Nr.'; T k*jts
eo crAo DUC vA DAo rao * nor roAN Hoc vrpr NAM
*PHUOIIG PHAP ilo! SUY UA GIG BAT TOIiII . . .
t^A.-4-:
*xuNs ounNH IUQT (ONG TH6C riUH THC TICH
1:,2^,A.2?t
*TIM HIEU THEM VE BAT DANC THUC LI.JONG CIAC
Thdv gido Y{i Hliu Binh vd cac h7c
sinh gi6i Thcinh pho', Todn quo'c cia
lop 9H trudng Trtmg Vuong, Hii N)t
7 <rrrt
1996
AM THU
TAP CHi RA NGAY 15IIANG TMXC
ToAI{ trQc vA TubI TR.E
MAT'HE.MATICS AND YOI.JTH
MUC LUC
Trang
o Ddnh cho cdc ban Trunghoc Co sb
For Lower Second,ary School Leuel Friettds"-
Nsuv\tu Van Vinh - Phtiong ph6p nOi suy va
'.i.iii t ii 'o "a" ainn aa tntc' 1
c Phan Tudn CQng - Dd thi t-qyd' sinh nhd
th6ng nang khi6u ti"f, ffai Hring ' 2
c Gidi bdi ki tudc
Solutions of Probtems in Preuious Issue
ba"uii cias6225 3
c Db r0 ki ndY
Problems for this issue
r{hi0,,..., Tlol2lg,L1l22s,Lzl229 I
o Nguydn Dung - D6y Fibonacci vd mach di6n 10
o Tim hidu sAu thAm tudn hoc Phd thing
To HetP Young Friends Gain' Better
tlnd.eritand.ing itt School Maths
Trinh Vinh Nggc - Xung quanh-mQt
.o;J\t i. ti"fi irra ticrr Jria trl di6n 11
o Hoc sinh tim tbi
Young Friends' Search in' Maths
Mdt phrrong PhaP chrlng minh c6c
ilai a'b"g tnlrlt aai so - 13
o Ddnh cho ctic ban chudn bi vdo Dai hoc
For College and Uruiuersity Entrance
Exam PreParess
Le Thdng Nhdt - Tim hidu th6m v6
fat ia"S"tlrrlc lrrong gi5c trong talar' g;t'itc L4
o Nsd Dat Tu - Gi6o sti LO Van Thi6m,
,"rfia toa" hoc cci c6ng ddu trong vi6c
;; ddt "e ph6t tridn n6n tosn hoc nddc ta
o Thi chqn dQi tuydn ho.c s-inh Vi6t Nam D>^ o
du thi lo6n Qudc td 1996 51a o
Tdng biin tdP :
NICUYEN CANH TOAN
Phd tdng biin ihP :
NGO DAT TTJ
HoANG cnfNc
ttOl oOllc elEH r4e :
Nguy6n CAnh Todn, Hoing
Chung, NgO Dat Tri, L6 Khic
BAo, NguY6n HuY Doan,
Nguy6n ViOt Hai, Dinh Quang
H6o, Nguy6n Xu6n HuY, Phan
Huy Khfri, Vrl Thanh Khidt, Lo
Hei Kh6i, NguY6n Ven Mau,
Hodng LO Minh, NguY6n Khic
Minh, Trdn Van Nhung,
Nguy6n Dang Phdt, Phan
Thanh Quang, Ta Hdng
Quirng, D4ng Htng Th5ng, Vfr
Drrong ThuY, Trdn Thdnh
Trai, LO 86 Kh6nh Trinh, Ng6
Vi6t Trung, DAng Quan Vi6n'
o Gitii tri todn hoc
Fun with Mathematics
Binh Phuong - GiLi ddP bni
Di6n s6 vdo hinh vu6ng
o Nguydn HuY Doan - Kdt quA trAn ddu'
Bia 4
458 Hirng chudi, Ha Noi DT: 8213?86 BiAn tQp ud' tri su: vu KIM THUY
2J1 Nguy6n V6n Cil, Tp IId Chi Minh Ot, ggSOf l 1 "* iri,n bav ; QU6C IIbNG
ffi €rE
=h
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ru,
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ru
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r=i
li----=
l-l
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t-:
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-2 1
e>
>o
Zlo
*g.i tr
oi o.
Z
Trong cric ki thi hpc sinh gi6i, thddng c6 cS.cbii todn v6 x"c dinh r46t
da thrlc.
Dd xdc dinh c6c hO sd ctia mQt da thrlc, ta thudng dtng ph6p chia da thfc
ho5c ding phrrong phrlp hC sd bdt dinh, phrrong ph6p 9;6 tri ri6ng kdt hop
vi6c giAi rn6t hQ phrrong trinh. Bii vidt ndy xin gi6i thi6u v6i c5c ban m6t
phuong phrip n6i suy cta Niuton (Newton) cho ph6p tim nhanh cric hQ sd
cria m6t da thrlc.
Kidn thrlc cdn thiSt g6m :
1. Dinh li B€zout (BOau) : Phdn drr cria ph6p chia da thrlc P(r) cho nhi
thrlc bAc nhdt r - o bing gi6 tri cria da thtlc tai didm o, t(tcld P(a).
2. Phuong phd.p nQi suy Newton
Dd tirn da thrlc P(x) bdc kh6ng quri z khi bidt girl tri ctra da th(rc tai (n
+ 1) didm : C 1, C2,.... Cn+ , ta cci thd bidu di6n Prxtdti6i dang :
Pk) : bo*br(x - Cl)+ br\x- Cr)tr - C2) +... +bn(r - C,)... (r - Cn)
Bing crich thdr ldn lrrot bang cac 916 tri C1, C2, ..., C,*, vio bidu thrlc
P(x)taldnluottinhduoccrichQs6b.'b1,b2,..-,,b;.
Dtr6i dAy lir m6t s6 bni torin vAn d"ung.
P(0) = 19 ; P(1) : 5 ; PQ) : 1995.
Gi6i : Dat P(x) : c * b(x - 0) + a(x - O)(x - l)
=c+bx+q"x(x-l).
Chor : 0, P(0) - c suyrac = 19.
Chor: 1,P(1) = 19+b suyra 19+b = 5,b: -14
P(x) - 1?-l4x+ax(x-1).
Cho r = 2, P(2) : 19 - 28]-2a, suy ra a : 1002
vQy P(x) = 19 - 14x + t002x(r - 1)
Rrit gon P(x) = t00zr2 - 10i6r + 19.
Bni 2 : Iirn mQt da thic bd.c 3, P(x) cho bidt :
P(0) = 10; P(1) = 12 ; P(2) : 4; P(3) : 1.
(D6 thi hoc sinh gi6i C.H.D.C Drlc 1979).
Giai : Det P(x) : d * Cx *bx(x - i) + ax(x - l)(x - 2).
Chor = 0, P(0t = d, suy tad, = 10.
P(D = 10 + Cx *bx(x - 1) + an(x - 1)(x - 2).
Chor = 1, P(1) = 10 +c, suy rac :2.
P(x) = l0+2n*bx(x - 1) + a$(x.- 1)(x - 2)
Chor = 2,P(2): 10+ 4+2b, suyra b = -5.
P(x) : l0 +?n - tu(* - 1) + o,x(x - l)(x - 2)
Chor = 3, P(3) : 10+6 - 30+6o
5
SUYTaO=r.
vay P(x): 10 + 2^r - hx(x- 1) + f,rA - \@ - 2).
5^ 25^
Rrit gon P(x1 = irt - i r, o lzx + lO.
Bii 3 z Tim mQt da thic bdc 3, P(x) cho bidt khi chia P(x) cho (x - 1), (x -
2), (r - 3) dbu duoc du ld. 6 ud P(-1) = -18.
(Dd thi hoc sinh gi6i Qudn I ldp 8 - 1994 - 1995)
Gi6i : Theo dinh li B6zout ta cci :
P(1):P(2)=P(3)=6
Do d<j d.6ltP(n) = d-rc(x - 1) +b(r - t)(r - 2) +a(x - 1)(x - 2)(x - 3) Cho
x: l,P(l) = d, suy rad = 6.
Chor = 2, P(2) = 6 *c, suy rac = 0.
Chor : 3, P(3) = 6 +zb,suy ra6 = 0.
P(x) = 6*a(x - 1)(r -2)(x-B)
Chor = -1, P(-1) = 6 - Z4asuy ra a = L
vQy P(x) : 6 * (r - 1)(r - 2)(x - 3)
Rlit gon P(x) = x3 - 6x2 * 11r.
Bai 4 : Cho da thrtc Pk) bdc 4 th6a mdn :
P(-1) : O vit, Pk) - Pk - 1) : x(x + l)(b + 1)
l. Xac dinh Pk).
e
2. Suy rit gid. tri, crta tdng sau ddy (n ld. sd
nguy€n duong)
S = 7.2.3 + 2.3.5 +... + n(n + 1)(2n + 1).
@6 thihqesinh gi6i Tb Hd ChiMinhlop 9 : L992).
Giii : Cho r : 0, suy ra P(0) - P(-1) = 0
mn P(-1) : 0, v4y P(0) = 0
Chor l2in lrrot crie giritr!r = -1 ; x = | ; x :
2, tanh{n duocP(-2) = 0, P(1) = 6 ;P(2) = 36.
D4tP(x) = e*d,(x+2) +c(x+2)(x + 1) +b(r +
2)(x + t)x + a(x + 2)(x + 1)r(r - 1).
Chor = -2,P(-2): e
suyrae=0.
Chor = -1 ta suy rad = O
Chor:0tasuyrac=0.
vSy P(x) = b(x + 2)(x + lh I a(x + 2)(r + th(r - 1).
Chor = 1. P(1) = 6b. vAv b : l,
Cho r = 2', P(2) = 24'+ X4a = 36
1
vdy P(x) : i,*@ + \2@ +2)
2. P(x) - P(x - L) = x(x + 1)(2r + 1).
Chor: lt2;3:n tac6:
P(1) - P(O\ :'1.2.3
P(2)-P(l):2.3.5
P(") - P(n - l) : n(n + 1)(2n + t)
suy ra 'P(n) - P(0) : 1.2.3+2.3.5+ ...+tu(n
+ 1)(22 + 1) I
Do dci : s = P(n) = )n(n + L)2(n +2).
BAi 5: ,
Cho bidt da thrtc bqr hai P(x) c6 3 nghiQm s6
phd.nbiQt a, P, y. Ching minh rd'ng P(x) = 0 udi
mei x.
Gid.i:
Ta cd P(a) = P(p) = P(y) = O.
Deyt P(x) = c I b(x - o) * a(x - a)(x - il.
Chor = a, P(a) : c,v4yc : 0.
Chor= p,P@)-b@-a):0
iF*asuyrab=0
Chor = y, P(y)' a(y - o)(y - p) = O
uiy*q,T*Psuyra
a = O.YQty P(x) = 0 vdi moir.
Dd luyQn tQp, cdc bpn h6y glhi cdc bii torin
sau diy :
1. Tinh eic tdng sau dAy
^ (x-a)(x4) , (x-b)(x--c) , (x--c)(x-a)
A = @-e)(c4)* (a4)(a-c)-@=7fr=d
(x--a)@a)@-c), (x -b)1, -c)(x - d)
b = (d-a)(.d4)ld,-c) - 1a= o11a 41a - a1
. (x - d)(x - a)(x - b)
- (c 4)@=q@ 4)
DapsS:A=l;B=1.
2. Tim mQt da thrlc bQc 3 cho bidt
P(0) = 2, P(l) = 9, P(2) = 19, P(3) = 95.
1
,v4ya= 2
nii rsr uuvdrq srNrr pHd TITONG NAxc KHICu roAN r,6p ro v0nc z
xAvt Hec lees - 1ee6 TINH nz(r nuNc
. Thdi gian: 180 phfi (hhOng kd giao db)
Cd.u 1.' Cho phrrong trinh x2 - (a - \)x - a2
+a-2:0 (1)
a) Chrlng minh phtrong trinh lu6n cd 2
nghiOm trdi ddu.
b) Ki hiQu m, n ld nghiQ^m crla phriong trinh
(1). Tim grt td ciaa d6. rnz +nt dat gi6 tri nh6
nhdt.
Cdu 2: Tim nghiQm nguy6n dtrong ctra
phrrong trinh
1 1 t t[44++
_f,_r r_
L2- 2^3 -1- "''rr(, + 1) = 644
Cd.u 3: P li tQp hgp nhtng s6 t1t nhi6n cd
tinh chdt : ndt 2 sd thuQc tflp P thi tdng ctia
chring cring thuQc P. Gi6 srl a - b ld. s6 nh6 nhdt
trong cdc sd dangr - y v6i x.y li nhitng s6 thuQc
Pvd.x>y.Ddtd.=a-b.
a) Chtlng minh b chia hdt cho d.
62
b) Ki hieu K : -2. Chrlng minh v6i s6 tg
d
nhi6n A bdt ki md k > K thi sd hd thuQctap P.
2
Cd.u 4: Cho o, b, c ld sd do 3 cqnh cria m6t
tam gi5c. Chrlng minh
,t 1 1
(a+b +") (o+a *o*"*"*")*
, Sobc
L ----- - E
' (a + b)(b + c)(c + a)
Cd.u 5; Cho tam gS.6c ABC.
a) Ldy 2 didm X, Y tr6n AB, AC. Chrlng
minh:
dtAXY AXAY
dtABC = ABAC
b) Gqi M, N, P ld chAn cric drrdng phAn gi6c
trong crla tam giric. Chfng minh ndu dtABC =
4 dtfuNP thi tam $6c ABC d6u vi ngugc 14i.
Cd.u 6 : ABC lBr tam gi6c d6u c4nh bing 3.
Ldy 2 didm E, F tr}n AB, AC. Chrlng-minh ,Etr'
qut tim tam grdc ndu c6 hC thtlc
11
nn + Ae = 1vi ngUqc lai.
PHAN TUAN CONG
B,ni'I11225. Gidi phuong trinh:
ra + 1srl5 - 7y2 + bz - 28t[b =
: (34 - t?/t[S - 3r21x2 + arIS
Ldi giai.
Dato : x2 + 4t[5,tac6a>ovd"a2-7a-
23: (34 - 3,o)lG-
e(a2 - 7a - 28)2 = (34 - 3a)2a
a(a4 - 23a3 + lg7a2 - 764a * 784 : O
e(a2 - lZa + 1&)(a2 - lla+49) = o
T^
*lr'-l2a +16=0 (1)
et 1",-tra*49=o (2)
Do? > 0 nOn (2) vd nghiOm vi vdtrdi bing
(a - 5,5)2 + L8,75, vi lu6n lu6n drrong. VAy
phriong trinh da cho trtong drrong v6i (1) vd ctj
nghi6mlA:r, - {5 - l ; xz: 1 - {5.
Nhdn x6t.
Ldi giei t6t g6m c6 : Trd.n Tdt Dat,8A Chu
v6n An - Tny H6 - Ha NQi ; Nguydn Vinh
ThuQn,8T NK T! Ha Tinh ; Triin Tudn Anh,
8T Le Qui D6n - Nha Trang - Kh6nh Hba ;
Chung Nhdn Phil, STrNguydnAn KhuongH6c
M6il - Tp Ho- Chl Minh ; Nguydn Anh Tudn, gt
Phbn Chu Tlinh - BMT - D aklak ; Hodng Phuong
Ddng,9APTCS C6c l6u - Lio cai ; Cao Xtdn Sinh,
9T Nga Li6n - Nga Son - Thanh Hria. ,
oAuc vl 6r.r
B,diT2l225
Gid,i phuong trinh nghiQnt nguyAn duong
xz + rz : 21lLlsssk+t116 - e)
Ldi giei. (cira ban Le Thi Tdm gAYinh)
Trd6c h6t ta chrlng minh bd d6 : Ndup li sd
nguy6n t6 dang 4h + 3 vdxz + y2: p thi x i p,
vi p.
ThAtvAyn6.tx/ pthiyZ p
Theo dinh li Fecma d -1 = x&+2 = I
(modp), y4k+2: 1 (modp). Mat kh6; "i
x2 : a2 (modp) ---+ y2(2k+1) = -A2(2k+t)
(modp) + 1 = -1 (modp) V6 lf .
Trd lai bni todn vi 2011 li sd nguy6n td dang
4h + 3 do do theo bd d6
x = 20llx1,
I = 2011y,. Dat 2n = lgg* + L
-x?+y?=ZOtt2n-zGO-z)
Tidp tuc nhtr vAy n l6n ta cci
*1+*=10-z (t)
vdi x = 2ollnxn, y = 201lnyn.
Bing cach thii trgc tidp ta thdy nghiQm
nguy6n drrong crla (1) ld.xn, ln, z) = (1, 1, 8),
(1, 2, 5), (2, 1, 5) vd. (2, 2, 2) V4y nghiQm nguy6n
duong cria phuong trinh dE cho li
z}tln , 2OlLn, 8), (2011n , 2.2}t7n , 5),
(2.20L1n, 20ltn, 5), (2.20L1n, 2 . 201tn, 2)
199* + 1
itd6n =
Nhfn x6t : Nhi6u b4n tham gia gi6i bdi ndy
vi cri ldl giai t6t ntrrt : Trdru Tud.n Anh (8, Nha
Trang) Biti Mq.nh Hitng (9H Trtlng Vrrong)
Phan Chi (9A chuy6n ngrl Hd nQr) Ngd Kien
Cudng (9, QuAng Ngai), Nguydn Minh NguyQt
(9, Hei hrrng), Nguydn Trung Ki€n (7 Hi n6i),
NguydnVd.n Quang (9, Thanh h<ia). MQt sdban
thta nh6n bd d6 md kh6ng chrlng minh. Cri m6t
vdi ldi gi6i sai'
DaNG HUNG rHiNG
BdiT3l225 : Cho a, b, x, y lit, cdc sd th4c th6a
md.n
(1)
(2)
lra \/4 1
4a b a *b
I
lx2+y2:1
ching minh rd*g ,.tl^l^1- ,"ll! = 2
Ldi siai. "t ^ Nsr',r!)t'*ol.Torfr:';'il"
ThuAn Thdnh, Hn B6c.
Thay 1 = @,2 +yz)z vdo (1) ta cci
x4 , y4 @2 + y2)2
a b a*b
+(bxa + ay4)(a +b) = ab(x4 +y4 +%ryr)
oa2y4 +b2xa - 2abr2yz = g
o(ayz -bxz)2 = 0
obxz - ayz
. x2 y2 *2*y2 1
Tt dd tac6; --"--- (3)
a b- a+b -a*b
11994 yt994 1
TrI (B) tac6: on, = "Un, = C .ifm
x1994 y1994 2
YQLy I assT * "* : (o + qssi (dpcm)
Nhan x6t 1. C6 fit nhi6u c6c b4n gui ldi
g,At. Tdt cA d6u dring. Cric b4n sau dAy cti ldi
giei tdt : Nguydn Danh Nam, 8T, NK B5c
Giang, Hn B6c ; Nguydn Hir. Duy,9T, Chuy6n
V-T Phf Xuy6n, Hd TAy ; Bili Vidt L!c, 8A;
Phqm Quang Vinh, 9A, Bd Ven D2rr, ; Dinh
Qu6c Vlnh, gAb PTDL D6ng D6, Nguydn Tit ng,
8CT, Tt Li6m, Nguydn Thi Minh Thoa, 8C,
Nggc Ldm, Gia Ldm, Hn n6i ; Nguydn LA Thily,