f r-{ m
';!: ;f;CIl1ru
'BO GIAO DUC VA DAO TAO
GS
* sor roAN Hoc vIFT NAM
#stC
ut^
TAP CHi RA NGAY 15 }IANG TIIANG
* vfi nAr roAN so sANH PHAN sd
* cfctt t{}tiN MEr Df NG ro6N ouY ricH
* "?tQ" 2%/4 77ZiW'.l/i?tE'l/E D4?
* GIG sd nnusEY
* l{i THr oL€rutPt( Tonu ouOt rd
Do?tn Viil Nam
tai IMO - 96 : Tir
trAi sang phAi :
Phqm L6 Hitng,
PGS Phan Duc
Chinh, NgO Ddc
I uan,
Nguy6n
thity
xndc
Minh, Nguy)n
Thei He, Trinh
Thd Huynh, D5
Qu6c Anh, Ng6
Duc Duy.
ToAN Hoc vA TUOr rRE
MATHEMATICS AND YOUTH
MUC LUC
Trqng
Ddnh cho cdc ban Trung hoc Co s0
For Lower Secondary School Leuel Friends
Nguydn H{tu Bd.ng - Vd bai to6n so srinh phdn s6. 1
Gini bdi ki trrdc
Solutions of Problems in Preuious Issue
Cdc bdi c.iua s6 226. 2
B iographies of M athematicians.
Nhd.n ki ni€m 170 nam ngity ra diti
cila hinh hoc Lobasepski -
Nguydn Cdnh Todn - MQt quA tnlng ving vi dai. 8
Db ra ki ndy
Problem.s in this Issue
T11230 ... TtOl230, LU230,L21230 I
Hoc sinh tim tbi
Young Friends Search in Maths
Ngd Minh Nghia - S*y nghi v6 m6t bdi torin. 11
Ddnh cho cdc bun chudn bi thi bdo Dsi hoc
For College and Uniuersity Entrance
Exam Preparers.
Hb Quang Vinh - C6ch nhin mOt dang todn
qu! tich trong kh6ng gian.
Hoitng Chrlng - C6.c s6 Ramsey.
Pharu Dtc Chinh - Nguydn Khir Minh - Ki thi
Olempic To6n Qu6c td ldn thn 37.
Gidi tri todn hoc
Fun with Mothematics
GiAi ddp bdi : Vidt sd trong bin cd.
Tudn Dang - Tim ctla vD.o vd dtrdng di.
t2
13
16
Bia 4
Bia 4
o
a
Tdng bi6n fip :
NGUYEN CANH TOAN
Ph6 tdng bidn tdp :
NGO DAT TIJ
HOANG CHUNG
nOr oOruc arEN rAp :
Nguy6n CAnh Todn, Hoang
Chring, NgO Dat Trl, LO Khic
B&o, Nguy6n Huy Doan,
Nguy6n Vi6t Hai, Dinh Quang
HAo, Nguy6n XuAn Huy, Phan
Huy KhAi, Vri Thanh Khi6t, Lo
Hei Kh6i, Nguy6n Ven Mau,
Hoing L6 Minh, Nguy6n Khic
Minh, Trdn Van Nhung,
Nguy6n Dang Phdt, Phan
Thanh Quang, Ta Hdng
QuAng, Dang Hung Thing, Vfl
Dtrong ThUy, Trdn Thdnh
Trai, LO Bri Kh6nh Trinh, Ngd
Vi6t Trung, D4ng Quan Vi6n.
Tru sd tda soan :
458 HDrng Chudi, Hn NQi
23f Nguy6n Ven Crr, TP Hd Chi Minh
DT: 8213786 Bi€n tQp ud. tri s4 : VU KIM THIfY
DT: 8356111 Trinh bay; QU6C gbNC
T
a
\
\,@
\\w
@
W^
*@
kil
Mf
tu
tffiil
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w
\@
@
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w
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14
\€A
w
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H
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r@
Wf,
S6ch girio khoa To6n 6, tdp 2, d phdn 6n t{p vd tinh chdt cria b6n ph6p tinh
trong N vi trong Q* cd bdi to6n :
So sanh bang nhi6u phuong phrip khric nhau xom trong hai phan sd f ,'a
13.,
* thi phAn sd ndo I6n hon.
Ddy lamOt bAi toan don giAn nhrrlgchfa drrngnhi6u ydn-dd trongchrrongtrinh
Torin 6. Trong bii niy tOi xin trao ddi vdi cac ban vdi di63 nhrJ sau- ; Tlti6c hdt xin
ndu tcim tSt nLrrng c6ch so uinh phAn s6 (trong tap hSp Q*) quen thuOc ld :
1. Quy ddng mdu cac phAn sd de cho rdi so s6nh cAc tit v6i nhau.
Z.iiAt caclhan s6 di cho du6i dang cdc phdn sd cirng trl rdi so s6nh c6c
mdu vdi nhau.
3. So sdnh phAn s6 theo tinh chdt : ndu ad, < bc th\t < 3
4. so ssnh ti s6 c.ic ol?t;: $u."f,tf,,r:,i"d Y$ "nu, ,
5. Vidt c6c phdn sd drrdi dang s6 th{p ph5n rdi so s6nh cric sd th{p phnn
6. So s6nh sd nghlch dAo cria c:ic phdn sd, theo tinh chdt
Choa,b,c, d* o, ndu ! . 1 thi,9 >,
o.cDd
7.. DUa vdo' tinh 'chdt b6c cdu cria quan he thf tu : ndu
nL m c .a c
< -ud - ( -thi-<-
ttndjd
z
{
o^
5S
ztrt
5Z
I
P
z
a
b
\^
N
\
{'
$
$
$<
8. So srinh "phdn bir cria c6.e phdn sd d6i
a c a c..-a c
b,A. 1 vit 1 - b. I -Athi b > A
vdi don vi", theo tinh chdt : ndu
Ti6p theo xin n6u th6m vdi c6ch giAi khic :
9. Ta ctj tinh chdt (d6 chrlng minh)
a c .a a*c c
N6uU <rthi O. O+a.A
Ap dung vdo bii to5n tr6n ta cci :
53583
YiV. Vndn7. O. Z,
585138
Tn r. rsuyrai. 16. g
513
va,i< 16
10. Tt tinh chdt d6 n6u d c6ch 9 ta do ddng suy ra tinh ch,dt sau :
a c a, afl+c c
Nau , < ; thi i . U"* O. A (o ='1, 2, 3, ...)
Ap dlrng vdo bdi todn tr6n ta cri
5 3 5 5.2+3 3 .5 13
Yi V.7 n€n 7<-2.2+Z <I "uyru i. lA
MQt vii nhin x6t vd c6c crich giAi
a. Do bdn sd 5, 7, 13, 1"6 ddi mQt nguy6n td cirng n,hau n6n khi ap dr;ng c6c
c6ch gi6i 1,2,3,4 vdo bdi to6n tr6n ta d6u quy v6 so s6nh 5.16 vi 7.13.
b. Khi so s6nh cic phan sd ching hqr#"a # ;ff"eY*,ffi"u
1993 i 6p dung cddcdch 6, P li nhdt.
ffi , ro rdng ta n6n rip dung cdc'cdch 6, 7 , 8, tuong rlng ld ho1
c. Dd vAn dgng c6ch 10 trongbiti to6n ban ddu ta cdn vi6t i
L3=5.2+3; 16 =7.2+2
.Tuong ttl ta cflng so sdnh duoc mQt crich nhanh ggn hai phAn sd ching hqn
89 895
95"" g54
89 5 89 89.10+5 5 -.89 895
"tac6 *< 4ndn 95< gs. 10 +4'a dooo gs' gsa
. (xem tidP trang 10)
\t
N
Bni Tl I 225 : Tim sd add. trong hQ ddm thdp
phd.n th6a mdn :
1) Sd6A x 4 tQn citng bdng cd,
2) adAn -EZ x 4ld. sd chinh phuong.
Ldi giAi : Tr) thidt ta c6 :
' a6ccl -Uc x 4 = rr2 x 100 (n e 7j)
nx 100+6ex 10+d -frx4-n2 x 100=0
*1AO(n2 - 10o) = 666+d (1)
Do 0 < b, c, d ( 9 n6n 0 < 668+d < 610
(2).Kdt hqp (Z) vdi (1) ta c6 :
o<n?-10o<6 (3)
- Nduy2 = I ta c6 4 = 2 +{199 -7= 2x
6y',-L99-x2-?*
*x2+2r-lgd=o
o(r-13)(x+15)=g
+t; = 13 vi.r = -15 (1)
- Ndu yz =4 ta c6 16 = 2 + \[109=7 - 2x
*196 : 799.- 12 - 2x
+x2 +2x -3 = o
o(x-1)(x+3):0
e+x=1vdu=-3 Q)
Thay cac gi6 triy = ! t, ! = + 2 vdcdc gl6ti
c'&a x 6 (1) vh (2) vdo phrrong trinh ta thdy
phrrong trinh duoc th6a mdn. YQy cdc nghiQm
cria phrtong trinh ld : (x,y) = (L,2) , (L,-2\ , (-3,2),
(-3,-2), (13,1), (13,-1), (-15,1), (-15, -1)
Cd.ch2 ciaPhamVan Tidn, gA, ; THCS bdn
cdng, Ddm Doi, Minh HAi.
. 4y2 = 2 *rtlsg -7 -%
o4y2 =.2 + r[r00= @ +IY
Dd phrro.ng trinh cd nghiQm nguyOn thi
rfm=@ +-IY = t[2 . lF:(x *1Y =
= {F4W -1:@ +:IY
phAi ln sd chinh phuong. Khi dd phAi cti :
ho6c (r + 1)2 = 102 hoac (x + L)2 : 22 holc
(r+1;z =142
- Ndu (r + 1)2 = 102 + 4y2 = 2+ 10=+y2 = 3
phuong trinh kh6ng c<i nghiQm nguy6n
-N6u (x+l)2 =22+r* l. + 2+x = lvd.
r = -3 (1) ttdd c6:4yz = 2*14+y = +2(2)
-Ndu (t+ 1)2 = 142+x+ 1 = + 14+r = 13
vdff = -15 (3) ttddcd ,42:2*2+y: + 1 (4)
Thay c6c c6p (x, y),6 (1) vn Q),6 (3) vd (4)
vdo phuong trinh ta thdy phuong trinh drroc
th6am6n. VAyphrrongtrinh cci 6cep nghiOm
nguy6n ld
(L,-2), (7,2), (-3,-2), (-3,2), (13,-1), (13,1)
(-15,-1), (-15,1).
_NhAn x6t : 1. Da sd c6c b4n gi6i theo cdch
1. Giai theo c6ch 2 cdn cci c6c ban : LA.Hod.ng
Drtc Khd.nh, 9T, Chuydn Nguy6n Nghi6m, Drl"c
Phd, QuAng Ndai ; to rnZn"n ViQt", 9As'Qr6c
hgc, Quy Nhon. Binh Dinh.
_ 2. Ce9 ban cci ldi giai t6t la : Phsm. Hd.i
Trung,9T, NK Ti6n son ; Duong Quang KiAn,
?!rq*_ Trung Drtng, 9T, NK B6c Giang ;
NguydnViQt Qttng, NgOVd,n ftic, 9T.NK L4ng
Giang, Hn B6c. Deng ViQt Dung,92, Chuy6n
Vinharrdng ; Anh'H"img'DAng,"9.0,, Sa Dric -
ryU-ThS ; Dsng Thu Hubng, 8T, Chuy6n cdp
II, Phri T\o ; Vu Mqnh Cuimg, 8A; Biri Dd,ng
Quang,9T, Chuy6n cdp II, Tam DAo, Vinh Phri.
N
gu_ydn Thi N
gqc Anh, 9T,L6 Hdng Phong Thi
x6, Y6n Bdi, T?d.n Td.t Dqt, Ng| Qu! Duong,
8A,, Chu Van An ; Nguydn Tl^rdn Anh,8An Ngoc
taffi, Gia Lam ; rrin"uuu Drlc, gC'i, Chuitjn
Hon nta, 0 < o < 9 (o € N) n6n chl c6 cic
qp @, n) sau day l} th6a m6n (3) : (1,4), (2,5),
(3,6), (6,8), (8,9). Va"ta cd c6csd tuong tlug sau
d6y : 1996, 2828,2832,3996,6664,8164. Tht
lai, ta thdy cA 6 sd ndy d6u ld sd cdn tim.
. Nh?r, x6t. Cri 84 bai glLi, tdt c6 d6u gi6i
{ring. Crig ban sau ddy crf ldl giai t6t: Trd.ntinh
?4ry $ To6n L6 Quy DOn NLa Trang - Kh6nh
Hda), _Br)i Dtc Anh (7A Trong Didm THCS
UdngBi - QuAng Nirrh), TritnThanV Son (g
Torin NEngKhidu T* Ninh Binh), NeuvdnVi€t
pQng (9ATo6n Nang Khidu Lane G"ia;e - Ha
lec),_Ngrqydn Trdru Ngoc euans (9 T;6n LO
Quy D6n LongKhrlnh: Ddne Nii).Vuonp Gia
Vtl (6TC PTCS TrungVuone, HoanKigm"- Ha
NQi), pd Qu6c Bd,o (gt frdir Dane Ninh Tn
Nam Dinh - Nam Hii, Btri Anh fn (9f Tfi
Li6m - Ha NOi), phqm Dinh phil (93 THCS
t_ong Qin-h ChEu Thdnh - Ti6n Giang), Vit
Tudn Anh (9 To6n THCS Nane Khid;'Th6i
llqoyoq - B5c Thai), Nguy La"Giini 1CX
Ighia-onn --Nghe An), yo Thanh itet ed
^Qlr6c hoc - Quy Nhon), Trdn Eic Son' (g
Qhyyen_lhi trdn Ba D6n (eu6ng Trach -
QuangBinh)'
oANc vr6N
Bari
nguy€n
EZIZB : Gid.i phuong trinh. nghiQm
.41 = 2 a {igg -- rr- 2*
- Ldi g1iri: Cd,ch 1. ciaNguyiln Thanh Tilng
8Ar, Hdng Bdng, Hei Phbng -
Di6u ki6n dd bdi torin crj nghia : -lb < r< 18.
Tacd:
4y2 = 2+r[1$s424; = 2a{26s= @Try
Viy ld sd nguy<in n6n tr) dd suy ra
I . 2 +r[2oo
,<J-<--4 <4
}J:ry y2 = lhodcy2 = 4.
2
Ti Li6m, He NOi. V{r Vd.n Qu!,91., KimAnh,
Kim M6n HAi Hrrng. Luong Si Tirng, 9T,
chuy6n Kidn Xrrong Phan Huong Thu, Hoitng
Thd Doanh, 8T, Chuy6n Thi x5, Thdi Binh
Nguydn Trgng Ki€n, Hit. Thanh T\.r.d.n ; Dito
Hodng Anh, 8T ; Nguydn Thi Hbng Dun g, I{ai
Nggc Kha,9T, Trdn D6ngNinh ;Nguydn Nggc
DiA.p,9B, Thanh Lrru, Thanh Li6m, Nam ltri.
Dinh Httu Todn, 8T, NK Trrrong Hrin Si6u ;
Trd.n Thitnh Soz, 8T, NK Th! X6, Ninh Binh.
Dodn. COng Anh, 84, NK Ha Trung ; Philng
Hd.i Anh,9A, Xi Mdng ; D6 Thi lloo, 7T, NK
Bim Son, NguydnTrgng Phong,8C, NKThdnh
ph6, Thanh H6a, Phan Thanh Trung, 8T, Qu6n
Hinh, Nghi LOc ; Phan Thi Nghia,8A, NKY6n
Thdnh ; Nguydn Tud.n Duong, 9ll L6 Mao,
Vinh ; Nguydn Anh Til,9T, chuydn Phan BOi
ChAu, NghQ An. Nguydn Thi Thtly Hqnh,8T,
NK Thi x6; Trd.n NguyAn Thq, 8T,.NK Ha
Tinh. Nguydn Minh Kien,6T'Vo Chi !hd.nh,
9T, Chuy6n LC Khi6t ; Nguydn Hdi Au, 8T,
Chuy6n M0 Drlc, QuAng'Ngei. Bi,Li Tidn Dat,
Trinh Duy Binh,9Ar, LO Lqi, Di Linh, LAm
Ddng. Nguydn Ngqq-Minh, 9A, Ltrong Van
Chrinh, Phri Y6n. Luong Trung Tud.n,9T, Bdi
dudng gi6o duc, Bi6n Hda, Ddng Nai. Chung
Nhdn Phil, 8"Ib Nguy6n An Khrrong, Hdc
Mdn ; Pham Mifuh Hilng, 9T, Nguy6n Du, Gd
Vdp, TP. Hd Chi Minh. Nguydn Chi Thdruh,
8T,n chuy6n Nguy6n Binh Khi6m, Vinh Long.
PhATh ViQt San, 6 Ar, Phqm Th{ Vd,n Giang, 8A"
chuy6n Bac LiOu, Minh HAi.
16 NcuyEN
Bei T3(226) : Tim td.t cd. a e N dd phuong
trinh x2-a2x*a*7=0
c6 nghiQnt nguyAn.
Ldi giai : Cach I (cria da s6 cric ban)
Dd phrtong trinh cd nghiOm nguy6n di6u
ki6n ld L, = aa.- 4a - 4ld sd chinh phrrong
Ydia:0, lthiA<0
Ydia:2+L,=4th6amdn
Y6ia > 3 ta cd L > (a2- 1)2 vl
L > (az - l)2 +,o+ - 4a - 4 > a2 - Zaz - |
= 2a2 - h - 5 > O *-2a(a - 2) > b
dringvi2a>6,a-2>1.
D6 thdy 6. (o2)2. V4y A kh6ng li sd chinh
phuonsv6io>3vi
(a2_L)2<L.<@212
Kdt luan i e = Zla gi6 tr! duy nhdt cdn tim
Cdch 2 (cria L6 BAo Toin, I Quy Nhon, Vfl
Van Quy 9a HAi Hrrng)
GiA st x, xrld. nghiOm nguy6n cria phrrong
trinh. Theo dinh li Viet
x1*xr: a2
' xrxr:a*l (1)
Tr) dd (xr- 1)(x2- 1) = - (a2 - a - 2)
Yi xr, x2 € N n6u tt (1) suy ra
x, 2 l, xr) l.Ydy at - a - 2 < 0
j0 < o < 2. Thtt trltc ti6p chi c6 a : 2 th6a
mfln dbi h6i bii torin.
NhQn x6t : Bdi niy cti rdt nhi6u ban tham
gia giei. Chi cti 6 ldi gidi sai : Ldi giai t6i ld cria :,
Nguydn IIit. Duy 9 Hd Tdy, Nguydn Thi Minh
Hod.ng 9 Hi Bic, Vo Chi Thd.nh 9 QuAng Ngdi,
L€ Hod.ng Anh I Ha NOi, Phqm Thd Anh I H.it
NQi, 7r&z Drlc.Son,8 QuAng Binh Pham Thi
Vd.n Giang 8 Minh tlhi Nguydn Trung KiAru 8T
Nam Hd, Trd.n Td.t Da, 8A Chu Van An Ha Noi,
Nguydn Vd.n Thirnh 9 Ninh Binh, Nguydn Thi
Thny Hqnh 8 Hba Binh, Le Thi Tdm I NghQ Arr.
DANG HI]NG TH-{NG
BldiT4l22S- Trong cd.c hinh thang cd.n c6
chu ui 2p, gdc kbdd.y l6nbd.ng a (o < 9?o),dqrug
hinh thang c6 di|n tich ldru nhdt,
Ldi giai vfun tf;t : Ggi ABCD ld hinh thang
(AB ll CD) c6 AB = 2a, CD = 2b, BC = c. O lit
trung didm CD. Ta c6 2p = 2a * 2b + 2c
+P=a*b*c
Mitkhrich=csina.
+ S = (a +b)h = (a *b).csina
Do dd S l6n
nhdt*(a*b)c
16n nhdt. Vi o *
6vdcldhais6
dtrong cd tdng
khOng ddi n6n
tich crla chring
l6n nhdt khi vd
chi khi a*b = D
" = f,. D6 thdy CK = c (K ln hinh ehidu ctraA
xudng CD). Suy ra cAeh drrng : DUng LBHC
D
vuOng d H e6 C = o, BC : i.Yu dudng trbn
(c, cB) c6t cH k6o ddi 6 K. K6 Bx ll CK vd Ky
t CK cdt nhau t4iA. Tt dri suy ra didm D.
Nh4rr x6t : 1. Bii niy nhi6u ban giAi rdt dii
do chtta chri f gie thid.t cho p vi a trlc li dE
kh6ng ddi.
2. Cecban giAi t6t bAi niy :
Cao Bing : Ld.m Mqnh. Trudng,9A THCS
Hop Giang.
B6c Th6i : Vfi Tud,n Anh, 9CT THCS Ndng
khi6u Thrii Nguy6n.
Hd B6c : Tfd.n l{6ng Quang, 9T Chuy6n B6c
Giang N
guydn NhU Chudz,,8 NK Thudn Thdnh,
Pham HAi Trung, 9CT Nang khiSu Ti6n Son.
' Vinh Ph:d : Trh.n Thi Tho 8, PTCS. Supe
Phong ChAu, Bili Dd.ng Quang,9T Chuy6n
Tam DAo, Pham Td.t Dat 88 Chuy6n YGn L4c,
"Anh Hilng Dung,9A PTCS Sa D6c, Phri Tho,
DQng Thu Huong 8T Chuy6n Phri Tho,
__.YCq Bdi : Nguydn Thi NSoc Anh, 9"1 Ld
H6ng Phong.
Hd Tdv : D6 Hoirn* DiQp, CII BO t6ng,
Chuong Mi.
Ha NOi : Ddo Phuong BilL,8A, BdVan Ddn, Vrl
PhrrongNhi, TH, L€ Thi Hodng 8H TlrtngVrrong
Tfr.n Phtic Long, 9APTCS Phan Chu lYinh.
HAi Phirno : Nquydn Thanh Tir.ng, 84,
THCS Hdng Bnng "
tQ \\
Hc(
<_-_