'1 ia t ' , il l 1l l, ,t ] aiii ,r' , l I l it ,,i I & ,iv t , i L , ,a i , H0 THt cAn s0, cIno vlEn THU vrin cffi ruAn qu6c un tttu nnt Nam 2005 hu6ng t6i CuOc thi c6n b0, gi6o vi6n thu viOn gi6i toin qudc ldn thrl hai, c6c dia ' phuong trOn cZ nudc rlang td chY HQi rhi cdp iirrt, tttann phd nhu Lang Son, Th6i Nguyen' Ninh Binh, HAi Phdng, Hh Noi, Vinh Phfc' TP Dd Ndng, Quing Tri, Tidn Giang Cdn Tho' 1? H6 Chi Minh Day ld hoat dOng thidt thuc nham nang cao trinh cl6 chuyOn mOn, nghi€p vu cira[.]
Trang 1t
'.:,
il :l
1l l, ,t:
:,a
Trang 2H0 THt cAn s0, cIno vlEn THU vrin cffi ruAn qu6c un tttu nnt
Nam 2005 hu6ng t6i CuOc thi c6n b0, gi6o
vi6n thu viOn gi6i toin qudc ldn thrl hai, c6c dia '
phuong trOn cZ nudc rlang td chY: HQi rhi cdp
TP.Dd Ndng, Quing Tri, Tidn Giang Cdn Tho'
1?.H6 Chi Minh
Day ld hoat dOng thidt thuc nham nang cao
trinh cl6 chuyOn mOn, nghi€p vu cira c6n b6,
gi6o vicn thu viOn trudng hoc vd dQng viOn d6i
ngfl c:ln b6, giSo viOn thu viOn trudng hgc trong
cf nudc ph6i huy s6ng kidn kinh nghiOm phuc
vq yeu idu nang cao chdt tuong day vh hoc
trong nhd trudng
Theo s6 liOu didu tra nam 2004 c:iua NhiL xudt
bin Gi6o duc, hi€n c617.459 trudng (trong tdng
s6 23.472 trudng tham gia didu tra) c6 thu viCn,
dat ti l0 14?o' Lrons d6 mdi chi c61.761) trudng
t)+q") ctat Tieu cnridn thu vi€n truamg phei thong
io gq Gi6o duc vi Dio tao ban hlnh Tron-s so
hcrn 24.000 c6n b0, gi6o viOn thu vien- c5 33r'
ld c6n b6 chuyOn tr6ch, cdn lai ld kiOm nhiem'
Tai hoi thi, m6i thi sinh sE tham gia 2 phdn thi
chinh: bii thi ly thuydt, tap trung vdo c6c rAn
bAn quy dinh vd c6ng t6c thu viOn trudng hoc:
hu6ng xAy ra trong hoat dOng thu viOn trudrg
hgc.
Hoi thi cdp qudc gia s6 cluoc rd chfc tai 3 khu
wc : cdc tinh phia Nam, khu wc midn Trung v)
Thy Nguyctt ueo cudi th6ng 312005; c6c tinh
phia B5c vdo ddu thtug 412005.
Bdi vd cinh.' BACH KIM
Hqi thi cdn b6, gi6o vi6n thu vi6n gi6i tai VInh Phtjc
Hgi thi c6n b6, gi6o vi6n thu vi6n gi6i tai Ha Noi
Tba soan tap chi To5,n hgc vd Tudi t€ r,,rla nhQn dugc cudn s6ch.Bdi todtt
hdm si qui cdc ki thi )lympic,Nhi xuA't bdn G6o cluc, 2004 cria tiic giiiNguy6, Trgng Tudn, GV-truinrg TF{PT Htng Vuotrg, Pleiku' Gia l-ai'Cu6n s6ch ldL m6t tdi liOu gifp rld phdn nlo cho gi6o vi€n vd hoc sinht.n"g uiC giing clay va b6i auOng'troc 'sinh gi6i' NQi.dung chinh cuacudi s6ch ia cai Uai to6n lion q,,un ttd' h)m s6, giii phucrng trlnh hhnt trOn tap sd thr,tc vd tAp sd rdi rac kh6c' Ngodi ra trong cudn s6ch-cdn c6
,"0, pfia, gdm c6c Uii vlgt tim hidu sau hcrn vd mOt sd bdi to6n hdm so
trong c6c ki thi to6n.
Cam on tdc gihdd tAng s5ch vi trAn trong gi6i thi€u vdi ban doc'
THTl'
Trang 3Viec giii phuong trinh (PT) c6 tr) hai c6n-thfc
chrla'biEn tro lcn thuong kh6ng dc dang' Ngudi
ta dd dua ra nhidu c6ch gi6Li nhu nAng I€n lfiy
thta dd llm mdt can, dai can thrlc lhm dn sd
phu, sft dung c6c hang ding thrlc phir hoo
Trons bli nly xin gi6i thi0u mdt phuong ph{p
kh6c dd gi6i phicrng rrintr vo ti d6 Ih sft dgng hi6u
thrlc lien hqp Ta de biot fJi+^lu>tJi-Ji)=o-b
lI hai bidu thrlc li6n hop vtli nhau Thu.c chdt
cria phucmg ph6p niy th neu nhAn mOt bidu thrlc
d,4ng a t Jn voi bidu thrlc li€n hop m) xudt
hiQn mQt nhAn trl chung vdi bidu thrlc kh6c cira'
ohuons trinh thi sau khi dat nhAn tir chung ta
Xin minh hoa bang cdc thi du dudi dAy.
Thi du l Gidi phuong trinh
n6n ndu nhAn cA hai v€ cira PT (1) vdi bidu thtlc
tion hqp v6i vd tr5'i (bidu thrlc niy luOn duong)
thi xudf hi€:r nhan tit chung li x+3 Ta c6 :
Thi du 2 Gidi ph*mg trinh
PHAM eAC pHU
(SY ldp CLC K52Todn, DHSP HA Nbi)
Hudns ddn gitii Didu ki€n :
llx + 13 > 0,4x + 13 > 0, x * 1 > 0 suY ra
x > - l NhAn ci hai vd clra PT (2) vdi bidu thrlc
Ii€n hqp vdi vd tr6i (bidu thrlc niy luon duong)duoc
8x = .,/x+1 tJtzr*n*"[4x+n)
TrU theo tirng vd hai phurrng trinh tren duoc
7x-1=2il;1x4r+13)
Binh phuong hai vd r6i giAi phuong trinh ta
tim cluoc nghiOm [d x1 t'n= 3 v) x2 = + Thir lai
ta thdy x, = -l] kh6ng th6a mdn (2) ve PT(2)
JJcri nghi€m duY nhAi ld x = 3.
Chn y rang PT(2) c6 tfrd gi6i bang c6ch binh
ohuons trai -ve hoac' chuydn mot can thric i.l ve
inli saig ve phrii roi binh phuong hai ve sau d6
dua vd eiai I'f bac hai Tuy nhi€n ir dAy vAn nOu
c6ch eiTi tren dd thAy rang phuong-phrip dilng
bitlu ihric li6n hgp vAn sir dung dd girii duocphuong trinh clang Jax+ b - J;x+d = G** f
Thi du 3 Gidi phtong trinh
Hudng ddn gidi; Didu ki€n 2r'+16-r+18 > 0,
f -t>osuyra x< -4-Ji , -4+Ji (x(-1,
Thrl thdy xt = 1 vd x2 = -1 ld nghiem ciia
f'f(4) GiA srl f * 1 Nhan cA hai v€ ctra PT(4)
u&'fie" thrlc liOn hqp vdi vd phii (bidu thrlcniy kh6c 0 vi x2 + 1) ta dudc
Ji#::-J4x+t3=Ji+r (2)
Trang 4Tidp tuc gi6i PT on[;1=z'-3 bang c6ch
binh phuong hai vd, ta tim du-o c nghiOm xz= -l
vi x3 = -25n NghiOm xz = -25[J khOng th6a
min 3(-x-1) 2 0 nOn bi loai Thrl lai ta thdy
PT(3) c6 hai nghiOrn lh xl = 7 vd xr= -1"
Thi dq 4 GiiliPhudng trinh
Et ar{^2 -b:rt= ',E;2 +2o+i*ff-r+z rsl
' Hudng dd.n gidi Didu kitu : C6c bitju ttllc
thrlc trong cin cbn lai khdng Am dAn riOn
"t u ch v{, tr{i v} vd phii ctra PT(6) (de thdy
cac Uldu thrlc niY ddu duong) duoc :
Th{r th{y x = -2 li nghi€.m crla trf (D ve
th6a m6n didu kicP ban dAu.
Ndu.r * -2 th\ chuydn PT (7) vd PT
* JzF *2"* =o (8)
Ta rhdy vdi x < -Ji tz hoac x > Q*Jn)tz
thi bidu thrlc & vd tr6i ciia PT(8) ddu kh6ng Am
va c6 ft nhdt mqt bidu thtlc duong nen vd tr6i
ctia PT (8) drlong, do d6 PT(8) v0 nghiOm'
V4y r = -2ldnghi€m rluy nhdt cfra PT(5)'
Thi du 5 Giei bdt Phudng trinh
C6c ban hdy srl dung phuong phdp dLng
thfc ti6n hqp dd giii cd,c PT vh BPT sau :
Ban chua lim duor c6c bii trong D€ ra ki
nd,y v\ kh6 qu6 ? Ban chua tham gia thi giriitoan Uang m6y tinh i Ban vAn c6 thd thzrm gia
;;;; ihi1rl va ciai tri mdi tr€n THTT' 'rat
nhien crl c6c ban cI6 tham gia hai cu6c thi trenu6tr duo tham du Cuoc ihi clinh cho tdt cri
moi nc.trdi Xudn ru dA - H? giii ddp sE ui' rl
cau trli d6ng tr€n hai sd ba-o th6ng 4'20O'5 t)
5.2005 Nhin bhi giAi ldn luot trong.thdng -i'
O.-Oap r{n vI dan'ir s6ch c6c b4n dugc giai
thucrn! s6 dang trong hai
-.-6 -3-7 (th6ng
1.2003) vh 338 (riri,s it.2005) Mong duoc
"6, ;; hudmg ,:mg r[ tlrong tin cho nhjdunsudi cirne biel Ban nhd dit b6o thdng
I-'zoos .re 6ia them cjhi ti€i vd cudc thi'
'fI{l-f
Trang 5A\f r)0 THr TUYEN sn{H vAo LoP 1o
T
TRUONG THPT CHU VAN AX{ YA THPT HA NOI AMSTERDAM 2OO4
Ngdy thi thit nhdt : Ddnh cho moi tht sinh
(Thoi gian ldm bdi : 150 philr)Bni 1 1z aidm;
1 Chrmg minh phuong trinh c6 2 nghi€m
phan bi0t vdi moi gi4, ti cila m.
2 Tim m dd, ti sd gita hai nghi€m cria phuong
trinh c6 gi6 trl tuyOt ddi bang 2.
Cho phuong trinh: x + 3(m - 3f)2 = m
1 Giai phuong trinh vdi m = 2.
2 Tim rrl dd phuong trinh c6 nghiOm.
Bni 7 (2 didm)
Gihibdt phuong trinh
1 Vdi gi6 tri n)o cfra /< thi duong thing (r/)
song song vdi dudng thdrng y = xJi ? Khi d6
hdy tinh g6c tao bbi (A vdi ria Ox.
2 Tim rt dd kho6ng cdch tir gdc toa d0 ddn dudrng thhng @) ld ton nhdt.
Bni 4 (4 didm)
Cho g6c ru6ng xOy vit hai didm A, B ten
canh Ox(A nim gitra O vd B), didm M bdt ki
trOn canh Oy Duong trbn (?') ducrng kinh AB c6t
tta MA, MB ldn luot tai didm thrl hai ti C, E.Tia
OE cat duong trdn (I) tai didm thrl hai lh F
1 Chung minh 4 didm O, A, E, M nam tr€n
m6t dudng trbn, x6c dinh tim cira dutrng trdn
d6.
2.T(r gi6c OCFM ld hinh gi ? Tai sao'/
3 Chrrng minh hO thrlc :
OE.OF+BE.BM=OB'2
4 X5c dinh vi tri cira didm M dd tf gir{c
OCFM ln hirh binh hdnh, tim mdi quan h€ gifra
OAvd AB dd, trl gi6c l) hinh thoi
Cho tam gi6c ABC c6 3 g6c nhon, k6 hai
dudrng cao BE, CF
1 Biot g6c BAC br,ng 60", tinh d0 dli EF theo
BC=a.
2 Tr€n nrla dudng trdn duirng kinh BC kh6ng
chta E, F ldy m6t didm M bdt ki Goi H, I, Kldn luot li hinh chidu wOng g6c cria M trcn BC,
CE, EB Tim gi6 tri nh6 nhdt cira tdng
- BCCEEB
\ - -r-r-MH MI MK
Bni 9 (l didm)
Cho m6t da gi6c c6 chu vi bang 1,
minh rlng c6 mOt hinh trdn b6n kinh r
chrla todn bO da gi6c d6.
NSdy thi thit hai : Ddnh cho cdc thi sinh thi vdo chuyAn Todn - Tin
(Thoi gian ldm bd,i : 150 philt)
chfngI
Trang 6crup BAN rqlou rnl
ofi oN LuvrN sd z
('[hoi gian ldm bdi : lB0 Philt)
CAu I 1) Khio s6t vd vE dd thi him sd :
x2 -x-2
2) T(nh phdn dicn tich hinh phing dugc gi6i
han b&i dd thi ctra hdm sd vd truc hoinh
CAu II l) Gia st a, b, c , d ld ciic sd th6a m6n
rling thfc : ab + 2(b+c+A = c(a+b) Chung
minh rang trong ba bdt Phuong trinh :
it nhdt m6t bdt phuong trinh c6 nghiOm.
cosx cos2x cos3x - sirx.sinlx inl, = ] )
2) Cho f(x) = (1 + x + I + xo; Sau khi khar
tridn vi rut gon ta dudc :
.f(x) = eo + a$ + a2* + + ar{tu.Hdy rinhgi6 tri ctra h€ so a,,,.
CAu IV 1) Trong mat phing vdi hO toa do
Dd-c6c r,u6ng g6c Ory cho elip (E) c6 phucrnu
.22
rrinn f+f=l (vdi a' b"
a>o.b>Os-Gii sr) A, B ld hai didm thal' ddi tren (0 sa'r
cho OA w6ng g6c v(i OB.
rl
a) Tinh j- + :; theo a vd b.
b) Goi H lh chAn duirng vu0n-g g6c ha tt 0
xu6ng AB Tim tAp hqp c6c di6m H khi '{ I
thay ddi trcn (O
2) Cho hinh lAp phuong ABCD.A'B'C'D'",'i
canh bang a Hdy tinh khoing c6ch gita c:rnh
AA'vitduirng ch6,o BD'theo a.
CAu V Cho r, y, z ld nhtng sd ductng thoa
mdn xyz = 1 Tim gi6tir nh6 nhdt ctra bidu thue :
2) Vdi nhfrng gi6 tri nio ctia a thi h0 phuong P
Him sd c6 cuc dai, cuc tidu khi PT y' = Q 96
hai nghi€m phan bict kh6c 1.
l!'=4-3m>o 4
lf()=t*-q+o 3
,9 +r9 ,9 +ro'
PHAM HtrNG
qgx'+y'
Aana'
DE ON LUYEN SO 1(Da dd ddng ffanTHTT sd 332, thdng 2 ndm 2005)
Trang 7Y\28A2 + 682 khong d6i ncn (2MA2+MN) c5
gi6 tri nh6 nhdt khi EM c6 gi6 tri nh6 nhdt,
2) Ticu didm ctra parabol F (-l; 0) Ducrng
thing (A) qua F c6 PT :
a(x+1)*by=Q
Dudng thing y=0 chi c5t parabol tai I clidm, khOng th6a m6n b)i to6n Cho.n a = 1 = x = -l
- by Tung dO giao didm cira parabol vi dudng
thing (A) ph6i th6a mdn PT
nghiOm lt * lz th6a mln : lJz= -4 .
Gii srl A(xr; yr), B (xr; y) ld c6c giao didm PT
ti6p tuyen vdi parabgl tai A,B tdn luot li
lJ = -Z(xt+ x) (dr) vdyry = -2(xz+x) (dz)
Ta c6 cdc v6cto phdp tuydn cira (dr), (d") ldn luot
Id nl = (2; y,,) ; ,z = (2; yr) + ntn,) = 4+yry, = O
hay (d,) L(dr).
Cnu V a) Goi s6 tao thdnh Ih ,brd, .
TIll 36'tao thdnh chfta cht s6'0
sin3x= 3sinx- 4sin3x = 4sin3x= 3sinx - sin3x
cos3x = 4cos3x-3cosx + 4cos3x = 3cosx+cos3x
BPT<+ 3 , 2
PT tr& thdnh
(3sinx-sin3x)sin3x + (3cosx+cos3x).cos3x =
3(sinxsin3.r+cosxcos3x) + cos'3x - sin23x =
3cosbx + cos6x = ! o +"o"'Zx = )
Trang 8P}IUUITG PHAP TACH BO PHAN KEP
Trong bdi Phuong phdp g()i s6' hang vans ran tap chi
TmT s6 262, th6ng 4 nam 1999 dd nOu hai thuat toiin dC
0
tim si6i han ":^' dane -;;-E I 0 cira mOt sd hlm s6 chrla cin thfc Na1
chring tOi xin trinh bly th€m Phuong phdp tdch bo phcut
kip-d6 li thuAt todn thf ba dd x6c dinh bidu thrlc chrla dn vang mat trong khi tim gi6i han cua phin thuc
ddi bang cSch thom
phAn thrlc phii tim gidi han
Luu !.- Bidu th1c h@) duoc x6c dinh tir c6c
bidu thrlc f(x), S(x) vi duoc goi ld bQ phan kep
trong bdi to6n tim gi6i han d+r-,g (*)
r = ' - ;:;e-a)k.e1G)' tim f{x) + tim
x-s,t1;1-qyk .eSG) x-a(y-a)* .QrG)
dang x6c dinh hoac dang quen thu6c
Dudi day td c6c thi du minh hoa.
Thi du 1 Tim gi6i han
toun
s{x)+[h(x)l"
Trang 9- MOt vai sd hang ctra bO phAn k6p h(x) c6 thd.
bi dn trong/,(x), g,(x), ta phii tim chring dd x6c
dinh chinh xdc bidu thftc h(x)
Thi du 2 Tim gi6i han
g(x) = cos3x +3 cos x -ln(1+;r)a
Mdi c6c ban thu st dung phumg phdp tiich
bQ phQn kdp dd, nOu dd tim cric gitli han sau
Trang 10GIMPS vA sd xct,vrx ro MERSENNE
Lol\ xnAt cHo pfx NAY
Ngny 18/0212005, Martin Nowak ngudi Drlc
Oa tl* ra sd nguy6n td ($'{T) ltrn thdt dugc bi6t
cho ddn ngly nay, d6Id sd 22s'e6a'es1'1 Sdndy c6
7816230 cht so vI l) so nguy€n td Mersenne
th1 12 duot tim thdy Ndu n6 du-o c in ra kin tr€n
rnat gidy A4 v6i cd font binh thudng thi cfrng mdt
tOi tron- 1700 trang! Day ln lAn fif 8 kf tgc vd
SNT Itu nhdt rluoc lap b&i Dt dn tim SNT
Mersenne li€n mang (GIMPS $IT Mersenne lh
cr4c ${T c6 dang M , = 2!:l v& p lit s6 nguyen t6'
vI cho ddn nay ngudi ta chi bi€t dtroc 42 sd nhu
vQy ViQc tinh to6n dd tim ra SNT nhLy d6 mdt
fron SO ngdy tinh to6n trOn m6y tinh cria b6c s!
Nowak (CPU Pentium IV, 2.4 GHz) $IT mdi
nly cluoc d6c tAp kidm chrlrig cdch sau
-5 ,gay
b&i Tony l{eix & Grenoble, Ph6p, c6 sr} dung
chuong trinh Glucas ctra Guillermo Ballester
Valor & Granada, TAY Ban Nha
Trudc d6, SI{T Mersenne thf 40 duoc tim
thdy tr 220'ee6'0tt - 1 vdi 6.320.430 cht sd vio
l7il?oo3 b&i Michael Srafer' Nely 15/52004
Josh Findley dd tim duo c SNT Mersenne thf 41
11 2zaor6sa:-1 g6m 7 .235.733 chfl s6.
Martin Nowak th m6t b6c s! nhSn khoa &
Micheltetd, Dr1c Ong bidt tdi GIMPS vio th6ng
4l7g9g khi doc mot bIi vidt & b6o dia phuong'
6ng le mdt ngudri y6u thich To5n hoc, kh&i ddu
vdi mOt m6y tinh c6 nhAn tham gia vio GIMPS'
Siu nam sau, 6ng c6 24 mdy vi t(nh cDng tham
gia vio GIMPS vl vinh dq gid dAy d6 ddn vdi
5ng Thuc ra Nowak kliOng tim ra mOt minh mi
c6-su d6ng g6p cira hing chuc trong sd hhng
ngin tinh
"g"V.* vi€n ctra GIMPS' Danh tidng
cf,o klr6m -pie *oi niy nhu viy thuoc vd
"Nowak, Woltman, Kurowski, vi nhidu ngudi
kh6c"
(The Great Intemet Mersenne ftime Search
vi6t t6t li GIMPS) duo c thi'nh lAp th6ng 1|1996
b&i George Woltman nham x6c lAp c6c k! lqc
m6i vd O-O fOn ctra SI'{T Mersenne' GIMPS kdt
hqp sfc manh cria hfurg chuc trong sd hi'ng
ngdn mriy tinh th6ng thuirng dd tim ra "cdi kinr
tr"ong cldng rom" Tien s! Cranclall 116 ph6t tritlnthuAt to6n FFT ding cho GIMPS Ngudi s6ng
lAp Entropia, Scott Kurorvski, dd phdt tndn h€ th^dng PrimeNet cho ph6p kdt hgp rt't nhidu m61'
tinh ch4y song song dd duclc mOt "siOu mdy tinh6o" c6 thd thuc hicrr 17 ti'ty ph6p tinh mot giay'Ch(nh rlidu niy d5 girip GIMPS lQp ki luc md chi trong 9 th6ng thay v) 1500 nim ndu nhuchay tren mdt m6y tinh don 16 !
Nhfng ngudi tham gia GIMPS don giAn l) do
yeu thich vd mudn tim ducvc SIrIT ivlersenne
m6i, mOt sd ngudi hy vong rang sC doat drloc
mOt phdn cria giii thu&ng 100.000 USD cira
Quy tei tro (Electronic Fronlier Foundation) cho
.rguOi ti* ra SNT ddu tion vdi 10 trieu chft so'
Ngudi d6 sE nhfln dur;c -50.009 USD, Hqi tirtlilen sc nhan 25.b00 USp i'a phdn cdn tai dd tii
trd cho chinh viOc tim ra c6c SNT l6n hcrn'
"NgutY tim duoc s6 niy c5 drd rudt hiOn v)io
tuiin tdi hoac cflng c6 thd k'r vii narn sau nfa, d6
li su hii hrrdc trong kh6m phd, To6n hoc!"
-George Woltman n6i Nlm 2000, mOt ngudi
tham gia GIMPS tru6c cI6 dd duo c nh0n 50'000USD Jho viOc tim ra SNT c6 1 tri6u chfl sd' "St
ki6n tri vi tlLm vi0c theo nh6m rdt cdn cho sr,t
thinh cdng ctra chring t6i Xin chric mtng vd
cim crn tdi b6c s! Nowak vi tdt ci 75'000 tinhnguy€n vi€n tr0n todn thd gi6i dA lhm nOn su
th-anfL cong niy" - Woltman b6 sung'
Bdt cft ai quan tAm v6i mOt m6y vi tinh ndi
mang ddu
"O tfrd tham gia vio GIMPS-dd ro
thlni m6t tay "sf,n sd nguyen td l6n"! Nhfrngphdn mdm cdn thidt c6 thd clownload miOn ph(,ui hrtrt//***.*"tt"t tt".otn Bqn cl-gc c6 thd
tim tt ay
GIMPS vd c6,c trang web liOn quan t4i trang web
nhy C6c ban c6 ihd xem th€m vd nhtrng sd
nguy0n td Mersenne trong THTT sd 318 th6ng
t2.2003
HAN NGQC DUC
(T he o httP:/www-mersenne'org)
Trang 11Nh^ h c^;o* rg"/ Quiil.lzPb * I 3
Nguyiin Quynh Huong hicn la sinh vi€n
trudng dai hoc St Cloud, bang Minnesota, Hoa
K! Huong dang theo hoc c[ng hic 2 nginh Kd
to6n vh Tii chinh vi s€ tdt nghi0p loai danh du
(honor graduation) vio gifa th6ng 5 ndy
Sbdt thdi gian hoc TIIPT Lc Quf D6n, TP H6
Chi Minh, Huong so nhdt li rnOn tidng Anh
ThSng 6 ndm 2000, Hucmg sang Hoa K)'du hoc
vdi vdn tidng Anh gdn nhu IIL con sd kh6ng- "Di
du hoc nhung t6i khOng bidt li minh c6 thd hoc
vi tdt nghi€p duoc & M! khong nta Nhtng
ngiy ddu ti6n sao mi chAt v6t thd Tdi so ph6i
giao tiOp, so n6i "thrl tiOng Anh" mi ngudi kh6c
kh6ng thd hidu ld minh dang n6i gi Tuy nhien
chuong trinh hoc, d6c bi€t li m6i trudng hoc tAp
rdt soi d6ng v) l( thf Nhfrng didu hoc duoc tr)
s6ch v& vh c6c gi6o su luOn c6 tinh thuc td',.'i
tinh rrng dgng cao Hoc kh6ng phii vi thhnh tich
cao md vi t6i thuc sr,r ham mudn duoc hi6u bi€l
sAu s5c hon nila vd chuyOn nghnh mi minh theo
hqc" - Co kd lai nhu vay N6 I$ ay dd girip
Huong li€n tuc giinh duoc hoc bong "Cultural
Sharing Scholarship" tt nam hoc thf nhdt cho
d€n nay.
Kdt thrlc 2 ndm dai cuing (general education)
v6i sd didm trung binh tuyit?6i 4.014.0 vi c6
tOn trong quydn National Dean's Lisl ghi danh
nhtng sinh viOn xudt s6c nhdt hang nam cita
hon 2500 trudng dai hoc ften toln nudc M!.
Huong nhAn duoc hoc b6ng "College of
Business Executive Coancit' vI duo.c k6t nap
vdo HiAp h1i Beta Gamma Sigma - Hicp h6i
Qudc td ctra nhtng sinh vi0n danh du c6c
nglLnh Business (kdt nap 17o sinh vi€n c6 thinhtich xudt sac nhdt cira nginh business tr6n toin
nudc M!) Huong cfrng ducrc chon llm Ph6 chit
tich Hi0p hOi niy tai trudng dai hoc St Cloud
C6ng chfng ctn Hi€p hdi s.inh vidn nh6m ngdnh
bdo hidm My Hgc ki vta qua Hucrng cfing dd IiOn kdt vdi mOt sd ban sinh viOn Vi0t Nam cDng
trudng thanh lap HOi Sinh vi0n Vi€t Nam tai
truirng dai hoc St Cloud (Vietnamese Student
Association at SCSU).
Mong mudn ctra Huong vi c6c ban sinh vi€nViet Nam ti day Ii c5 thd gi6i thicu vin h6a
Vi6t Nam ddn ban bd thd gitli cflng nhu girip dd
lAn nhau trong hoc tAp vlL cung c{p th0ng tin du
hoc cho c6c ban sinh viOn Viet Nam
Huong cdn li tro gi6o m6n Toiin (Math Tutor)
& Trung tdm kI ndng tod,n hoc ciua trudng (SCSU
Math Skills Center) ngay ttr nam ddu dai hoc.
Vdi vdn kidn thrlc kh6 rrlng vd mOn ToSn tir
thdi trung hoc, Huong b5t nfrip vdi cOng viOc
nly khrl d€ ding
"Ngudi ViOt Nam rdt gi6i, c5c ban sinh vien
Vi0t Nam li nhtrng ngudi c6 chi cdu ti6n cao vi
nhidu nhi€t huydt trong cOng vi0c Chi cdn cdg6ng vd tin tuimg vdo bin thAn, ban sE dat duocnhtrng kdt qu6i mlL ngay ci ban c[ng khOng ngd
tdi" Qu!,nh Hucrng dd kdt thric bu6i trb chuy€n
vdi kinh nghi€m nit ra tt chinh minh nhu thd.
(TheoTin hoat dbng cdc H6i KHKT 2.2005)
GIUP B1tN (Ti€p trans 5)
Ta c6 4 c6ch chon vi tri cho chri so 0, sau d6'c6
,4/ c6chchon hai trong bon vi tri cdn lai cho c6c
chfr sd 1,2 Tidp theo, sd c6ch chon hai trong bon
chfi sdkh6c 0, 1,2 cho hai vi tri cbn trdng ld 4 ,
Theo quy t6c nh8n, ta duo c sd c6c so lb :
TH2 Sd tao thdnh khdng chta chfr sd 0.
Ta c6 ,$ c6ch chon hai trong 5 vi tri cho c6c
cht so I, 2 Sau d6, sd c6ch chon 3 trong 4 chfr sd
kli6c 0 1,2 cho ba vi trf cdn trong ld fr .
Theo quy t6c nh0n duoc sdc6c sdl): ,4.fi=q&O .
Theo quy t5c c6ng thi so c6c sd tao thinh li :
576+480=1056
b) D4t ct = x + l, b = y + l, c = z + 4 thi
a,b,c>Ovda+b+c=6.
- a-l b-l c l - (l I 4) O- t
r. -=3-!-+-+-!
a b b \a b c)
11444168 S=l+-+'> +-> nOn
a b c a+b c a+b+c 3
8l O=3-S<3-
.,J
r3
VAv maxo - - 6' a= b', a+b= c a o=b= - :
3216'=].qAX=V=a',2=-L. 2
NCUYENANH DUNG
Trang 12cAc NHA toax Hoc uEr NA,\I
cap c0 oAu xuau
Nhan dip Xuan Ar DAu chi6u 2.2.2005 taiViOn To6n hoc Viot Nam, Hoi Todn hoc VietNam dd td chrlc cu6c gap mat c6c nhl todn ht'e
Tdi rlu c6 GS TSKH Pham ThdLong, Chu tichHOi Todn hoc Vi€t Nam c6c Ph6 Chir tich
Tdng thu ki HQi Toiin hoc Vi€t Nam, c6c nhi
to6n hoc 16o thdnh, cdc cdn b0 nghiOn crlu vi
giing day toSn tai ViOn To5n hoc ViOt Nam, c6c
i*Ofrg 5ai hoc, Tap ch( To6n hoc vd Tudi tr6,
TSKH Lo Trdn Hoa, Ph6 Chi tich kiem Tcing
Thu ki HOi To6n hoc Vi0t Nam, dd thOng bdo vd
cdng t6c cira HOi trong nam qua GS' TSKH HnUuy Xnoai Ph6 Chfi tich HOi, Chu tich Hoid6rig Gini thuirng L€ Vln Thi0m dd cong b6
danh s6ch 4 hoc sinh v) 1 giSo vi0n todn duoc
nh0n Girii thu&ng L€ V[n Thi0m nam ]0C+
Tidp d6 c6c nhd to6n hoc 16o thdnh vd l6nh dao
Hoi To6n hoc Vi€t Nam dl trao GiAi thu&ng LeVin ThiOm cho c6c gt6o vien vd hoc sinh
Ngay Z.Z.ZO05 tai H6i trudng Nguv Nhu Kon
Tum 19 LO Thdnh T6ng, H) Noi dI di€n ra HOi
thAo 30 ndm Vi€t Nam tham du Olympic TodnQudc td r'6i gAn 200 d+i bidu do trudn-eDHKHTN - DHQG IIe Noi cing IIOi To6n hoc
Viot Nam td chrlc Tham du HQi th6o cti GS.TSKH Trdn Van Nhung, Thrl trucnrg B0 GD-
DT, dai di€n ctra Vu GD Trung hoc, Cuc.khioth( vd Kidm dinh chdt luong GD, HOi Todn hocVi€t Nam, c5c nhi toiin hoc I)m viOc lai c6'ctruong DHKHTN He NQi, DHSP Hd NQi, T4P
chi To6n hoc vh Tudi tr6, nhi€u Truong dolLn,
Ph6 tru&ng dolLn vd hoc sinh Vi0t Nam dd tham
tinh, thlnh phd trong ci nu6c'HOi th6o dd nghe cdc b6o c5o v) th6o luAn vd
viOc'td chfc phit hifir, tuydn chon, bdi dudngnlng khidu to6n hoc v]L chuong trinh n6i dung
ki6n thrlc to6n & cdc l6p chuyOn to6n TIIPT
Hdi th6o cfrng ki6n nghi v6i BO GD-DT gdp
rut chudn bi cho'vi€c Vict Nam clang cai td chrlc
rhi Olympic Tor{n Qudc t6'v}o ndm 2007'
P.V
HA HUY KHOAI(V i€nTodn l'toc)
Dd khuyen khich thd he tr6 say rh€ hoc tip
m6n to6n v) co thd lua chon to6n hoc lIm nghd
nghiOp tuong tai cta minh, dd ghi nhAn cdng lao
"[u rirrng ngudi thdy day to6n mn tuy vdi nghd
nghiOp, Flof toan hoc Viet Nan trao giAi
thu0rrg h)ng ndm mang t€n Gidi thuong Ii Van
Thiem cho mOt sd hoc sinh xudt s6c vi thdy
gi6o day to6n gi6i trong ci nudc
Gi6i thurrng ddi vtli hoc sinh duoc trao cho hai
ddi tuorng: cZc hoc sinh doat kdt qui xudt sac
trong c6c k]' thi Olympic to6n Qu6c td, vd c6c
hoc iinh hodn cinh kh6 khan nhung d5 won lOn
dat thhnh tich cao trong hoc tAp m6n to6n
Giii thuong L€ Van Thi€m 2004 duoc trao
cho c6c hoc sinh vi thdY gi6o sau dAY:
I HOC SINttr
l Pharn Kim Hilng, sinh nam 1987, l6p 11,
Khdi Pf ChuyOn to6n, DHKHTN - DHQG HN
Qudc gia THPT 2004, HuY chucrng ving
Olympic loiin Qudc t€2004
2 Nguy€n Kim Sorr, sinh nam 7986,l6p 12'
KhOi Pf ChuyOn to6n, DHSP Hd NOi Thlnh
t(ch : GidLi nhi ki thi hoc sinh gi6i to6n Qu6c gia
TIIPT 2004, Huy chuong vhng Olympic to6n
Qudc td 2004.
3 Nguy€n Minh Trudng, sinh nam 1986, l6p
12, trudng TIIPT TrAn Phri, HAi Phdng Thbnh
tich : Giai nhdt ki thi hoc sinh gi6i todn Qudc
to6n Qudc te 2004.
4 Pham Vidt HuY, sinh ndm 1988, l6P 12
Tru&tg TIIPT ChuyOn LC KhiCt, QudLng NgIi
hoc, i997-1998, Giai khuydn khich ki thi hgc
sinh gi6i to5n Qudc gia TIIPT 2004
II GIAO VIEN
Thac si Trin XuAn D6ng, sinh nam 1955,
gi6o vi0n trudng TfiPT Chuy0n L€ H6ng Phong,
Nam Dinh Thinh tich: Dd giAng day chuy€n
todn !2 ndm, g6p phdn v)o thAnh tich ctra dOi
tuydn Nam Dtnh vdi 66 gi6i t1{c ei3, 't gi6i
trong c6c ki thi Olympic torin ChAu A.- Th6i
Binli Duong, 1 giii trong ki thi Olympic to6n
Qudc td (2003) Vidt nhi6u bii cho t4p chi To6n
h-oc vi Tudi tr6 Duoc Bang khen crla BO tru6ng
B0 Gi6o duc vh Dio tao.
Trang 13o0lr u TAI rR0 Gllil'lH
A Df, DANH cHo rHCS (3 os)
Bni 11 CS Tinh gdn dring gi6 tri nh6 nhdt vh
gi6 tri lcrn nhdt cira phin thrlc
Zxz -7 x+l
1=
x2 +4x+5
Bni 12 CS Tim nh6m ba chfr sd cudi ctng
(hlng tram, hing chuc, hing don vi) cria sd
12 + 23 + 34 + 4s + 56 + 67 + 7t + 8e + 910 +
1011 + 1112 + 1213 + 1314 + 141s + 1516
Bni 13 CS Tinh gdn dring g6c nhon x (dO,
phrit, giAy) ndu
sinx cosx + 3 (sin-r - cosx) = 1,.
Bni 14 CS Didm E nam trOn carrh BC cria
hinh r,u0ng ABCD Tia phAn gi6c cl,aa cdc g6c
EAB, EAD cdt cdc canh BC, CD tttong rlng tai
M, N, Tinh gdn dfng gi6 tri nh6 nhdt ctra ti s6
YY rtrt gdn dring (d6, phrit, giafl g6c EAB
Bei 15 CS Hai dudng trdn b6n kinh 3dm vh BAi 15 PT tfinh chSp S.ABCD c6 dudng cao
4dm ti6p xric ngoii vdi nhau tai rlidm A Goi B SA = 5dm D6y ABC D li hinh thang vdi
mOt ti0p tuydn chung ngoii Tinh grin dring dierr CD = 6dnr Tinh gin dring diOn lich toi\n phiin
Quy udc chung cho cd 2 nhdm bdi : Ghi phrong phdp gi(;i vdn tdt, c6ng thtlc, phim bdm c.ila
tung bdi Neu kel qrui ld s6'htu ti thi phdi ghi ket qud duoi dang s6'nguy€n hodc phdn s6' Nei keiqud ld s6'vb ti thi phdi ghi ket qud duoi dang sO'thdp phdn gdn dfing cd I 0 chir s6',
NhQn bdi gitii gtti trthc ngdy 15.4.2005 theo dd'u Brtu d.iAn Cdt'ban hoc top t0 duoc rtt thiphdn THCS, Gli r.5 s* dung mdy tinh loai ndo Ngodi phong bi ghi rd ; THl GIN T1AN TREN
MAY TINH BO TUI thdnrg 3 ndm 200,5.
T10/333 Prove that where A, B, C are angles of a triangle
7721333 Let ABCD be a tetrahedron
inscribed in a sphere (O) with center O,Iet G be
the centroid of ABCD,let M be a poinr lying inthe interior of or on the sphere with diameter
OG The lines MA, MB, MC, MD cul again (O)
respectively aI Ar, Bt, Ct, Dl Prove that
volume
ll
where xl, x2, , xn are n positive numbers
satisfying the condition irr + xz * + x,1 L
Tlll333 Find the greatest value of the
QUA TAP CHt ToAN HgC VA TUor TRE
B Dd DANH cHo rHpr (3 os)
Bni 11 PT Tinh gdn cfting giri tri ldn nhdr cira h)m s6 y - Jx + xt +
Bni 12 PT Dd thi him sd y - ax3+bf+cx+d
di qua c6c didm AG a;3), B(7;5), C(- 5; 6),
D(- 3; - 8) Tinh gi6 tri cira a, b, c, d vi tinh giin
dring khoing c6ch gifra didm cuc riai vd didmcuc tidu cira dd lhi d6.
Bni 13 PT Tfnh gdn ching cdc nghiOm cria
hd phuong tdnh
Bii 14 PT Trl gi6c ABCD c6 AB = 4dm,
BC = Bclm, CD = 6dm, DA = 3clm, EID =
80", M li trung didm c,,ia AB vit N lh didm nam
tr€n canh C D sao cho M N chia tr1 gi6c thdnh hai
phdn c6 di€n tich bang nhau Tinh gdn dring dO
ddi MN
t E+=[,.1)
i=r ! xi \ n)
Trang 14SO KET CUOC TIII GtrAI TOAN BANG
DA thi dd ddng bAn THTT so 331 (l.20AS)
C6 tC le vio dip Tdt ncn s6 luong bdi giii gui
vd it hon dot thdng 11.2004 Didu d6rrg mimg li
c6c tinh phia B6c d6 c6 ti l€ cdc ban tham gia
cao hon dot trudc Hoc sinh Ha NOi cdn it tham
gia Dd thi dot ndy cfrng kh6 hon nOn so b4n d3t
iidm toi da khong nhid-u Didu dring n6i li ddu
bei da ghi 16 : N€u kdi qud ld s6'hrtu ti thi ptuii
ghi kel qud duoi dqng s6'nguy€n hodc phdn s6'
ntrmg nhidu ban van ghi k6t quA bli d4ng sd
th4p phan trong trudng hqp lh phii ghi phAn sd
nhu bli 9.PT.
Ldn nliy, c6c ban sau d6y duoc nhQn qui tang:
Khdng Hod.ng Thao, 8D, THCS LAP Thach,
LAp Thach, Vmh Phric ; NguyinThi Th€u,9D,
THCS D6ng Hung, Th6i Binh ; Cung Vdn
Dfr.ng, l}t, Tltr'f Qu6c hoc, Thira Thi6n ' HuC
Ngiy\n NhQt Linh,10 To6n, THPT chuy€n Hir
Vung II, Diing Th6p ; Le Bili Ti|h Duy, l0
chuycn To6n, TIIPT chuy6n Lc Hdng Phong,
chuy6n Hi finh ; Nguy€nThanhTiing,l2AT0
THPT, Ngo S! Lian, Pham Vi€t Dftc, 11A2,
THPT tan Yen I, TAn YOn, Bdc Giang ; Trdn
Hfru Huydn Trdn, 12 To6n, THPT chuyOn Tidn
Giang
Mdi c6c ban tham gia tidp cu6c thi dot 3
(th6ng 3.2005) c6 dd dang trOn sd b6o niy Kdt
quA cuoc thi duo.c so kdt vio th6ng-5.2005 Bai
idn ghi ddy dtr c6c th6ng tin vd bin than nhu
didu lC cu6c thi quy dinh (ho tOn, Idp, trudng,
Sau day td ddp 6n bei thi th6ng 1.2005.
Bni 6 CS TU c6c rling th(tc a+b+c = 3 v)r
ab = - 2 ta bidu thi duoc b vh c qua a Thay c6c
bidu thrlc d6 vlo ding thfc b2 + c2 = 1 ta dugc
mQt phuong tdnh bAc 4 ddi v6i a TU phuong
trinh d6 ta tim duo, c (bang phuong ph6p l4p) hai
gi6 tri gdn dring ci;,- a Vl b vi c d6 bidu thi
duoc qua a nOn M li m6t phAn thrlc^ ctra a Thay
cdc gi6 tq gdn dring ctra a vio bidu th(rc M ta
duoc hai gi6t4 gdn dring c:0;a M
Bni 7 CS Thay gi6 tri ctta x vho da thfc P(x)
ta du-o.c 5 phuong trinh bAc nhdt ddi vdi c6c.dn
a, b, c, d,-e Dirrg phuong ph6p cdng dai sd ta
duoc ba phucrng tiinfr U4c nhdt ddi vdi cdc dn a,
b, c, mOt phuong trinh bAc nhdt ddi vdi 4 d,n a,
b, c, d vi mOt phuong trinh bAc nhdt ddi vdi cri -5
dn Gi6Li hO ba phucnrg trinh bAc nhdt ddr v6i cdc
dn a, b, c ta dudc cdc gi6, tri cfra a, b, c Sau dti
tinh duoc gi6 tri cfia d vh e tt hai phuong trinh
cdn lai
Thay c6c gi6 tri tim duoc c:&;a a, b, c, d, e vdtt
da thrlc P(r) rdi ding phuong phAp lap ta tinh
duoc gdn dfng hai nghiOm cira da thrlc d6
Bii 8 CS DC dang tinh du-o.c CD Goi F li
didm <l6i xrlng cira C qua O Tt su ddng dang
cira hai tam gi6c wdng COD vd CEF ta tinh
c\toc CE Goi H ld chAn ductng ru6ng g6c ha tit
D xu6ng CE vir M lh trung clidm cira OB Tuquan h€ gita dudng cao vd canh cria tam gi6c
CDM ta tinh duoc DH th d6 tinh duoc di€n
tich tam giSc CDE Vl dA biet CD, DH, L'E n€n
iltoc g6c CDE
= !ro^,, CDE *88" 12'36
Bei 9 CS Tf gi6c ABCD vUa nOi tidp duong
trbn vta ngoai ti€p rludng trbn n0n di€n tich c&a
n6 bang cin bAc hai cria tich bdn canh vh cflngbang tich cira b6n kinh duitng trdn nOi ti6p vtli
nira chu vi Tt d6 tfnh duoc b6n kinh dutrng trdn
n6i ti6p
G6c fuCD li g6c nh6 nhdt cfra tr1 gi6c ABCD
nOn n6 ph6i ln g6c nhon Tam gi6c ACD c6 g6c
Trang 15bang die.n tich trl gi6c ABCD suy ra s6 do cta
g6c ADC vd sd clo cira g6c l6n nhdt fre .
Tam'gi5c ACD c6 hai canh AD, CD viL g6c
ADC d6, bi6t n€n ta tinh duo c canh AC Bdn
kinh dudng trdn ngoai ti€p duoc tinh tt c6ng
AC
thtlc R = -:=, .
2sin ADC
n < 10 Tinh ddn ddn gid tri cfia b, ra sE duoc
tdng 10 sd hang ddu ciia ddy sd b,l?t 7120643
Tt d6 suy ra gi6 tri cfia S1e.
1124643
S,o =
104976
CAC LOP TRUNG HoC PHO THONG
z.,fi > o nen o < / < ? *, d6 bicn ddi A
4thhnh m6t him s6 cria r Tinh dao him cira A
theo , Tinh gdn dring nghi€m cria dao hhm.
Thay nghiem d6 vlo bidu thrlc cira him sd ta
duo c gi6 tri gdn dring cdn rim
in A = 5963,4176
Bii 7 PT Gqi C vd D ld c6c giao didm cfia
hai dubtng trdn d6 chb, x li sd do g6c CAD.Tinh
diOn tich phdn chung ctra hai hinh trdn vi di6n
tich hinh trdn tdm B theo x vd AB Khi so s6nh
hai diQn tich d6 ta duo c m6t phuong trinh cria x.
Gi6i phuong trinh d6 bang phucrng ph6p lap ta
duo c gi6 tri gdn dring cira x Sau d6 ta tinh duoc
AC theo AB rdi suy ra ti s6 di€,n t(ch cira hai
hinh trdn
x 1.34265167
Bni 8 PT Tt quan h€ gitta c6c g6c ctra tr1
gi6c ABCD ta tinh duoc c6c g6c d6 Tiep d6
tinh duo c di€n tich tam gi6c ABD, canh BD, g6c
ABD, g6c CBD, diln tich tam gi6c BCD, diAn
tich tf gi6c ABCD
bn= 312.2s a,= 17006172 a,
{hl b,,., = ! b,,.,* ! r,,,* o,
3 "'' 2
x 25,10056702 mBni 9 PT Dar
Bni 10 PT TrU tlng vd hai phuong trinh dh
cho ta duoc phuong trinh
[21
(x-y)l L (x+/Xx" +y')- xv) _ l=0.
Trong trudng hqp x - ) = 0, thay y = x vdo
phuong trinh xa + xy - 2 = sta duoc phuong
v
trinh bAc 5 ddi vdi x Giii phuong trlnh nhy
bang phucrng phep lap ta cluoc ba nghiCm gdn
dring ddu ti0n ctra h€ phuong trinh d5 cho.
,)
Trong trudng hqp (, + y)(x'+ y') - - - 0,
xy
khi dat S = x * y'td P = ry thi phuong trinh d6
tr& thinh phuong trinh cta hai dn S, P CQng
m6t phuong trinh cria hai dn S, P Ti haiphuong trinh d6 ta tinh duo c S, P Sau d6 ta timduoc hai nghi€m gdn dring kh6c cira h€ phucrng
trinh d6 cho nhd giAi hC phuong trinh
vdi moi n eN* vd b,, ld sd tr,t nhicn vdi moi
n a 15 T(nh ddn ddn gi6 tri ctra D, ta sE duoc
hs= 884020909 Til d6 suy ra giftt' cliua arr.
l*r 1,733381464
ly5 = 1,604483557
13
Trang 16cAc rop rHCS
Bli T1i333 (Ldp 6)" Vict c6c phAn sd vdi trl
sd, miu sd duong theo thrl tu nhu sau :
1' l' 2'1 ' 3' 7'2'3' 1"'' l' 2
2l
, , sao cho trong ddy khdng c6 phAn
Bni T61333 Cho tam gi6c ABC vdi truc tAm
H (H l<hdc A, B, C) vd M ldtrung didm crta BC.
Dudng thing di qua H ru6ng g6c vdi HM
c6t dudng thing AB & E vI cat clulng thing AC
& F Chung minh rang tam giiic MEF cdn vdi
,JAy EF
VI QUOC DUNG
(G/ DHSPThdi Nguyen)Bni T7l333 Ldy didm M nim trong hinh chf
nhat ABCD cho tru<1c Ke CE L BM tai E, ke
DF L AM tai F Goi N lh giao didm cua CE vir
DF Tim quf tich trung didm cia MN khi didm
M di chuydn bOn trong hinh_chff nhat ABCD
NGUYEN XUAN HUNG
(GV THPT Lam Son,Thanh Hl.ta)
Bni T8/333 Chung minh r[ng vtli bdt ki sfi
nguyen duong n thi hieu ,, = (t t+ll- t#t
I't", Ltl, L
l) sd ngul'On chdn, trong d6 ki hi€u [x] chi phlin
nguyOn c&a x.
TNAN NeV DUNG
(GY khoaTodn, DHKHTN TP Hd Chi Minh)Bii T9l333 Giai he phucrng trinh
Bni T10/333 Chrrng minh ring
H6i trong ddy rr€n phan sd 209 6** 6 t'i
tri thri bao nhieu ?
"u, THE HUNG
(GY khoaTodn, DHSPThdi Nguydn)
Bni T2l333 (Ldp 7) Cho tam gi6c ABC vdi
dudng cao AH Ggi M vi N ldn luot li ch0n
duong rnr6ng g6c ha tt H ddn AB vd AC Chfng
minh rang n€u BM = CN thi tam gi6c ABC ca,n
v6i d'ly BC.
PHAM HOANG HA
(W THPT chuyin ngri DHNN Hd Nai)
Bni T3/333 Tim moi clp sd nguyOn duong x,
-4 -)
TA HOANG THONG
(tr Todn 2001, DHKHTN TP Hd Chi Minh)
Bii T4l333 Giii phuong trinh
l6xo +s = 6.f;r1,
HOANG HAI DI.XJNG
(&' THC S C hu Manh Trinh, V dn Giang,
Hung Y€n)
Bni T5/333 Chung minh bdt ding thrlc sau
trong d6 a, b, c ld c6c s6 thuc duong
PHAM VAN THUAN
ba g6c cira mdt tam gi6c
QUACH VAN GIANG
(6/ THPT LtrtngVdnTuy, Ninh Binh)Bni T12l333 Cho trl diOn ABCD nOi ti€p m6t
mat cdu tam O vi goi G II trong tAm cfra trldi0n Ldy didm M nim hOn trong hoAc tr€n m61
2F*=('4) Dc.u
Ki nay