i{.-i ' i'.i:,}P"./' i. i
BO GrAO DUC VA DAO TAO * HOr TOAN HOC vrET NAM
RA HANG TUAXC
"-.-- +.1 - 4 --!--. ac 4 ? q a
;* wffi F{#T ffiNr{ €HffiT Ttr{tr rf$ (ffiffi ffiilffiffi w{roNffi
m xAy DIJNG e0ruCI TF{uc rfrurt nQ nru TRUnTG TTJyEN
M UwE DUI\tg n Cf cytfiy nywr E ror Wy
7
m BlilH ffGHIffi B Bu#$rs (ffi${r( Trqoffie ffimT prlfirss BFr$*
0-3
At!'?t
m BE rHt QU0G SIA SH$r.I H$C Srr'Ilt Gror rm $
obd/tl x,g@ lsss - tes6
cDodn ngdj sinfi Xac dinh tam
duon6 tron NhCrng doy s6
ki lo
Ldp 12T bddng chuyAn LA Khidt, Qud.ng Ngdi ndm hoc 1995 - 1996
ToAN HQC VA TUbI TRE
MATHEMATICS AND YOUTH
MUC LUC
Trang
o Ddnh cho cric ban Trung hoc co sb.
For Lower Secondary School Leuel Friends
LA Qudc Hd.n - Vd mQt tinh chdt tht vi
ctia hinh vudng. 1
e Gidi hdi ki trudc
Solution of Problems in Prouious Issue
Cdcbdi eias6229 2
c Db ra ki ndy
Problems in This Issue
Tu233,..., T10/233,LU233,L21233 8
e D6 Nhu Ngq, - Xdy dung c6ng thrlc tinh
dO dai trung tuydn tam gi6c I
c Dinh cho cric ban chudn bi thi vdo dqi hac
For College and UniversitY
Entrance Exam PrePaPers
o Nguydn. Thanh Giang - Uttg dgng tich phin
10
tinh gi6i h4n
o Nguydn Tltrtc Hd'o - Dlnh nghia 3 drrdng c6nic
trong m[t phing a{in 12
o Nguydn Htu Thd.o - Dd thi qudc gia chon
hoc sinh gi6i tornn ldp 9 ntrm hoc 1995 - 1996 14
o Gidi tri todn hac
Fun with Mathem'atics
.Binh phuong - Gi6ri d6p bdi : Do6n ngdLy sinh Bia 4
Va Kim HuQ - X6c dinh tdm dtrdng trbn
Thanh Tud.ru - Nhirng deY s6 ki 14.
Tdng biAn tdP :
NcuvsN cANH roeN
Ph6 tdngbi6n fiP :
NGO DAT TU
HoANG cIlfNG
nOl oOruc etEH rAP :
Nguy6n CAnh Todn, Hodng
Chring, Ng6 Dat Ttl, LO Khdc
BAo, Nguy6n HuY Doan,
Nguy5n Vict Hai, Dinh Quang
HAo, Nguy6n XuAn HuY, Phan
Huy KhAi, Vfl Thanh Khidt, L0
Hai Khoi, Nguy6n VEn M{u,
HoingLO Minh, NguY6n KhSc
Minh, Trdn V6n Nhung,
Nguy6n Ding Phdt, Phan
Thanh Quang, Ta Hdng
QuAng, Dang Htng Th5ng, Vr1
Duong Thuy, Trdn Thdnh
Trai, LO 86 Kh6nh Trinh, Ng6
Vi6t Trung, D+ng Quan Vi6n.
Trq. sd tda soqn : -
45B Hlrng chu6i, Ha Noi DT: 8213786 BiAn tQp uit. lri sy: vu KIM THIrY
2Bl Nguy6n vrn Cil:?;'h Chi Minh DT: 885G111 "- --' irirn oav , QU6c sbNc
Ddnh clro cdcbgn fH6
u( ful
Trong srich gi6o khoa hinh hoc l6p 8 dE n6u
l6n c6c tinh chdt co bAn cria hinh vu6ng. Trong
bni brio niy, chring t6i xin n6u th6m m6t tin[
chdt khric ctia hinh vu6ng vd c6c rlng dung
phong phri ctla nri.
Bhi torin I z Cho hinh uu6ng ABCD uit. cd.c
d:6y M, N, P, q tuong ilng tr€n cd.c dudng
t!*"19;,1.c; A r,t fi
ryrT[rrytt ffimTTtt$u$ c*m ,m]]*Huu0r$G
,\l
lE oudc HAN
(Nsh? An)
tll
tlPc
' Illnh 1
Gi6i: Goi P t
Ii didm d6i "
xingciaMqua 0
o thi P thu6c
canh BC. Tt N
k6 NI{ t MP vit
l6y tr6n dudng
th8ng NI/ m6t
didm Q sao cho
NQ = MPthlQ,
thu6c eanh AO
(xem hinh 3). ,A
Gqi ^E li didm
d6i xrlng cria Q qua O vdl ld chdn dudng vu6ng
g.rg ha tit O xudng EN.Ldy B vd C tr6n dtrdng
thing.EN sao cho : .IB = IC = IO.Ldy A vd. D
ddi xrlng v6i C vi B qua O thl ABCD li hinh
vu6ng phAi dung.
K6t qu6 sau d6y li su tdng qu6t htia cria bii
to6n 1.
Blri to6n l' : Cho hinh chit nhet ABCD c6
AB = a, BC = b ud. cd,c didm M, N, p, e nd.m
tr€n cd,c duimg thd.ng AB, BC, CD, DA. Cfulng
ruinh MP a NQ khi utr. cni nhiffi:I
Chring minh bii to6n 1' tuong tu nhrr crich
chrlng minh bni to6n 1, xin dinh cho ban doc.
Blri todn 4z Chofi giacABCD. Dung hinh
chit nhQt MNPQ ngoai tidp fi giat ABCD d6,
bidt tt s6 crta hai canh hb nhau bd.ng h (k td, s6
duong cho trudc)
Gi6i: GiA
/Y :.,I A, B-,9,D
theo thrl tg
nim tr6n c6c
cgnh MN,
NP, PQ, QM
^
(xem hinh 4)
" vd,MN1NP =
K,KAAH T
fc
Htnh 4
p DB, AE k6o
ddi cdtPQ tei
E thi theo k6t
CD, DA, ChTNg
m.inh rd.ng : MP
= NQ khiudchi K
khi MP.r NQ.
Gi6,i: Dd
chrlng minh ta ,A
K6 MH II AD,
NK ll AB rdi
chrlng minh hai 0
tam giric vu6ng
MHP vd NKQ
bing nhau (xem hinh 1).
BAy gid, ta hiy 6p dung kdt quA crja bdi to6n
1 dd giai crnc bii to6n sau :
Blri to6n 2 z Cho hinh uu6ng ABCD eanh
bdng a ud. mQt didm M chuydn d.6ng ffAn canh
BC. Phdn gid.c crta g6c DAM cat CD tqi N.
Chtng minh AN < Z . MN. D&ng thtte xa.y
ra hhi ndo ?
tY
I
l,l
Giii: Dudng
th&ng kd tr) M
vu6ng gdc vdi
AN citANtaoH
vi cdt drrdng
thing AD tat L
OGq hinh 2) S
DAN = NAM
n6n D
1
NH=HI=;AN
. Theo bdi to6n
1, ttAN t Ml,tacd AN = MI = ^,IH > Z. MN.
D&ng thtlc x6y ra khi vi chi khi ff = tf <+
a
CM = 7@ DN = alL).Ban doch6ychrlngminh
ta! q9{ 1-ay (Dua vdo su dong d4ng crla c6c tam
$6e ADN, NCM vd ANM). -
BAi todn 3 z D4tng hinh uu6ng ABCD bidt
ui ti tam O cila hinh uu6ng uit. ui-tri hai didm
M, N theo thtl ttt nd.m, tr€n hai cgnh AB uit. BC.
quibiitodn 1',tacdffi = ffi= f nenFhoen
todn x6c dinh, tr) dri xdc dlnh dugc c6c dinh cria
hinh chfr nh |MNPQ.
(xem ti€p tang7)
P unns
Iltnh'2
BidiTVzLg Cho x > 0, ! > 0, z > 0
Chilng ntinh
(xyz + D (;++).:.; +L > x *y * z * 6
Dd.ng thtlc xd.y ra khi nin ?
Ldi gini : (c{ra b4ri' Nguydn Hoqch Tnic
Sinh, 8A. Qu6e hoc Quy Nhon). Ta cti vdtrr{i li
A = ( nrf\+ /rv+I\ + (xzf\*! *+.+
\r-'yl \--r' xl \ zl x y z
z
Ap dsng bdt ding thrlc cQng ta a Y * i '
,v,,x,
2z , (xt *;) .- 2t , \xz + -) > 2x.
VOyA > ?'x +2y + 22 ** *1 +L = x I Y *
xyz
, 1. l, , 7,
z+ (x *;) * (, *r) + (z +;) r,1y*z*6.
Ddu bing xAy ra khi vi ctri kfri *y = y;, r, =!,
z 1 '-, " =1. ,ru' niry tuong
drfongv6ir=Y:z=1.
Nhan x6t : Bii to6n niy drrgc hing tr6m
ban grli ldi giai d6n. Tuy6t d4i da s6 giAi dtng,
ngiriggn, "o batt nhtr tr-on. Chi cci mQt s6 it ban
giei h-i dii. Trong sd nhi6u ldi giei tdt c6 -: Dinh
fuam Duong 9A NghQ An., Le Anh Tho,9-A
Thanh H.6a, Trd.n NguY€n Thq I Hh Tinh,
Nguydn Viet Hd 9 Hh B6c, D-d lhity Chi 8A
H-ai f nang, Trd.n Luu Vd.n 8C Ngqc t>am, Bt4
Thanh Hilie 9H HA NOi, Yd Anh Tud.n,9T,
Quing ginh, Nguydn. Tud.n. Trung 8T HII
EA", Nguydn Thd.i Soz 9! Thanh H5a,
Ngiydn"HiyVu, ST Ninh Binh, In lnhVinh
sil, Ha, N[i, rvguydz Dtc Hdi 98 Vinh Ph(r,.
Nguydn Thi Thi.FIa 8A Quing Ninh, La Thd
fnd:ng 8H Hn NOi, Ld Trung Ki€n 9T }Jt6,
Dinh-Trqng Quang 7C Hn NQi, Trdn Tq Dpt
8A, Ha NOi, Ha Thu Hibn Y6n Bdi'.'
B4n D6 Nggc Dtlc (6H Trttng vrronglld NQi)
de phrnt bidu va chrlng minh bii to6ntd-ng.qu6t
sau : Cho tu >- 3 at, an > 0. Chrtng minh ring
,1 I' a2
(at...an+l) (-+...*4) * %..*,+
&r, al
+ * ----:-- 7 ar* ...
d1..an_2 a2... an-l
tsdi TZl22g. Tim nghiQm. nguYAn cia
phuong tinh
x2 +f +i +f :27144a
Ldi gi6i : .c:iua Nguydn Hdi Hd,9b, Chuy6n
Van - Torin Ung Hda, Hd TAY.
x2 +F +xa +f = 277440
+x27x2 + 1)(x + t) = 24 .92 .s . lg . _29 (t)
Tt (1) ta suy ra ngay nghiQm r phAi I6n hon
L vd x2Id udc chinh phttong ctra 271440' Cdc
rioc chinh phrrong ctiZlt{qo c<i thd le'22,24,
22.g2. 2+.92. X6t rr6e chinh phrrong l6n nhdt
24B2'- (zz.Uz = !22.
Thay x2 = 122 vio phttong trinh ta thdy
phuong trinh dugc nghiQm ding
x21x2 +111x+1) = 122 $22 + 1) ( 12 + l) = 271440
YOix2 : (23)z = 62 tathdr
* (* + tl(6+ 1) < 122 $22 +D $2+1) = 27L440
VQyr : 12ln nghiQm duY nhdt.
Nhin x6t : Hdu h6t cac ldi giai grli ddn d6u
dring. Song ldp lufn ddi dbng. C6c b4n au dAy
cO tai giai t6t : Hlr B,6rc : Nguydn Danh. Nann,
Nguydi Hilng Cuitng, Trd.n Thi Hd Phuong,
9T, NK BEc Giang. Lho Cai : Nguydn Hbng
Quang. Vinh Ph6 : NguYdn Dtc Minh, 8A,
Chuy6n Tam Dtro ; Hd.Vd.n Son, 9T, ChuY6n
Phri Thq. H}r Tey : D6 Anh Tud.n,9T, Thudng
Tin; Nguydn. Mq.nh Hd, 9K, La Lqi, Hd' DOng'
HDr NOi ; Diling NgQc Son, 9CT, Tit Liam'
Qu6ng Ninh : Etrt Ann Dtc 8A, TD Uong Bi'
Hii Phdng : D6 Thity Chi, 8Ar, Hdng Bdng'
Ttranh H6a : Hd. Xudn Gid.p, 6Tr; Hoit'n'g Thi
Hd, Hd Thi Phuong Thd.o, 8T, IIK gim Son.
I0r6nh H6a : BitiThanhMai 9T, L0 QuyD6n,
Nha Trang. TP Hd Chi Minh : Nguydn Cd'nt
Thgch,8r, HdnBEing, QuQn 5,.
rd NcUYEN
BAi T3/229 " Gidi Phuong trinh :
@ -sr+2)(x2 * 15x *56) +8 = o
LA,i giai. Ta cci :
@2 - s, + 2)(x2 * 15x * 56) + I =
=x4+12f+fi*-fiBx+120=
= 1x4 t of - ts#1 + @f + 36P - eo) -
- (8r2 14Bx - 120) = x21x2 + 6x - 15) +
* 6x(x2 * 6x - 15) - 8(x2 * 6x - 15) =
= (x2 * 6x - 15\(x2 + 61c -8) = (r + 3 - 2r[6)
(r +3 + 2\[6)@ + 3 +{17) : o. vQY -Phuong
irinh c6 4 nghi|m. lit. : x1= -3+2'[6;
xz = -3-2r[ 6,' rs = -3 + {I7 ; x4 : -B -'ln '
Nh$n x6t. C<j z4lbdi gi6i trong d<i cd 8 bni
giAi sai. C6c b4n sau ddy cd tdi 4Aitd-t "
Phsm
-Nguyen Thd.ng (D} Lgt, I To6n PTCS Chuv6n
t[ane Long, Lam Ddng), Trd'n. Tud'n' Anh
a3
1
-
I ...
a4... anal
*an|2n.
2
oANc suNc ruANc
(Khr{nh Hda, 8 To6n L6 Quy' D6n, Nha Trang),
Nguydn D6 Thdi NguyAn (Vinh I.ong, 9T,
Chuy6n Nguy6n Binh Khi6m, Tk Vinh Long[
Ng4ydn Hbng Quang (Tx Lho Cai), Nguydn
Khanh Linh (Ha NQi, 9c THCS Ngoc LAm, Gia
L6m), Hdn Minh Trung (Thanh H6a, 6E
THCS Nang Khidu, Tp Thanh H6a),Vtt Mqnh
Cudng (V[nh Phti, 8A Chuy6n CII Tam DAo),
Dinh Trqng Hilng (Virng Tdu, 9T LO Quy D6n,
Tp Vung Tdu),'Nguydn Cd.nh Tod.n (Iuydn
Quang, 9 To6n Nang Khi6u Le Quy Ddn), ?a
Xuyan Hung (Y6n Bdi, 9T LO Hdng Phong),
Nguydn Ngoc Quang (Hi NOi, 9H THCS
TrungVrrong),Ld.m Manh Truisng (Cao Blng
9A THCS Hop Giang, Tx Cao Bing)./.
oANc vrEN
BdiT4l229.Cho tam gidcABCu6i cdr cqnh
a = 5 ; b = 6 ; c = 7- Tinh khod.ng cd,ch gitta
td.n dutmg tritn nQi tidp uit. trqng tAm cfta tam
giac d6.
Ldi giai. Gqi M
lA trung didm cria
AC vd G li trong
tinr tam g16c ABC,
ta c6 G nim tr6n
doat BM sao cho
GM:GB=1:1(1).
Goi O ld tAm drrdng
trbn n6i tidp tam
$ec ABC vi I lBr
giao didm clfr'a AC
Nguy6n Hrru Quy6n (Vinh Ph(, 9T Chuy6n
Phri Tho), HdVdn Son (Vinh Phrl, 9T Chuy6n
Phri Tho), Nggc Bich Phuong (Tidn Giang, 9
Torin NK huy6n Cai LAy), Le Chi ThAnh (Hu4
9I Nguy6n Tri Phrrong), Nguy6n Hoach Tnic
Sinh (Binh Dinh, 8A Qudc Hgc Qui Nhon),
Nguy6n Minh QuAn (QuingNg6i,9T Chuy6n
Nghia HAnh), Nguy6n Hoing Chrtong (B6c
Th6i, 9 To6n THCS Ntrng Khi6u Tp Thrii
Nguy6n), Trdn Ngoc Cudng(Tp Hd Chi Minh,
BT, Nguy6n An Khuong, Ho<ic Mdn), Vfl Thanh
Ha (Tp Hd Chi Minh, BT, Nguy6n Du), Vri
Anh Tudn (Ha NOi, gAt TH^CS Thdnh C6ng).
oANcvrEN
Biri T5/229 : Tran mQt ph&.ng cho g6c xOy
c6 dinh ( xOy = 600). MQt tgpl<gidc cdn MAB
(MA = MB= a khdng ddi ; AMB = 1200) thay
ddi ui ti sao cho hai dinh A, B chq.y ffan cdc
tia tuong tng Ox, Oy. Tlrn qu! t{ch cia didm
M. Ldi gi6i vin
t6t: Gqi Mrld,tdm
vbng trdn ngo4i ti6p
LOAB, M) ld trung
didm cta -cung nh6 a
AB ct&a drrdng trdn
dd. Dd tim tAp M ta
cdn tim tAp
{MJ u {M2}.
a) ?4p {Mr} :
Tgg2 OM, = M)A:' BM, - a.
MpA ( 90o, MrOBag0.
+M, thu6c cung nh6 M'M" cria drtlng trdn
(O, a) (6 ddy OM' t Oy, OM" t Ox
OM'l,Jtflcphiavdi
Oy so vli Ox
OM" khrlc phia
Ox so vdi Oy).
b) Tap {Mz} la
do4n MrM, trong dci
Mp M2 thuOc phAq
gtr6c Oz c$.a g6c xOy,
OMt - a, OMr= to,.
VOy tQp @didm
M ehinh ld. M'M".\)
MrMz'
Nh$n x6t:
. MQt sd bpn chi tim drroc 1 trong? t4p Mr
ho1cM,
o C6c ban d6 giai t6t bii ndy :
Vinh Phri : Mai Thu Thd.o, Ilit.Vd,ru Son gT
Chuy6n Phri Tho, Nguydn Trung Lqp, 88
Chuy6n Vinh Lac.
Hlr Tay : Nguydn Manh Hd, 9K', THCS Lo
Lqi, Hd D6ng.
v6i tia BO, AO lA cric ph6n giric clioa cdc g6c
tudng (tng ABC, BAC. Ap dung dinh li v6 tinh
chdt drrdng phdn gi6c (hinh hqc 8), ta cri ;
IABATIAIATT
IC: BC: E- AC= IA+rc= b+7 = 12'
YQy IA: 1AC :"L2 = 3,5. Vdi phAn gi6cAo,
. u Ar 3.5 1._ __
talarcd, OB=M= 7 :rQ).K6thqp(t)
v6i (1), ta cd GO /i IM (dinh li Tal6t dAo). V6y :
OGBG222
IM: B*: t, hay OG =1r*:; (IA_AM) =
21
= 5 (3,5 - 3) = I, ve khoAng e6ch cdn tim
i^1
rdo'
d
Nhan x6t. Cd 96 bai giAi trong dcj crj 5 bdi
giAi sai. Nhi6u bdi trinh bdy dni ddng cri bdi
dtroc trinh bdy hdt bdn trang grdy ( !). Dec biet
cdban Hdn Minh Trunghgcldp 6E THCS Ndng
Khi6u Thdnh phd Thanh }jr6a c6ldi giAi tdi
cirng cric b4n sau ddy : Trdn Tdt Dat (He NOi,
8At PTCS Chu Vdn An), Trdn Thi He (Thanh
f,tr6a, B Torln Irlang khidu Bim Son), Phqm Thinh
NSU (Vinll Ph6, 9A, NK thi xa Vinh Y6n),
$t