aa). @2, 9), @y a6)
kh6ng th6 ddu le. hay
c6 it nhdt mdr cip mi
ci hai sO ddu chdn.
Ndu c6 hai trong ba c6p n6i trOn li cep chdn, gii
s& d6 Ih : (ar, aa); (az, a) thi ar, a6, ar, ar, arIE.
Y\ at+au+a., = 15, vi au+a, ld, so'chdn n€n a., 16
(v0 10. VAy chi c6 dring m6t cip trong 3 c4p (ar,
ao), (a2, a5), (a3, au) ld c6p.chdn. GiA s& d5 li : (a,,
a),litc d5 a2, a3, a5, a6ld cdc sd 16. Mat khdc: a, + oz + as = 15, a3 ! aq +ds = 15 * as, aeli cdc sd
chdn + a, li sO 16. Cfrng til gii thiOt bii toiin, ta
c6 : (a2+a\+q) + (au+ar+ar) = 30
+ (ar+ar+ar+au+ar) + a7 = 30
p+ (1+3+5+7+9) + at = 30 * a7 = 5.
Ttt (1) ) at+a4 = 10 = aB + as. Vt a1, a4, as, ae
li c6c sd chin, nghia ld ap ao, as, as € 12, 4,6,81
nOn c6 4 c6ch didn co brin nhu tr€n, (ndu ta quay quanh Am tam gi6c ara*a, cdc g6c ct = 120o, F =
240" thi duo. c tat ch24 cdchdidn s6).
NhAn x6t. Bbi to6n nly duo. c rat nhidu ban tham
gia gi6i. Da sd chi dua ra mOt ddn hai ci{ch didn s0
mi kh6ng phdn tich c6ch didn cira minh. Cric ban
c6 ldi gi6i tdt hon ci ld : Thdn Anh Dt:rc, 8H,
THCS Le Quf DOn, Hi NQi ; Irrrn Manh Cudng,
98, THPI L0 Hdng Phong, Phqn Thi Kim Hoa, 11A1, THFI chuy0n Nguy0n Tdt Thinh, YOn
BAi; Nguy,in Dtng, l2A| THIrI YOn Phong I,
YOn Phong, Bic Ninh ; Khdng Minh Thdo, llT,
TTIPT Nguy0n Trdi, Tp. Hii Duong, Hii Duong.
HONG QI]ANG
sd cuAN [rAY sd r,E :
Ban Thuong viet tat ch c6,c sd nguy6n tii I ddn
2002. Sau ddy thay hai s0 nio d6 bfli hieu c(ra
chring. Trong nhfrng sd m6i niy, ban A'y lai thay
hai sd nho d6 bing hi6u cfra chring. Crl thay nhu
vty mdi thi cudi cing chi cdn m6t s0. Satr I phtit ban c6 thd cho bi0t sO cudi cirng Thuong nh6n
duo. c ll sd chdn hay 16 khOng ?.
NGIIYEN VA}I THIEM (Hd Noi) I j :j: jt::i:t!aa:at:::::a : : i iiiririli)ji.I):,i Bg. +,tiii;::l'i:i.ill . ',?.ii:n'.i!"iiitliii,E.Ni: 23
TCI,;ln frse va* f,ud,g fn#
Ml atta e m ati *s amd Yoe*ffx
Nam thtr 39
sd'303 {s-2a02}
Tda soan : 7878, pnri Aiang Vo, Hd Nfli
DT-Fax:04'5144272
Email : toanhactt@Yahoo.corn
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TRONG SO NNY
Dinh cho Trung hoc co s6 - ForLower Secondary Schools Lower Secondary Schools
Nguydn Ngoc Khoa - Tim cltc tri m6t bidu thrirc bing nhi6'u c6ch
Dd thi tuydn sinh l6p 10 THPT chuY6n
To6n-Tin DHSP He NOi 2002
Tieng Anh qua c6c trhi to6n ' English through Math Problems
NgO ViQt Trung - Bei sd 57
Chudn bi thi vio Dai hgc - University Entrance Preparation
Nguydn Anh Dtrng - Nhfrng luu i' khi
elAi dd thi tuy6n sinh dai hoc ' cao ddng
kh6'i B n[m 2002 €hAn Dai s6', Giai tich) Tim hiiu sAu th6m to6n hoc so cdP '
Advanced ElementarY Mathematics Pham Qudic Phong - Phtiong Ph6P dat dn phu dd giii phttong trinh c6 hai
ph6p to6n ngrrdc nhau
#
@
Di ra ki niy - Problems in This Issue T1/303, ..., T10/303, Ll,L2l303
GiAi bni ki trudc ' Solutions to Previous Problems
Giii c6c bdi cria s6'Zgg
&
ffi @ Cau lac bd - Math Club
Sai ldm 6 dAu ? - Where's the Mistake ?
GiAi tri to6n hoc - Nlath Recreation
Bia 2 : Kdt qui ki thi chon hoc sinh gi6i qudc
gia mOn Todn loP 12 THP| ndm hoc
2001 -20a2
Bia 3 : To6n hoc mudn miu - Multifarious
Mathematics
Duong di cfia qudn I{Au
ripr ntue ToAN FIQC T-uoI TRfr 0 $Au ?
rxdN;;i; 8lTrrin FSmg Dao, t8?B GiangV6,25 Hln Thuy6n,23TrhngTidn, Z-i?TaySon.'. Dd Ndng; 15 Nguyen Chi Thanh, ... Tp. Hd Chi Mirch; 231 Nguyen V[n Cil.v] 240 Trdn Binh Trong, d. S,.". ia. f,leu srich ldn trong .i nudc. Ban c6 thd d4t mua ci nam hoac tutng quf & c6c tsrru
dien ila pfru*e Cric dai 1i s6ch brio mudn ph6t hhnh tap chi THTT xin li6n h0 vdi :
Trung tdm Pltdt hdnh sdch tlnm khdo, NXB Gido duc' 57 GidngVa,Iid Nr.i, ilT : A4.5141253 24
l 3 Hinh 5 ,iji,ffi i " {. '.:'l' ! :a-f, rEat rtrt,W g{ W,/t lutl4'trujil
r, Tr0n bhn cd vua kich thudc 8 x 8 6, m6i bu6c di
6 lir 6 nhy.sang.o khdc cfia.quAn HAu duoc bidu thi
= i :r' ,:. 'r' E rl boi mOt doan thing theo hing doc hoac hing ngang
hoac dudng ch6o cria ci{c 0 nhu o hinh l.
DANH CHO BAN DOC
1) H6i quAn HAu xuat phiit tir mor 6 g5c bln cd (h. 1) cdn di liOn
tidp it nhat lh bao nhi€u budc dd tuOt qua tdt cit c6.c 6 cira bdn cd (m6i o ft nhdt m6t ldn) vd trd vd vi tri ban ddu. HIy vE tdt cb cdc duong di
nhu thd mi ban tim duoc.
\ HAy pnai Uidu bhi to6n tren theo ngOn ngfr hinh hoc.
cidi ddp..niruu ruAo NHrCu cANH ruuAr a
Vdi m6i hinh H, (gdm n didm khOng thing hdng vd moi doan
thing kh6ng cit nhau noi tung cap di6'm iO ttrd ndi Aubc), ta goi hinh
bao cria H,,ld da gi6c tao thlnh tU cdc canh cia H,, sao cho moi dinh
vh moi canh cta H, ddu nam b6n trong hoac trcn bi0n clra hinh
bao nlLy.
MAnh dd 1, N €ir H,, c6 s6' carth nhi6u nhdt thi hinh bao cia n6 ld hinh tam gidc.
Gii sir trdi lai ta sE dmg duoc hinh c6 nhidu canh hcrn : ndu hinh bao li da gi6c l6m rhi vO them I
canh d phdn l6m, neu ld da girlc ldi nhidu hon 3 canh thi di chuydn tuong duong 7 hai canh kd nhau
vdo phia trong da giric dd trd thlnh da gi6c lom r6i v6 thOm 1 canh.
MAnh d0 2. 56'carth nhidu nhdlt cila H,,ld S(n) = 3n - 6.
DcthayS(3)=3,S(4)=6.Xdtrr>3.NOutdntaiS(n-l)=3(n-l)-6thitathemldidmvhoben
trong m6t tam giiic
ndo d6, didm niy ndi
vdi 3 dinh cta tam
gi6c d6 tao thlnh 3
canh mdi n6n tdn tai
hinh H,, c5 sd canh
ln3(n-1)-6+3=
3n - 6. V4y S(rr)
>3rr-6(1).
Gii sit tdn tai hinh H,, c6 sr5 canh nhidu nhdt ld S(ru). Khi bd
di 1 dinh nho d6, dinh d6 phAi ndi vdi it
nhdt 3 dinh (do yOu cdu kh6ng dd s5t canh
ndo) nOn S(n) < S(n - 1) + 3. LAp luAn
tudng tu c6 S(rt ) < S(n - 1) + 3.1 < S(n - 2)
+ 3.2 <... < S(3) + 3(n - 3) = 3n - 6 (2). TU (1) (2) suy ra S(n) = 3rt - 6. Ta c6
5(6) = 12.
Goi sd canh xudt phr{t tt dinh A ld d(A) rhi d(A) kh6ng ddi (bdt bidn) qua ph6p
tuong duong A. C0 tdt ch 5 hinh H, kh6ng Hinh I Hinh 3 E Hinh 4 CCG Hinh 6 tuongduongT.Otrintr2th\d(A)=d(B)=d(C)=d(E)=d.(G)=(t(m=4.Cdn0ci{chinh3,4,5,6 th\ d(A) = d(B) = 5, d(C) = d(E) = 4, d(G) = d(H) = 3.
Ci{c ban sau c6 ldi giii tdt duoc nhAn ting phdm :
l)Vdn Dd Phlic Anh,1116 THPT Nguy6n Dinh Chidu, Ben Tre.
D Bni V dn Dai , 1 2T, mCT, DHKHTN - DHQG Hn Ndi
3) Neuydn Bd Dai, x6m Dio, x6 Nghia Hda, Nghia Ddn, Nghd An.
PHI PHIN N I \ !r,l{ * Hirtlt 2
Hrrong d6n
lim biri tir
..,i.,1&fuu:4g i*a frV;a
ffir.*t
no B0E0ufr crlgll tg(
Tllfrll
roAN TRTJNc Hoc co sd
li - Hon - srNH TRTJNG Hoc co s0
Tin sdch Gid bia (d6ng)
LOP 8
To6n nAng cao Dai 8
MOt sd vdn dd phtit tridn Dai sd 8
Mgt so vdn dd phdt tri6n Hinh hoc 8
Oe hqc t6t Hinh hqc 8
Ve thern ydu td phL.r d6 giai Hinh 8
6.500r0.rJ00 r0.rJ00 1 1.000 21.000 9.000 LOP 9
Todn nAng cao Eai 9
Rdn luy6n ki nang giai toln H6a 9
Tdi li0u bdi dLrOng hoc sinh gioi Ilinh hoc 9 Trr lictr h,ii Jtr.rn3 lt.r. .tttlt.:ioi Der ii Q
56 ta;- Todn Li HtiaN{ot so phrit triin Dai sd 9 N{ot so phrit triin Dai sd 9
\4ot so phat tri6n Hinh iroc 9
Dd hoc tot Dai so 9
De hoc tot tlinh hoc 9
Chc bai toin di6n hinh Hinh hoc 9
Xrl dung lrrnlt hoe battl plttrutt,.: plrip tien di'
1 1.0009.000 9.000 12.,+00 15.000 .1.200 I 1.50t) L 3.,100 l 3.000 t5 000 10.200 7 tt00 .a : a Cs .H .*
Tin srich I Gia bia
1 (dons)Bdi tap Vat li nAng cao 7 | 5.700 Bdi tap Vat li nAng cao 7 | 5.700
Huong dAn vd 6n tip Sinh 8 | 5.000 Rdn ki ndng.giii toiin Fl6a 8 i 10 200
Tdi li€u b6i duOng Vat li 8 I 7.000
Bdi tQp VAt li ning cao 8 L 7.500
0n tdp Hoa o I Lq00
Biii iluo-ng H6a lroc trung hoc co so I l;.+ttg
Giri:3000dBa nghin d6ng :fi Ba nghin d6ng :fi ,s;*$.ult.l iiraii..l|'i. :!r.lail,i1i,..5:. . _:a:ll::i::,ii:!ri l i ".s I #i t,, i' ISSN:0866-8035 Chi sd :12884 Mn sd : 8BT05M2
Che ban tai Toa soan
In tai Nh2r may in DiOn Hbng, 187B pho Giang V0 In xong va nQp luu chidu thiing 9 ndm2002
ON TAP\. \. HINH HqC @ j:=l:r; ni! *a -.' -"omruc Il!
TAn sdch Gid bia (d6ns)
L6P 6
Phuong ph6p day hoc todn o truong THCS