HOI - SO NAY ry ftAcd, &k4& b4d
f .(5.04). Trong mOJ cudn sdch c5 mof bdi tip
nhu sau :
"Cho c6c sd 2141,1345,4620,234.Ttong c6c
sđ5:
c) Sd ndo chia h6t cho ci 2 vi 5 ?
d) Sd nio kh6ng chia hdt cho ci 2 vd 5" .
C6 hai f kidn nhu sau tri ldi cho cAu d :
l) 2l4l v\ 2141 kh6ng chia hdt cho 2 vi
khOng chia hdt cho 5.
2) Cau d) lh phu dinh cira cAu c) (d6p r{n ciu c) lh 4620) nOn dr{p 5n c0u d) lil 2141, 234,
1.145.
f tien nho lh dring ?
(DdnS Duy C, Binh Dinh)
2. (5.04). Tren thd gidi c5 rdt nhi6u bii tho,
cAu vdn girip tri nh6 v€ cdc con sd bidu thi sd n.
(So chfr cili cira m6i tt trong c0u vin ring vdi mot chfr sd trong ci{ch vidt s6 n theo thrl tu).
Ban c6 tim duọ c nhfrng c6u nho bang tidng
nu6c ngoli vi tidng Vi6t girip nhd s6 n ?
(MOt dbc gid qudn Ddtg Da, Hd Nl.i)
Ti6n day ld tidn kim loai hinh trdn, det, đng chAt, bdng nhaụ
Bii to6n df,t ra li : Lim sao s6p x6p nhfrng
đng ti6n nam canh nhau (tr€n mat phing) đ
duoc nhidu đng ti6n nhat tren m6t dicn tich cho trudc. Cau tri ldi li x6p 6 đng titin xung quanh m6t đng tidn.
SU m6 rQng tg nhiOn cria bii to6n ftcn trong kh6ng gian li xdp ci{c hinh cdu sao cho thd tfch trdng gifra chring ld it nhdt. Ddi vdi hinh cdu
trong khOng gian 3 chi6u thi x€p 12 hinh cdụ C4c ban xem th6m bii Ph6ng dor{n Kepler vi đ 6n Flyspeck trong THTT s6 321 (3l2OO4).
Ddi v6i khOng gian c6 sd chidu l6n hon thi bni to6n chua c6 d6p sd, trt kh6ng gian 8 chidu vi
24 chida,vdi c6ch s6p xdp rdt "ki qu6i".
Hai nhiL toi4n hoc Henry Cohn 6 Dai hoc
Washington vh Noam Elkies & Dai hoc Harward
vla cii tidn d6ng kd c6ch d6nh gi6 vi6c x6p đng tidn cho trudng hqp kh6ng gian 8 chidu vI
24 chidụ
Ddi vdi kh6ng gian 8 chidu đ cii tidn d6n do
chinh x6c 10-6, cdn d6i v6i kh0ng gian 24 chiiiu dOn 10j.
PHAN THANH QUANG
(Theo Recherche sd 1112003)
TU NGAY Ị5.2A04, PI{AT UANU SGK L6P 3 M6IrU NcẠv 1s.s.2004, pHAr uANH scK LOp 8 MoI rU NcẠv 1s.s.2004, pHAr uANH scK LOp 8 MoI
B6t đu til ngiy 1.5.2004 sE ph6t hlnh bO sr{ch gi6o khoa l6p 3 mdi gdm 6 cudn : Ti6ng Vict 3 -
mp l,Tidng Vict 3 -t4p2,To6n 3,Tunhien-xdhOi 3,Tapvidt3 -tap l,TQpvidt 3 -mp2. Gi6bo
srich l6p 3 ln 38.700 d6ng.
B6t đu tD ngiy I5.5.2OO4 sE ph6t hanh bo s6ch gi6o khoa ldp 8 mdi gdm 13 cuOn : Ngt van 8 -
rap 1, Ngf v6n 8 - mp 2, Dia li 8, Lich sft 8, Gir{o duc c6ng dan 8, Am nhac vi Ivfi thuat 8, To6.n 8 - rap 1, Tofn 8 - 4p 2,Vqt lf 8, Sinh hqc 8, COng ngh6 8, H6a hgc 8, Tidng nu6c ngoii (Tidng Anh 8. Tidng Nga 8, Tidng Ph6p 8, Tidng Trung Qudc 8).
Dd phuc ru nam hoc 2OO4-2005 NXB Gi6o duc dim bio dr{p tmg đy dri, đng bO (siich hoc sinh
vi s6ch gi6o vi6n), kip thdi trudc nghy khai giing.
(Phdng QIXB' NXBGD)
Gidi đp so 316
rrd cac vr6c rHoI GIAN
Cic cOt mdc thdi gian 1i0n quan đn Bp chi
To6n hoc vi Tutli tr6 đ nOu li :
1964 : Ra sd đu tien 15.10.1964. Tru s6 tai
39 Trin Hung Dao, Hh NOi thuOc Uy ban Khoa
hoc vir Ki thuAt Nhd nut1c.
1968 : Ra hai thr4ng 1 kị
1970 : Td chrlc cu6c thi giii to6n dac bi€t
chdo mirng c6c ngly ki nicm l6n.
tr975 : Truc thu6c Vi6n Khoa hoc Vi€t Nam.
Tru s& dat b6o 70 Trdn Hrmg Dao, HA Noị PhAt hlnh tron toln qu6c. 56 luong 15000 bin / kị
1984 : Duoc tlng FIuAn chuong Lao d6ng
hang Nhị I
1985 : D<ii trinh biy mingset
1992: Truc thu6c Bo Gir{o duc v} Dio tao, do
NXB Gir{o duc quin lị Tru s& 8i Trdn Hrrng Dao, Hi Noị C6 dai diOn tai mitin Nam.
1993 : Chuydn thinh tap chi, tr& lai ra hdng thdng. Dtii mangset.
1994 : Txng lOn 2A tang, c5 biạ Tru so 45B
F{ing Chudi, He Noị
1995 : Duoc tlng Hu6n chuong Lao d6ng hang Nhi ldn thrl haị
1997 : Tru s0 81 Trdn Hmg Dao, HIL Noị Ra Tuydn tap 30 nam mp chi To6n hoc vi Tudi tr6.
1998 : Tang len 28 trang. Dtii mangset. Tru s6 25 Hin Thuy6n, He NOị
2000 : Tru s& 1878 Giing V6, HA Noị Ra
phu truong Tor4n Tudi tho, sau niy tr{ch riOng
thlnh Tap ch(.
20AZ : NhAn bang khen cria Thtr tu6ng
Chinh phir.
2003 : Ra Tuydn tAp 5 nam tap chi To6n hoc
v) Trrrli tr6'
BNH
Qkm ridp trang 8)
&i trf, todn hoe
Kdt quti&rli; TiN{ NGAY SINF{(THTT'sd 321, thdng 3 nam 2004) (THTT'sd 321, thdng 3 nam 2004)
(Dua theo ban Nguydrt Qudc Hing,4D, TH
TAn D6n. Vier Tri, Phf Tho)
Gia su a, b ldn lucrt lh nghy sinh, thdng sinh cira cO g6i (1 < a <31, I <b < l2). Theo bdi ra
ta c6 phuong trinh : l2a + 3lb = 244 (l) hay
3lb = 4(61 - 3a), suy ra 3lb i 4. Nhung 31 l) sd 16 nen b i 4. Tir (1) ta thdy 3lb <244 nan
b < 8. Vay b = 4virdo d5 a = i0.
Sinh nhAt crta c6 gdi iiy ld ngdy 10 thdng 4.
Bii niy duọ c rdt nhitiu ban trOn tohn qudc hutrng rmg (c6 hon 700 bhi gui vti tda soan). Tdt
cir cdc ban đu cho kdt qui dring. Mot sd ban
cbn giii bang tho kh6 vui, khr{ h5m hinh. C6c ban it tu<ji hon ci tham gia giii dring vd gon bhi to6n trOn duoc nhAn tang phdm ki niy : Nguydt Dfic Minh,7H" TTICS Lc Quf Don, Cdu Gidỵ
Hodng Mai Khanh, 8D, trudng He NOi -
Arnsterdam, Duong Manh Tudn, 8A, THCS
Giing V6, Ba Dinh, I{A NQi ; NSd Thfiy Hdng,
4D, TH Nguy6n Vidt Xuin, Hi D6ng, Hi TAy ;
Nguydn Manh Drtc,9Al, TTICS Chu Van An, Thanh Hn, HAi Duorrg I Trdn Thi Qu)nh Mai,
7D, THCS Dang Thai Mai, Tp. Vinh, Ngh€ An.
HONG QUANG
eriu rnfc cHfA G0NG mor eutr su
Bqn hdy dirng dring 10 chfr sd gidng nhau (tri I đn 9) v6i ci{c ph6p tinh c6ng, trir, nhAn, chia,
lfry thta vd đu ngoac ndu cdn đ vidt bidu thrlc
c6 gid tri li 2004.
Ching han vdi chfr sd 1 ta vi6t duọ c
(1+1)t(11-1;t*t*t + I + ll =2004 NGUYEN DTJC TRLONG (W THCS DaTdt,Gia Ldm, HA Ndi) i ! l l: 6
W*Pa Mlt 2,22khi4c theo BDI Na-so-bit :
! ,* ! ,--!-r1, suy ra ur--) ,.
ý ủ
^2 rr2 i * y- z 1at b)tDang thric xiy ra khi vi chi khi x = ! =:. T5m Dang thric xiy ra khi vi chi khi x = ! =:. T5m lai gi6 tri nhc nhat cfra I,I li -l (a+b)2^ .
C{c ban sau c(r dr{p 6n tOt, giri bii vd TS s6rn
hon ci : Nguydn Huy KhOi,7D, THCS Dio Vi0n,
Qu6 V5, Bdc Ninh ; Nguv€n ','dn Ngoc, 9A, THCIS Ldng COng, LQp Th4ch, Vinh Fhric :TdnE
VdnVuong,gc, TI{CS Cao XuAn Huy, Dien Ch&u, Ngh€ An ; Nguydn Iy, i0CT, THPI- Qudc hnc
Hu6 ltud ; \'u 7'h! Thu liurmg,62, THCS Cam Nghia, Cam R"anh, Kh6nh Hdạ
r
NCTC}C HIEN
f sex ruf xAo r
Mot ban d5 thuc hren loi gi6i bii torin :
op
Gidi dap bdi :
PHAI CHANG ... LA DUNG ?Trorrg l0i giai ban d6 da sri dung khr{ nhi6u Trorrg l0i giai ban d6 da sri dung khr{ nhi6u
bát ding thrl'c (tsDT) nhrrng chi xdt dau ding
thuc xiy ra & BDT
't1a
.xvz3
J ---1--1-- 5 --
1 1 1 1 ) 1-^
.v- I z' x'+z' ý +x- z
rni kh6ng x6t đu ding rhrlc xhy ra it c:6c
BDT cbn laị lheo cdch giii d5 ding thfc
1
M- . ; xityrachikhix=y=z=a=b.
2(á+b' )
Nhrmg thco gi6 thidr a, & lI hai s6 duong tily !, nenv6i a*bth\u,
--j_--.
2lá+b')
C:5 thd giii lai bii toi{n nhu sau :
'fa c6 : n* rm (Jr'**r": i7;A nhu san : Tac6: 11* f.,frz *n'*3 vÍ? -,*i} *>-\ l x2+3x+3-x2 *x-l -' ."3r"3 * Ut 1-.r1t (ay+bz)(az+by) < {ay+bz+az+by)Z ị+ 1a+b12 71,2 +221 2 x _> {at+bz)(az+by} 'Iuong tu cfrng c6 : 2 y__> (az+bx){ax+bz) 2 A grf_g:* 4 suy ra : (a+b\2 (yz *r2) 2)'2 1a+bf yqx?' rz21 222 2x+2 'k'$rl3* J"'Z *r*1 *./;2 *,; -2), ,) )^ x- +ir+3 I ,tl2 \ ,) L-,í x 2x2 = lim ,',_>a = lim r-)6 =**[ l7-'.3,.3 "l- ^ + \,, 1 .,L L-r-' x = lim f-)6
Y Ư, thdnio ? Cd trao Odl gi th€rn r€ ldi giii
tron kirong'?
Hd colrc D{ING
(A/ TI-(ff chuy4nTrdn lltrng l)no, PhanThi0i, BinhThudn) - 1" {ex+by)(ay+bx) @+b)2 (*2 -ry2) Dod6: ^i222'\ iLd> L | ' -, ) *-'a-i (a+b)2.y2 *r2 ' *2 *r2 ' ,'ný ) 27
Todn hgc vd Tudi trd
Mathemafics and Youth
NAM THU 4I
sd 323 (5-2004) Toa soan : 1878, Phd Gidng Vó HA Noi
' DT - Fax:04.5144272Em ai t : toa n h octt@Ya h oó com Em ai t : toa n h octt@Ya h oó com
@@ @ @ @ @
Thri tt ban mu6n ndi
Dd ra ki nhY - Problems in This Issue
Tlt 323, ..., T8l 323 ; LL, L2l 323 -
Cuoc thi giii toin ki niem 40 nam THTT GiAi bhi ki trudc - Solutions to Previous Problems. Giii c6c bdi cria sd grg' X h6i ? Y,Ztrhldi
Ban c6 biét - Do You know ?
Pian Thanh Quang - Bdi to6n xdP tidn kim loai
o
@
@ ca" hc bo - Math clutr
GiAi tri toin hoc - Math Recreation @ Sui lim & dAu - Wherés the Mistake ?
Bia2 : Tnho4t dQng ciia NXB Gi6o duc
-
gi"1' . cioi irtieu "oii todn hoi cao cdp -
Introduction to Higher Mathematics
DdmVdn Nhi - Chu6i vh fng dung dti nghiCn cfu dIY sd
Bia4 : Giii thuong L0 Van Thicm 2003
DON DQC THTT Sd :Z+ (612004)
r tfinh lQp phuong magic hohn toln bac nh6 nhdt đ duọ c xiy drrng- . Ctlmg minh dudng thing di qua mot didm c6 dinh'
. Dudn; cong HypoxiclOiisteiner vh ho dubng thing Simson cira m6t tam gi6c' o Hu6nB din giii Di! tu on thi DH, CD sd 4, sd 5'
. Chu6ivi rlng dung đ nghiF,n criu day sd (ti6p)
Mdi c6c ban dit mua THTT tai cic ca s& Buu di€n !
THTT
TRONG SO NNYO Dhnh cho Trung hoc Co s& - For Lower O Dhnh cho Trung hoc Co s& - For Lower
SecondarY Schools
Ta Pham Hdi -C6ch tim c5c chfr sd cria
rndl so
Hrr6ng dnn giii đ thi tuydn sinh vio 16p 16 tnidng THPT Chu Vin An &
rutPt Ha N6i - Amsterdam nim hoc2003 - 2004 2003 - 2004
Di6n dirn day hoc to6n - Math Teaching Forum
i;';;s Cbng Thd.nh - Gi6i thi6u PhAn
hinh iioc kh6ng gian trong s6ch giSo
khoa ToSn 8 (Ti6P theo sd tnr6c)
Chudn bi thi vio Dai hgc - University
Entrance PreParation
Nguydn Anh Dung - Hrr6ng dln giii d6
trr 6n thi sd 3
Phqm Hing - D6 tU on thi sd 5
NgO ViQt Trung - Giai thrt8ng Abel ndm 2004
Phuong ph6p giii toin - Math Problem Solving
ia CEi Dqt -D6nh gi6 c4n ad giii mot sd phtrong trinh lrrong gi6c
/f-],
(E4.
F
\)i rh
TOAI\ HOC CAO CAP
CffiU#E va URIfi [}Ultlffi
đ R]EI*I&RI CTJU IIAY STi
DAM VAN NHI
Bdi b6o nly nham gi6i thiOu khrii niOm chu6i
vd mOt so fng dung vdo nghiOn cfu d6y sọ
1. Chu6i vi c:ic ph6p tofn :
X6t hai da thfic cua bidn x
f(r) = tt,, l a1x+ ... + a,,x" = f,,.r'
i-0 !{.i)= b,,+ byt+... + b,,.\'" = f,O,r'
i-o
trong d6 a,, b,ld ciic h€ so thuc (khOng nhdt
thidt kh6c 0).
Khi d6 ta x6c dinh ciic ph6p todn cua da thfc
nhu sau :
fl.r) = g(r) <> ẹ = b; vat moi z = 0. 1,..., ir ;
ll
crl(x)- Z{o",)r' ; cr e R.i-0 i-0
cluoc tring flx) = Lo,*' . Mai tdng nht vdy
i=0
cltcrc goi lii mdt clru6i luy tlu:ra litth thitc.
X6t hai chu6i :
.u cc
flr) = Zo,r' vi g(x) = Zbix' .
i-0i=0
Ta dinh nghia ci4c ph6p to6n cfia cdc chu6i
nlLy tuong tu nhu cta da thrlc :
f(r) = g(r) neu ai= biv6i moi I > 0 ;
TS Dirm Van Nhi
sirilt ngdy lB tlfitrg
3 rrum 1952 tqi
Ki€'n XLtortg, Tlfii Binlt. Sau khi rc't
ngli€p kh6'i PTchuy€n todn DHSP chuy€n todn DHSP Hd ^16i, hoc DH Tdng hctp Todn tai Halle CHDC D{rc" Tn 1976 dqy t(ti DHSP Hd N6i 2. Sau ct6 chuyitt v€'
day rai CDSP Tlfii
Biilr vd tray gicing
day tai khoaTodtt - Tin DHSP Hd NOị Bao t'e
Itrdn dtrTi€h sl nam 2001 taiVi€rtTodrt lrcc'. Da
c6 nhidu bdi bdo dang ó trong nltoc vci tttrctt' rtgoiiị Vdn y€u rhich gidi vd scirtg tric cdc brii
todn so cd'p tctp chi THTT. 6 f(x) + g(-r) = \(ai+b;)r' ; r=0 a crJ'G) =l(aai)x' ; cr e R. r=0
Hon n[a, m6i hIm sol(x) c6 dao h]m moi cap
tai r = a thi ta c6 su bidu di6n sau vdi dao hhm cdp i cfiaflx) ki hieu ta l(1)(-r) . , ,,,r lo) . i (rt= f L (.r-d) r4;l r0 l^ vh ' - l-.r+.rr +... v6i hl < l. 1-x
2. Ong dung chu6i vio nghien crlu mOt so day so
a) Chring ta bit dau bang viec xet dly
Fibonacci duoc cho nhu sau i a,, = 0, a1 - l. At*2= At,+l * A,,,Ytt) 0.
Tim cOng thfc tudng minh cho a,,. X6t chu5i/( x) = au * a 1x * agz + ...
Khi đ xf(x) = a,ý * af2 + a2x3 + ...
,2 f(r) - a,rY2 + o1r3 * a2x4 + ... Trir theo tirng vd
f(t;_ xflr)-*2111=
= a,, * (a1 - arr)x + (az - a1 - n,r)r2 + ... +
* (an+2_ att+l_ art)r"*2 + ... = 1.x f rt.str)= f[ ; ,,,,,,'lr' i-0(7"7 ' ) ft,,,t + s(.i 1 = ft u; - b, l.r' : i-0 ,,( \ Il | ,,,b,, l, i =0(t,+q=i ) dung, cho rr tidn
/(r).9(r) =
Thu h)nh ra v0 tAn, ta nhAn
\,!
?
Fioi đng GiAi thuong L0 Van Thi€m 2003 da hop vio đu th6ng 412004 vd nhat tri quydt dinh trao giii thuong L€ Van ThiOm 2003 cho ci{c gi6o viOn vd hoc srnh sau dAy:
1. GIAO VIEN.
l. Nguyin Ludng, sinh nam 1959, girio viOn t.rudng THPT ChuyOn Nguy0n Du, BuOn Ma Thuot, Dak Lak.
'fhinh tich: Tir 1981 day CDSP, tit 1995 day
tai ciic trudng THPT Buon Ma ThuOt. Dd dho tao nhidu hoc sinh gi6i: tir 1991 đn 2003 cd 2
giii Nhat, 2 giii Nhi, 5 giAi Ba vh 3 giii Khuydn khich tai ciic ki thi chon hoc sinh gioi to6n
THPT toirn qu6c. Dl duoc t[ng nhidu bang khen
cira Ujr ban nhAn dAn tinh Dak Lak vd th)nh tie h giang day xuat :ac.
2. LA Sdng, sinh nf,m 1952, gi6o viOn trudng
THPT Chuyen L0 Quf DOn, Nha Trang, Khdnh
Hoh.
Thlnh tfch: 2-5 nam tham gia giang day v) boi
dLr6ng hoc sinh gioi cta tinh, c6 3 hoc sinh doat
giii Olympic todn Qudc te (2 HCB, 1 HCD), nhidu hoc sinh doat giAi tai cdc ki thr chon HS
gioi to6n TIIPT tolLn qu6c. Lh t6c giA 2 cuon s6ch bdi duotrg hoc sinh gioị Duoc tang Hu),chucrng
vi su nghiOp gido duc r,) nhi6u bAng khen cila So GD-DT vd Ui, ban nhAn dAn tinh Kh6nh Hoir.
2. I-IOC SINH :
l. LA Hing ViAt Btio, ldp 11 A, Khdi PTchuyOn Todn-Tin DH KHTN, DHQC} Hd NOị chuyOn Todn-Tin DH KHTN, DHQC} Hd NOị
Thhnh tich: Giai Ba cuOc thi chon hoc sinh
gi6i torin THPT toln qudc 2003, tluy chucrrg
Vdng (didm tuy6t doi 42112) tai Ol1,mpic toiin
Qudc td Tokyo 2003.
2. l{guyAn Trong Ctinh, 16p 12, Khdi PTchuyOn To6n-Tin DHSP HA NOị chuyOn To6n-Tin DHSP HA NOị
Th)nh tfch: Giai Ba cuOc thi chon hoc sinh
gi6i torin THPT todn qu6c 2003, IIuy chuong
Vdng (ditlm tuy€t doi 12142) tai Oil,mpic todn Qu6c te Tokyo 2003.
3. Nguydn Dang Hop,16p 11,'frLrdng THP|
Le H6ng Phong, Nam Dinh.
Thlnh tich: giai Ba cuOc thi chon hoc sintr
gioi todn THm toln quoc 2003, Hu-v chucrng
Bac tai Olympic todn QLroc te Tokyo 2003. Hohn cinh gia dinh c6 nhi6u khd khan.
4. IVguyan Minh Hrii,l6p 12 'frudng THPf
Chuy0n, Phan Thidt, Binh Thuan.
Thhnh tich: Huy chuong Bac Ol1"mpic toiin 3011 cdc nim 2001 , 2002. Cidi Nhat cuOc tl.ti chon hoc sinh gioi todn TIIPf todn clLroc 2003. Hohn cinh gia d)nh gAp nhidu kh6 khan.
SIAI TtrIIUIiIS XE UAIII TffiIEtrX,I g$il3
'l-lúr
-l ti trdi stutg .
Nlli, girio Ngirrcir
[trdtt g, N gut'itt Dcitt g
Ílr4t, L€ HirtgYiir Bcio, N grtt,irt Tt'ottg
Ccittlt, ciotg c'cic ttltit
toitt hctr'. ISSN r 0866-8035 Chi so : 12884 Mn sd : 8BT25Mzl Gi6 r 3400 tl6ng Ba nghin bdn trlm tlOng Cheban lai Toa soan
In tai Cdng ti CP in Di0n Hdng, 1B7R phd Giing Vo In xong r,I nOp luu chidu thiing 5 narn 2004