BQ GIAO DUC VA DAO TAO
12evt
.. [ , ).
| / ( dll(,,
1. \ g-'
* ugr roAN Hgc vrFr NAM
TAP CHf RA HANG UTANC
* VB NrQr rruIq DUNG e0* DINH r"[ Mur
'l' riru cndr orty circ ri s6 shxs ilHRU
* odu,To\lcuul c'(iA'I7i otFN
* y3, sd aauven rd tclm rurudr
{. Lfi firir sd THr Qrric t;rA orr0N tr0{r$rurr crCIr To[N TnpT il,[M II0o rsss- 19g6 IBING B]
{. suv ncHfr v€ mrQr BAil ToAm HAnn sd
{. ori:x aej vAo nir.tn l.Uc crAc
* erprl a-mtr? ez6a<* cngete
Thdy tro dQituy6'n thiqu6b gia kip I (1995 - 1996)
tinh Thanh Hda.
ToAN rroc vA rubr rRf
MATHEMATICS AND YOUTH
MUC LUC
Trang
Ddnh clw cdc bgn Tnrng hoc co s&
For Lower Secondary School Leuel Friends
Nguydn Dilc Td.n - V6 mQt rlng dung cria
dinh ly Viet 1
Gidi bdi ki trudc
Solution of Problem.s in Preuious Issue
Cdc bdi crla sd 230 2
Db ra ki ndy
Problems in this Issue
Tu234, ..., TL01234, Lu234, L21234 10
dngkinh cdi cdch day vd hoc tuin
Kaleidoscope : Reform of Maths Teaching and
Learning
Huynh Van Trgng -
Tinh chdt day Jc ti sd bing nhau 11
Tim hidu sAu thim ndn hgc phd thdng
Helping Young Friends Gain Better
Understanding in Secondary School Maths
Nguydn Minh Hd.
- Didm Toricelli ctra trl di6n Lz
Nguydn Khd.c Minh - Ldi giAi d6 thi qudc gia
chgn hgc sinh gi6i to6n THPT
n6m hsc 1995 - 1996 (BAng B) 14
Hqc sinh tiln tbi
Young Friend's search in Maths
La Quang Nd.m - Suy nghi v6
m6t bni todn hdm sd 16
Ban cd bidt
Do you know ?
Nguydn Tltdng - V6 sd nguy6n td ldn nhdt Bia 3
Tdng bi€n fip :
NGUYEN CANH TOAN
Phd tdng bi€n tQp :
NGO DATTU
HOANG CHONG
xOr o6xc etEN rAp :
Nguy6n Cinh Toin, Iiodng
Chrlng, NgO Dat Trl, LO Kh6c
BAo, Nguy6n Huy Doan,
Nguy6n Viet Hai, Dinh Quanpi
HAo, Nguy6n XuAn Huy, Phan
Huy KhAi, Vri Thanh Khi6t, L6
HAi KhOi, Nguy6n Vdn MQu,
HoAngLO Minh, Nguy6n Kh6c
Minh, Trdn Van Nhung,
Nguy6n Dang Phdt, Phan
Thanh Quang, Te Hdng
QuAng, Dang Hing Thing, Vtr
Dtrong Thgy, Trdn Thdnh
Trai, LO 86 Khtnnh Trinh, NgO
ViQt Trung Dang Quan Vi6n.
o Gidi tri todn hoc
Fun with Mathematics
Binh Phuong - Giai d6p bdi :
Di6n sd vio hinh lgc gi6c
NgO Hd.n - Trb choi chia di6m
Bia 4
Bia 4
Tru sd tba soan :
45B Hhng Chudi, Hn NQi
231 Nguy6n Vtrn Cu, TP Hd Chi Minh
DT: 8213?86 Bian fip uit tri s4 : VU IgM T$IIY
DT: 8356111 Tinh bay; QUOC HONG
Ddnh cho cdc bgn trung hec co sO
r
[a e (r) 22
lrn=, *YA:Z=
[a e 1ry
Pu=, =/s
["n
v^
Vfl M0T Iroq DUNffi UUM $ffiffi Ii VIflT
Dinh U Vi6t &e thrlc Vidt) : N€u phuong
rtnhbQntwia* +br+c = Odtninghifin\ i 12
thi tdng uir. tich hoi nghiQm d6 ld
bc
S = rl l-xr= -;; P = xtry= i.
V6n dung dinh U dd giei to6n li vdn dd rdt
hdp d5n.
Tr6n Tqp chl T.HvT.T e66 (204)11994 the'i
gi6o Nguy6n D6 de vi6t v6'Sfr dung dinh U Vi6t
dd giai mQt bdi todn v6 bdt dlng thric'. Trong
bdi vidt niy, t6i xin drrgc trao ddi cirngbqn dgc
v6 rlngdgngcria dinh ll Vi6t trongviQc giAi mQt
s6 bdi to6n vd him s6.
X6t c6c bAi to6n r6t quen thuQc sau :
Bli todn I : Cho parabol (P) y = 12. Gqi A
ud.B lit.hai didmthuQc (P) c6hod.nhd| [An fuqt
tit. -1 ; 2. Yi€t phuong tinh dudng th&ng AB.
Dey li bei to6n d5, hdu h6t hgc sinh vd nhi6u
tii li€u to6n d6u cho ldi giAi sau :
fee,pt
t";-='j =/A = (-1)2 = 1' v4YA(-1 ; 1)
= 22 = 4.Y4y B(2;4)
Phuorig trlnh.dudng th&ngAB cdn tlm cd
dangy = an+b (AB)
leerarD lt=<+b [-go=-3 -
1r. i*i:1n = ?.a +b :1, = -a. +ba
la=L
1b=z
t
Vflyphuongtrlnh dudngthlngA8 lay = x * 2.
N6ueuy nghi ddn viQc xrl dUng dinh ll Vi6t
ta cti lUi giAi'dgp'sau :
Phuong trlnh dudng thlngAB cdn tlm cri
dAng:!=o.x+b(AB)
Phuong trlnh xdc dlnh hoinh dQ giao didm
crla (AB) vi (P)
12 = a$, +b €x2 - a:r -b = 0 (*)
aA ='-1,frn = Z li hai nghiQm cta (*)
M4t khec theo hQ thrlc ViCt ta c<i
lxa=o
'xB- 4
0
NouvENoOctAx
(T.P Hbcht Minh)
doddo=L,b=2,
' Phuongtrlnh dudngthflagz{B ldy = aa2.
;2
Bli to6n2 r Cho porabol y = i fel, en
d,idm tran @) c6 ttod.tlh dO ld 2. fi.m phuong
rtnh fi€p tuyeh 4.i Au6i (P).
COng nhrr bAi todn 1, hdu hdt hqc sinh vit
nhi6u tti liQu to6n d6u cho ldi giAi sau :
1V4yA(2; 1)
Phuongtrlnh duBngthlagcCn tlm ctf dqng
! = o*+b (D)
A e @)a+f = 20+bs5[ = L - 2o.
V0YY = Q,**L -2o (D)
Phrrong trlnh x6c dinh hoinh dQ giao didm
crla (D) vn (P) ld
1
1x2 = at t l - fut afi - kt -4 f 8o = 0
4
[,i=4a2+4-fu=4(o-t)2
(D) tidpxfc (P)se[' = g4(a - 1)2 = 0<+
a=trc=1'+b=1-2.1 =-1
Phuong trlnh dudng thlng cdn tlm li
Y=r-L.'
Sau ddy li ldi giei ndu xfi dung hQ thrlc Vi6t.
Phuongtrlnh dudngthlngcCm tlm cd dgng
! = o.x+b (D)
Phuong trlnh xdc dinh holnh dQ giao didm
crla (D) va (P)
*2
i=*tb€*-4ar-4b=0(**)
4
x = 2ld,nghiQm k6p crta (**)
M{t khdc theo hQ thrlc Vidt ta cd
[x, *rr= la
Jr
lx'r,,= -46
I r.
f,l = fZ= 2
doddo=1,b=-1
Phtrong trlnh dudng thtng cCn tlm lAy = a
-1
C6c bqn h6y tlm th6m nhitngbii to6n tudng
tU, bei todn v6 him sd mi rlng dung dinh U r/i6t
cho ldi gi&i ng6n gpn.
* + 2.9 *ff = 4500udi x < y < z.
Ldi giai. (DUa theo La Hbng Linh,9T,NK
Tx Ninh Blnh). Tt phrrong trlnh d6 eho, ta cd :
* < 45OO< 56, mi5 > 1 n6ne ( 5. Hon
n{Ia,r<y<zndn:.
ff +2.A +ff < 53 +2.54 + 55 = 4500. Suy
ra, nghiQm cria phrrong trlnh li : x, = 3, ! = 4,
z = 5 ikhi vd chl khi dd thi Mu bing mdi xdy ra.
NhAn x6t. 1) Cd 305 bei giei, tdt eAddu gi6i
dring, phdn l6n d€u giAi d{a vio t{nh duy nhdt
cria d4ng phin tlch ra thita s6 nguy6n td ctra sd
nguy6n, tuy nhi6n thudnglir ehtra thSt ch{t ch6,
ching h4n kh6ng n6u 16 t4i sao bidu thrlc cht
dang x6t li sd nguy6n.
2) Cdc b4n sau dAy ctf ldi giAi t6t : Phqm
Dinh Phrt (Ti6n Giang, 93 Long Dinh - ChAu
Thinh), Trd.n Dtc Son (Quing Binh, 9,
Chuy6n thi trdn Ba Ddn - QuAng Trach), Phg.m
Ngqc Huy (DdngNai, 9T, NguySnBinh Khi6m,
Bi6n Hba), Duong Thiiln Drlc (Thta Thi6n
Hu6, 9r, Nguy6n Tri Phuong), Phgm ThiVdn
Giang (Minh Hai, 9A, Chuy6n B4c Li6u), Dd
Thi Di ThiQn (Khdnh Hda, 9 Tornn L6 Qul
D6n, Nha Trang), Ngd Qude Anh (Dek Lnk, 8
Chuy6n To6n PTTH Nguy6n Du, Bu6n Ma
ThuQt), Bil.i Tidn Dq.t $-em Ddng, 9Az THCS
L6 Lgi, Di Linh), LE rlbng Linh (Nin[ Binh,
9T Nang Khidu, Tx Ninh Blnh), L€ Quang
Tud.n (Thianh H6a, Bl PTTH HAm Rdng, TP
Thanh H6a), D@ng Td Nhu (Qunng Binh, 9
Torin THCS NK HAi D!nh, Ddng H6i), Bni
MinhKhoa (TrlVinh,9oL. f TU Trgng, TXTrn
Vinh), Phq.m Qudc Trd.n ioOttg Nai, 9E Lf TU
Trgng, T6n Bi6n), Nguydn Xudn NguyCz (Hi
NOi, gAl THCS ChuVEn An, Tdy Hd),Nguydn
Hoqch Trrtc Sinh (Binh Dinh, 9A Qu6c Hgc
Quy Nhon), La Trudng Giang (Thanh H6a,
10A, PTTH Lttong D6c Blng, Hoing Hda),
Nguydn Thanh Tdn (IJ'i,i Hung, 8 NK Phfc
Thinh, Kinh M6n), Dd Mq.nh Hitng (Th6i
Binh, 222 -Td 4 - Chu Minh Ti5n, Thi trdn Vii
Thu), La Tru.ng Kian (ThitaThi6n - Hu6, 9A
Nguy6n Tri Phuong), LE Trtn Trung (Quring
Nam - Dir Nllng, Td 61/6 Phudng VT, Tp Dn
2
Ning), Phq rn Bda Chau (Bt Ria - Ving Tdu,
8 Chuy6n To6n, Chuy6n Le Qul \6n), La Dilc
"ri (Vinh Long, 7T Nguy6n Binh Khi6m),
Nguydn Ngqc IId (Minh HAi, 9Al THCS Thi
trdn Ddm Dai\, Phan Thanh aizh (QuAnB
Ng6i, 9 To6n Chuy6n I€ Khi6t), Nguydn Thd.i
Phil (HdTinh, 9 To6n Nang Khidu Drlc Tliq),
Nguydn Vd,n Hqp NghQ An, 9T Phan BQi
Ctrau), Nguydn Hod.ng Girip (Vinh Ph6, 9T
Chuy6n THCS Tam DAo), Dodn COng Anh
(Thanh H6a, 8A THCS Ning Khidu Hn
Trung), Luang Thanh IIo&i (H}r Tty, 9 To6n
Chuydn Honi Drlc).
oANc vtEN
Bli T2l230 z Cho bidt rndi phuong tinh xz
- mr*p = 0 uit.xz - ttx * q = 0 dbu c6 hai nghiQnt
d.uong. Chrtng tninh rd,ng cor bd.t dd.ng thilc m
< n, trltl < p + q uit, mq < np hh.6ng dbng thiti
xd.y ra.
I,dli giai : GiA sr1 ddng thdi xAY ra
,rL < rL (1)
mn<p+q (2)
tnq < np (3)
Tt dinh II Vi6t d6 thdy n7, n, p, g duong.
Tt (1) -ntp < np. Tt Q) mzn < mp *mq
< mp + np < znp -ntz < 2p. Vt^phrrong trlnh
x2 --mr+i = o ui nghiQm rr6nm2 >- 4p.YSyhp
>4p-p<0.VOI{.
NhAn x6t : Tdt cA c6c bqn tham gia gi&i bei
niy d6u ldm dring v6i nhi6u c6ch giAi kh6c
nhau. C6c b4n c<f ldi giai t6t : NguyCn Hrtu
frnh, 9, Hn Tinh ; HuYnh Minh Son, 9,
Quing Ng6i ; Nguydn Min h Hodi,9, H}r NQi ;
Nguydn Vd,n Long,9, He B6c ; NguYCn Hd,i
Hd, 9, HDr TAy, Dd Ngqc Dtc, 6, HI NQi;
Hod.ng Minh Phrtc, 9, NghQ An ; Ng6 Thi Hd.o,
9, Hb-Bic ; Bili Phiong Thti'o,g, Th6i Binh;
LucVd.n Hd,o.9. Tb. Hd Chi Minh ;Vu Ngqc
idi, g, Thdi'Blnh ;vfi. vdn Phong,9, VInh
Ph6 ; Nguydn Vidt T\i,9, He Tinh ; Duong
ThiOn Dic, 9, Hud ; Biti Dile Giang, 9-, Hir
Tny ; Ngiydn Vd.n Binh,9, Qu6ng Binh ;
Nguy6n Hbng Quang,9, Llo Cai.
DANc HUNGttrAr,lc
Bid T3/230 : Gid.i phuong trinh (nghie.n7
thttc)
xa_4'[Sx-5=0
Ldigiei.ra-4r[3x-5=0
ay4 I 2-rz + L - c2r2 - 4r[5 r- 6 = 0
e(x2+1)2-2(x+{5)2 = 0
a(rz*Er+1 +{)(r2-Ex+ I -r1T.)=0
lrz+r[-Zr+1+{6=o (1)
€-l
-l*z-,[-zr+1-{6=o @)
L
fari tiTZaO . Tim nghiQnt nguyan duong crta
-
{z + 1416 -2
*l- 2 ,*2-
Ta thdy (1) v6 nghiQm vl cti A = -(2 + 4{G-)
<0
Giei (2) ta drrgc :
,t, -
AB +BC 2{z
v{y chu vi MNPQ , 2.--{, - =ff = 2
(Ddu = x6y ra khi MNPQ li hlnh chfi nh4t vi
ABCD ld hlnh vu6ng).
Nh+n x6t : Nhi6u bqn gi6i rdt dii dbng, Sau
day li danh srich cric b1n giAi'dring vi ng6n
gqn:
Vinh Phf : NguyCn Dilc Minh, NguYdn
Hod.ng Gid.p, Vu Mq.nh Cubng, 9T Chuy6n Tam
DAo. Hlr B6c : Nguydn Nhu Chudn, 9NI!
Ddng. Dinh Hanh, 9A, NK Thudn Thinh;
Hodng Titng, S Chuy6n, NguyCn Thi Hdo 9B,
NK Ti€n Son. Hdi Hung :Vtt. Tud.n Long,Phi
Ti6n, Vr7 Thdi Son 8T NK tlnh. HI NQi :
Nguydn Thi Minh thoa, 9T Nggc L6m, Gia
I.dm, Trfi.n Tdi Dq.t, Nguydn Xudn N
guyen, 9Al
Chu Vdn An, Dd.o Thd Vtt, 9A1, GiAng Vo,
NguyCn Quang Dftng, Ire Thd Thd.ng, 9H
TrungVtrong. Hii Phdng : Trtn Trung Hidu
9Dr, Lec Vi6n, NgO Quydn. Nam Dinh :
Nguydn Trqng Kian, Trh.n Dinh Hilng, Dirc
Hod.ng Anh, Vit Trd.n Cuong, Phgm Ngqc
Hung, Phq.m Thu Giang, D6 Qu€ Huong, LC
Nam Binh 9T, Dodn Nant Thd.i, Phq.m Dinh
Qu6c Hung; 8T, Trdn DangNinh, Hod.ng Ti6n,
Ll 9 NK HAi H{u, Nguy€n Dilu Quynh 9T NK
Yyen. Hi Tay zNguydn Htu Dinh 98 Chuydn
Ilng Hba, Nguydn Hd. Duy,9T Chuy6n Phri
Xuy6n. Th6i Binh : LuongVdn Huynh, 9T L€
Quf D6n, HrrngHi. Thanh H6a: NguyCnTd.t
Thd.ng, Dd Thi IJoo, 8TNK Bim Som, Dd.ng
Dilc Trung, 9A Ba Dinh, Bim Son. NghQ An :
Hodng Minh Phtic, 98 Nghi Li6n, Nghi LQc,
Nguy€n llbng Drtc, 9A1 I.IK YOn Thdrnh,
NguyCn Cd.nh Thd.nh, 9TB, Phan BQi Chdu,
Vinh. Hi Tinh: Nguydlt Httu Tl.nh,gTl NK
thi xe. Qu6ng Binh : Dl.ng Td Nha,9T NK
HAi Dlnh, Ddng H6i, Tfh.n Drtc Son,9 Chuy6n
Ba Ddn, Qu6ng Trqch. Thfia Thi6n - Hud:
Phq.m Nguy4n Qu! 9L, Nguy6n ?ri Phuong,
Dinh Trung Hi6u, 8A cdp 2-3 Phf Bai, Huong
Thrly. Binh Dinh z Nguydn Hog.ch Trilc Sinh,
9A Qudc hqc Quy Nhon. Qutng Ng6i : Mai
Hd,n Giang, 9C L€ Khi6t, ffi Ngo" Nam,9T
Chuy6n M0 Dr3c. Kh6nh Hda : Dinh Thd,i
Minh Tdm 92, THCS Cam LQc, Cam Ranh.
D6c L6k z Nguydn Thanh Nhd, 9T Chuydn
Nguy6n Du, Ban MB ThuQt. I,om Ddng z Phan
Thi Thanh f/4, 8A1 Quang TYung, De l4t. Blr
Ria - Vfing Tiu : TTd.n Giang, TNZ Nguy6n
An Ninh. Ddng Nai: Phg.m Ngq" Huy,9T
Nguy6n Blnh Khi6m, Bi€n Hba. TP HCM:
Chung Nhdn Phil, 9Tl Nguy6n An Khuong,
V{y phuong trinh d6 cho cti hai nghi6. m tr6n.
NhAn x6t. Cd rdt nhi6u b4n gui ldi giAi ddn,
trong dd da s6 cd ldi giAi t6t.
16 NcuvEN
Bni T4l230 : Cho mQt hinh chft nhQt c6 chu
ui kh\ng nh6 hon 2r[2 uir. tnQt ffi gid.c c6 cd,c
dinh nd.m trdn cd.c cg.nh hhdc nhau cila hinh
chit nhQt d6, Ching minh rfi,ng chu ui crta fil
giar khing nhb hon 2.
Ini girili. (V6n t6t)
Cd.ch 1: /
0Pc
Trong LtuIDN c6 : MNz = DMZ + DNz
+ ?.IuIIt2 - (1 + t)(Dr,tz + DI{P) > (DM + DN)2
IJay ?.I4IN2 >- (DM + DN)z
+{zMN >. DM +DN
Ttrong tU {ZNP>N C +C P, r[I
P q r- 3 g a3 r,
'{zuq >- AM + Aq.
Do dd : rliluw +NP +PQ + QItt) >
> AB +BC +CD +AD >2:{Z
VQy chu i tit STflIMNPQ khdng nh6 hon 2.
Cd.ch2t / A ,
l,6y 8,.[ G ldn lugt la trung didm MQ, MP,
PN. Suy raEF =f,Or,re =f,ux
I
96 =,NP
Do dti chu vi MNPQ = Z(AE + EF + FG +
GC)>2AC=2,[WTEA
MAt kh6c (AB - Bc)z ) 0 =+ 2,(AB2 + Bcz)
> (AB + BC)2 * {W+-Ge ,Edff .ao
Lqicr2 =e = 90o=+ * =**,
4r[6 -z