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BO GrAO DUC vA DAO TAO * HOr TOAN HOC VrF.T NAM
{ =O€.
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rap cHi RA NGAY 1b rIANG uraNc
MgT Ctrqr sfil{G
160 ro mQr afir
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KI THI HOC SINH
1
GIOI TFfiPT 1995
tee6 @ He
DUONG CONG
t{
TIEP XUC VOI
MQT DUoNG co
DINH@ . vE vOr
PHUONG PHAP
CHUNG MINH
o
DANG THUC VEC
ro 6-i o€ THt
^ \--l
A.
TUY€N SINH VRO
C6C LoP CHUV€N
^?
Dnr HQC TONG
HQP @ slEnrr
1X
cun uOr Ncuor @
7/qt7 so 7Ro"rE
ei"teo @.
Th$t vd tro truong THPT Qu6:c hqc Hud.
TOAN HQC VA TUbI TRE
MATHEMATICS AND YOUTH
MUC LUC
Trang
s Ddnh ctu cdc bqn Trung h7c Ca st
For Lower Secondary School Level Friends'
Nguydn Drtc Td'tt
- MOt chrit sSng tao tr] mOt bdi to6n' 1
t Gidi bdi ki tudc.
Solution of Problems in Previous Issue
c6c bdi cias6 224 2
c Db ru ki nd1
Problems for this issue
"t11228, ..., "ltol228,Lll228,L2l228 8
o Kdt qud. hi thi hqc sinh gi6i THPT 1995 - 1996 10
o Ddnh chn cdc ban chudn bi thi vdo Dqi lqc'
For College and University Entrrance Exam Preparers'
Trd.n Phuortg - Hg drrdng cong tiSp xtic
v6i mQt drrdng c6 dinh' 12
o Tim hidu sAu thim todn hoc phd th6ng'
Helping Young Friends Gain Better Understanding
in Secondary School Maths
Nguydn Viet HAi - V6 m6t phrrong phSp
chring minh ding thfc v6cto. 15
o Db thi fiq&t sinh uin cdt l6p cfuryen DSt hP tdng h'Sp
o Gidi tri todn hqc
Fun with Mathematics.
Binh phuong - Didm cria mdi ngUdi
Ngd.n IIA - Vidt sd trong bin cd'
Tdng bidn tdP :
NCUYEN CANH TOAN
Phd tdng bidn tQP :
NGO DAT TIJ
HOANG CI]ONC
HOt DONG elEN rAP :
Nguy6n CAnh ToDLn, Hoing
Chring, Ngd Dat Trl, L6 Kh6c
BAo, Nguy6n HuY Doan,
Nguy6n ViOt Hei, Dinh Quang
HAo, Nguy6n Xuin HuY, Phan
Huy KhAi, Vtr Thanh Khi6t, Lo
Hei KhOi, NguY6n Ven Mau,
HodngLO Minh, NguY6n Kh6c
Minh, Trdn Van Nhung,
Nguy6n DEng Phdt, Phan
Thanh Quang, Ta H6ng
Qu6ng, D4ng Htrng Thing, Vo
Drrong Thuy, Trdn Thnnh
Trai, LO 86 Kh6nh Trinh, NgO
Vi6t Trung, Dang Quan Vi6n'
Bia 4
Tru sd tita soqn :
45B Hirng Chudi, Ha NQi
231 Nguy6n Vin CiI, TP Hd Chi Minh
8213?86 BiAn @P uit. tr[ s4; VU KII TEII
835611 | Trinh bd'Y; QUOC HONG
DT:
DT:
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-...qS"hE
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frs
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in F.
Z
Chring ta b6t ddu tt bdi to6n rdt quen thu6c sau ddy- : ^
Blri to6n. Chrtng minh rd.ng udi rtgi sd nguy€n n thi nt *n * 7 khdng
chia hdt cho I
Chdc rang nhi6u ban da bi6t cric ldi girli sau :
Cach 1 :
c X6tn = 3h (k e Z) thi n2+n+l = 3k@+ 1) + 1/ 3 non n2+n+L / 9.
x6t n = 3h + I (k e z) th\nz +n + | : th(h + L) + 3 / 9.
X6t n : 3h +2 (k e Z) thinz +n * ! = 3(3kz + 5k + 2) + 1 / 3 nen n2 + n + 1 / 9
Yqy n2 * n -l I / 9 v6i moi n e Z
Cdch 2 :
C6 n2 +n* 1 = (n +2)(n- 1) +3
(n + 2) - (n - 1) = 3 n6n z * 2 ; n - 1 ddng thdi hodc kh6ngddngthdi
chia hdt cho 3
n*2i 3;n-li Ssuyra (n+2)(n-l)i gsuyra (n+2)(n-l)+32 I
n+22 3;n- 1/ Ssuy ra(n-*2)(n-7\ / 3 suy ra (n+2)(n - l)+3/ 3
+ (n't 2\(n - 1) + 3 / 9Ydy nz * n * I / I v6i moi n € Z
Cach 3: (PhAn chrlng)
Gi6 str r*'+ n + | i g. bat n2 + n + | = 9m (m e Z) n2 + n + I - 9m : 0 (*)
L, = 36m - 3 = 3(72nt - l) i 3 nhrrng 3(l2m - l) Z I
A ld sd khdng chinh phuong n6n (*) khdng c<i nghiQm nguy6n. VO li !
Ydy n2 -r n * I / 9 vdi moi n e Z
4@2+n+7): (2ru*1)2+3
c 2n*li 3+(2n+D2i gdodci (2n+L)2+3/ I
o 2n*l/ 3+(2n+l)2/ 9dod6(2n+1)2+3 / 3+(2n+D2+3/ I
Ydy4(n2 +z+1) / 9+n2+ru+L/ 9 v6imoi n €.Z
V6i ni6m dam mO hgc to6n chhc cieban cd thd tim thdy nhi6u bdi toSn
"hg hdng" v6i bii to6n tr6n.
Chdng han:
Bdi I Ch.ing minh rdttg udi m7i n e Z
a)rt2+1lnt39/ 49
b)n2+3n+5/ L21
drf +sn*16/ 169.
Biri? Chilng rt inh ritrug cd.c phuong trinh sau khOng c6 nghiQm nguyan
a)xt+3x*4=49y
b)x2+25y2-2xt6:0
Bei 3 Tim. x e Z dd 25x + 46 ld. tich cia hai s6 nguyan lien fidp (Bii
'tUz0l D6 ra ki ndy Tap chi THVTT 3(201) 1994)
.CA bdn cSch gi6i tr6n d6u cd thd girlp ta giAi dugc c6c bdi to6n ndy vi
ndu ta "md m6m" s6 tim th6m thAt nhi6u bdi todn hay nita.
Nhung... v6i bii todn sau :
Bii 4-Chnng minh rd.ng grl3 + gnz * 3n - 16 khlng chia hdt cho 343
uoimgin€Z
Vdi cdc cach giAi I ;2 ;3 cci 16 ta phf,i b<i tay, khai thdc crich gia! 4 (chf
y ring 343 :73)-ta ddn v6i ldi gini that'd6 thudng" sau : 3(9zr +9nt +3n -
-16):(3n+l)3-49
o Ndu 3n*1: Tthi (32+1)3i 73md 49/ 73 non (32+1)3- 49/ 73
+9n3+9r*+3n-t6/ 343 oNdu \ru*l/ 7th\(32+1)3 -492 7
+ (32 * l)3 - +g / 73 +gn3 +gnz +3n - 76 / u3
Yay 9n3 +9n2 +3n - LG / Ytg vdi moi n € Z
Suy nghi sdu hon mQt chrit v6 cdch grAi 4 s6 girip ta ddn v6i bdi todn .
tdng qurit
E;,i toan tdng qu6t. Chrtng ruinh rang A(n) hhong chia, hdt cho pk u6i
mei n nguyAn, p nguyAn 6 ; h rtguyAn duortg.
C6ch giAi li tim cho drloc d&ng thrlc
a.A(n) : lB(n))K *pm
YdiB(n) li dathrlcbi6nn, a € Z;rz nguy6n drrong;h > m D6n dAy
bdi to6n xem nhrr de giei quydt xong.
T6c giAbdib6o ndy, mudir ifran agn cacban ringnhi6u di6u cci-v6 th4tbinh
thudng, that quen thudc, nhtrng d d<i c<i thd cbn nhi6u di 6u hdp dAn dang d<in
chd b4n vd.. . sin sdng dtta cric b4n ddn noi md c6c bqn dang cdn ph6i d6n.
i:.;?r- -- *. --,--
J
..".--affi
:*--ffi
'.=_- ;
ri@
-"-ffi#
:='?ffi
l-.--<;**.d
" "#ffi
-+:. -: t
ii_"ffi
i''- _-si%l
.-. -1ffisi
,-=:- --*.1
. -i+Yr$"'
t -T:r-S
,-'''- r' ' z
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ffi
Bidi TLl224. Tim cd'c chil sd hhd'c hhOng a'
b, c, thda md'n a66Z : dE x ac x 7 '
Ldi giai zCd.ch 1:Tac|ffifr =d6 ' a'c '7 (l)
(1) <+10Q27 +62 = 7@6)(At)
ea6(7.or-100):62
6A
e7.ac -100: -.
a6
vio < E. ro<+Q a 1.ac -1oo< 10
a6 100 110
Yqyfr = 15. Thay vdo (1) ta drroc
T665: T6 x 15 x 7<=+1005+110b = 10504
+ 105.b
alb = 45:b :9.
Thti lai 1995 : 19.15'7
Yirya=1,b:9,c=5.
Cach 2 (ciaVu Huong Gian 9,6,{, Phan Chu
Trinh, Ha NOi).
(1) +a66Zi a6
+a6Tc:ab =T[E (0 < ft < 9)
=TOE: aE .7
+TOEi 7+5Ei 7+k--5
+ac=15
Thay vdo (1) va suy ra b : 9 theo c6ch 1.
Nhd,n xdt : C6 rdt nhi6u ban grli ldi giAi dfng
kh6c nhau. Chi c<i bar, Trd'n Thi M! An,9A9
Lrrong Van Ch6nh, TX Tuy Hba, Phri YOn cci
nhdn x6t ring bdi T11224 chinh IA bai T1/213
d6 ra tr6n s6 b:io 213 (3-1995) vd de dang ldi
giAi tr6n sd b6o 2t7 (7-t995) (cach 1). Chic ld
' uhi6u ban kh6ng bidt di6u ndy ho4c bidt md
kh6ng n<ii ra.
T6 NGUYE,N
BdiT%lzz4. Cho a,b, c, d aZud. ne, n € N*
nit. nt uir n. khOn'g phd.i ld, sd chinh phxong.
Chung minh riing dibu kiQrt cd.n ua dit dd
a + btim - c + d.fr ld, a = c ud.btli = d\lrl
Ini gl.i: TltJdchdttachrlngminh: Nduz € N*
thi fi Jtri co thd Ia s6 nguy6n ho4c sei vo ti (1).
Thit vdy gi6 stnln ::, (r, s) : 1, s > 1.
Goi p ld m6t u6c nguy6n t6 ctra s. Ta cd
s2ru : P -* i p2 -r'. p.'}nii giethidt (r, s) : 1'
Trl siA thidt o +br[ru = c + d'{n (2) suy ra
(o. - c\l : d.Zn. + bznt. - zbrl{rnn (3). Ta phnn
biet hai tnldng hoP :
2
a) Ndu bd, = 0 ch&ng h4n b 0' Tt (2) ndu
d-c
d * 0-,[n :; € Q miu thudn v6i (1). VaY
b=d,=0---a=c.
b) Ndu bd * O. Til (3) 1'l*" € Q' -
Tt (2) + o{n +b{tl1n : dn * c'ln +
O -c\r[n = d.n -b{nxn € Q. Ndu a * c+
irt e a tr6i gia thi6t, Y4v a = c +
br{m : d,[n.
Di6u kiQn dtr ld hidn nhi6n'
Nh4n x6t : 1) Didm mdu chdt cria bdi toSn
ln khing dinh (1). Hdu hdt cric ban de kh6ng
"frrr"g ,ii"i, ttl miL coi nhrr mOt sU kiQn hidn
nhi6n (xem th6m bai \E - 3 : ; tr6n THTT
41226 6 d6 6c t6c g1il cd n6u mQt y kidn xung
quanh khing dinh (1))
- 2) C6 m6t s6 ban de sai ldm khi cho rang
hi6u cria hai sd vO ti kh6ng thd la sd nguy6n'
3) C<i thd chrlng minh d6 ddng ring {17 chi
c<i thd li s6 v0 ti hoac sd nguy6n vd tdng qu6t
hon, nghiOm cira Phtrong trinh
rk + ou_rrk-1+... + atx - n = Q (ara Z)
chi ctj thd la sd vd ti ho6c s6 nguy6n'
$ Cec ban c<i ldi giei t6L : Nguydn Minh
Hd.ng (9, Hei Drtong, HAi Hrrng), Nguydn Thd
Vtni (9,D6ngNai), LatuLhTho (9,Thanh Hcja),
Phanr Quang Vinh (9, Ha NOi), Nguydn Tud'n
*rn Q,oaclicl, Nguydrt NguyQt Minh' (9,Ddc
L6c), N
g uy 6n Tr d.n M an h Q udn (9, Thr)a Thi6n
}J:ure), irh.n Nguydn Thq (8, Hd Tinh), Nguydn
Ngqc Minh (9, Th6i Binh), Nguydn Hd'i Dang
(g] Quang "bi), Cao Thi Ly (9, Th6i Binh),
Nguydru Trgng Kian (8, Nam Hi)'
DANG HUNG.THANG
Bidi Tgl224. GiAi phuong trinh' nghi4m
nguyAn:
x6 + Bx3 * 11x2 * 28x * 12 + 3y2 : o (*)
Ldi giai. Ydi x > 0 thi vd tr6i cria (*) lu6n
lu6n drrong n6n khOng c<i nghiOm'
VAvr < 0. Ta c<i (*).*(r3 + 4)2 + ll(x+2)2 -
- 16(r + 3) + 3y2 = 0. (**)' Nhrr viy, v6i r < -3
thi vdtr6i cria (**) lu6n lu6n drrong, n6nr < -3'
Suy rar li s6 nguyOn thu6c khoAng (-3 ; 0), n6n
chi c<f thd nhin hai giri tri la - 1 vd -2, vd phuong
trinh c<j cric nghiQm nguyOn (x ; y) sau dAy :
(-l; -2), (-l;2), (-2 ; 0).
Nhin x6t. Cd 132 bai giei, tri mdt vii bii
n6u s<ii nghi6m, cbn tdt cA d6u gi6i dting' C6c
ban sau day cO ldi giai t6t : Pham Thi Vd'n
diorg (8A Chuy6n Bac Li6u - Minh H.i,j), Mai
XuaiTrxang (9T Chuyon DOngAnh - Ha NOi),
Trinh Hoiti d.n (PTCS LO Loi Tx Hd D6rrg --Hd
"tdy), Dinh Cao Cuimg (8 Chuy6n Ba Ddn
Quang Tr4ch - QuAng Binh)' Nguydn Ngsc
Miruh (9A Luong Vnn Ch6nh - Phf YOn), Ddo
Th.d Vu (BAr PTCS.Gi6ng V6 II - Ha NOi),
Nguy1ru Dic'Xud.n Binh (9T PTNK Hd Tinh),
Mai Hdn Giang (8T Chuy6n Le Khidt - QuAng
Ng6i), Nguydn Thd.i Son (9T NK Nga Son -
Thanh H6a), Trdn Van Nghia (9A Luong Van
Ch6nh - Phf YOn), Nguydn Tud.n Anh (9 Todn
Phan Chu Trinh, Bu6n MO ThuQt - Dek Lak),
Nguydn Thd Vinh (9II THCS Tnrng Vrrong -
Ddng Nai), Trd.n Khoa Vaz (8A NK Quj'nh Lrru
- Nghe Ant, Nguydn Ch; Thdnh (8T, Nguy6n
Binh Khi6m - Vinh Long), Vo Ch; Thd,nh (9
Toan Chuy6n L6 Khidt - QuAng Ng6i), Hoitng
Phuong Dong (9A PTCS Cdc L6u - Lio Cai),
Hod.ng Qudc Hoa (9A Chuy6n Lrtong. Vdn
Ch6nh - Pht1 YenS, Truong Quang Trung (84
Trung Hoc Chuy6n Kontum), Phant Thu
Huorug (9Ar THCS Hdng BAng - HAi Phdng),
L6 Tdm (9 To6n THCS Chuy6n Phri Thq, Vinh
Phu), Hoirng Trung Tuydn (11A PTTH Ha
Trung - Thanh H<ia).
DANG VIt]N
BiriT4l224 z
Ch.o tam gidc ABC
ctin tai A AB : AC).
TilBKABMTAC.
Chtng minh rd.ng
AM ,AB. I
MC=2\ud--t
Ldi Siai
Bdi to6n chi dring
trcngtrudnghop gdcA
nhgn. LJy E ddi xrlng
vdi C qua A. D6 thdy
LBCE vudngtaiB.
ThuQn, 8T Chuy6n Nghia Hdnh, Vd Trd.n
NguyAn LQc,9T Chuy6n Le Khi6t, La Hoitrug
Dic Khd,nh, 9T Chuy6n Nguy6n Nghi6m, Drlc
Phd, Qu6ng Ngdi, TrdnVidt NhQt,8/1 Chuy6n
Nguy6n Khuy6n, Dd Ning QuAng Nam - Di
Ning, Le Ch; Thd.nh,9l Nguy6n Tri Phrrong,
Hud, Thrla Thi6n - Hud, Nguydn Thi Sen,9A
Trrrdng Thi, Trd.n Khoa Van,8A NK Quynh
Lrtu, Nguy4ru Tidn Trung, 9T, NK Vinh, NghQ
, An, D6 Thitnh Trung 8T Nguy6n Hi6n, Nam
Ninh, Vi7 Thi Thiry Duorug, 8T NKY Y6n, Dodz
Nant Thd.i, 7T, N
guydn Trgng Kian, Phqm Thu
Giang, Nguydn Tu(in Anh, 8T Trdn Dang
Ninh, Nam D!nh, Nam }Jd, Mai Xudn Truarug,
9T Chuy6n D6ng Anh, Nguydn Ti^tn g, Vu Manh
Tud.n 8CT Ti Li6m,.Pfram QuangVinh 9A,Bd
Ven Den, He NOi, Bili Hdi Nant,9A Le H6ng
Phong, Ngd Quy6n, HAi Phdng, Biti Thu Nga,
9A Chuy6n Phong Ch6u, Vinh Phri,Nguydn Thi
Hdi Ydn,9 Neng khidu Bdc Giang, Hd Bic,
Nguydn. Nggc Ti,88 Trong didm, Ha Long,
QuAng Ninh, Ngzydn Hoiing Tilng,8T cdp II
Nang khidu Thrii Nguy6n, Ld.m Mq.nh Tudrug,
9A THCS Hgp Giang, Cao Bing.
VU KIM THUY
Bdi T5l224. Cho hai hinh chft n hQt c6 chu
ui bd.ng nhau. M\t hinh c6 cd.c canh duoc t6
md.u d.6 ; linh kia c6 cd.c canh duoc to mdu
xanh. Ngudi ta chbng hai hinh cht nhQt'lAn
nhau sao cho phL.n giao cia ching lit mQt hinh
bd.t gid.c. Ching ntinh riing : hinh bdt gidc c6
tdng dQ diri 4 caruh md.u d|bang tdng d0 diri 4
canh md,u xanh.
Ldi giai. Trong
hinh vO b6n, hinh
chit nh4t canh d6
ABCD vd hinh chir
nhQt canh xanh
AtBlCtDt cirng cri .
chu vi lit p, c6tni
nhau tao thdnh
hinh 'b6t gtdc .
MNEFPQIK. D4t
m.=MN* EF+PQ
-t IK vb,n : NE + FP + QI + KM.Tacd cf;ctam
gr6c sau dAy d6i mQt ddng dang vi ld tam giric
vuOng vd cci mOt gric nhgn bing nhau : LAJM,
LBTNM, LBNE, LCTFE, LCFP, LDTQP, LDQI,
LATKL Do dti, dat
KA+AM
n = --@-' ta c6
JA+AM NB+BE FC+CP
h=.p-:--NE--:--FP :
fr
BC2 RC2
Suy ra MC = CE : ZAC
Mat kh6c AM : AC - IvIC =
BC2 21192 - Bgz
- a/1 -zAC _ LA.C
TiI (1) vn (2) tac6:
AM Z,q.Ca - SC2 ^ , AC,2 -
MC: BC, :'\Ad -1'
(1)
(2)
NhOn x6t:
Giei tdt bdi niy c6 cdcb4n t Pham Thi Vd.n
Giang,8A Chuy6n Bac Li6u, Minh HAi, DQng
Minh Thanh, I An Lac Tdy, Kd S6ch, S<ic
Trdng, LucVan f/do, Nguy6n Manh Cttdng, 8/1
H6ng Bhng, Q5, TP HCM, Tr1.n Anh Hitng, S
Chuy6n Nguy6n An Ninh, Vflng Tdu, Bd Ria -
Vung TAu, Nguydn Thd Bd,o,8A, Hi6p Thinh,
Thti Ddu M6t, Sdng B,6, Trd.n Dic Sinh, 8B
LttongVan Chrinh, Tuy Hba, Phri Y6n, Truong
Quang Trung,8A Chuy6n Kon Tum, Phan Thu
lfa, 8Ar Trdn Van On, KrOng Phc, Luong Hrtu