.
(C{c ban tu giii) C6 dd thi nhu sau:
2l C6 cdc cdch
gi6i sau:
Cdch l:

c phuong
(2) trd thhnh
,.:tlj3):1,
trinh
-'
duo.

-x3+3x2=m(3)

P.hgong" trinh (PT)
Phuong
phAn biOt
nehiOm phan
bi,
!rl$, trrl (3) c6 3 nghiem
khi vi chilsd y - -x3 + 3x2 tui : Oid. phAn bi0t.Tt aO tm

ta duo.c 0 < m < 4. Gfii0 < -k3 + 3k2 < 4 <+

j_1fr'ta-r)ro ._.
-ro [k*o'k*2

kt*t)ft-z)2

Cdch 2: Bi€n luAn sd nghiOm phuong trinh.

PT niy c6 nghi€m x = k n€n
f(x) = xz - (3 - k)x + k2 - 3k = o phii

c6

TacS A =-3k2+6k+9>0e -1
f(k)= 3k2-6klo<+ klo, k+2.

6

e

-

(k3 - :t2xt3
3k2 + 4) < o
<+ k2(k- z)2(k+ 1)(k - 3) < o
Til d6 cfrng suy ra kdt qui trcn.
3) Ta c6 Y' = -3x2 + 6mx + 3(1

-

m2)

Bi0t thrlc A' = 9 n€n PT y' = 0 luOn c6 2
nghiOm phAn bi0t Xr, Xz vh ddi ddu qua 2
nghiOm d6. Dd thi hhm sO (l) c6 2 didm cqc rri
M1(x1, yr) vi M2(x2,y2).

Tt

y'-

0 c6 x1 = m

vio (tr) ta tfnh duoc

!-Yr -Yz-Yt
x-Xl x2-xl

dugc

-

L,x2= m +

1.

y1, y2 rdi thay vho pT

pr y =zx-m2+m phii tim

Cdch 2. Thuc hidn chia da thrlc

,=,'[*-t')+2*-.n'*rn
' '[3 3)'""
Vi tai c6c didm cgc tri Mr, Mz cd y'(x,) =
y'(xz) = 0 qghia li c6 y1 = 2xr - m2 + m vi
yz= Zxz-m2+m. Qua trai aldm Mr, Mz x6c dinh
duy nhdt duong thing c6PT y= 2x_'m2+m.
Luu y .' CAu h6i tildng til vdi hdm sd y cfrng giii duo.c bdng cdch2,
v(x)
Gii srl dd thi hhm sd nhy c6 2 didm cuc tri thi

{riilt.,

Kdt hqp lai c0ng duoc kdt qui trcn.

Til (s) c6 y(0).y(2) < 0

-a'x+b'

2 nghi0m phAn bi6t khr{c k.

kion:

(5)
lYco.ycr < 0
Tt (4) c6 y'(x) = -3x2 + 6x = -3x(x - 2) = O

c6_ 2 nghiOrn ph0n biQt lh xl = O, x2 = 2 r,I y' ddi

ax2+bx+c u(x)
---.-= ''-'

-x3+3x2+k3-3k2=o
<) (k - x)[x2- (3 -k)x + k2 - 3k] = g

Didu

[y'(x)=0 c6 2 nghiOm phan bi6t
jvd y'(x) d6i ddu qua c6c nghiOm d6 (4)

Thay

m=-k3+3k2

, {-T

Cdch -1. Irf (2) c6 3 nghi€m phan biet nghia ld
dd thf hdm sd y(x) -- -x3 + 3x2 + k3 - 3k2 ctt
truc hoinh tai 3 didm ph6n bi0t, hay li

Cdch 1.

St dung dd thi
Det

oI._,

Nci-ryBN eNu oUNc - DANG THANH HAr

ddu qua chring.

3x2 + k3 --

_.31

ta

v

uu'u Luc do y _
Y'=0=vu'-uv'=0= _=_.
=
VV'U
u'
v'

/axl.-b

a'
hai didm cuc tri.

PT dudng thing di qua

Bhi tAp 1. Cho hlm sd y=

x2-(lm+l)x++m

2x-l


Ad nam sd c6 cuc tri vi hai didm cuc
ctra do thi d6i xung vdi nhau qua
tidu
dai, cuc
duong thing x + y + 1 = 0. (D/s: m= 1).
CAU I Cho phuong trinh:
.
t--7
logj x+{logix+l -Zm-l=0 (1 )
(m ld tham s6l
U Gleiphuong trinh (1) khi m = 2.
2lTrm m dd phuong trinh (1) c6 it nhdt mot

Tim'm

nshiom
--o--- :' thuoc doan

vdi t >

Tadu-o.cPT: t2 +t-Zm-2=O

(2)

1.

(2)trd thdnht2+t-6=0

Ta

e0
*<.6

€)

I it nhdt

II
t*+:lt
y=lx'-z

v=X*3
.

Iludng ddn gi.rii:
1/ Didukicn sin2x*-f

.

2

5(sin +2sin
rtre--ffi

x sin2x +cos 3x

+sin3x)

mOt nghiQm x thuOc

Q)

Bidn ddi ttl thrlc vd trii (2) duoc :
sinx + cosx - cos3x + cos 3x + sin3x =
= cosx + 2sin2xcosx
Thay vio (2) vI bidn ddi vdphii (2) c6

o

5(cosx+2sin2xcosx)

vi

chri y crtsx < 1, ta

Trong (0,2n) c6 cic

:

=Zcosl x+3

l+2sin2x
2cos2 x-5cosx+2=0Gi6i

v6i cosx

.

c5 t
Do d6,Pf (1) c6

dudng:

o

,r

logj x=+16, x=3Jv3

(1)

Tinh diOn tich hinh phing gidi han boi

(2\

V6i t = 2, ta c6 logl x=3 til d6

2l

cdc

t+Zsin2x )

=cos2x+3

Dlt ,=lEg3 **l

ll VOim=Z,Pf
ft=2
e[t=-3
(loai)

I
2l

x

[rtl6l
I
L

Hudng ddn gidi:
Didu kien: x>0.

5[rir**"ot3'*tin3*)=cos2x+3

hq

Irf

bQc2 ddi

duoc cosx=|.
2

rc 5t
nghi€m x=-i,x=r-i 1[fr3

mdn didu ki€n 1 +ZsinZx+4.

2l

Xdt phuong trinh holnh d0 giao didm

e PT (2) c6nghiom t thuoc ll;21.
[r':f
Ll' I
=x+3 (3)
l*'-+*n{
(2)
thlnh
tr&
PT
thi
(0
2
+
t
t2
D4t
=

-Ndux<1ho4cx2

f(t; = 2*

vdi i
f(t) dong bidn, do d6 f(1) S f(0 S f(2) = 0 S
Zmi4. Ti duo.c 0 < m < 2 th6a mdn y6u cdu
cta d6 bii.
Lxu it: Phuong irinh f(x) = k (k ld tham sd) c6
ft nhdt mQt nghicm x e (o; F) ttri vi chi khi k
thuOc tAp si6l tri ctra hdm s6 v6i x e (cr; F). Do
d6 ia c6^ it e rd dung dao hdm dd x6t su bi€n
thiOn cira him sd, tir d5 suy ra tap gi6 tri cira k.
Bdi tAp 2: Tim m dd h9 phuong trinh sau c6
sd

nshi€m

{$*Jll=^
+V3-x =m

Huong ddn: Tril ttmg vd cira c6c phuong trinh
r6i chfnrg minh x = Y.
Hq c6 nghiOm khi Phuong trinh f(x) =
{-z* *Jz-*= m c6 nghiom. S[r dung dao him,
til d6 suy ra gi6
tim tAp gi6 tri cira hdm sd

f(!

m.

(D/s: .6
cAu ul:
U nm nghiOm thu6c khoing (0, 2x) cta

phuong trinh:

ex2-5x=o
<>x=0,x='5.
-NOu1Pr (3)

e

x2-3x+6 = 0.

V6 nghiOm.
Ta cdn tinh
l.
S

= {x+3{x2-4x+3))dx

l5\

[{2Y

tri

PT (3)

!x+3+x2 -4x+3Ex

13

1.

+

+ (tx+3-tx2 -ax+3))dx
3.

s= (sx-x2)ox+ i(x2 -3x+ 6)dx+5j(s*-*')a*

0l

+) [.[+ +.,.)
{+
(s^'
*') It io9 (dvdt)
.[?-T)1,=;

3

l:


- A /\.
--44MgTSOrry

,

,\

,\

Tr{UCLTIINHE
grfra DUdhrGrHANo vh DUoI.IGTRON
(fi€p

theo ki trudc)

LE HAo
(GY trudngCDSP PhrtYen)

II. He -thrric
dudng trdn

li6n

hQ gi8a dudng

thing

vh

- (* + HI)EF =,4a
) EH.FF =,qE> Ptl.W =,r\D

Bii todn 1.

Cho

dutnrg
O
b6n kfnh R vi

Til d6 vh dp dung (3) c6

trbn tam

ddy cung AB
c6 trung didm

M.

T}rr€

didm

thi 4
thing

hnpg A, B, C, D li hnng
didm didu hda khi vn cfi
tui oe .oD
(5)
Chungminh.C1t:df ring OM LCD(h.3)
Theo hC thrlc Niu-ton cd

Hinh 3

=rt

* .*:(ou

* *o)
"r)(oil
= Ofr+ tWe .-tti =OLf +MAz=fr
+

Bii

toSn 2. Cho dudng rrdn tAm O b6n kinh
{._Ceq dy*g rhang chrla hai d6y cung AB vi
CD cit nhau rai didm E sao cho-E khic trung
didm AB-vi khong'nam trcn d;dn;;d;. N,Io;
{qqg $ing 5 qua E cit c6c ducrng thing AD,
BC Hn luot tai p vd e. Ggi Il li hinh chidir cira
didm O lOn dudng rhAng 6 rhi

II2EH
:-l-:=-

EP EQ

(6)

,"/(E)

trong d6 ,4E) = Oe - np U ptruo,rg tich cira
didm E ddi vdi dudng rrbn A^b.
Chang minh. Giit srl c6c duons thi.ns AD vi
BC ciit nhau tai didm Ug di€rri l,t trcil AB sao
"r.
chg A, B, E, M h hnng didm didu hba, Dudng
t\4"e lU-cat PQ tai F Grmn 4 vc truhng hqi
didm E nern trong dudng rrbn). Vi JA, JE, JE,
JM ld chtm di6u hda n0n p, e,, E, F h hnng
didm di6u hba. Theo (5) c6 OE . OF = rt

) oE(oE**)=R2 )E6.EF=7E2 -R2

8

1t2zEH
:-::::EP EQ EF
HQ thrlc_ndy
trilng v6i D, C.

D

.7(E)

vin dring khi p, e tuong ring

He qud I. Khi
dudng thing 6 di qua
giao di6m J cia AD vit
BC. Ggi 11 h hinh

chi6u cira tim O len 6
thi,q\= tn.At .
HQ qud 2. H0 thrlc

(6) vin dfng khi A
trirng vdi D vi B tring
vdi C, h(c d6 AD vir

BC ddu li ti6p tuy6n
cta duirng trdn (h. 5).
Tt bli to6n 2 khi
cho dq.r,rg thing 6
thay d6i vi trf nhung
v6n di qua E ta du-o. c
ci{c truhng hqp ri0ng
du6i dfly.

Hinh 5


ne
tai

suoi fridudng

rhing 5 cit dudng

trbn

S vd

Chitng

*Ti#ffififffijl"r"Jtr#l,rrH:?#;;i

tri ti6p tuydn v6i dudng trdn (oz)'
c4c b4n hiy sir dung c6c k.t qui trcn dd lim

l'n' , 1
l+i=:+:
EP EQ ES ET
mintt. Apdune (6) (h. 4)

c6

1 1 es* gr
:+:=:+==

"T}ii:t#:ru*

u kh6ng thuQc dudng

trd.n

1 1
zEH::+:n
dudng trbn d6 ldn luot tai A, B vh C, D. MOt
ES Er ES .E-r '"@) EP EQ il# ffis ;rar ,org *brg.roi AD cht BC i+r
H0 quri 4.IoE tai E thi E tI trung di6m ciua PQ.

trong d6 H 11 hinh chidu ctra O lcn EQ.
Bei 2. Ctro hai dudng trdn tim Op Oz vh 5 li
Chrtng minh. VrE trirng va H ncn EH = 0
truc dang phuong cira hai dudng trbn d6' N'ftit
+
tq
=gc6t tuydn song song v6i 5 c6t dudng trbn t6m 01
- f,f
Arn o b6; kinh R k6 hai i6t tuy6n tuy f_c6t

Ciic ban hdy vE didm E nam trong vi nf,m
ngoli dudng tiOn & hQ qui 4 sd tht'y d6 chinh th
"bdi todn con bddm".

tai f< va f', mQt cit tuydn song song vdi 6 cit
dudng trbn t&m 02tqi G vi, H. E lb didm tren E
khonl thing hhng vli O1, 02. EK, EF cat
tai Q vd P ; EG, EH cit
Bhi to6n 3. Gqi 5 li ruc d&ng phuong ctra hai ducng trbnlam o1
02 tai M vd N. Chung minh
dudng trbn (O1) vI (O2). Qua didm F thuoc 6, duhng trbn tam
vd 6 d6ng quY.
kb caltuydn cit (Or) tai P, Q virk6 c6t tuydn c6t rilng PQ, MN
Biri 3. Cho tir gi6c L6i ABCD noi ti6p dudng
(O) tai J, G. Gai E 1n didm thu6c 6$t*g
tam O bdnfi"n n. (hc dudng thing AB vh
trbn
ciit
kh6;g'nam tren cdc dubng trbn. EP vd EQ
thing 6 blt ki qua E,

(O,) idn nfra tai Xvd Y, cdn E/ v>a EG cat (Oz) CD citnhau tai E. Dudng
vir BC tai P' Q' At{mg
AD
th&ng
duirng
c6c
cit
ldn nfra tai M vd N. Chfng minh rang 6, W, MN
2-Eo
d6ng quy.
l*I
Dang thrlc
t
minh rine
EQ
EP
OE'-R'
^.
Chfing minh. Gih sir 6 ldn lugt cit W, MN vi
xiy ra khi vi chi khi 6 di qua O.
OrO2tai Kt, Kzvi l{ (h. 6)- Theo (6) vn do 5l}
truc dang phuong nan OlEz - R? :982 - fi

zEH ---- zEH =
1 1
EF EKt OtE' - Rf ozE" - R;
!1
suy ra EKr = EK2 haY K1 tring
:
+i

=
EF EKZ

'

DON.$ ,,

,,"',. ".,'
.''
,. ,,,, ., THTT SO SOe (8/eW2l '

,

" *'+n,ceng
tam gir{c

t{nh chdt,dudn8'phfln,gi6c ctra

vor,K2(dpcm).
o Nhtrng didu cdn luu f khi giii dd thi tuydn
sinh dai hoc kh6i A nam 2002

o V€ mQt b6t ditg ''th$q'trodg 'ki"thi to6n
, t:- ': :.,' 1 1 ,1:.,:,i r :r; '
,' qude#
e

Lich strbintrtnenh ttrainiQ-m-him s6

0{c b4n sE'nhfn' dryc tdiglni$Pcfria

' ' :Shtl:16,6;gdu phuonghinh'r&n; . "
Chi 6len dtng thu6c vi comPa,

:

Cho mdy didm ? ...

C{c ban nhd dat mua 'I rITT t4i c6c co sd
':,:
buu.dieii fu"hd;.

.

Hinh 6


siclt(uot#o* roAN6 toqi trotr
rON rsAN
B0 s6ch gii{o khoa (SGK) To6n 6 mdi gdm hai
du-d. c chinh thrlc dua vdo day hoc b c6c
trudng THCS tt ndm hoc 2002*2003. C6c ban
nh6 ticn khip moi midn Adt nudc ldn ddu ticn
bu6c chAn ddn trubng THCS sE duo. c lhm quen
v6i cudn SGK To6n 6 v6i bia in nhidu mhu-, c6

iap

nhidu hinh v6, nhidu brlc inh dep. Nhi6u didu
^

vi li thri dang chd cdc ban.
ki6n thrlc to6n hoc duoc trinh biy trong
sr{.ch ddu don,giAn, dC'hidu.'-Nhfng n6i aun[
lryrg 14p v6i SGK To6n & Tidu hoc"duoc gi6fi
Ugt (yi dq quy t6c thuc hinh cdc ph6p tinh C6ng,
Uti fch

Cr{c

trir, nhAn, chia sd.tu nhiOn vh sd ihap phan). Sau
tAp vi b<$ tfc vd sO tu nhi'eri, cdc ban
dyO-c
!r9c ngay tAp hqp c6c sd'nguy€n v6i cdih
trinh biy.rdt nhe nhhng, th1ng-qia cdc vf du
thrlc t€ gdn gfii vi phr) hgp vdi sfc tidp thu c[ra
c6c ban, ching han sd nguy€n 6m xudi hien khi
x6t nhi€t dQ dudi 0"C, dO cao du6i muc nu6c
bidn, do cAn thi, s6 tidn no, rhdi gian trubc cong

khi 6n

nguy6n...

Tidp theo sd nguy€n, SGK gidi thiOu vd ph6n

^.4 v6i a, b e Z vd b
s0
;

b

phAn s6 duoc trinh

*

0.

CAc ph6p

tinh vd

biy dua trOn c6c kidn thfc vd
cdc ph6p tfnh nhy md c6c ban dd bidr 6

Tidu hoc.
Cac ikidu tidp c6n quy nap, rU quan si{r, thir nghiem,
do, v6, n6u nhAn x6t, di ddn ddn kidn thrlc m6i.
D6 li c6c kh6i ni0m mo ddu cira hinh hoc phing
nhu : didm, duong [h8ng, mat phing,.iii ,iri
mit phing, doan thing, trung didm-cira doan
thang, g5c, tia ph6n gi6c c&a-g6c, dudng trdn,
tam gi6c. Khbng y€u cdu chtnis minh cidt che

ViCc giii cdc bdi tap ney cdn girip ndng cao
w
,J
mat bdng vdtt hda chung. Vi du bidt duoc : Binh
NgO Dai C6o cira Nguy6n Trdi ra ddi nam nlo ?
COng d6ng c6c dAn tOc Vi€t Nam c6 bao nhiOu
d6n t6c ? Loi ich cira kOnh dio Xuy-e. 6to,

mdy bay ra ddi nam nho ? DO cao ctra niri
Phan-xi-p5ng, D6 sAu cira vinh Cam Ranh ? Hai
di tfch 6 nudc ta duoc c6ng nhAn ld di s6n van
h6a thd gidi vho ndm ndo-? TOn nhh roi{n hoc
Vi6t Nam ndi tidng 0 thd ki XV ? Cong thrlc
mu6i dua cAi vI cdch ldm m6n "dta kho thit"?
Quy ddi dQ C ra d6 F nhu thdnlo ? "Ti sd ving"
trong kidn trfc vh hQi hqa lh gi ? Tidn l6i tidt
kiOm duoc tfnh ra sao ? ...
.Nhtrrg kidn thfc to6n hoc gldu tinh ftng dung
ddu duoc s6ch chri f khai thic. Ching hin, khi

hoc "Ti s6'ctia hai sO' -Ti s6'phdn trdm -T{ t€
xich", ciic ban du-o. c y6u cdu gi6i thich thd ndo li
vdng 4 con 9 (.9999) ; tinh ti sd phdn rram mudi
trong nu6c bidn,.tfnh luong nu6c trong 4 kg dua
chuOt ; tinh chidu dei rhar cira chidC m6y bay
Boeing 747 l.hi bidt dO dii cria n6 trOn m6t bin
vE ; Gn bin dd ri lc xich 1 : 20000 thi cay cdu
M! ThuAn duo. c vd vdi chidu dii bao nhi0u
xentim6t ? v.v...
C{c bhi qp du-d. c rhd hicn du6i nhi6u hinh
thtc : didn sd thich hqp vlo 6 ruOng, didn tt
vlo ch6 tr6ng, tim ch6 sai rrong ldi giii, quan
s6t hinh vE vi n6u nhin x6r, thuc hdnh, do dac,
....Cdc ban cflng {uo. c l}m quen vdi c6c bdi t6p
Lryc_nghiem. Nhidu bei tap duoc vi0t du6i dang
"Dd \ui". Vi du : Nhin hai hinh v6 sau do6n
xem hinh nbo c6 chu vi lon hon ? H6y do dd
kidm tra du do6n :

vd "didm ndm gifra", "tia ndm gifro,,.

Cr{c dinh nghia, quy r6c, tinh chdt ... duoc
di6n dat chinh x6c nhung don gi6n, -f,td,r^sCr<
16 rlng, it
dnng nhrng thu6t ngt to?" tf""tiiO
Todn 6 coi trong hinh v€, xem k€nh hinh c6 tdc
dl{g ldn trong vi€c giitp hoc sinh dd hinh dung,
d€ nhdn ilfic cdc khdi ni€m tritu tuortg.
Sau.m6i bhi hoc, trong sdch c6 khoing 5 bli
tap de c6c ban luy€n t6p vin dung kidn thfc vi
rEn luy€n ki nang.
Voi y€u cdu tdng thuc hdnh, luy€n tdp, sdch
T_odn 6 rdt chti trong xdy d{ng hQ thdngbdi tAp.
C6 nhfrng bhi r4p rdn kl n[ng tfnh to6n, ki nang
suy luin, ki n[ng vE hinh, c6 nhtng bii tap rbn
ki nang v6n dung to6n hoc vio c6c mOn hoc
kh6c vd vio ddi sdng.
10

Trong sdch cdn gidi thiOu nhi6u rrd choi todn
hoc : "DLta ngua vd dich" khi hoc "I_bc vd b6i" ;
"c6t bia gh6p hinh" khi hoc "Tfnh chdt co b6n
cira phAn sd" v.v...

.Muc "Cd thd em chua bi€i" hdp d6n cdc ban
bing cdc c6u chuyOn li thii : Gauxo dd tinh tdng

l+2+ 3+... +99+ l00nhuthdnlo?

CAu

lich vd Lich can chi, SO Am, Cu6c
hAnh ffinh 20 thd ki, Ph6n sd Ai Cap ld gi ? ...
Si{ch to6n 6 cdn c6 nhi6u bhi rap hudng d6n v}
rdn luy€n hoc sinh ki nang sit dung mdy tinh
bd tili.
chuyOn vd


Vf du : M6t quvdn sdchgiri 8000d' Tim gi6
m6i cta quydn shin AO sau khi glhm gif l5Vo'
Ldi giii I Bdm Phim lion tidP nhu sau :
ta duo.c kdt

EEEEEEB@EEl

qui

li

6800(d)
Mdc di duoc vidt cho ddi tuong chlnh lqdong
ddo'hoc sinh lop 6 trong cd nild-c, song cdc ban

hoc sinh v€u iodn, khd vd'gidi todn cfing c9
tiai dd dao sdu suv ighi, ndng cao hidu
"hiai-ii
bi€t vi todn vd rin luy€n ndng luc tu duy

todn hoc.

Vi du f. 6 lai

$10. Tinh chdt chia hdt ctra
mot tdne (To6n 6, tap I, tr.34--35), silch c6 nOu
iiui tronZnh chia hdi cira mot tdng nhung khong
chrlng minh:
Tinh chdr f

: a.imvd bi.m = (a+b)im
Tinf chdt 2: a/m vd' bim* (a+bVm
Cic ban hoc sinh yOu to6n hdy thtl tg minh
chrlns minh'hai tinli chdt trcn dua vlo dinh

nntii

.iai

ph6p chia hdt, ph6p chia c6 du vh tfnh

pnan phdi

ciri pnep nhan ddi vdi

ph6p cOng.

thing (To6n 6,tap II,
bly vd d"oen thing
trinh

tr. 114-115), s6ch c6
t!,.utg S*g
duong
cit
tia,
cit
c6t doan thing,
g4p
:
l),Hai doan
thuong
hqp
trudiig
v6'ba
chi
th&ng AB vd Cd ci*'nhau ;2) Do4n thing AB vh
ia dx c6t nhau ; 3) Doan thhng AB vi dudng
thbngry c6t nhau'
Ban c6 thd v6 tidp cic trudng hqp cbn lai
kh6ng ? (cbn 6 trudng hqP nta !).

Vi du Z. 6

Vi

Uai $6. Doan

dU 3. Bdi mp sd.72 (To6n 6, mp I, tr'31)

Sd chinh phuong

*oi

CI

iJnhien

rii

li

s6 birng

du : 0,

b4h

p-huo.ng c-qf

l:4,9,16, ...). M6i

tdng sau c6 lh mOt sd btrintr phuong khOng

?

ujf+23;

b)13+23+33;
c)13+23+33+43
Sau khi giii xong, chac hin b4n ndoy€u to6n

ohii neu e'ii thuvdi vh cd tim c6ch chrlng minh
Ino ruor! hqp t6ng qu6t sau daY chf ?
!3

+z3i :i-*

... +

n3-

(1 + 2 + 3 + ... + r)2

v6in€N*

Vh ban cflng c6 thd tt bdi to6n c6 trong SGK
rarg 1a" rZ cac dd to6n tudng tu hoac tdng

AC

qu6t hcrn.

Cdc r6rc gii Todn 6 hi vgng cu6n s69h tly ry
fa mot cudi s6ch gi6o khoa to6n gdn gti vdi ddi
.0"". a6-t o", vui,'hdp d6n, girip c6c b4n nh6 16p
ekfid; .q toan ioi tign toivcu to6n vi hoc tdt
mOn hoc nIY.

A['ll Qt/f, CAC g/l: TOAI'J

TIi,,NCI

nil sd ss
Problem. A man whose clock had

stopped

running wound it up, but did not have access to
the correct time to reset it' Leaving the clock at
home, he walked to the house of a friend whose

clock was correct, stayed fdr some time and
then walked home (in the same time as he took
earlier). Upon arriving home, he set his clock to

the correcf time even though he did not have his
friend's clock with him! Explain why he could
do that.

Solution. Upon arriving home, the

man

compared the reading of his clock on departure
and on return. So he knew how long he had
been away from home. He also knew from his
friend's ciock how long he was in his friend's
house. Taking the time difference the man got
the time he spent walking. Now !9 added half
of this time to the reading of his friend's clock

when he departed the friend's house' The
obtained time is just the corrected time for
resetting his clock.

Tir mdi

:

clock = ddng hd
stop = ngirnS lai' dimg (dQng tt)
run - chAY (dQng tt)

wind (uP) = cuQn, xo6n, lon daY ddng hd
(dQng tn)
access = ldi vho, su tidP cf,n
correct = dfng.chinh x6c (tinh tt)
reset = d4t l4i, didu chinh (dQng tit)
leave = rdi kh6i, b6 lai, dd (dQng tt)
walk = di bQ (dQng tit)
stay = & lai (dQng tit)
upon = 6 tr0n, vio ltic, trong khoing

(gidi tit)

arrive
ser

= ddn, d4t t6i (dQng tt)
= bd trf, d4t (dQng tt)

=

dtr rang
(dQng tit)
chi6u (dQng
ddi
s6nh,
so
=

even though

explain = giii thfch

til)

"omp*"
departure = s{ khdi hdnh

return

= su tro ve
away from home = xa nhi
spend = trii qua, ti6u (dQng tit)
dlpart = xudt Ph6t, kh&i hhnh
NGO VTE] TRI.]NG

11

Chfngminhrang:
a) Ddy sd trOn c6 v6 sd sd ducrng vi v6 sd

sdam.

b) azwz chia hdt cho

1

1.

NGUTENTRoNGTUAV

cAc r,6p

tncs

Bni T1/301. Tim t{t ch cde cip sd nguyCn x; y
sao cho

t+xy+1?+t4x + t4y+ 2018 chia h6t cho 101
r-EeuaNcuAu
(SY khoaTodn
-Tin DHKHTN _
DHQGTp. HdChi Minh)

Bni T2l301. Giii phuong trinh
(;r- 1 8X"r-7) (x+35)(x+90) = ZO}ti
vOrrqHUEprnIJNG
(SV K33B khoaTotdn

DHSpThdi NguyAn)

Bni T3l301. Tim gi6 t4 nh6 nhdt cira bidu
2
thrlc
* | *a4+b4

ob a2+b2'

2

q$g d6 a, b ld. c6c sd thuc duong th6a m6n
diduki€n a + b = l'
NGU'ENT#BINH
60GD-AI HdGiang)
Bni T4l301. Gii sir a, b, c li dO dai ba canh
cira m6t tam gii{c vi 0
f"

.l_

\ b+c-ta

+

u *^l-g->-2ffi
; \ a+b-tc
c+a-tb

DOANH

(9 DH34 Hoc vi€n eudn y, HdTdy)
Bei T5i30f. Ticn mOt dudrng thing lay ba
didm A,B, C theo fhf, ru sao choAB * BC. TrC,

mdt nrla ma1 phing bd AC, dung c6c tam gi6c
vuOng c.dn ABE (AE = EB) vi BCD (BD
= DC).
Ggi S 11 giao didm c:iua AD vd. CE, gei H ti giao

li

didm cria DE vd BS. Gqi F' ruig didmlira
DE. Duong thing vu6ng g6c v6i eC tai n cat

DE W K. Chrftrg minh rang: g-ry vi
" KDBD

*+
KF

BF

nm

he thrlc

li.n

hQ

sifta hai ti sd nry.

PHAMHUNG

(Hd N4il

cAc I6p THPT
Bni T61301. X6t dfly s6 (a) duoc xr{c dinh
&i: q = 5, ez= 1l vi an+t =Zan- 3ao*1vdi
moi n = 2,3,...

t2

(G/ TH?T HnngVuon!, pldyKu, Gia Lai)

Bni T71301. Bidr rang vdi. a, b, c xric dinh thi
phuong trinhx3 + oi + bx + c= 0 c6 3 nghiQm
ph0n biOt. Ch,lng minh rang
t27c+za3

-gabt.

zJe 1D,

NGUYENVANHIEN

(W THP| U Quy D6n, Qwing Tr!)
Bni Y301. Tim gi6 tr|lon nhat cta bidu thrlc

3(a+b+c)-22abc
trong d6 cdc sd thuc a, b, c thba m6n didu ki0n

a2+b2+12=l
TnANxuANDANG
((ilTHPT Le Hdng Phong, NsmDinh)

Bni T91301. Ctro tam gi6c ddu ABC c6 canh
AB = a. MOt dui,ltg thing di qua rro.ng mm G
cria tam gi6c, cit c6c dudng thhng BC; CA, AB
ldn luot t4r M, N,P. Chfng minh rrng

lll
_____;+_;+_:;
GM" GN4 GP4

ld khOng dtii

NGUTEN VAN TTNH

(PhdngGD-il| Htrng Nguy€n, NghA An)
Bii T10/301. Cho rrl dien ABCD c6 thd tich
y. MOt mdt phing di qua rro.ng tim G cira trl
diQn d6 citcdccanhDA, DB, DC ldn luot taiA,,
B', C'. Tim gi6 tri nh6 nh6t cria tdng cr{c'thd tich
trl diOn
T

=

Ve,s,s,c, +Vsl, a,c, +Vct,s,c,
NGI.TYEN VAN rrroUC
(C/ THPT U Quj, Dan, Dd Ndng)

cAc uC vAr r,i
Bni L1l301. MEt khdu sring dit trOn xe lrn c5
ndng nghiOng g6c ct' so vdi phudng ngang ; khdi

s

o1:5__Ir

,l

i.

_-----"@
ifit


vi

sting ld M-Xe c6 thd
chuydn dOng trOn mit ngan$ vdi hQ sd ma s6t /c
= 02. Xe ding dtmg yOn thi sring b6n ra 1 vi6n

lugr,rg tdng c6:rg ctra xe

dan, khdi htong m - l0g. Ngay khi d4n rdi

nbng, xe giflt lni vdi vAn tdc V = Zlcmls- I-€n
d6n-didmiao nhdt, dan xuyOn qua m0t tdm gd
nbb m1= l,l2 kg treo vio ddu day nhq dhi / =
2m; iAV" nang lucrng d4n chuydn thhnh dQng
nrng 96. Sau d6, 96 len ddn vi trf day nghiOng
goc P = 30" thi day b6t ddu bi chirng. Thdi gian
d4n roi ddn ddt bnng 1afi0 thdi gian day bf
chtng. Hdy tim M,a.Bi*trang mong nbng sring
gia tdc ciri Oan rdt 1&r so v6i g. 86 qua srlc cin
tnOrrg khi vd kfch thudc xe, sfng so v6i qu! d4o
cfia dan. LdY

g=

10m/s2-

Bni L2l301. Mo.t tu dic.n phingkh0ng khf c6
cdc bin nam ngang cd dinh, duo. c ndi vdi ngu6n
die.n khong Adi c6 hicu dicn thl€ U- Ben trong tu
d[t mQt t*n f.i* loai bd ddv a kh6i
Ai1,
"O
(hinh
vE).
m
lucnrg

6

tnOi didm ban

ddu tdrn kim 1o4i tidp
xric v6i bin trOn cria

tq, sau d6 n6 duo. c

ill ;;,-nm van tdc
cfra tdm kim loai khi

FOR LOWER SECONDARY SCHOOLS
T1/301. Find all couples of integers x, y such
that I + ry + f + l4x + 14y + 2018 is divisible

by [01.
TZt3Ll. Solve the equation
("r- I 8X-r-7Xx+35Xx+90) = 2AOl
T3/301. Find the least value of the expression

I

where a, b,

a+b=1.

c

are lnsitive numbers satisfying

T4l301. l-et a, b: c be the lengths of the sides
of a triangle and 0 < t < 1. Prove that

" Ll-L
b+c-ta ' \ c+a-tb
When does equality occur

^fI-rz$+t
I a+b-tc

KD

KF
ratios.

BF

FOR UPPER SECOI\DARY SCHOOLS

T6/301. The sequence of numbers (c,,) is
defined by : ar= 5,az= 11 and
a*1= Zar, - 3or-t(n = 2,3' "')' Prove that :
i) the sequence has an infinite number of
positive terms and an infinite numhr of
negative terms.

arn"is divisible bY 11.
T7l30L"I-et a, b, c be real numbers such that
the equation f + al + bx + c = 0 has three
distinct real roots. Prove that
l27c +2a3

-gabl <

2

(a2 -3b)3

T8i301. Find the greatest value of the
expression 3(a + b + c) - 22abc,whete a, b, c
are real numbers satisfying a2 + b2 + c2 = lT9l301. The equilateral triangle ABC has side

?

T5/30f. Let A, B, C & three points arranged
in this order on a line and AB * BC. Construct
the right isosceles triangles ABE (AE = EB),
BCD (BD = DC) on a same semi-plane with
boundary AC-l-et S be the point of intersection
of AD and CE, ff be that af DE and BS- lrt F be
the midpoint at DE. The line perpendicular to
n9,16
AC atB cuts DE atK. Prove 11ru1 4d =

KH BH

ffi

11)

a4+b4
|

2
a2 +b2

f b-

r__I

NGUrENNHATI!m'III
(Hd Nai)

Neh€ An)

ffiswss ffi }}ffi

2
ab
-L-r-

JJ

n6 cham vio ban tg
dufi. Bi6t di6n tfch
m6i ban tu brlng diQn
tich tdm kim lo4i vI
bing S, khoing.crich gifra hai ban tq lA d.

NGTTYENMANHmnqc

(6/ DHVinh,

A

d1

BD

Find the relation between

these

AB = a, A moving line passing through its
center of gravity G cuts the line BC, CA, AB
respectively at M, N, P. Prove that the quantity

111
- +

+."--"----- is constant.

GM4 GN4

GP4

T10/301. The tetrahedron ABCD has volume
V. A variable plane passing through its center of
gravity G cuts the lines DA, DB, DC
iespeCtively at A', B', C'. Find the least value of
the sum of the volumes of tetrahedra
T

= Vte's'c'

+Vna's'c' +Vca's'c'

t3


ban dd sri dung didu kiCn giii duoc cria phuong trinh
bdc.,2 6p dung vio yT (2) dn a li A = (b - l2)' 4(b'-l2b+47 ) 2 0
3(b-4)'< 8 suy ra gi6ttri cia b.

ll,

=

Mot sd ban giAi nhu sau
(a

- !@-$(a-5)
tri
vdi gi6,

r.it. ., ", ,.tr
ii;ir' ii-:. r.;- -" l :fl-

. ,'f,., .r., .,
.-i. .1./ .;

Ldi giii. Dd bei y€u cdu tim 3 gi6 rri nguyOn
phAn bi0t a, b, c ctta bidn "r sao cho bidu thric

x3 - l2x2 + 47x c5 cDns mot s.i6, tri k.
o3 - l2o' + 47a ="u' -' ti.u' + 47b

=r3-luc2+47c=k

(1)

Cdch 1. Tir ding rhrlc b6n tri{i ctra (1) c5
o3 - b3 - r2(a2 - b2) + 47(a- b) =o
* (a - b)(a2 + ab + b2 - I2a - llb + 4T =

A

Y\a*bnln

o2 + ob + b2
Tuong tu ta c6

- lLa-

r2b + 47

+o,+12 -I2a-l\c+4'l
b2 +bc+ 12 -l\b*IZc+47
o2

=o

=o
=0

Trir tirng v€ ctra (2) vd (3) vd bidn ddi

(b-c)(a+b-c-12)=g
Y\b*cn€na+b+c=IZ
Tr) d6

vi

(5) c5

-

az +b2 + 12

>

=

(4)
duo. c

duo. c

24(a+b+c) + 141 = 0

21a2+b2+c27 + ab+ac+bc

Tn d6 vd (a+b+c)2

(3)

(5)

C6ng theo timg vd cia (2) (3) (4)
21a2+tt2+c21 + ab+ac+bc

(z)

=

147

144 c6

-(ab+:ac+bc)=3

(6)

li 3 nghiOm ph6n

bi€t ctra phuong trinh

*3-!z*2+47x*k=o
c)=0

vh(x- a)(x-b)(x-

(g)
(9)

Khai tridn (9) vn so s6nh vdi (8) suy ra

a+b+c=12
ab+ac+bc=47

o3

-

l2a2 + 4Ja =
cing gi6

ti a bing 3. 4. 5. C6ch eiAi ndv ih"ua
"60
khang dinh duoc ngo.hi cdc gi6 rri d6 rhi cdn criigiri tri
nio kh6c th6a min-dd bii. phong
_.Ban . Ng!16n Trudng Tho, 9A, THCS Gir{y,
ChAu, Phi Ninh, P^hrfi Tho dd n€u bii to6n tdng qu6"t
d6i vdi bidu thrlc o3 - 3bo'* (ZUz - t)a
2) Cdc ban sau c6 ldi giii gon :

jtho Tryin Hfru Hidu,94, THCS TT. S6ng
;
Thao, Nguydn Quang Huy, 9A, THCS Giay, Phon!
CtrAu, Phi Ninh ; Vinh Phrlc : Vfi Vdn euang, 8Ai,
THCS Wnh Tuong ; HA Noi t Vfi Nhdt Miih,8H,
THCS Lc Quj DOn, Nghia Tan, Cdu Giay, Trinh Dtc
Linh, 9A, THCS Ha NOi - Amsterdam ; Hda Binh:
_Phri

Nguydn Thi Minh Phuong,9A3, THCS Le Quf DOn,
Tx" Hda Binh ; Bdc Ninh : Nguten Dftc biy, 9F.,
THCS YOn Phong ; Bdc Gihng : Phan Tud'n enh, Sl.,
IHCS Dric Giang, Yen Dfrng ; Hii Duong : Hd Tuan
C-ttdlxg:8/3 THCS Le Quf D6n, Tp. Hhi Daong, Trdn
'Qudc Hodn,9A1, THCS Chu Van An, Thanh Hn;Hii
Tud'n Anh, 9A, Pham Huy Hodng, 88,
IlpryiAa;
THPT TrAn Phri ; Nam Dinh : P'hanVinThdni, Ddo
Manh Chiah,9A2, THCS tt' Qui Don, Y ycn, Ltru
Ly, !4qryVan Hdi,9C, THCS Dio Su Tich, Truc
U,
ycn
Ninh Ninh B\nh z PhamThi

Huong,8A, THCS

;

Phong, Y€n

M6 ; Thanh H6a z Ltttt Thi Nhdn,

9F.,

THCS l.€ Dinh KiOn, Pham Vcin Quang.8C, THCS yOn
Trudng, Y€n Dinh, TrinhThdnh Ki€n,8A1, THCS Trdn

Mai Ninh, Tp. Thanh H6a ; Quing Tri; Hodng Kim

Minh,7E, TI{CS Hei I-ang ; Dac Ldct: Ngo-Thtiy
Qaong, 8Al, THCS Trdn Hung Dao, T1. SuOn i\4i
ThuQr ; Quing Ngai : Bdl MinhTue, gI, THCS trdn
Hung Dao. Bii Ld Trgng Thanh. gDl. TI{CS Nguy€n
Nghiem ; Binh Dinh : Nsuydn Philc Tho,9D, ?ACS
Ngd May. PhU C6t : Tp. Hd Chi Minh : Tr()n Hcii
2dng._7A9. THCS Trdn Dai Nghia ; Ddng Th6p : Ly
Duy Khi€m,9A1, THPT Tx. Cao Lanh.

VIET HAI

(7)

Gii sil a < b < c, til (7) suy ra a = 3, b = 4,
c = 5. Cdc sd nguyCn nly th6a mdn dd bdi v6i
k=60.
Cdch 2. Theo (1) th\ a, b, c

,

6O nOn bidu thrlc tr6n c6

(a-b)2 + (b-c)z + (c-a)2 = g -

=12+12+22

(s)
(10)

Til (5) vn (10) suy ra (6) rdi lim nhu 6 c6ch 1.

NhAn x6t. 1) Hai cdch giii trcn chi dirng kidn thrlc
lop 8. Bei niy c6 rAt nhidu ban lop 8 gini dfng. Mot sd

t4

+

{t'<;tiy titl;

s, i; iti jLui ;o'tiiL.ti.;lt{c'ttx;'iiit;

ii,rj,i i,;rt; - r,n 1-

ittt[t,

i'

1
Lligiii. DaLM= !-*!-+
l-a l-b a+b +a+b
a2b2l
' _2
l-a
l-U'
a+b
-1l4a+-+l+b+
lll
a--L

,1

|-a l-b a+b
Ap dqng BDT Bu-nhi-a-cdp-xki c5

(t
I *- l)
_ +_
. l(tt-o)+tt-b)+(a+fi)
\1-a l-b a+b)
I

>


> (1 +

a

I + 1)'3

o5

*

I .9
1 I
2
a+b
r-a l-b
-.u--+-->-

khi a = b = Il3giii bii

to6n

* a2+ "'

+ arl

NhAn x6t. 1) Nhidu b4n dd ph6t bidu vi
tdng qu6t sau v6i c6ch gi6i tuong tV tr€n :
Cho a1, az, ..., an> 0vdi didu kiOn a1

< 1. Hiy tim min ciaM, trong d6
M=

l- au

or* "'*

(D6Ps6:minI

l,:.
=4#,

MQt s6 ban kh6c d6 giti bii to6n tdng qu6t tren v6i
di6u kien a1 * a2+ ... + an< k (hing s6 k duong)'
2) Rat nhidu ban

giii tdt bii niY :

Huynh Thi Thiy Lam,8C1, THCS flt

I-lm,

Tuy

Hba, Phri Yin ; Hodng Tudln Dilng.8A7. THCS Phan
Chu Trinh, Tp. BuOn Ma Thu6t. Ddc Ldc :Trdn Quang

Quy, 8A4, tgCS Ctru Van An, Tp. Th{.Nguy€n,
frgryen Ditc Duy,98, THCS Y€n Phong, Bdc Ninh ;

tl'gry6nirung Kian A, Ngrl VinhThai;8A, .Nguydn
THCS Giav, Phong ChAu, Phi Ninh'
ril$ngrho,91^,
"Hnu'Hie'u,9A2,
THCS thi trdn Song Thao, Phti
Trdn
Thg ; Bnl Qudc Hing,gAl, THCS Trdn Qudc Toin,

Uong Bi, QuAnS Ninh ; Pftari VdnThdnlt,9A2, THCS
Lc Quj Don.'i Y€n. Nam Dinh, Ngaldrt NggcTfi,9B.,
TI{CS Le Dinh Ki€n, YOn Dinh, Thanh H6a

16NcuYsN

'

Bi.i T3297 " Ch*ng mirth

bd'r ridng

iliil'r

,4,

,i
at * a3 * du
llr;
.rJ

{l-1

+ al + A5

_,1i

o1

9B.,

THCS L0 Dinh Ki6n, YOn Dinh' Thanh H6a).
Dat A li vdtrillcria BDT cdn chimg minh.

+ al@z+a++a5) +

B

=

+

o! 1aa+o5+a1) + af,(a5+a;a2)* a! @1a2+a3)

+ a?

@

4+a

+ ollq+oo+a5)z +

5+a)2 + af, (a 5+afta2)2

* o! {ot*or*or)'

a!) +
+

+al+al+af,+a!

Do 3(i+y2+r21> 1*+y+212 ding cho moi sd
(1)
thqtcx, y,zchon6n DZC
2rnz

-o = foltr*A-Aaf .\e-A44)'*

+(a\

+ al,

-

- o?)' +(al + a! - ol - o|)z +

o?

+ (a!+al -o3-o?)' +(o! -ol)2 +(ol -o2.)z
+ (,| -,tr)' *(,3
chonon

2e'
5

Sfi dung bdt

+

-,?)' +(ol - "1)2 > o

,o

Q)

ding thrlc. Bu-nhi-a-c6p-xki ta dC

(3)
ding chimg minh duo. c A.B > E2

Vin dung bdt ding thrlc Bu-nhi-a-c6p-xki ta
thu cluoc 82 < EC. TiI (1) vlL (2) ta c6

E
< ED

<

n.Zn'=Zr',
s5

(3) vh (4) ta thu duo.c

l. Dinj

a4=45

-r!f

Ldi giii. (cira b4n Nguydn Ngoc Tri,

a? (a2+a3+aa)2

E = a?

E>

aq * a<, 't- tt1

ol*nluoi+oj+oi>t

c=

+

+a!(al +al+a\1)

rir

3
0i*A1 iLl4 A,,*tt2*(\
tioiig 116 o1, tt2, u1, tt.1, a5 lit 5 s6'rluong tli1ct
trrdtt rliert Liett

a? (a2+a3+aa)

azo

Bang bidn Cdi aon gi6n ta c6

:

o?Ao?,1
"t + -: *...t.--:!1_t

l- q l- az
+ ar+az+...+a,t

3(o? (a] + a?. + al,) + a| (a?, +

+al(afi+a! +al)+al(a2t +al +aL)

GTNN cira bidu thtrc M lit

" ;-r=i.uft

512 dqtduo.c

o=

nen

=

thrlc chi

xiy

o.

(4)

+Ji.f

ra khi at = a2

, oo

-

a3 =

1

",6.

NhAn x6t. Day ln bni to6n kh6, c6 12 ban giti ldi giii-'

giii

thidu chinh x6c do gi6 srl
1 aaS 45, hof,c st dung cicbdt ding thrlc
at 3 at I ctz"xac
hoac it dung sai tinh b6c cdu cira bat
Jtrint
"iuu
ding thrtc'
vu D'NH HoA

vn phdn ldn c6t ldi

Bni T41297. Duittg trin tiirtt O biit kirtir ll i'it
dtrong tr,in tdrn O'ttdn kitfi R'(fi > it')tttis.t::t
ngttii rai didm t\. Gic vttdttg "tAy' ciit ititi tittir'itg
tron d ctic didm B t'd C {khdc A). Gq)i I{ ia iiilt
cltiifu ct);s A * An :iaon'g iltCng BC. I'!ay xtic diult
vi tri rdc tlidm B, C dd do ddi AH lon nlidi ';i
tinh gia fi iii theo R, R'.
Ldi giii. K6o ddi AB citt duong trbn (O) 6 D'

Do

fu=6iD

= 90o n6n C, O', D thing hing'

l5


Tucrng tu tt ADPFDP.BP. Me E'N = PF, FN = pE n0n tt (2), (3)
c6 AN.CN = DP.BP
(4)

(4\6

Tr(1)

Tt t1e

AM

.CM =

AN CN

BM DM

BP DP

(s)

ffi*ok

ra
= 90o.
Ncn ffi+6dA =90". Tt 66 ffia(fin =
180". Chrturg tb OB\IO'C. Tt A k6 duong thing
song song vdi OB cit BC r?r K. Theo dinh lf
AK'-BA
AK
Ta-l6t ra c6 :
suv ra
=

90o suy

CD BD-OA.
OO' '

bay AK
?RR' .
=4=
R+R' ' = R+

2R'

=

AIr < AK nanAH

171iL

Dudng thing qua M song song v6i EF, cat OB
b T, cit CD b R, citt AO 0 G. Duong thhng AO

cat BD

6 S. Ap dUng dlnh tf

Ta-l6t c5:

AM MG. CM MR BM MT . DM MR
AN NO, -=-.
CN NQ, BP PO , DP PQ
B, C lil tidp didm ctra 2 duong rrbn v6i ridp ruydn
-:chung ctra chfng. Didu niy xiy ra vi khi BC h Thay vio (5) vn chri f NO = PO ta c6
MG
tidp tuy6n, k6 tidp ruydn chung ,4x cfut BC thi
^ MG NQ
(6)
-MT
NQ
PQ
MT PQ
frr =Gr, 6ir=fr, suy ra gcx, BACvuong.
311.
= R+R'

Diog rhrlc xiy ra khi

I/ = K rrtrc li

khi

_ N!r4o x6t. Giii tttt bi{ niy c6 cdc ban : vinh phric :
Bni Hrtu Dttc,gA. THCS V-rnh Y€n :HiiPhinpz pham
Anh Minh,gA, TI{PT NK Trdn Phti : Hi Nari' : Zldz
Phan Binh,gB, TIICS Trdn Phf ; Nam Dinh z Nguy4n
Dfic Tdm, Hodng Ii Son,9A7, TI{CS Treir Oang lVinh,
P_hryn

Kim Hnng,

Quing Ngai

:

9M,

Bni

b

TI{CS Le

euf

Don,

f

yen;

Trong Thinh, 9Dl, THCS
Nguy€n Nghicm, Tx. Quing Ngfi.

VLIKIMTHOY

Bdi TSl2g7. Clto dudng rrdn tdm O

dudng

k{nh EF. LAy hai didm N, P tr€n dudng thdng
EF sao cho ON = OP. Til didm M ndo d6 ndm
b4n tong dud'ng trbn md khdng thu1c EF, kd
dudng rhdng MN cdt dudng rbn tai A vd C,
dudng thdng MP cdt dudng trdn tai B vd D sao
cho B vd O ndm khdc phia ddi voi AC. Goi K ld
giao didm cila OB vd AC, Q ld giao didm cfia
EF vd CD. Chrtng minh rdng cdc dadng thdng
KQ, BD vd AO ddng quy.

Ldi giii. Tn MMB AM

-DM >
BM CM

(l)

AM.CM = BM.DM

Tir AANE,"
EN

AN.CN = EN.FN

l6

CN
(2)

rhay. thd

g:ry
NOKN

d

!g=!+
PO

vio (6)

SP

KNSM ----=NO

- PQ

-

rIUQc
_-.-.KMSP

(7)

Gii sri dudng thing KS vit EF cit nhau tai X
Ggi /, J, H ldn lugt li chin ducrng vuong goc ha
ti N, M, P ddn duimg thlng KS ta c6

KN SM NI MJ NI NX
(8)
KM SP=--=
MJ PH PH PX
N?
So sr{nh '(7) vn (8) suy ra
=NX

PQ

g:g

PX

=

) PQ= PX>Xtrtng vae,ngfra

PQ PX
li ba didm K, S, Q thang hfurg, hay lI ba dudng
thing AO, BD, KQ ddng quy tai

Oi-dm S.

Bii to:6n drflng khi /V ni.m trong, nam
ngoii duhng ftdn (N thudc dudng thing EF

NhAn x6t. 1)
trOn, ndm

k!6c di6m O). Nhieu b+n di

chrtorg

mffi

bA,i

todn Ecn

bing cdch 6p dqng dinh li Men0lauft thuAn (chffi

li

a{ch chrlng mtuh (8) & tren; vi dinh I Menelauft dio.
2) MoJ vii b4q dd nghi phdt bidu vh ch&ng minh bni
toi{n du6i dang tdng qudt va cen dcii hon nhu sau :
Cho dudng tron (O) vi dAy cung EF cfia n6. H li

trung diern cia EF. N. P thu6c EF sao cho HN = HP.


- li

Dio {ai gii sr} m n sd khong dgp. Theo
bu6c 1 c6m-n=ax+byvo.Oe Z. Yt m n lI s6 kh6ng de.p nOn y < 0 hay

G{c didm A, B, C, D thu0c (O) vi khdc E, F. Gii stAC
cx BH W K, BD cat AH tar S, CD cat E F tat Q. &frmg
minh ri.ng : K, S, Q thing hang.

Ta vin c6 thd giii bei todn tdng qurit !,ang PhuS,lC
ph6p dd ding 6 trcn C ctng v6i nhflng hidu biet v€ do
dii dai sd.
3) G{c ban c6 ldi giii dring :
Phri Tho z Nguydn Trudng Tho, 9A, THCS Gidy
Phong Ch6u, Phn Ninh ; Yinh Phric z UTludi Son,9B,
ffiCS V-mh Tuimg ; Hn Ttnh z Hodng Nguyin ViQt

Ir

Hi

Tinh

-

,

;

Thanh
H6{z Nguydn Ngpc Til,98, TTICS Lc Dioh Kien, Y0n
Dinh : Bti Gianc : Phan Tudn Anh,&A, THCS Dtlc
Ciang, yen Dfrn! ; Hii Phdng: Trdn Tudn D-ilng,
Phari Mtnn enh, 9e, TI{PT NK Trdn Phri ;, Nam
Diih z Nguydn Dtrc Tdm,9A7, THCS TrAn_DangNinh,
ff. Nam"Ointr ; Phan Vdn Thdnh, Dodn Duy Thuydt,
sA2, THCSrc Quf Don, Yen.

Dfrns,

TEIC-S

Van Thi€m, Tx.

f

.NGUTENMINH

HA

BiiT6l297. Cho a, b ld cdc s6'nguy€n duong
nguy€n fi'cilng nhau, chfing minh rdng c6 dilng

L,rO - a - b + I ) s6' tu nhian khbng bidu di6n
2'
duo.c thdnh dqng ax + by voi x, y ld cdc sd

nguy€n khdng dm.
Ldi giii. (cira ban PhanTudnThdnh,-Nguydn

Tidn Thdnh, 10 Todn, THPT Nguy0n TrEi,

Hii

Duong).

Bdi to6n duo. c

giii

theo ba budc sau

:

Butc 1. Vdi m5i n e N tdn t4i x, y e Z thba
mdn 0 < x <
ThQt v4y

b- I

sao cho n = ax + by.

v\(a,b) = 1 nOn tap laxlx = 0, 1. ...,

b-l l 4p thinh mot he th+ng du ddy dri (mod b).
Vly 3-r v6i 0 < x < b-l sao cho crx = n (mod D)

e

n= ax+ byv6iy e z.
Budc 2. Mgi sd n ) ab - a - b ddu bidu di6n
duo.c du6i darrg ax + by vdi x,y e N (khi d6 ta

goi n li sd de.p).
Chrlng minh : Theo bu6c 1 t6n t4i 0 < r <
b - 1 sao cho n = ax + by.Ta c6 by = n - ax>
ab - a'- b - a(b'l) = -b )Y > -1
= Y > 0.
<
Budc 3.Vdi m6i s6 n m = ab - a - b th\ n
li sd dep Q m - n li sd 1fi6ng de.P.
Chfng minh : Gii sir n li s6 de.p e n = axo t
byovdiro, yo € N. Khi d6 m - n = ab - a - b -

byo.N6u m-n lisdde.Pth\m-n=
ax1* by1(tr, yr e N). SUY ra
(*)
ab=a(xo*x1 * 1)+b(yo+yl + l)

-oxo-

=(ro+rl

+

l\ib =ro*.r1 +l>b.

Tuong tII 1lo + )r + I > a. Ti (*) c6 ab > 2ab
vo li. Vfly m - nld. sd kh6ng deP.

'v<-1.v0y
n = m - u - by = ab - a - b - ax - bY
- a(b .1 -.r).+ b(-l - y)
Tacd b- I )xvi -l >y. Suy ra n 1i sddep.

TU ba budc tr€n ta suy ra sd cdc sd khOng dep
bang mQt nrra s6 cdc s6 nguyOn trong doan

fo, a.b - a -

tu-":b+l

bi ndt-

LL

Nhin x6t. 1) C6 thd tinh tryc tidp

-r,r E[f]
hphdnnguyOnciax.

tro+n [0. ab

Ir]

2)

-

a

C6o b4n sau c6

ldi

-@

giii tdt:

=

sd

-

94?

cic

t)!b

sd de.p trong

-r)

rong d6

Nguyin HodngThanh,

11A1, DHSP Hn Noi ; PhanThdnh Naz' TIIPT Luong

Vin

Chdnh, Phri YOn

;

Nguydn Vd Vinh

I4c' llT,

THPT Sa D6c, Ddng Th6p ; U Dfrc Trim' 10T1, TIIPT
Lam Son, Thanh fi6a ;Trdn NgocTrung, llT' THPT

Ouy Nhon, Binh Dinh : Nguydn lim Hung, ll't,
ff|f OffQc Tp. Hd Chi Minh I lni Nguvan Mai
Ngdn, ll Todn, THPT Le Quf D6n, Khint Hda ;

f iinn ui

cnau, 11 To6n, TIIPT Luong van Tgy.,-Ninh
Bihtr ; i-a fnuong,1l Todn, THPT Luong Thd Vinh,

DdngNai'

DANGntrNcrgANc

Bhi T71297. Tim gid tri lon nhdt cila bidu
thirc a! + ol + ...+ a7, fu > 1), trong d6 cdc sa'
thuc a1, a2, ..., a,, thudc {0; 2l vd thda mdn
A1*a2+...+ 0n=tt
Ldi giii. D{t -q - ai l, i = 1,2, --., n-Ta c6
lrrl < 1 vl;r1 + ... * x,, = S.
Khi d6 +4 + ... + a], =
"?
= (x1 * 1)2 + (x2+ 1)2 + ... + (xn+ l)2

-n+ i?+4+...+ x?fZn.,

*) Trudng hgp 1 : n chilr^, n = 2m. Bdt ding,
thfl tren tr6 ttrinfr ding thrlc khi vi chi khi la sd
.q blng I vd m sd x; blng -1, trlc ld m sd ai
bing 0 vI m s6 aibiing2.
*) Truimg hqtp2 : nLE, n =2m + 1 (m e N.)
Khong mdt tinh tdng qu6t gii sir.r1, ..., x1s 0 ;
Xpal , -.-, xn) O.
< l*tt + Lr2l + .-. +
+... +
Ta c6

il*$

ii

4l (do h/< I Vr)

- x2- .-. - xkt xk+l* "'+ {n

=Z(xt*t+...+x))
- 4(xt + xz+ ... * ,r)
= -x1

(1)
(2)

t7


Tr) (1) suy ra

(2) suy ra *?

Tt

+4

+ ...

x]

lim

<2k.

+4 + ... + x2,
Do d6 *?

hlm nghich bidn trong [0,1']. oo

YBy f(t) lh

xl+x| +...x? <2(n-k)

,-+0

n - kl <

2m. Dhng thrlc xiy ra khi vi chi khi trong cr{c
s6 x1, x2, ..., xn c6 m s6 xi blng -1, m s6 xibing
1, m6t s0 bang 0, trlc li c6 m sd a;bingO, m s6

a;bdngZ, mlt sd a; blng 1.
Trong ci hai trudng hgp tren gi6 tri t6n nhdt

tn1

thtc al +a]+...+a], bang n + Zl:
Lt.l
x6t. Toa soan nhAn duoc ldi giii cfra 250 ban

cfra bidu

'NhAn
hoc sinh, hdu hdt c6c ban giAi dring. C6c ban sau c6 ldi

/(r)

=

1

vifl/)

\'4)

nghich bidn ndn dd phuong

trinh (2) c6 m6t nghiQm

I < a= "/fI)=+

ki€n cdn vd dir lh :

KdtluAn ,

,."rr, (o,f] thi didu

\4)

n2

\|t'
NhAn x6t. Toa soan nhAn duo. c trOn 100 bni giii grli
ddn (ri0ng truong THPT chuyOn VInh Phric c6 ddn 20

bdi)' hdu hdt ddu giii

d{ng'

NGU'EN vAN MAu

Eei Tqi297 " Drriltg trbrt tdru I bir tlr{rt/i ,'
.ttir vit be csnh. B{' = $, CA = ls, AB = c

ti€'yt
Giang ; Bdc Ninh z LuuThi ThuTrang,SA, THCS Y€n
Phong ; Hba Binh : Dd Thu Huydn,8A4, THCS H[u
Nghi ; Hmg Y€n z Dodn Thi Kim t/ad, 8C, THCS
Pharn Huy Th6ng ; Phrfi Ydn : Hujnh Thi Thny Lom,
8C1, THCS Phri LAm ; Cdn Tho : Nguydn Minh Ludn,
8A, THCS Nguydn ViOt Hdng, v.v..._

NGUYENMINHDUC

Bri

'I-8,'?1)? - "f

ph#l*tg t"inh
,-.)Ji,:,

- ,.,i

,

in

nrci gid

tri e ia

tltanr sb' a clc'
+ 2ro,c,t = .1 t
c.i

i-r
fl;l

l,r,,q j

cLia tom ,sidc ABC kin !u'a o cdc dii'm
,S lc) di€n rich A,AflC i'r\ h,,.|t1,. h,

Goi

dirdng ctto cria A,4BC titoug ri'ttg
B, C. Chi:ng minh rring
152

,l'ip2 T-_, pM2
Ir.h6 lt1,l4 lr,.11

,1,11V.

cua 1C voi

Khi d6 ,t/,\'r = J.\t K2 = 4.41Csinr e

.

2

oi +Zcosx=Z
c6 dfng m6t nghi€m tr,rOc

A

(1)

(0,1]

Vidt (1) dudi dang:

/ . r2
ISIN/}
l-i=a-t=_
VAy cdn tim a dd

t,MC
J

[r/

2

Q)

'[',;]

e)

c6l nghiom,nrO" (o,i]

[ull ,,. [r't;
\,/

= zY<,-rgr)msr

=

*t,-rv)

Mli2 =2(p

-

c)zQ

- cosC) =

.q'-4)l
'L\ Zab ))
" -.1[r-[g'

= z(n

ab

Yay ab.MhP = 4@-c)2(p-a)(p-b).Tuong

f ',(t) = L_tt-' 'cos'-sin/

i2

}Jray

_ 4(p-c)2(p-a)(p-b)

/ , r2

18

!

I

Ldi- giAi. a) Goi K lir giao didm

NhAn x6t rAng x = 0lu6n lu6n li m6t nghiOm
ctra phuong trinh. Viy ta cdn xr{c dinh cdc gi6
tri cira a sao cho phuong trinh

Ta c6 :

= ab.MN2 t- br'.i',tP2 + ca.PM2

-,, MlJz

L)

Ldi giii.

x6tf(t)=

ri6 dii
tric dt'tili .ti.

;

l)

i

2l

vr.i'i

M, N, P

li

bc.NP? = q@-a)z)(p-b)(p-c)
ac .M Pz = q@-b)z (p-a)(p-c).

=

< o, /e

(r,;]

Tld6

tu:

:

ab.MNz + bc.NP2 + ac.MP2

= 4(p-a)(p-b)(p-c)t@-a)+(p-b)+(p-c)i =

49


MNz Np2 Mp2
_+-+hoho hth, h,l6

=

aU.tttNz bc.NP2

ac.MP2

(rli)'
i
_*tgp.tgy

_ *-tgl.tguf
I

Nguydn Trung Chinh, 12 To6n, truong Hn NQi ; Phri Tho z Nguydn Trudng Thp, 9A,
THCS Giay, Phong Ch6u, Hodng Ngoc Minh, llAl,
THPT chuyOn Hing Vuong ; Hii Phbng: NguydnVfi
Ldn,8A, Pham Huy Hodng, Dudng Hdi Long, Nguy,in
Thi Thanh Loan, 8B, THPT NK Trdn Phri ;Th5i
Nguy€n: Trdn Dfic Phong,10T, TI{PT chuyen Th6i
Nguy€n ; Trdn Dtc Phong, 10T, THPT chuyOn Th6i

Amsterdam

DinhTrrdng,llT,

TTIPT

NK Han Thuy0n ; yinh Phric : Nguy2n Phil Cudng,
98, THCS YOn Lac, Ii Thdi Sarz, 98, THCS \fmh
Tuong, Hoiutg Long, llAl, Ddo Ddng Hda, llA3,
TIIPT chuyOn Vinh Phtic ; Hda Binh z Hd Hfru Cao
Trinh, llT, Nguyin L)m Tuydn, 12T, THPT chuyOn
Hoang Van Thu ; Hh Thy : Phqm Ngpc HQii l0Tl,
TIIPT Nguy6n Hue, Hi D0ng, Nguydn Thanh Son,
11A1, THPT Ngoc Tio, Phrlc Thg ; Hii Duong:

I4c, llT,

+

Itgcr.tgp
Ldi giii

452'452'452

Nguydn Thdnh Nam, Nguydn Thd

cua

bie'u rhftc

NhAn x6t.'Da sd cdc ldi gi6i gti vd tba soan ddu srl
cic tam gi6c APN, BPM,
dung dinh li hlm sd sin cho
-CMN hodc dinh li Pt0-10-m€ cho c6c ti gi6c nQi tidp
APIN, BPIM, CMIN dd, bidU di6N MN, NP, MP qUA A,
b, c dd tt d6 suy ra kdt qui cau a). TS nh{n dugc rdt
nhi€u ldi gi6i cho bii to6n niy, tdt ci c6c b4n ddu giAi
dtlng. Clc ban c6ldi giii ggn hcm cd li ;
Hi NQi : Ii Minh Thdng, l0AT, Vfr Quang Thanh,
l.e Hing ViAt Bdo, l0A, DHKHTN - DHQG Hn NOi,
Nguy,in Chi Higp,11A1, Khdi PICT-T DHSP Ha NQi,

Nguy€n ; Bdc Ninh zTrudng

tri ktn nhd't

OAB,voi mdt ABC. Tim gid

b) Ap dung kdt qui c6u a) ta c6 ngay

Nguydn

Thi Thanh Hdng, l0T, THPT chuyOn Nguy6n TrIi ;
Nam Dinh z Nguy,Sn DtcTodn,9A7, THCS Trdn Dang
Ninh, Pham Kim Hting,9A2, THCS lJ Quy Don, f
Yan. Trdn Vfr Di€u, llTl, THm chuyOn L€ Hdng
Phong ; Ninh Binh z Trinh Tlttiy Nhung, 12T, THPI
Luong Vin Tgy, Tx: Ninh Binh ; Thanh H6a t BiliVdn
Nghia,9A, THCS Trdn Mai Ninh ; Mai QuangThdnh,
llTl, Nguyin Xrdn Hda,12T, THPT Lam Son ; NghO
An : Dinh Xudn Hdi, Ta Quang Dilng, Nguy,Sn Dfic
Nam, Trdn Ki€n Trung, llAL, Nguydn Vdn Du, k

Qudc D0, llAT, Khdi FICT-T DH Vinh, Nguydn
Danh Hdo, Phan Hodng Phuong,1041, TF{PT Phan
BQi Chau ; Hi finh : NguyAn AnhTudn, I0T, L€ Tdm,
11T, THPT NK Ha Tinh ; Quing Ngai t NguydnThi
Minh Chdu,9T, TTIPT Tx. Cao Ldnh,Trdn Minh Ki€n,
lOTl, Nguydn Huy Cung, 12T1, THPT chuydn [f
Khidt ; Kh6nh Hda: Nguydn Hodng Dny, 10T, TIIPT
Ir Quf DOn, Nha Trang ; Ddng Nai z Nguydn Hodng
Minh Dqtg, l0T. Bi€n Hba : VInh Long : Nguyin
Trudng An,1lT, TFIPT chuy6n Nguy6n Binh Khiem ;

Tp. Hd Chi Minh: TrdnVd Huy, LlT, TIIPT NK Tp.
Hd Chi Minh.

ud eueNcvnur
Bni T10/297. Cho fi di6n OABC vudng d O

(OA, OB. OC vubng gdc tittg el6i mbt). Goi a,
p, y ldn lrot ld g6c hon bdi cdc mdt OBC, OCA,

tg2cr.tg2p.tg2y

l. (Hodng Ngoc Minh, 1141, THPT
chuy6n Hirng Vuong, Phf Thg ; Pltan Thdnh
Nam,ILT2,T[IPTLucng Van Chdnh, Phri Yen).

Doo

AO

L (OBq

ke

OD T BC thi

AD LBC
) BC L(OAD).
Ke OH I AD,ta
c6 OH I BC nOn
oH L (ABC). Tir
AO LODvit
OH LAD

A

s

s6 frF=ffi =
Cflng vAy, f,jfu = B vit
+
ddE =y. Do d6 : cos2cr +
= 1 (*)
"or2B

hay

li

:

1+tg2cr

1+tg2.p

!+tgzy

"os2y

1;

TiI (1), khai tridn rdi rrit gon ta duoc :
tg2a +tg2p + tgzy + 2 = tgza.tg29.tg'y (2)
Mdt kh6c, theo BDT Bu-nhi-a-c6p-ski, ta c6 :
(tga + tgB + tgy)2 < 3(rg2o + WzP + :4zy) (3)

TiI (2) vd (3) suy ra

:

+ tgB + tgy)z + 2 < tgz.'.tgzg.te2y
] r,t"

hay:

(tgcr+tgB+tgy)2 +6

<3

(4)

tgzc,,.tgz$.tg2y

VAy ta duoc

:/<

3

Ding thrlc xiy ra khi vi chi khi ding thrlc xiy
ra 6 (3), nghia
: tgo = tgp = tgy (0 < cr, 9, T <

li

nll)ec=B=y<)HA=HB=HC
€OA=OB=OC o AABCddu

T6m lai : Bidu thtc f dd cho dat gi6 trf lon

nhdtfi* = 3.khi vi chi khi tr1 diQn OABC lit
vu1ng cdn b O, vi do d6 OABC cfrng ld mQt
hinh ch6p tam gidc dA'u k6 cdc mdt vudng cdn
d dinh chdp).
Ldi gi6i 2. (cta Hu)nh Thi Thtiy Lam,9Cl,
THCS Phri L0m, Tuy Hda, Ph[ YCn).
X6c dinh OH L (ABC) vi o, B, y nhu ldi
giAi 1, H lI truc am LABC. Tt d6 ta duo.c:

t9


cotgo.cotgp =

oH2
AH.BH

!?
=r*frEz =
BH

"or.I&

AH.HD
AH.BH
= cosc.

(i)

2

cosA.cos8.corc <

rtd6suyra

1
8"

,f=A*f

(ii)

=,

Ding thrlc xiy ra e (i) (ii) khi vi chi khi A = B
= C nghla li khi AABC ddu.
Vi-f,"* = 3 <) LABC ddu e HA = HB = HC
o OA = OB = OC <> trl dic.n OABC vu6ng

cdn b O.
NhAn x6t. 1) MQt sd b4n giii sai gi6 tri max cia f,
hoic c[ng cbn nhi€u ban chi dtng & kdt luAn/-* = 3

<) c = p = 1 mi

chua chi ra duoc dac tinh hinh hoc c&a
trf diQl OAIIC, tham chf cdn kdi lunn voi vang OABC te

m6t trl dicn ddu !
2) Ngoii c6c ban duoc neu t€n 6 trOn, c6c ban sau
dAy c6ldi giii tucmg d6i ggn gang vi ddy dri :

Hn NQi z Nguydn Chi HiAp, ttAt, Nguydn Anh
llA2, PTCTI-DHSP He
NQi ; Bdc Ninh : Pham Thdi Som, l0T. Trdn Kiiu
1yne,-llT, THPT NK Han Thuy0n ;0 Nhung Diu,
\Suydn Vdn Hoan, 11A1, TIIPT Y€n Ptrong I"; Bdc
Tudn, Nguydn HodngThanh,

9iang z_PhamTudn Anh,8A, THCS Oric Giang, yen
ptrng,lguyanTudn Huong, LzM,TIryf Viet Yin S;
Trdn Bdc, lOAt, Phan Bd k Bian, phins
Jlg\rnng;
Thi lnn Phuong, Hodng Vdn C6ns, LLAI. li Vd;
Quynh,1143, TIIPT Li6n Son, Lap Thach, TrdnVdn
Dfing, llA4, TIIPT XuAn Hda, Mtlinh ; phri Tho :
Dinh Thdi Sar, llAl, TIIPT chuy€n Hing Vucmg,
Nguydn Cdng Son, 11A1, THPT pnt Nintr ; Thii

lgqV-co : Bni Ollng Hdo, 11 To6n, THPT chuyCn Tp.
Thtii Nguyen _; Hi TAy : D6 Vdn Tiep, ItTt,'flryt
gltuy€{r Nguy6n
Ngri Vdn Cudng, llB3, TIIPT
-HuQ,
XuAn Mai. NguydnVdnTrung, I IAl, THPT Ngoc Tio,
$t[c Thq : Hba Binh z Vil Hiu Phtnmg, t-l Toan,
TIIPT Hoang Vdn Thu, Hda Binh :IiliiDucog z Dinh
Hd Chdu, Phan Tudn Thdnh, t}T, Nguy4n fnt qc,

Phqm Thdnh Trung, U Huy Tnrdng,-TIJtrT Nguy6n
Tfri: li Dinh Crdng,11A2, TIIPT Chi Linh ;-Nam
Dinh : Dnqe Dd Nhudn,9A3, THCS Le Quf Don, Y
\c1 lleWan Duy Minh,10T, TIIPT l,c Hdng Phong ;

Ninh Binh

20

z

{1tdn_!rdng, 11A, TIIPT Le Ldi, Ttro Xudn, Nguydn
V d n*I odn g, U V dn N guy€ n, Ii V dn T rin h, | | A, Ti{yl
Luong Dic Bing, Hoing H6a, Trdn Manh Tudn, 9C,

Son, TrinhThiVang, tlA,
TI{PT Lam Kinh, Tho Xuin ; Ngh0 An z frdi Dinh
Trung, Nguydn Vdn Du, 11A T, PICT-T DH Vinh,
Phqn Hodng Phttdng,lOAl, TIIPTPhan B6.i Chflu ; Hh

TI!C! Triau Thi Trinh, Tri0u

Chfng minh tucrng tu, ta dgo.c :
cotgB.cotgy = cosA vi cotgy.cotgcr = cos.B,
trong d6 A, B vd C li, cdcg6c cria LABC.
Do d6, bidu thrlc/bay gid c6 dang :
/= (cosA + cos8 + cosC)2 + 6cosA.cosB.cosC
Mit kh6c, trong mOJ tam gidc ABC btitk\ ta
c6 cdc BDT quen thu6c sau dfly :

cosA+cosB+"orc.1

lhny Nhung, l2T, THPT Luong Vdn Trly,.; Thanh
H6a z Mai QuangThdnh,1lT1, THPT tarn Srln, ttlai

liVdn Nghla,Ii Xtdn Quiin-llT,Trinh

Trnh: PhamVdn Chidn,llT, THPT NK,Truong eudc
llB, Phan Dinh Phing, Tx. Hi Tinh ; Quing
Eloh t Duong Trung KiCn,.llA, THPT Dio Duy T[

DOng,

lT1, ITIPT chuy€n Quing Binh ;
Qu3og Triz Truong Song Hdo, lLT, Dodn QuangTri.
Phqyn Hnng Cudng, I

llT

Bach Nggc Bdo Phtrilc, HodngTidnTrung, LW,

THPT chuy€n Le Quy DOn, Phan Qudc Htng. lZB,
THPT Hei Lang ; Dh Ning : Mai Ti Na, tU9 TlJtrt
Hoing Hoa Thi{m ; Quing Ngei : Ngayin Huy Cung,
1272, Dodn Vdn Lufin, ll Li, Nguy4n Vdn Thdng,
11T, TI{PT Lc l(hi€ti PhamVi6n,llAl, THPT 1 Tu
Nghia, Quang NgIi ; Binh Dinh : Nguydn Hdng Hdi.
llA4, Qu6c hSc Quy Nhcrn ; Phrfi YGn z Trdn Viit
Quac, llA Phan Chu Ttuth, U Hodi Vrt,10T2, TFIPT
Luong Van_Ch6nh; I(henh Hira: Nguydn Mai Ngan,

LLT, N_guydn Tidn Vidt, l0T, THPT-chuycn Ic Quf
Ddn, Nha Trang ; LAm Dtdng z Nguyin iruong Khi,
1lT, THPT Thang Ircng, Tp. Dn L+t ; Binh Thuin :
Aeuydn Anh Dtc,llA2, THPT Trdn Hung Dao, Phan

Thidt ; Tp. Hd Chi Minh : Ng6 Minh Tridt, tctrt,
Trdn Vd Huy, Nguydn Pfuonf Nam. Nguyan lim
Hurtg,llT, PINK DHQG Tp. HC\,I ; Ddng Nai : Il
Phtmng, N7rydn Qudc Thing, llTl, TI{PT chuy€n
Luong ThdVinh; Tny Ninh : Ngaydn AnhTudn,llT,
TIIPT Hoing L€ Kha ; An Giang z Hd Trung Nhdn,
1^1T, THPI chuyOn Thoai lrlgoc Hdu ; Cdn Tho : Ha

llAl, TIIPT chuy€n Lf Tr; Tieng, Tp. Cdn Tho ;
pac_Li€u z Nguyin Danh Dfing,llTl, TIIPT chuyOn

4n,

Bgc Li0u.

NGI]YENoANc pHAT
Bdi Lll297. MAt doan mach di€n c6 so cl6
nhu hinh dudi. Cho bi€t u7a1,1 = I27JZ sinl\Om
(V ) ; SO' chi ct?a v6n kdV ldn gdp d6i s6' chi
1
cilavdnk€'V'2; R= 50t); C = l03ln(pF).
Hdy vi1? bidu ilutrc cila cudng cl6 cldng di€n
ttc thdi i di qua doan mach di€tt tr€n, bi€t rdng
uypvu6ng pha ,-di upy.

tq
l

tC

V

l-.ii giii.
uu, wong pha vdi upp nOn cu6n
day c6 die.n ffi r. VE gian dd v6cto nhu hinh bOn.
Ta c6 U2rr +Ufu, = Uhu = 16129. Mit ktri{c,

:

theo dd bdi Il MD = 2IJ py,suy ra Up1g

=

#

rr.


c6 diAn td thudn R n€n dao dbng tdt ddn' Dd
duy tri dao ddng ngildi ta ldm nht sau : Vdo
tnh didm tu tici dian cqc dqi ngildi ta thay ddi

Uttn

khodng cdch 2 bdn tu mil laong Ad, vd khi diQn

ttch cfia tu bdng kh6ng thi dua cdc bdn tu vd vi
tri ban d,iu (cdch nhau d). Cho idng thdi gian

rtd thay rtdi khodng cdch girta 2 bdn tu ld rdt
nhd so vdi chu ki dao dbng.
Hdy xdc dinh ttd bieh rhi€n tilong
dao d1ng duoc duy

UDN

Bidt Zpy

-

R'*4 =

uon o
1,114 (A)
zox
trO pha hon

giin

-

2=

oo

.E6oo,

suyral=

Io= IJi= 1,584.

=

i g6c tPz mI
v6cto

tEQz =

R5
=

tgcr

tg(gt + Qz), rflt ra tg
er x 0,01 rad. Bidu
=
j 1.5Ssin(100tr 0,91) (A)

uotu

tga-tqqz _
l+tgg2tgd"
thrlc cira i

NhAn x6t. C6c em c6 ldi

giii

7

gon vh dfng.

Mau: Ldm Phtic Duy,1287, THPT b6n cOng Ci Mau ;
Hi TAy z Nguy€n Anh Tud'n, 11A11, THPT Thach
Thdt ; Phf Tho; Nguy€n Kim Ngoc, 1181' THPI
chuyOn Hing Vuong,-Vict Tri ; Phri Y0n z Daong Dd
Tai: ll Li, THPT Luong Van Ch6nh, Tuy Hba ; Hii
Phdng: Dinh Ngoc Ting, ll Lf, THPT NK Trdn Phf ;
Bdc Giang : h Nho Thily, l0B, THPT NK NgO Si
L;En; Hodng Ngoc Duong, 12A6, THPT s6 I' HiQp
Hda; B6c Ninn : frinh Dd,ng Phuong,lZl, NguyAnVdn

tl

TftrT

H6a, THPI NK HAn ThuyOn, Trdn Ddng Qudn'
Thufln Thanh l ; Quing Ng6i z Nguy'inVdn
Thang, LlT, THPT l,€ Khidt ; Nguy4n Thi ThiAn Trang,

1181i, THPT Trdn Qudc Tuan ; Yurh Phric ; Nguyin
Duy Htrttg, llB2, Chuong HA, LlA3, Hodng Anh, Trdn
Vdn Ddng,l 1B3, TIIPT chuy€n Vinh Phtic.
MAI ANH

Bii

LZl297. Cho
mach dao dQng di4n
til nhu hinh v€. L =

c Ln,c=

Ae-otsirr(at
/r\

+

(D).

vol a =

R

to-oF,

4Tt
R = 5Q. Do mach

a

2L
-.

=

--_ .6aaymc6co-0,,=
1

=

..lLC

diOn tich cta tu dat
diQn tich cfia tu dat

tri
tri

ki 7 =

4 = *trl
(,1 100

s6 cuc dai hai ldn' Gii su
sd cuc dqi Q,,o thdi didm

,., khi d6 ning luong dao d6ng lh
v6i

finh : Le Dftc Dat, lL Li, TIIPT NK Hd Tinh ; Ci

12A1,

a7-o2

la

- Ti6n Giang . Nguy,5n L€ Qudc Dfrng, l2A7 'T[7PT
Nguy6n Dinh Chidu, Tp. M! fho ; NghQ Arr:. L€ Cdng
Tiudng, K29 L| THPT Phan BOi ChAu, Vinh ; Hh

Trinh,

=

9

-

=

thitc q

200rad/s. Trong m6i chu
Ul,to

d

tri.

Lbi giii. Di6n tich tfc thdi ctra tu diOn c5 bidu

r'rp7v

Z'=L . Theo

n/

d\'i a ad

c = I9's
d2

W,,

=

Ddn thdi didm r = t,, +

q
2C,,,

l,

dien

tich cira tu lai c5 tri sd cuc dai Q vit nang luong

dao dQng le N = *2Cn
(
w.
t, _t o.\2
-u I _-ut
W \O)

fa
. SUV fq

TiI

d6

Lw w -w

W
-=-')

W

-

| - e'r < 0. Dd bir vho phdn nlng luong hao
hut AW thi ta phii thuc hiOn cOng A = -AW dd
ti{ch xa hai bin tu diOn mot luong Ad, sao cho
ndng luong dao d6ng v6n bing !y., khi d6 W,,=
112
ee..S A W^-W Ld
= r'OiC- "
=---:-=-.
2Cd+LdWWd
Ttdd tac6 & =eoT - l:r0,1.

d
x6t. Bhi to6n hoi phrlc tap ncn chi c6

mQt sd
em c6 ldi gi6i ciling hoic gdn ctring. D6 lh cdc em :
Ngh€ An z Trdn Quang Vrt' fiA7 ' THPT.chuyen
Pha; BOi Chdu ; Hh finh : Ll Drtc Dqt, 1l Li' THPT

NhAn

NK He Tinh ; VInh Phfc : k Vdn Qu)nh,- llA3'
THPI chuyen VInh Phric ; Hii Duong z Nguydn Qudc
Long, Il Li, THPT Nguy6n Trdi ; linh^Di.nh t Pham
QuingTrung,ll Li, TI{m chuyen l.e Quy DOn' Quy

Nhon'

MAI ANH

2t


Vai,r

rlni taat :

l/S/tr/t/#/il/#0/t/ ?

ryen s& nqnr mrnD snne

eDE

Kd tU khi c6 mf,r tr6n triii dAt con ngudi kh6ng
ngung ph6t mintr, iane che nhfine duneiu. thiet b'i
plrgc _uq doi sOng, sin xuat. B+n hay itrd Uii* nam ra
ddi c[ra m6t so tliiet bi van phbng, nhe phrit minh, qu€
huong c[ra nhh phrit niinh A^O Uafig ca.fi Oidu itri,rtr'tai
bang thdng k0 chua dfng du6i 0a] nfre t
Thiat bi

Ndm ra ddi

Brit chi
Brit m6v

Bft bi
Mr{v tfnh ouav tav
M6y chfr

Nhd phdt minh

1884

L.E Waterman (Oa-to-man)

1867

L. Bir5 (Bi-rO)

1642
1794
1939

Pascal Blaise (Pat-xcan)
C.L Sholes (SAu-lo)

coxc

thrl.nFI gt u AL KACHI (c6 tii li0u ghi
mdt namr 1429) sinh 6 Ka-san nay thuOc kan (Vucng
qudg Ba Tu ct), hm viQc & Sa-mi-can (nay thuOc UI
do-be-kr-xtan).
COng trinh

Cdng trinh

a2=b2+c2-Zbc.cosA

tf

ldi nOi ti0p c6 cdc canh a, b,
blng

giric

c, d vit nira chu vi p

@

Neu A, B, C, D lh bon didm thing hing rhi

_ _;

BD' + AB.CD' +BC.CA.AB = 0

G{c cf,p canh ddi (k6o dni) cira mOt luc eii{c n6i
tidp trong mot elip cit nhau tai ba'didm ihing irang
C6c ban sau duqc nhAn tang phdm ki nay

Phr{n

Huns-sa-ri
HA ANH

rr

COng tlnh thf hai cfia BRAHMAGUPTA sinh ra &
Multan (fui DO cfr) nay thuOc Pa-kr-xran.
Dudi day h bing thdng k0 d[ng.
tu6c

giti

QUA

An Ka-si
(AL KACHD

huong Ndm sinh, mdt
Tu

?*t430

An D0

598-660

Xcdt-len

1717-t785

Ph6p

1623-1662

Ba

Bra-ma-gup-ta

(BRAHMAGUPTA)
Stiu-u6t M.

_ _:

BC . AD- +CA.

MV
MV

TRiNH Cue

TAn

Trong mOt tam giSc ABC c6 cdc canh ld a, b, c thi

huong

Ph6p

N. J. Cont6 (Cdne+O)

Gidi ddp;

Di0n tich mOt

QUA

:

Vdn Thiet, K25A. Toi{n - Tin, DHSP He NOi
II, XuAn Hda, M0 Linh, V-mh Phric ; NgrydnTr,in
Qu)nh Di€p , 9A3, TIICS Luong Th€ Vinh, Quy

fui

(STEWAR.T

Mathew)
Pat-xcan B.
(PASCAL Blaise)

Nhon, Binh Dinh ; Trdn lim Bdo, l0 Toin,

t€ Quf Ddn, Nha Trang, Kh6nh Hda
; Trdn.Thanh Giang, I2A2, Ttl thuc hinh - DHSP
Tp. Hd Chi Minh.
THPT chuyOn

HOAI THUONG

TaI wd
t,

,2.,?

a

LOI GIAI

PHAI CHANG
LA DUNG ?

Nhidu ban chua chi 16 ch6 sai trong ldi

giii. MQt sd b4n gidi t4 bii todn vin chua chi,nh"x{c.
Tt 2 + 2(cosx + sinx) >2 - 2J, = 20

- Jil

22

khOng suy ra

du-o. c 12

+ 2(cosx + sinx)l

>

lz-ZJil

= zdl-D
Sau dAy lh m6t ldi giii dring : Vi f(x) = ll + 2cosxl
+ 11 + 2sinxl > 0 n6n dd tim GTNN cia him so
y = f(x) ta tim GTNN cfra hhm g(x) = [f(x)]2 =

6 + 4(sinx+cosx) +

Dat t = cosr

11

+ 2(cosx+sinx) + 4sirxcosxl

+ stnx GJ, < t s Ji). Khi d6 ta

l4i x6t GTNN cira hi.rn h(t) = 6a4,1212f+2t-L

.-


MOr nAt roAx cua EINsTEIN

*i,

r.,...--

=L/

I

i .,

Qirii iltist: xEp cAc rcn ruot

Qu,i

Nhi€u ban cho rans Ba n6i dring vi c6c ban dd

P*#iffi ttr#r'*#J,il"H'af#:iffi
ili;il:Atie;odllri.#E;,]ffiei ;ilil ;d'khds

3iX"r)Lt[T"?'ffi'ffi:XT#ail;1,tsit,Trilffil,ffi
xOp ein t"i" -ong M c6c cOng trinh cira minh' c6 nhi vAt
c6
co

chi
nhv ctu
tint ndy
nann tmh
da d6nh gi6 : q9 tE tren hhnh
l( lon &:d{f-+";r;-t.s,_,igen

lait"fikd,hi"h tltttruneUln-gan6i iir-d,,
.aii3i",,[8J;ffiitrJ8ffili'ti#li:fii+i;.4."4O
Einstetn.
dugc
hinh
hi6u
theo
kieu
ngudi
nim
sai ddy ! Vdi crich xep
C6 iau chuy6n sau d6y gifia Einstein vh Sa-li SaZ itil{gp duoc dring 41 lon. lfinh chft
c6
2
&
hinh
4i
lon
,irai t]i'o;r.*h
m-rin iCt*ti6'Chaphn),'tf,'uong dgo.c goi Ib Yc-10
chidu rOns'bane 25 (cm) = 5 (cm) * fCtrartot), mQt nhi hai Ectr carn tO_i lac.. M6t hOm'
5 (lon) bO"n chiSu dhi bane AB + DE + iem xonq mot phim cria Charlte Chapln! bmstem
vdr
bilg thu
--,..,.1
mol btlc
qCD
Chaplin mot
Chprlie-Ch"p.l4
< 36,95 < thich th(nidt'clio

th(nidt cho Chprlie
4.2(5cos3d)
5 + 4.2(5cos304
4iO = i+
tu.yd'
'i,r,Xi
Hinh
H inh lt
dbng.
")ns
ihtng
ldm
:
tu,
ax
u
ic,
nt
Jo
ct
r-av
;6i;;fi
id."I{"h".hr
''fr t4Lvd
nh6 hon mQt chrit so v6i ddy irinh'hgp giay kich fic ratbin4airrd'r ddi.thilgltc.a.i cfing !r:!y,!,ry?:
nsudi trl trd dbh gid. ddu fhai cuoi;.vd vi the nw ang
thuoc 25 x 40.
rdi khdm phuc vd xin

-

n\i

ileng kluAp rite'gioi.'Tai

chilc mtng 6ng".

Mav neiv iau Einstein nhAn duo.c thu phfc d6p
ctra-drarfie'Chaplin : "Xn cdm m 1ng dd -qud khen'
fai
-ria ldi nhtng itidu ai ai cilng hidu va nai1iing th€

iit

,:d ni

fni to I chinh ois,6ng ldm nhfrng didu

Tiang ai rtdu c,i'na ndi iengkhditha'gioi mrti thdt
ld di€u la ling !".
Einstein aiaUng nhi€u phut thu gidn dd ra nhfrng
a6lo,en vui cho 54n dqc igng rdi cira td br{o Phdi*g:piir8i6i*tlirtti'.sau a5v li mot trong c6c dd

DE

C

,{B

todn d6.

Chin vdng trdn
dnc ddt d cac dinh

Hinh 2
NhAn x6t : C6c ban sau c6 ldi giii dring, giii thich

crta 4'tam gidc ddll
nhi vd 3 tam gidc diu

nrong d6i 16 rang duCrc nhAn t4ng plidm :
Nsuvdn Npoc Hd,lm, THPT Cam Duitng, Tx. l-do

ldn (xem hinh vd).

I
dii 9 vdo 9 vdns trdn

Hdy vi1t dfi 9 s0'tn

Cai,"Lio Cit' ; Phan Thi hm, 12A1, TIIPT NgO Gia
Tu.'Tir Son, Bdc Ninh' ; Hodng Tnmg Ki€n, l0A,
ffff'f ga Truns. Hd Trung, Thanh H6a : Nguyin
Hdnp Dtlrc, Td i, Khdi 10,?. Trung D0, Vinh. Ngh0

sao cho tdnp

Neuydn Minh Hdi,814, THCSTh6i Nguyen, Nha
Trang, Kh6nh Hba.

HOANG QUY (r2. H6 Cht Minh)

sdNc eUANG

ttcn

hlm

y = lr(r) trcn

,pylB,
Do d6
max
xeR

l-Jz,Jil'
=

Lip bing bi€n thien cira

FJi,Jil

lmco

*"{',[-+)'[9)]

min/(x) =.6 -

=

('6 -t)',

1. Ta cflng th6Y

reR

/(x) = Z(Ji + l)

tdI

CIAI DA HOAN UAO CHUA

?

Trong mOJ quydn s6ch "Phuong pMp.giai tor{n
khio sit hirn sE'-c6 biri to6n : " Tim gid tri nho nhat
(GTNN) vd gid tri lon nhnt (G|LN) ctia hdm sd

y= cosx+2sinx+3 (1), x e (-n, n)" v6i ldi gitiLi
2cosx-sinx+4
nhu sau

:

Vi 2cosx - sinx + 4 > 0, Vx. Ncn tt (1) c6

.

Nhiing ban sau duqc nhQn t4ng ph&n-: .n-q Viet
Crdng,-qe, THCS Nam Cao, L! Nhan.^H"a Nam ;

Nsuyfu Vdn Nam,11A, THPT Duqng Quang Hem,
Vin" Giang, Hmg YOn ; Nguy.dn Vdn Tfurong-LQ
To6n, TlFi'NK Ean Thuy0n, Bic Ninh ; UVdnTdi,
11F, THPT Lam Sc,n, Thanh H6a ;
THPTNghi t6.c I, NghQ An.

sd' d

d€'u bdng nhau.

Lnl

r

{

dinh crta mbi tam gidc

k

DuyThtlt,llAl,
NGOC HIEN

(2+y)sinx+

(t -ZY)cosx= 4Y -3

PT (2) dnx c6 nghiQm

- 3)' <+

t2

e

(2)

(2 +v)2 + (1 -ZvY > (4v

!-u=2"U r*
i ,t :i
Cdnldi giaiciraban?

l,di ginidnadhoinh6o

TRUONG VAN TU
(ThbnVrln Cdn,TT Sla, Quting Didn, Thna ThiAn Hud)

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