Tạp chí toán học và tuổi trẻ số 299 tháng 5 năm 2002
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Nam thO 39
Todn hoc vd fudi trd
Mathemafics and Youth
s6 zee (5-2002)
Toa soqn : 1878, pho'Gidng V6, He Noi
Em
ai
I:
DT - Fax :04.5144272
a n h octt@Yah o o.com
to
TRONG SO NAY
O
o
o
Dinh cho Trung hoc co sd - For Lower
'
Secondarv Schools
Nguy|ru ViAt Hd.i'- Ddu hi6u chia hdt
trong hO thAp phAn
VKT _Vd H6i nghi To6n hoc to).n qu6'c
lAn thir 6
Chudn bi thi vio Dai hgc
-
Trdn Xudn Bang - Bdi to6n v6'dridng
th&ng c5t d6 thi ham bac ba
D6'thi tuydn sinh l6p 10 chuydn To6n Tin trttdng Dai hoc Srr pham He N6i
n[m 2001
Nguydn Vdn Th6ng - Tim cric tri cta
hdm s6'nhidu bidn bnng cSch khAo s6t
lAn luot trlng bidn
O
Lich srl to6n hoc
W;;@iCA-
-
History of Math
Ramanujan -M6t rhdn
ffi-nsTaiTqc
;:s'
* Thi giii
to6n tr6n
Tq.
@
D6 ra ki niy - Problems in this Issue
Til 299, ..., T70 I 299, Ll, L2l 299
m6y tinh b6
iiri
bni ki trudc - Solutions to Previous
@ Giii
Problems
Giii
University
Entrance Preparation
Duy Phuqng
@
c6c bdi cria sO'ZgS
@
Ban c6 bidt ? - Do you know ?
Pharu Thanh Quang - ChuY6n thri vi vd
s6'nguy6n td
@
Tieng Anh qua c6c
bii
to5,n
-
English
through Math Problems
Ng6 Vi6t Trung - Bei s6'53
CAu lac b0 - Math Club
Sai ldm 6 dau ? - Where's the Mistake
Bia 3 : Gi6i
Bia
4:
tri
to6n hoc
-
?
Math Recreation
Chi nh6nh Nhi xudt bin Gi6o duc tai
thdnh phd Hd chi Minh
_;;;"
fru6n*,.uslit
i
fi1 srri+Vt' 4
oai4!*ratl
rsd
. Biii cdn ddruh
r,B GIJI BAI cHo rAP
cHi ToAN HQC vA rudt rnB
may hodc uiAi tuy khbng cldp x6a tAn m6t mdt gidy. Hinh ud rd rang. Kh1ttg
n.hdn ban photocopy
t t ons qu(i 2000 chtt httac khong qud 6 trong ddnh may. Ei ra c:fin gtii ltitrt liti
di ci)a hoc sinh ph6 thong.
nhdn
gioi.
Khong
"cni
aiiy di hct ua ftn thdt, dia chi dd tian ti€n ha. M6i bdi chi gii mot ltin.
ghi rtt
4nh g*i cho tap chi phai la anh mdu, cd nh6 nhdt ld.9 x 12, hh6ng dp plastic. Sau rinh
, ll6t tai dii
.
.
.ful, "i:tu::!,;i:7,;rr,.*;n{J'",u,
ffan m6,t tit sidv rians tui m6i bdi do 1 sido uian chdv)
phtu tuAn bAn tai gii tA- rii tai, tAn phai ghi io tAn ua tOp, trtttng, hqtQn (OuQn), tinh
phd). Bdi siz;ri; i,2i ud iat dii chi iTap chi 7oan hgc-.ttd, Tudi t4 187r.,.ph6
?r,nari
'Giitng'V6,
Hd N6i. Ngoii phong'bi ghi rd: ort'ini gicii todn s6'tap chi... Kh6ng gxi bai
cia nhi€u s6'tap chi trortg ci)ng m6t phong bi'
. Thdi han nhdn bai gi(ii Cidi bdi hi trttdc ld hai thang tlnh tit cu6i thdng ra s6'tap chi d6'
Ctic bdi giai cia chuy|n muc khac ld 1 thdng'
THTT
DAU HIEU CIIIA HET
TRONG HE!a THAP PHAN
Odnh aha eae.
IRIIIIG IIIIG GII
Vr
NGUYEN ViET HAI
SIT
Trong'IHTT s6291(9120CI1) ciic ban Nguydn
Bon, Nguyen L€ thOu dA rinh bly "rnOt ti0u
chuAn chia het" cho c6c sd 16p kh6ng lh bOi cira
5. Phdt tridn ! tuong ndy c6 thd xAy dtmg mOt
tiOu chudn chia h6t tudng tu cho sd 16 p kh6ng li
boi 5 dd d6n dOn m6t ddu hiOu chia hdt chop mOt
ci{ch tcing qudt trong hC thap phan
Trong bii ndy ta lu6n ki hiOu
p lh sd 16 lon
hon I mh kh6ng ld b6i cira 5, nghla lh (p, 10) =
1. Vdi p € N* vi ciic s6 m, n, t e Z th6a mdn
m = pt + /r ta clng vidt ld m = n (mod p)
M6nh d|l.Vdi m6i so' p = l}s + r vdi r = I,
3, 7, 9 ludn tirn duoc s,3' rtguy€n k (phu thudc
vdo s, r) dd 10k = I (mod p), nglia ld 10k - I
chia h€'t cho p.
Chung minh :
. Ydi p = 10s + 1 ta chon [ = -s (mod p) hay
k = pt - s (r e Z) th\ I1k - | = l1(pt- s) - 1 =
10pt - (10s + 1) =p(10r - 1).
. Ydi p= 10s + 3 ta chon t = 3s + I (modp)
hay k = pt + 3s + I (t e Z) thi I1k- 1 = 10(pt +
3s+
l) -I--I}pt +3(10s+3)- p(I}t+3)
o Ydi p = 10s + 7 ta chon /r -(3s + 2) (mod
=
p) hay k = pt - 3s * 2 (t e D th\ 10/< - I =
l0(pt-3s-27 - | = llpt - 3(i0s+7) =p(10r-3)
o Ydi p = 10s + 9 ta chon k = s + 1 (modp)
hay k = pt + s+ I (r e Z) th\ 10k- 1 = 10(pr + s
+ 1) - | = llpt + (10s + 9) = p(llt + 1) (dpcm).
Tt c6oh chon trOn d0 thAy rang
cYdip= 10st l thiltl P-\l
= 10
oYOip=10s+3thil/d3
Trong bing du6i day d6 chon sO k c6 gi6 tri
tuy6t ddi nh6 nhAt tuong trng vdi s6 16 p kh6ng
lA bOi 5 mh p < 101, trong d6 p h sd nguy0n td
thi duoc in d0m.
M€nh dO 2. 56' tr.! nhi€n N = a1d2a1...a1
c6 cing s6' du voi s6' ngultdrt lTt-t .D =
lT'-t(a, + ka2 + tloj + ... + kt-ta,) khi chia
cho s6' p = /0s + r vji r = l, 3, 7, 9, cdn k ld
s6'thda mdn
l)k = I
lmod p).
N6i ngdn gon ld N = l7t-tD (mod p).
Chthng minh : Ta chrlng minh bang quy n?p
theo t.
Vdi r=1 thi N = LxD = a1 mOnh dd (IvD) ding.
o GiA sri MD dfng vdi r. X6t sd N, c6 r + I
chft s6. N1 = a1o2...etet+t = 10N + a,*1 vdi
o
N=
"r"r-q
.
Theo MDI c6 10fr - | - pv vdi v e Z.Tn d6
Nr = 10N * at+t = 10(N + ko,*) - (l0k - l)a,*1
= 10(N + ka,*) * pva*t hay N1 : 10N +
llka,*1(mod p)
Theo giA thidt quy nap thi N = 10-rD (mod p)
(1)
nOn N1 = l}t D + lOka,*1 (mod p)
X6t i0' Dr = Ld @t +ka2 +...*t-' q
= l}t .D + lot kt a,*,
+
t
q*1)
k'-r - l)a,*, + loka,*1
v6i r > 1 thi lO'-rtl-r - I chia hdt cho 10fr-1
- pv nOn lOt D, = lOt .D + lOka,*1 (mod p) (2)
V6i r=1 thi 10'Dr = LOt.D + lOka,*1cfrng c6 (2)
So siinh (1) ve (2) rflt ra N1 : l}t Dr (modp),
nghia li MD dring vdi r + 1. V|y MD2 dfng vdi
moi, e N-.
TU MD 2rfrtru ngay h0 qui sau.
MQnh dd 3. Sd- tu nhiln N = oroz-o, ,lrio
hti cho p khi vd cht khi D = at * ka2 + ioj +
=
lot .D + 10/e(10'-1
... + k'-1a, chiq hi:t cho p, trong db p = l1s + r
vdi r = l, 3, 7, 9, cdn k ld s6' thda mdn l)k = l
tloI
uE
NGHI
cloi cua sd
c16
nho t *
1
'l 2
. C6c day sd ndy
tudn holn, goi chu ki tudn hoin ld lr.
di=
- (1, 1, l,l,
ds
d.7 = (1,
-2,
4, -1, 2,
"'), h = |
-4, l, -2, ...), h = 6
d1=(1,-1, 1, -1, 1,...),h=2
ds= (1,4,3, -1, -4,'3, t, ...), h = 6
d.s = (1,-5, 8, -6, -4,3,2,7, -!,5,
dl,) = (1, 2, 4, 8, -3,
...), h = 16
-6, 7, -5, 9,-1, ...)' h =
18
Menh di 4. Ddy sd chia cho s6'ld p kh1ng ld.
b1i 5 td ddy s6'tudn hodn c6 chu ki ld uoc sd
phAn
tich p ra thira sd
tp(p)=fl,r(ri,) =
[
pi,-t (p,
nguY6n t6,
Hue trohg cdc nghy 1 , 8,9, l0 thring 9 narn 2002.
I Ioi nghi chia thiLnh 8 tidu ban : Dai so Ilinh hoi
- Topo, Ciai t(ch. Phtrong trlnh li
'l'oi uu vl Tinh toltn khoa hoc' Xdc sr"rdt
"tionIi; ft""s tie toon lroc.'lorin hoc roi lac vl ]-in
t'lirg clung torin hoc. Grang day
li rliuv,-'t.
'Du kieri Cliuong trinh khoa h.oc eua
tohn hoc.
Hoi nshi sE baio c[o gorn -l h[o ciio mol toltn
rhe t6b phritl, cric btio cio mdi tai cdc ti€u han
ia0 bh,irl;t;iic thong btio ngin (ts ptrutt'
Ilan tti chfc HOi nghi sE x6t vd tii tro mOt
rrhan kinh phi clto ,rni to cin bO nghie n criu tid
la sinl, r'ien toiirt rtrrii sric d0 thrLrn itu Hoi nghi'
Thoi han nop tdm tat biio ciio lir trudc ng)y
31.5.2002. Htii nghi phi m5i ngudi. drtt. Iir
100000d, Trong v)ftau ihoi gian hoi nghi c6 to
chric tham quail danh lam thing cinh- HQi nghi
sE tii tro ph^uone tien di tham quan cdn ciic dai
bidu tu trii cric clri ptri tnac.
Dia
chi li€n h0 :
'Ban
hoc
td chirc' HOi nghiTHTQ1
/r.
Tir bing tfnh k tr€n, ciic ban c6 thd tinh duoc
ddysdchiachom6isdp< 101 mh(P, 10)=
Tim so du trong ph6P chia cho P
1.
Ktri cta tinlr scitt cltcrc tldy so'cl,rla c5 thd tim
duoc khd clon giin vi nhanh (d{c biet uot-L]r.ha
lon) sd du R trong phdp chia sd ly' = a1a2...as
cho p nhd MD 2, c16 lI R = l0'-'.D (mod p) vdi
o
Vi
dU 1. Tim gi6
t
it
p' ll*tir*.
Vi du
cira
a
sao cho
sd'
2. Tim sd du ctra sd N = 35265148 khi
chia cho 13.
math.
ac-.
vn/c
o n ie re nc
e/hnthtq6/
Gidi. Tilhnhu tr6n duo. c D -- 36 =-3 (mod 13)
Til r = 8 c6 107 = (-3)7 = 93(-3) : (-4)3(-g) =
16.12 = -3 = 10 (mod 13).
Suy ra R = 10(-3) = -30 = 9 (mod 13) nOn s6
duR=9.
Vi du 3. Goi N li sd vidt boi m chir s6 a nhu
nhau. ihung minh rang :
a) Ni13 a mi6; b) NilT e mil6
c) N:19 o milS
Gidi.Y\ (a, p) = I vdi P = 13,17,19 nen N =
axMipQ Mip, trong d6 M viat b6i tohn chfr
sd 1. Goi do = (rr, 12, ..., r11) ld chu ki nh6 nhat
ctra ddy sd chia cho p. Ta thdy
L,, = 0 (mod p)
i=t
=
o Dil3 e a=3.
z
t
h
tri
35a65148 chia hdt cho 13'
Gidi. D = 3+4.5+3.a - 1.6 - 4.5 - 3.1+ 1'4 +
4.8 = 30 + 3a = 3(10 + a) (mod 13) non Ni13
N
Hbp thu 631' Bd Ha. Hd Noi
e - ri a i I : hnt ht q6 @ r I t t' r' i r h' t t c s t. u c' . r' t t
bidu d[ng
Ban td chrlc khuy€n khich c6c dai
'
ki-truc tigp iierr-muhs thco dia chi :
-t)
= 0, 1, ..., h-l,suy ra ddy sd 1,k,k2,...,k"-r,1,
ki
VidnTodn hoc
VKT
('f heoTh)ng tinTodn hoc)
Goi lr li udc s6 nho nhdt cira
ta c6 kt = pt'4+r = r''q. H' : //' (mod p), vdi moi r
... ld day so tudn hohn c6 chu
ItlAN tluoc
Hoi nstu To[n hoc toirn qtroc lan thf 6 do [l
crta s6'Ole rflp).
Chitng minh: Sd t duo. c chon th6a mdn 10k =
1 (mod p) nOn (k, p) = 1' Theo dinh li dle thi
po@) :1 (mod p), rrong d6 p = pil ...p.1, ta rU
HOG
LAI.I THU 6
(mod p)
Nhu viy dau hiQu chia hdt cho sdp = 10s +r (r =
l, 3, 7','9) tuong trng vdi mOt ddy so^, ggi
ld, ddy sd chia cho p,kf hiOu l dp= (1, k, kt, k', "')
Du6i day 6p dung MD3 cho cdc s6 P 7 3,7,9,
11, 13, ti, tg vI str dung bing giri tri k tmg vdi
p 6 trcn ta tim dugc ddy sd chia cho q, trong d6
cac s6 ctra ddy duo. c tinh sao cho gid tri tuyOt
TtlIN
J
cbn
f 410 (mod P) vdi moi s = 1,2'
i=1
"',
$r MD3 suy ra didu phii chung.minh'
Til c6c kdt qui trcn c6c ban c6 thd phr{t tridn
thOm nhidu van Ad thri vi hon nita, l.
ft
-
1,
[rrx Torrn
vf uUs'ng Tnmnc sKT ud ru
nnm Efre EK
TRAN XUAN BANG
(W
Trong BQ "D"d thi tuydn sinh vdo c6c trubng
dai hoc, cao ding vi trung hoc chuyOn nghiOp
m6n To6n" (BQ DTTS) cfing nhu mOt so dd thi
v2ro ciic trudng dai hoc vd cao ding gdn dAy c6
nhlrng bhi todn"vd dq*g thing cat dO thi hdm
bAc ba i y = ox'' + bx' + cx + d (a + 0) (l). Goi
f(x)ld vdphii cita (1).
Bhi vidt niLy xin trinh bdy mOt s6 tinh chat
(TC) crla dcAp thOm c6ng cu cho ci{c ban chudn bi thi vho
dai hoc khi giAi c6c bii toi{n thuOc loai ndy.
TCI: D6 thi hdm s6'(l) cd mdt tdm d6'i xftng
duy nhdlt m aidm udn cia n6.
Chtng minh. Bd dd sau cdn dd chrmg minh
TC1.
Bd dA; Dd thi him sd y = f(x) x6c dinh tr€n
R c6 tAm d6i xrmg T(a, u) khi vi chi khi
f(x) + f(Zu-x) =2v vdi moi x e R.
That vay v6i m5i didm A (xr, /(xr)) bdt ki thi
lu6n t6n tai didm B @2, f@)) cfrng thu6c dd thi
ln didm d6i xrlng ctra didm A qua didm T
thi FrI him sd nly dOi v6i h0 truc TW c6 dang
(
,z\
y=ax3*1.-Llr
I to) (2)
c6 tam ddi xring 7 vdi
d6 thi cira hhm sd
toa d0 m6i lI X= 0, I = 0.
TC 2z Di€u ki€n cdn vd dil dd m\t dudng
rhdng c6 h€ so'g6c k di qua didm u1it ctla d6 thi
hdm s6'(1) vd cdt d6 thi hdm sd (l) tai ba didm
,..,.t
pltafi Dtet ta
l"
__b
=o,vxe R<+ 1^o-
3a
U" = f (x")
Vay d0 thi hhm so (1) c6 tAm d6i xrmg duy
nhat chfnh Id didm udn c[ra n5 (dpcm).
(, = x.. +Y
\r=ri*,
7(;ro, yn) lh tAm ddi xtmg ctra dd
q2g
3a
I
0.. Lrdt o.o
L3a-
Ch{tng minh: Tinh ti6n h€ truc Oxy ddn h0
truc TW, trong d6 I(xo, y,J lh didm udn ctra dd
thi him sd (1) thi ddi vdi h0 truc mdi, hdm sd
(1) c6 phuong trinh dang (2).
Hodnh dQ giao didm c[ra do thihdm sd (2) vdi
duong thing f = kX lh nghi€m cira phuong
(
(rz)
Xl
[
aXz
,z\
, of + lr-?lx
= kX
roJ
I
*r-L-t
3a l=o
<+
(3)
)
Duong thhng Y = kX cit dd thi hlm sd (1)
phAn biet khi vh chi khi PT
tai ba didm
ax2 +
( ,z \
l r-?-k I = o c5 hai nghiom phan
(.3r)
bi€t khr{c 0, nghra ld
:
!(
a[
o-..4)
3o
>o.rir
)
d6 suy ra dpcm.
(Ta cflng c6 thd chtmg minh TC2 bang c6ch
t vdi dao him cira hlm sd (2)).
TC3z Vdi ba didm A, B, C phdn bi€t tudng
hdng cing thudc dd thi hdm sd'(I ) didu ki|n cdn
vd dil dd AB = BC ld B rring vdi didm uoh cila
so s6nh
I
eua ph6p bidn ddi
ln2
lk>r-"-n61
|
<
["ft*'.1 + f (x2)=2v
e/(;rr) + f(Zu- rr) = 2v vdimoi x e R (*)
Xj dd duoc chfng minh
Chf i rang khi rz thu6c tap xdc dinh cira y =
f(x) th\f(u) = v (theo (*)) non 7 thuoc dd thi.
Ap dung bti dd trcn ta thay I(x.,, yo) lh tam ddi
xrlng cfia dd thi him sd (1) khi vd chi khi :
f(x) + f(Zx" - x) = 2yo, Vx e R
e ,r3 + bxz + cx + d + a(2x,,- x)3 + b(Zxo- x)2
+ c(Zxo- J) + d =Zyo, Vx e R
e (b + 3axo)xz - 4xo(b + 3axr)x + Zf(x") - 2yo
e)
vi
rrinh (PT)
- lrr+xr-2u
tnoa man
THPT Ddo DuyTrt, Qudn7 Binh)
trong d6
thihdm sd (1)
d6 thi hdm s6'd6.
Chung minh: Qua ph6p bidn ddi truc:
x = xo + X y = )o * Y thi dO dli doan th&ng
khOng ddi, ncn chi cdn chfng minh TC3 ddi vdi
of + eX (2') ln dt. Vi
B(&, Y), C(&, Ir) (v6i Xt * Yx)
tliu6c dd thi hdm sd (2') nOn : Iz1 = aXl + eXr,
(4)
Yz= aXl + eX2, Yt= aXtr + eY\,
AB = BC e 2X2 = Xt + Xt ; 2Y, = Y1 + Y7,
thay vio (4) suy ra (X, * X, )' = O(*? +. xi
dd thi hlLm s6 dang Y =
(\,Y),
A
)
e(&
+X:X& -&)2 =0=> Xt+Yn=Q;Yr+Yt
Tit
d6 X2 = Yz = 0 nghia lh B tring vdi didm
= 0.
u6n I. Dio lai ndu c5c didm A, C kh6c B(0, 0)
lh giao cira dudng thing I = lcXvdi dd th! hnm
s6 (2') thi tir (4) suy ra aXl + e = k = aXl + e
> X? =X? * & = -Xt=; 11 = -Ij n6nX1 +
& = Y r * Yq = 0, nghia ld AB = BC
Tt TC2 vi TC3 cho ta :
TC4: Di€u ki€n cdn vd dfi dd mbt dudng
rhdng c6 h€ s6' g6c k cdt dd thi hdm sa'(l)
tai ba didm phdn bi€t A, B, C sao cho AB = BC
ld dxdng thdng d6 di qua
udn cfia
d6
tli
ldm
s6'
(t)
vdi k > c
"didm
- !3a
W,i a > 0,
khi a < 0.
TC5: NCu y = f(x) md f(x) = f '(x)q(x) + r(x)
thi cdc didm cac fi (n€u cb) cila d6 thi hdm s6'
tr€n thudc dudng thdng y = r(x)
Chftng minh. GiA sit M{\, y), M2@2, y) lir
cdc didm cuc tri cira dd thi hdm sfi y =/(x). Khi
d6 /'(xr) = f '(x2) = 0. M4t khi{c do Mv Mz
thu6c dd thi hdm s0 (1) n6n
'(xr)qQr)+ r(x1 )
! \ = flrr) = f
\yz = f Gr) = f '(xz)q?z) + r(x2)
(-.
-':''1 t e Ml, M2thtoc duong thing
o l'' -.-r--
llz = r\x2)
y = r(x) (dpcm).
Vi0c v6n dung ci{c tfnh chdt cira dd thi hdm s6
(1) vlo cr{c bdi to6n khio siit su tuong giao gita
m6t dutrng thing vI dd thl hlm sd d6 thuc su c6
hiQu qui. Cdc ban thfi so s6nh hai c6ch giAi cho
cdc bhi to6n sau
B)ri
toin
1**'
'22 -
1. Cho hdm sd v =.r3
*
! *'
v1i m lI tham sd thuc. X6c dinh m dd duong
thing y = x cit d6 thi him s6 tai ba didm phan
bi6t A, B, C sao cho AB = BC.
@A thi DH Hu€'-2000)
4
. rI xi 3 t +-L,nr=r
- - -.mx
22
cua
- 2 zx+ m
laco,^7
zx - 5ffLr
=
*
x)(x
x)(x
x)
2(x
=
= 2x3 - 2(x1 + x2 + x1)xz + 2(x1x2
-1
* x{1 *
- 2x-x2x7
TU d6 xt+xz+xr=! i x{2 + xfj *
x2x1)x
x2x3
,/.
,n'
=-llxt.x2.x3=^.
L
Kdt hqp v6i 2x2 = xt * .r3 suy ta m = o :
m= +Ji
Cdch 2. Duong thing y = x c5 hd sd g6c /r = I
,2
^ 2
Jmo
-- 43a= c - -:- (a= L > 0). Didm udn ctla
>
dd thi hhm s6
I
chdt 4 suy ra
,2
hodc k., - !'3a
Ldi gitii. Cdch I . Goi x1, x2, x7 theo thrl tu li
c:tia cdc didm A, B, C. Ta c6
2*z - .r1 * Jj vi x1, x2, x3 ld ci{c nghiOm
hoinh do
n r(Y. -]. o,
[2 4 )
dune trnh
thuOc duong th&ng y = x.
Tt
d6
*?m em-\):m=+ Ji
42
-=Bii to6n 2.Arc hlm sd
! = x3 - 3(m+Di + ZQnz + 7rn + 2)x 2m(m+2) (rz
li
tham sd)
Tim rn ad nam sd c6 cuc dai, cuc tidu. Vidt PT
duong thing di qua didm cuc dai vd didm cuc
tidu d6.
(Dd thi w Kl thudt Mdt nti - 1999)
gitii.
Ldi
Hdm sd c6 cuc dai, cuc tidu khi vi
chi khi PT y' = 0 c6 hai nghiOm phAn bict hay
L' = 9(m+l)' - 6(*' +'lm + 2) > 0
a
*2 - 8m m>4+ i7
1 > o c> m < 4
-
+
- Jtz r,oa"
Df
+ Z(m2 + 7m +
ph6p
chia da thrlc
2)x Zm(m + 2). Thuc hiQn
'(x).q(x)
'(x)
+ r(x).
ta
duoc
/(x) -/("r) cho
Dat y
-
=flx)
= 13
3(m
f
)^
Trong d6 r(x)'33
= lt i -&n-l)x * | \*' + 5m2
f
+3m+2) li PT duong th&ng di qua 2 didm cr..tc trf
(theo TC5)
Bii to6n 3. Vidt PT Parabol di qua c6c didm
cuctrictrahhmsdyLdi gidi. Ta c6 y
!' = f
'(x) = 4x? -
4x
(*-l)2 (x+
=.
1)2
f(x) = *o (knt
2i + l,
ti€.p
trang g)
TIN BUON
Nhd gi6o uu tri, Ph6 Gi6o su Hohng Chfng
sinh nghy 28.01.1931 tai Di6n Quang, Di6n Bhn,
QuAng Nam, nguyOn Ph6 Cuc tru&ng Cuc Dbo
tao Bdi dudng B0 Gi6o duc, nguyOn Hi0u truong
trudng DHSP Tp. Hd Chi Minh, nguyOn Ph6
Tdng bi€n tap qp chi To6n hoc vd Tudi tr6, dAng
viOn DAng COng sAn ViOt Nam, HuAn chuong
Khdng chidn chOng My crlu nudc hang Nhat,
HuAn chuong Lao d6ng hang Ba, Huy hiOu 40
nam tudi DAng, Huy chuong Vi su nghi0p Gi6o
duc ; do tudi cao bOnh trong de til trdn hdi 23 gid
ngdy 4.5.2002taiTp. Hd Chf Minh. L0 truy dieu
vh h6a tdng PGS Hodng Chring da td chric tai Tp.
Hd Chf Minh ngdy 9.5.2002.
Tap chi To6n hoc vb Tudi tr6 xin grii ddn gia
ldi chia budn
quydn PGS Hodng Chfng
sAu s6c.
HQi ddng bi€n tAp tap chi To6n hoc vir Tudi trd
Tda soan tap chi To6n hoc vi Tudi tr6
rIEu sO pcs HoANG cHUNG
NGUyEN pHo r6r,rc BIEN rAp
Ph6 Gi6o su Hodng Chfng sinh
ngay
28.01.1931, qu6 & Di0n Quang, huyOn Di0n
Bdn, tinh QuAng Nam. Tir ndm 1951 ddn 1953
ld sinh vi0n truitng Su pham Cao cdp Khu hoc
x6 Vi0t Nam tai Nam Ninh, Trung Qu6c. Tit
nam 1953 ddn 1959 6ng lh gi6o vi€n cdp 3
QuAng Ngai, sau d6 Quydn Hi0u tru&ng
trudng Phd thdng c{p III Hd Noi. Nam 1959
lh crln b0 giAng day vh Ph6 Cht nhiOm khoa
Toiin Trudng DAi hoc Su pham Hh N6i, ddn
i969 6ng nhAn cOng t1.c tai BQ Gi6o duc. Tt
1975 ddn 1983 le Ph6 Cuc tru&ng Cuc Dio
tao Bdi dudng BQ Gi6o duc. Nim 1983 la
Hi0u tru&ng Dai hoc Su ph4m Tp. Hd Chf
Minh ddn nam 1989. Tt 1990 ddn 1997 ld
Chuy0n viOn Cao cdp cria Trung tAm B6i
rAp cHlToAN Hoc vn rudr rnE
dudng gi6o vi0n tai Tp. Hd Chf Minh.
PGS Hobng Chfng lh mOt trong nhirng
ngudi ddu ti0n lhm b6o Todn hoc vd Tudi tr6,
Thu k( Tda soan ti 1964 ddn 1983 vd Id Ph6
Tdng biOn qp qp chf To6n hoc vh Tudi tr6 tr)
1992 ddnnay.
Do nhirng cdng hidn vdi nginh Gi6o duc,
PGS da duoc Nhb nudc tdng thucrng hai HuAn
chuong Lao dOng hang Ba (1968, 1986),
HuAn chuong Kh6ng chidn ch6ng M! cuu
nudc hang Nhdt, duoc phong danh hiOu Nhh
gi6o Uu tri.
PGS Hobng Chfng th t6c gi6 cfra nhidu cudn
siich tham khio vd To6n so cdp vd nhidu bii
trOn tap chf Todn hoc vb Tutii tr6.
THTT
or ru ruvfru $NH tdp t0 cHuvrn ToAN - TIN
rnUdruG DAI Hoc sUpHAt\{ HA nfi nAm 2001
UON roAN - voNG 1
Tldi giart tdm bdi : 150 phtit
CAU 1: X6t da
thfc
P(x) = (1-x + *2
-
CAu 6. V6i m6i sd ft nguyOn duong, d5t
*3 + ...
-
xreee
+r'ooo) *
x (1 + * + ?+ ... + xteee + ;2ooo)
Khai tridn vi u6c luong c6c sd hang ddng
dang c6 thd vidt
P(x) = ao 4 a1x +
o2i + ... + oonn fioo'o
Tinha6111
Cin2. Gi6i
phuong trinh
CAu 3. Tim ba
chf
sd
&=
7. Cho n
CAu 8. Tim tdt ci cdc s6 nguy0n duong
p > 1 sao cho phuong trinh sau c6 nghiOm duy
I
hing don vi, chgc, trim
nhdt x3 + p*2 + (p
I=0
- I + :a;lr+
p-L
hC phuong trinh
( I
3
2,(! + z\= xYz +. t^
14
l r' * y'' I x'
Cdu 9. Giai
I
1
Chung minh rang
,
9= +27b >28
b3
B', C'ldn
y' * r'
+
Y2
(z* x) = xYz -21
I
+13 +121*+Y)=xYz+7
lr3
l.
c6 3 nghiOm (khOng nhdt thiet phdn bict).
lucrt
lI
trung didm
c6ccung 6a,4, G khong chtacdcdinhA,
B, C crta duong trdn ngoai tidp tam gi6c ABC.
C{c canh BC, aAvd AB cat cic cap doan thing
C'A', A'B' ; A'B', B'C' vd B'C', C'A'ldn luot 0 c6c
cap didm M, N ; P, Q vi R, S. Chfng minh
:
1. Truc tam H'cita tam gi6c A'B'C'trirng vdi
tAm l dudng trbn nOi tidp tam gi6c ABC.
2. Cdc duong ch6o MQ, NR vI PS c[ra luc gi6c
MNPQN ddng quy 6/.
3. Ba doan MN, PQ, rRS c6 do dei b[ng nhau
khi vi chi khi tam gir{c ABC ld m6t tam gi6c
6
li
m6t s6 nguy6n duong. Chrmg
minh rang tdngTn= 15 + 2s + 3s + ... + ns chia
hdt cho tdng ctra n sd tu nhi6n ddu tiOn A,, = 1 '1
CAru
*3-i+3ax-b=O
ddu.
6l-z -t)k
n.
CAu 4. Cho a vd b ld hai sd duong. Bidt rang
phuong trinh
rAng
+
Chrrng minh ring 1 S-*, + S*='n = S*.S, v6i
moi m, n nguyOn duong th6a mdn didu ki6n m >
A=26u'nnt
CAu 5. Goi A',
(J2 + 1)r
2+3+...+n
^{;} -7x+3 - J;2 -2 =
---== J3*' -5x -t - tl*2 -3x + 4
c[rasd
MON TOAN - VONG 2
Tldi gian ldm bdi ; 150 pltut
C0u 10a. (Ddnh ri€ng cho thi sinh da fui vdo
ldp chuy€nTodn)
Cho tam gidc ABC cdn b A. Kf hi6u x, y, z ldn
lugt li khoAng cdch MA', MB' vit MC' tir mOt
didm M nAm trong mlt phing cira tam gi6c ddn
cdc dudng thhng BC, CA vit AB. Tim qu! tich
nhffng didm M nam trong g6c BAC sao cho
2
Y
=\17
CAu 10b. (Ddnh ri€ng cho tht sinh du thi vdo
lop chuy€nTin)
Trong mOt bang k6 0 vu6ng 2000 x 2000 O (m6i
6 c6 kich thudc 1) dd v6 mOt duong trdn vdi b6n
kinh 10 khOng di qua dinh nlo vi cflng khong tidp
xric vdi canh nio cira cilc 6l'uOng.
1. Duong trbn dd vd
cit ci{c canh cta cdc 0
vu6ng tai bao nhiOu didm
?
2. Chung minh rang duong trdn dd cho c6t
kh6ng it hon 79 0 vuOng.
,l*
?rrl
\-I
TIM CUC
TRI
CUA HAM SO NHIEU BIEN
7a^l
\.J
BANG CACH KHAO SAT LAN LUOT TUNG BIEN
(Tidp theo
kj trudc)
(GV
Bii to6n 4.Xit cdc s6'thuc duong a, b, c thda
mdn didu ki€n 21ab + 2bc + 8ca S 12. Hdy tim
gid tri nho' nhdt cfia bidu thrtc
t23
= 1+ -
P(a' b, c)
abc
Sau
khi tinh g'(y) ta c6 : g'(y) = 0
e(8y2
_
s)@7;i
liy'z
t3-5ot- ll2=o e(/-8Xr2+81+14)=Q
1,r= ! ,r= 1 thiddbei 3/=8 oy=1.
bc
rir
dd
chuydn vC bai tudn sau:
Xit cdc sd thuc duong x, y, z thda mdn didu
ki€n 2x + 8y + 2Iz 3 12ry2. Tim gid tri nhd
nhil crta bidu thftc P(x, y, z) = x + 2y + 32.
Loi gidi. TU gii thidt
z(lZxy - 2l) >2x + 8y > 0
,r 2:3!-voix> !
l2xy -21
4y
Suyra P(x,y,z)2x+zr+
" 4xy -7
Tird6
-
Ta c6
f
:r2)
-
56xy
-
32yz + 35
lren I -v-+oo ) thi/'(x) = 0 c6 nghi6m duy
[4'' )
Jn" *t'q rr)
, thi
*
qua xn
vi n,ro
+
4v T
nhat
lr *"=
/'(x)
ddi ddu til Am sang duong n6n/(x) dat cuc
f(x) 2flx.) = 2xo -
I
,ry
tidu tai xo.
Tt
P(x, y, z)
f(x) + 2y > f(x,) + 2y = g1y1
>
X6t hdm s6s(y) =2y +
+.+
4v 2v
4y
l,
4'
1icd6r[ll=]1.
"\4)
(
l3) suy ra P(x, y, z) > g(y\
t)
f
= I [+]
,
t2y2
(
E
.
2
Phuong ph6p khio s6t ldn luot tirng bidn cho
thdy duong ldi giAi r5 rhng hon so vdi c6ch vdn
dung bdt ding thr1c, ddng thdi c5 thti dirng di!
giii hnng loat bii to6n tim cuc tri ctra him nhidu
bidn nhu du6i day.
gAI rAP
I
d6
-\
minP(a. b. c) =
(4xy -7)2
(t
/
=
3332
---------------:-
l6x2 y2
thi s'1y) ddi ddu tir
15
5
:xayravdiy- 1,x=3,
= Z . Dau dine_thfc
,=? havllL v6ia=!.a= 4..= 3 thi
2x +
'(x) =
:
=o
Am sang duong n6n g(y) dat cuc tidu tai yo
TU d6 vn
-!!3!
8y 4*'y -5x + 8y
4*y -7
4xy -7
vdi bidn *, ! vh'y li tham sd duong.
4y
- L
Vdi y > 0 vh qua y",4
=
""'[l)
\.4,
2
(11)
X6t him s0
*/ v\
-28=o
+ 14 vdi r > 0 thi phuong trinh
Dar I =
tr6n tr& thhnh
+:
(D€ thi chon dbi tuydn todn qudc gia 2001)
Ldigi6i.Ddtx=
NGUYEN VAN THONG
THY| chuyAn LA Quy Ddrt, Dd Ndng)
x, y, z th6a mdn
d
v.
,u
13)
htrlc
t23
-+-+x'),,)y-
z-
(Dd thi HS gidiTHPT todn Edc bdng A -2001)
Bdi 2. Cho hlm s6
f(,x, y, z) = xY + YZ + zx - ZxYz tron midn
D = {(x, y, z) : O 1 x, !, z vd, x + y + z = ll
Tim min f (x,y,z) vi max f(x,y,z)
DD
Bdi 3. Chung minh rang ndu x, y, z lit ci{c sd
thuc d6i mOt khec nhau thi
i
.,tr.
t
R
?ffiH"?^xi
sinh nghy 22-12-1887
t3, *0t llng qu€ crla
ci{ch thinh
Ma-do-rat
kfioang
400km. Sdng rong
7)
O
xy
1843
2 yz
2 zx
2 -"'
xy
--l--+->'{
y-
canh thidu thdn, bOnh
tAt, khong du-o.c hgc
hdnh ddy dir, thidu
silch vd,
;
nhung
Ramanujan dd. c6
nhfrng cdng hidn rat
d6c d6o, rdt
d6i
ngac nhiOn ddi v6i
H6y tim gi6 tri tdn nhdt ctra bidu thrlc
*r* 9,r'
2' 27
F(x. v. ,\ = L
*E
r'
8-
Runrunujan
toiin hoc.
Nlm
.
ffiAl TOAN Vf" "". i'l'rr:$ rruti1 -i)
D0 thdv hlryn s0 c6 ba didm cuc tri. Thuc hiOn
ph6p chii da thri'c /(x) cho f '(x) ta'duoc'flx)'=
'lx).q(x) + r(x) , trong d6 r(x) = -*' * 1
Tir d6 suy ra Parabol di qua ba didm cuc tri li y
= -x' + l. Mdi c6c ban hdy sir dung cilc t(nh chat
cira dd thi hem sd (1) bd gili cilc blltap (BT) sau.
f
: Cho hlm sd y = x3 - 3? + mzx + m
Tim tdt ch cdc si6 tri cfia tham so m dd hem sd
cuc tidu
c6 cuc dai, cuc tiEu vi'c6c didm cuc dai,
'qua'duong
cira ild thi hdm s6 thi d6i xfng nhau
BTI
15 tudi, Ramanujan da giAi duoc phuong
trinh bic 3 vi bAc 4 theo ci{ch riOng cita minh.
Trong sudt thdi gian hoc trung hoc, Ramanujan
chi tu hoc. vdi mOt cuon s6ch duy nhat cua Ca
(G.S.Can) "T6m tht cr{c kdt qud do cap vd toiin
hoc thudn triy", thdi gian ndy Ramanuj4n dd bat
ddu c5 nhftng kOt qui nghi€n cfu sdu sic vd c6c
chu6i, vd hang sd O-le (Euler), vd c6c sd Becnu-li (Bernouilli), vd hdm eliptic.
Nam 21 tudi, Ramanujan dm ning vd phAi trii
qua mOt cuOc phiu thuat. Di vAy, Ramanujan
v5n tidp tuc nghiOn crlu toi{n, khOng c6 ngudi
huong d6n, kh6ng c6 thOng tin gi vd thirnh tuu
m6i cira to6n hoc. Sau mOt th6ng b6o v6 k6t qui
nghiOn citu cdc s6'Bernouilll'), Ramanujan d5
ndi tidng trong gi6i to6n hoc an OQ nhu mot tdi
nang hidm c6.
"x5
thdnsv=
---!
22
(D6 thi DHQG He Noi - 2001)
BT2 : Cho him so J = x3 - 3? - gx + m
X6c dinh m ad ad thi him so dd cho c6t truc
holLnh tai ba didm phan bict c6 hoinh dO Bp thenh
m6t cdp s0 c6ng.
(Dd so 4
-
B0 DTTS)
BT3 : Cho hlm sd y = x3 - 3mi + 4m3
X6c dinh rn dd dudng thing y = x cbr dd thi tai
8
An ilo
HOANG CHUNG
(Madras)
Bdi 4. Cho ,r, y, z e R th6a m6n ba didu kiOn
ba didm'phdn
..{{r$ lilfuaa: F}iing Tndn Hoe
fur m,
phd
t.".' 'trG
,F.
ffi&K&WW#&W
bi.t A, u:?r;",u1,!^l!{rDrrs)
Rdt may m6n ld Ramanujan dd li6n h0 duoc
vdi nhi to6n hoc Anh ndi ti0hg Hac-di (G. H.
Hardy) (1877 - 1947). NIm 1914, Ramanujan
duo. c-hoc bdng sang hoc dai hoc Trinity 6 Kembrirgio (Cambridge) nu6c Anh.
Hardy ctng vdi Lit-ton-uot (Littlewood) d6
quan tdm girip Ramanujan hoc t9{1, v1,nna1
thay rang "nhlrng han chd trong kien thric ve
toi{n cta anh ta c[ng ddng ngac nhiOn nhu su
sau s6c cta chfng". DOi vOl Ramanujan, m6i sO
nguy€n duong lh"mot "ngudi ban". Mot ldn den
tham Ramanujan o b6nh vien, Hardy n6i dia
rang dd di xe taxi mang bi(!n so lI m6t "sd vO
v1" 1729. Ramanujan n6i ngay :"O khOng, dd ldL
mOt so Ii thri, lh so nh6 nhdt trong c6c sd vidt
duoc dudi dang teing cua hai.lAp phuong, theo
haic6ch kh6c nhau : tlZg = 1t+i23 = 9t+103".
Nam 1916, Ramanujan ldy bang ctr nh0n khoa
hoc. Nam 1918, nhd nhidu cong-trinh xudt sic
Ramanujan duo. c bdu vdo II6i Khoa hoc Hoing
gia Anh. mOt td chfc khoa hoc uy tin nh{1 cira
nudc Anh. Nam 1919, Ramanujan tro vd An DO
trong nidm vinh quang tOt dinh. Nhung srlc
vI 6ng qua
kh6e cira Ramanujan ngdy clng kdm
ddi ngdy 26-4-1920.
Trong cuOc ddi ngan ngiri cria
minh,
Ramanujan dd c5 inh huong khOng nho d6i vdi
m6t sd linh vgc ctia li thuydt s6, ddn mrlc c6 nhi
toiln hoc dd coi Ramanujan ld nhh so hoc l6n
nhat c[.ra thd ki 20.
Ramanujan dd lei 3000 cong thrlc vd dinh li,
trong d5 nhidu kdt qui khong cd chring minh
hoic chi din vd chung
minh. Nam 1p76, ngudi
ta tim thdy thcm "mOt cudn sd tay -ctra
Ramanujan, bi thdt lac trong thu viOn ctra
trudng dai hoc Cambridge, v6i 600 c6ng thfc.
C5 nhh todn hoc dd so si{nh cuon s6 tay n}ry vdi
bin giao huong bl thdt lac vi duo.c tim'thdy cia
nhac si thiOn tdi Bet-t6-ven (Beethoven) !
Nhfng cOng thrlc cira Ramanujan "thudng chrla
drmg nhi6u hon gdp bQi nhfrng gi md thoat ddu
chring ta tudng" (Hardy). Ddn nay, nhidu cOng
thric vd dinh li cira Ramanujan cdn duo. c c6c
nhh toi{n hoc nghi0n cfu, vd mOt so ket qui v6n
cbn llm nhidu ngudi sung sdt, chua bidt crich
nho dd suy ra chring !
Sau dAy ld mQt vii c6ng thrlc cira Ramanujan
ltEf;-
-r
.
il; - il; il; = llltz
,f 2n+ "f 4n+ .l =u=i/^h -31,11
i/.o'7 dcos- flcos-
f- z"+{cosI 4"+{cost- 8"=il
"E6=
,
V.o't
(1)
(2)
(3)
Cdng thrlc (1) don gian nhat vi c6 thd chung
minh khOng may kh6 khdn, ddi vdi hai c6ng thrlc
(2) vn (3) thi kh6ng phii vAy. C6 nhe todn hoc cho
ring "cdc cOng thrlc dd holn todn so cAp, nhung
rdt
iau s6c, vh"chi c6 mOt nhd to6n hoc hing ddil
c su tdn tai ctra chfng".
Ramanujan de tim ra mOJ kdt qui li.,rn kinh ng4c
nhidu ngudi, ld c6ng thrlc tffi gi6 tri cira n sau dAy:
moi du doi{n
duo.
I _ .6 € t4n)t(t to3+26390n)
L,^, f,Do sx*
" ltot
Sd hang ddu tien cta ting nav ra 1=
n "o=f
9801 '
,
tt
d6 c6 n chfnh x6c ddn 6
cht
sd thAp phAn
;
thOm sd hang thrl hai ta c6
l= fi
7t
[r ^rc3+@tl274s3\
-3964 )
9801 [^
vh duoc gi6 tri ctra n chinh x6c ddn 12 chfr sd
thAp ph6n.
Ramanujan cbn tim ra 14 c6ng thrlc kh6c nfa
a
1
cho
mh kh6ng hd c6
It
giii
thich, ching han
r-,t _(Zn)l
1_T (r, \3 42!:_5 ..ai
vot Lin
i= 4\ti,, ) ,**
@*
Dua vio m6t kdt qui ctra Ramanujan li0n
quan ddn tich phAn eliptic, cd thd xAy dmg
thuAt to6n dQ quy sau dAy d€ tinh gi6 tri gdn
dring cfia n vdi dO chinh x6c rdt cao.
Giisit0o=6
,
-4J, viyo= Ji*t,
.tll4
t-(t-Yl)
Jt1+l
rtttt
t t)
r+(r- rl,)"0
TiI d6 gir{ tri gdn dring cira I
lt
duo.. ,5"
dinh boi
cr,,*r
=(l + !n*r)a
d,, -22!*3 y,,*,
(t * ),,,r
+
tnr)
Chi vdi cr1, ta dd c5 n chfnh x6c ddn 9 chfr sd,
vdi u2 c6 n chinh xi{c ddn 40 cht sd rdi. Tting
qudt, an cho gi6 tri cfia zr chinh x6c ddn hon
2.4' chit s6.
Ramanujan c6 m6t d6ng g6p quan trong trong
viOc tim cOng thrlc cho p(n),li sd c6c c6ch vidt
mOt sd nguy6n duong dudi dang tdng cta cdc
nguyOn duong kh6ng ting. Vi du sd 4 c6 5 cdch
vidtnhuvly : 4,3+1,2 +2,2 + 1 + 1, I + I + 1
+ 1, nghia ld p@) = 5. D6 thdy rang p(1) = 1,
p(2)=2, p(3)=3,p(5)=7,p(6^) = 11,... vdp(n)
tang rdt nhanh khi
r tang. Chdng han p(200) =
3.972.999.029.388
|
Nam 1918, Ramanujan cing v6i Hardy d[
ldn ddu ti6n tim ra cdch tinh nhanh p(n) v6i
moi n, d6ng chri f li cOng thrlc gdn dfng
,rG
7 n .l,l2
p(n)
- :-----, trong" aO , = "l(3/
il
4nJ3'
COng thrlc ndy cho p(200) - 4.1012. Nam
1937, Rademacher da bd sung vho cOng trinh
cira Ramanujan vd Hardy, vi ldn ddu tiOn tim ra
mOt cOng thrlc x6c dinh p(n).
Nam 1962, nhAn ki niQm ldn thf 75 ngly sinh
cira Ramanujan, buu diQn An D0 dA ph6t hinh
m6t con tem c6 hinh inh cira nhi to6n hoc thi0n
tii niy.
THf
qil TO/N TR6[I I'LAY TINH gO TUI
TA DUY PHUOI'{G
(ViAnTodn hoc)
Ngdy 25.3.2001, Vu Trung hoc Phd thOng dd
td chfc ki thi khu wc "Giii toen EOn m6y tinh
di0n tt b6 tfi" ldn thrl hai. 38 tinh, thhnh phd da
c&doi tuydn tham du tai mOt trong 4 dia didm :
Phri Tho, HAi Phbng, Db Ndng vi TAy Ninh.
M6i doi ddy du g6m 5 hoc sinh THPT vd 5 hoc
sinh THCS. C6c tinh c6 nhidu thf sinh doat giii
h Bac Ninh (10/10) ; Phri Tho, HAi Phdng,
Nam Dinh, VInh Phdc, Tp. HCM (9/10). Vdi su
thi tro cira COng ty xudt nhip khdu Binh Tay, tdt
ci thf sinh ddu duoc quh luu ni6m, 507o sd thf
sinh m6i cap cira timg khu vuc duoc trao giii
(nhdt : 800000 ddng, nhi : 500000 d6ng , ba :
200000 ddng, khuydn khich 150000 ddng).
Bdi vidt nly gidi thiOu m6t sd bdi trong hai dd
du bi cria Phii thOng Trung hoc co s6 dd ban doc
tham khAo.
uz=2 vi un+t=
Bdi t. Cho \=li
3u,, +u,r-, vdi moi n>2. L4p quy trinh tinh
Ltn
tr)nm6y Casio fx-220 vh tfnh \siu1ei1t7s.
Gitii: Khaib6o: 2lMinlE
Vi lip lai:
lE t E
Errllx*MlE@ErE
Kdt
qui: urc =1396700389;
uD= 4612988018; ziro = 1523566443.
Ldi binh: Ddy ut=ai ltz=b vi
il,+t = dltn + pu,-, vdi moi n> 2 lh suy r6ng
cira d6y Phi-b6-na-xi (Fibonacci) : ur=l;
l', u,+t = lln + ttn-t (xem quy trinh tinh al,
trcn TH &TT sd 611.993 vd 411994) vd d6y
Luy-ca (Lucas) 1 1,t, = s; uz = b ;
uz =
ur*t = l,tn + ltn-t (Thi khu vuc ndm 2001)'
Gidi: Ta c6:
21999
,roee
+
22000
+22oot
(l+Z+22)=7 x2e x}to xzteso
=7 xZe *2ro *(22n
Yi (dilng mdy):
2e
-
=512;
)"
2ro =1024;220 =(21u)z
= 7 xTe x 2r0 x (2'o )" vh sd c6 hai chfr sd tAn
ctng li 76lui thira bAc bat k!'vdn c6 hai chf sfi
mn cUng ld 76 (dilng mdY ki0'm ffa) n6n
(220)e' cfing c6 hai cht sd tAn cing li 76.
Do d6:
ztsse +22ooo +22oot -_7 x29 xzto ,(220)ee =
7 x5l2x1024 x (...76) =...16.
V4y hai cht sd cuOi cDng cira
2teee
*22ooo *22001
s6:
ld16.
a) Tim sd tu nhien nho nhdt mi binh
phuong cira n6 lI mOt sd b6t ddu bang chf s6
19 vi kdt thric bang chfr sd 89.
Bdi
4.
b) Tim tdt ch cdc sd tu nhiOn n sao cho
n' ld mot sd gdm 12 cht so c6 dang:
n2 =2525 * {< 'r * {< *89 (trong d6 s6u ddu
bidu thf s6u chfi sd, c6 thd khdc nhau).
*<
Gilii: a)Dd a2 li mot sd bat ddu bang chfr sd
19 thi n6 phii c6 dang:
19 x
10" < o2 <20 x lO"
. o2 ,l}-n <20
Ndu n = 2k thi
hay 19
.
4.3588989 *...hg
--7 x}e ,
Ndu
210
,
mdY):
(2'n )n'
n = 2k
(dilng
+l
.
thi (dilng
Cho \ =l: uz =t vd 13.78404875 - Jtq0 < a x lO-k
. Jzoo xt4.r42r3562 .
un+t = ul +u'-, v6i moi n>2 . Ldp quy trinh
mdY):
Bdi 2.
tinh un vd quy trinh tfnh phdn du cira u, khi
chia cho 7. H6i uroo, c6 chia hdt cho 7 khong.
Bdi 3. Tim hai cht sd cudi cing cira sd:
)teeg )_12000 *72001
10
Cho t =0;l;2 ta duoc: a=14: 44 138;
139 ; l4O; l4I ;436; ...;447 ;1379:1380 ; "'
;1414.
Y\ a2 c6 tan cing ld 9 non a chi c6 thd c6
tAn cirng li 3 hoic 7. Thrt ffAn mdy ciic s0:
13; 23: ...; 83; 93;
l7; 27; ...; 87;91 ta tim
a
phitic6
duoc: dd a2 c6 tan cing le 89 thi
hai sd mn cilng li m6t trong cdc sd sau: l7: 33;
67;83.
VAy trong c6c sd
thira bAc ba
trt,
n
sd c5 tan cilng
du-o.c
ld
11. Tuong
(tht ffan mdfi dd n3 c6 tan cing liL 111 thi
phiti c6 tan cing lb: 471 vI dd n3 c6 tan
li 1111 thi n phhi c6 tAn cing ld: 8471.
a c6 tdn cDng lh 17; 33;
Gii sir lle lI sd chfr s6 drirrg gifa cdc sd 111
a' ld *6t sd b6t ddu bang chfr sd vd 1111. Ndu m =3k,/c > 0 thi 111 x 103ft*4
cing
67; 83 vi
19 thi s6 b6 nhAt ph6i tim li: 1383. Tht lai
(dilng mdy): 13832 = Lgl26Sg
b) Ta c6: 2525x108
.*< J2526 x 10a- 50.25932749x
Yay 502493 < x < 502593. Theo c6u a) thi
.
tim c5 mn cing lI: 17; 33; 67', 83. Suy
ra cdc sd cdn tim li: 502517; 502533; 502567;
sd cdn
502583.
Bdi 5. Tim sd tu nhien n ( 1000 < n 1 2000 )
dd a,, nt . zs,, cDng lh so tu nhion.
'm
Gidi: Y\ 1000 < n<20A0 nan (dingmdy):
- G7t2l + 35x 1000 se,=
,t57 rzr. 35, <'m r2r * 35 2w *356,54032.
"
303,51441
Suy
ra: 304 < a, < 356
Vi
a?,
.
=5712t+ 35n nen o?, -l = 35(1632
+ rr) phii chia hdt cho 35 = 5 x 7. Chrmg t6
hod,c a, +1 hoac a,-1 phii chia h6t cho'7,
hay
thi
304
=7k+l<
356 hay 44< k < 50.
chiah6rcho5 nen k=44;45;49
"?,-1
50. Tt d6 ta duoc 4 s6: k = 44, a,, = 309, n =
1096; k=45, a,, =316,n=1221 ;k=49, a,,
= 344 ; n = 1749 ; k = 50 | a,, = 351, n= 1888.
Tuong tu, ndu an=7k -1 thi ta duoc th€m
4s6:k=45, a,. =314,2= 1185;k=46, ar=
321 ; n = l3l2 ; fr = 50, a,, 349 ; n = 1848 ; k =
5l;a,r=356;n=1889.
Bdi 6. Tim sd tu nhiOn n nh6 nhait sao cho
n3 llL m6t sd c6 ba chf sd ddu v] bdn chf s6
W
cudi ddu bang
Suy
ra (tinh
10,35398805 x
tor*r - Vt t to * 1ot*r <
. 13 = 111...1 ItI
.
bing s0 103, nghia ld n=103...8471. 56 nh6
nh6t trong c6c sd d6ld: n= 1038471.
Khi rn =3k+l vd m=3k+2,k>0 ta du-oc
c6c sd dang nhy ddu vuot qu6 sd 1038471.
VAy s6 nh6 nhdt trong cdc s6 d6 ld: n =
lO3847l: n3 =l llgg}gggl2Sg 361 111.
Chri f : Khdng thd dnng mdy tinh truc tidp sd
n = 10384713 = 1.1 19909991('8) v\ trdn mdn
hinh. Ta phili kdt hqp gitta tinh ffAn mdy vd
ffAn gidy nhu sau:
l}3847f
=(1038 x 103+471)3
=
10383 x 10e +
+3x10382 x106 x 471+3x1038x103 x
x
4712
+
4713
=
(tinh tren m6y) =
a,=7k*1 hoic qn=7k+1.
Ndu a,,=7k+1
.,r3 =111...1111<1l2xl}3k*4.
ffAn mdy):
= 1 1 18386872000000000 +
+ 152242837200000 + 690812874000 +
+ 104487111
= (tinh tren gidy) 11199099912893611n.
1.
Gitii: Dd n3 c6 mn cirng l[ 11 thi n c6
lI l. Thft ffAn mdy c6c s6 lI,2t,3l, 41,
51,61,71,81, 91 duoc duy nhdt 7I khi lu!
dudi
11
DC RO
r99, 409,619,829, 1039, 1249, 1459, 1669,
1879,2089.
Chung minh rang khOng tdn tai cdp sd cOng
gdm 11 sd nguyOn td v6i c6ng sai d th6a mdn
I < d <23t0.
III ]IRY
DANG HUNG THANG
(DH KI]TN _DHQG HA Nbi)
CAC L6P THCS
Tll299.
Giai phuong trinh
Bni
B
nghiOm
lon
)oo2
^
ngu,en:x +Y =2OO32nn,(*, * yr)
|*"
lErHAcgoe
(S/ 42C4 DHThily Ini Hd Nli)
B iT2l299. Cho hai sd duong m, nvdi m < n.
Chung minh r[ng
]r'n
l
t
=l89Ox+
x2oor
*
x5 = 1890y
*
y'nn'
l+fl1-xr)(l-x)...(l-x,)
NGO VANTHAI
sinAsin-BsinC
PHAN HOANG NiNH
Todn B -K33, DHSPThdi Nguy€n)
vO uEN vrEr
Bni T41299. Cho tam gi6c cAn ABC (AB =
AC). Ltiy didm P ndo d6 trcn dudng thhng BC
(P khdc B, C).Goi M, N ldn luot lh didm d6i
xrhg ctra P qua AB, AC. Dung hinh binh hlnh
MPN). Chung minh rang didm Q luOn ndm
tren m6t- duong thdng khi P di chuydn tr0n
sa
(GV THCS chuyinVinh Bdo, Hdi Phdnl)
Bni T5/299. Cho dudng trdn tam O duong
kinh MN cd dinh vi m6t didm A nf,m trong
doan MN. Goi 5 lI tidp tuydn cira dubng trdn
v6i tidp didm N. Ducrngirbn mm f nho d6-thu6c
5, di qua A, c6t dudng trdn tAm O t1i E vb, F, cht
6 tai B vi C. Chrlng minh rang khi didm Z di
chuydn tr€n 6 thi :
EFlu6n di qua moldidm cddinh
2) Duong thing PQ luOn di qua mOt didm cd
dinh, trong d6 P, Q c6c giao di6m cila MB,
MC vdi dudng trbn tAm O.
1) Dudng th&ng
li
NGUYEN KH,A,NH NGITYEN
THCS Hdng Bdng, Hdi Phdng)
(6/
CAC T,OP THPT
Bdi T61299. Bidt ring 10 s6 nguyOn'td
sau
day lap thdnh cdp sd c6ng vdi c6ng sar d = 210
t2
zs
(G/ THPT BC Qu)nh Phu,Thdi Binh)
BdiT9l299. Chung minh rang trong tam giilc
ABC lu6n c6 :
sinA + sinB + sinC > sin3A + sin3B + sin3c +
lrrt'-48y+64=zl
t.
- 48: + 64 = x' .
ll2:'
.
rnAN
*
trong d6 n > 2vd,O a r,, x2, ...t xr, < 1. Ding
thrlc xiy ra khi nio ?
THPT chuy€nVinh Phrtc)
BC'
z2oot
n
PHAMHITYHOANG
Itzr'-48x+64=y3
dudng thang
=l89oz +
Bni T8/299. Chung minh bdt ding thfc
Giei he phuong trinh
(tr
Y5
(GV THPT Vdn Giang, Hang Y€n)
-T-T-l
trong
d6 ci{c s(i a, b, c th6a m6n m 1 a, b, c I n.
Ding thrlc xiy ra khi nio ?
B iT3l299.
r"
*
TRIEU VAN HTNG
b
c -3 ln-m)2
a
b+c c+a a+b 2 2m(n+m)
(6/
iT71299. Giai he phuong trinh
:
(DH An Ninh, Hd Nbi)
Bni T10/299. Goi O vd I ldn luot li tam cr{c
mdt cdu ngoai tidp vi nOi ti€p trl diAr, ABCD.
Nam mit cdu tAm Qr Qz, Qy Q+, Q5 br{n kinh
bing nhau, s6p xdp b6n trong t(tdicnABCD sao
cho c6c m5t cdu tam Qr, Qz, Qy Qatheo thri tu
tidp xric vor cdc m5t ctra g6c tam di€n cfia trl
diOn c6 dinh A, B, C, D, cbn mit cdu tdm Q5
tidp xtic vdi ci 4 mdt cdu tam Qt, Qz, Qy Qq.
Chtrrg minh rang c6c didm g,l,Qtthang hang.
(K
h,, r d?,Lo;'ffh,
"
cAc pd vAr
N sh € An )
li
Bdi Lll299. Hai vAt hinh tru cdu tao d6ng
chdt, cing chidu cao c6 dudng kinh tiOt di€n
li
D, d, d$ac drt nim, tidp xric nhau tren mat
phEng ngang. H0 sd ma s6t 6 moi mat ti6p xric
ld k. Quan m6t soi dAy mdnh kh6ng ddn quanh trq
l6n vh ti{c dung vdo ddu dAy mOt luc F truong nam
ngang (hinh ve). HQ sd ma sdt ft phai c5 gi6 trl nhu
thd nho dd c6 thd di chuydn 0q lon wor qua rrq
nh6 ? Tfnh gi6 ri cdn thidt cta lqc F dd thgc hiQn
duo.c di chuydn tr6n. BidtD =M,?,hdilugng trq lcrn
M = l]kg.Udy g= 10m/s'.
NGUYEN XUAN QUANG
(GV THPT chuy€nVinh Phrtc)
Bii
thi nghiOm IAng, khoing
lmm.
MIn quan s6t E song
=
song. ciich.mat phing chrla hai khe td D = 2r.r^.
Di0m A n5.m tr€n dudng trung truc cta S,Sr,
L21299. Trong
cr{ch hai khe ld a
ci{ch trung didm 1 ctra S,S, ld d = 40cm. Ngudn
S phr{t ra 6nh s6ng don sic, chuy(ln dong trdn
ddu xung quanh A v6i bdn kinh qu! dao r =
2mm (coi r << d) vh vAn tdc v = 3,14mm/s
trong mAt phing chtra N vh l,uOng g6c vdi mlt
phing chrla hai khe ; ban ddu ,S o ::ren lA.
a) H6i vdn sdng trung mm dich chuydn nhu
thd nho
?
b) Ngay sau S,, S, dAt hai bAn mit song song
c6 bd dhy vi chidt sudt tuong fug l, = 10pm,
tt1 = 1,7 vd, l, = 5trtm, n, = 1,6. Hoi vAn trung
tAm dich chuy0n nhu thd nho ?
rnANuaurHI.ING
(GV Khdi chuy€nTo,dn - Tin, DHSP Vinh)
res$ffiffitffiNW
FOR LOWER SECONDARY SCHOOLS
FOR UPPER SECONDARY SCHOOLS
Tll299. Find all integer - solutions of the
equation' x2oo2 +,2ooz = 20032001(xt * r')
TZl299. Let m, n be two positive numbers,
m < n. Prove that :
T61299.It is easily seen that the following ten
prime numbers form an arithmetic progression
with common difference d = 210 : 199, 409,
6t9, 829, 1039, t249, 1459, 1669, t97 9, 2089.
Prove that there does not exist an arithmetic
progression consisting of 11 prime numbers
with common difference d satisfying 1
a
b
..,'........,','.'......._a-r_
c
3
a _!
b+c c+a a+b 2
(rt-ml2
'
'
2m(n+m)
where the numbers a, b, c satisfy the conditions
m1e,b,c
When does equality occur
[*t'*vs=18902+22(nt
t"
] ,t' * zs =l89Ox + x2(x)l
?
T31299. Solve the system of equations
ln*'-48x+64=y3
ltzy'-48y+64=23
t"
Itzr'-482+64=x3
t
I
T41299. Let ABC be an isoscels triangle
(AB = AC). P is a point on the line BC (P
distinct from B, C). Let M, N be rhe mirror images of P respectively through AB, AC.
Construct the parallelogram MPNQ. Prove that
Q lies on a fixed line when P moves on the
line BC.
TSl299. Let be given a circle (O) with center
O, its fixed diameter MN and a point A on the
segment MN. Let 6 be the tangent of (O) at N.
A circle with center I on the line 6, passes
through A and cuts (O) at E and F and cuts 6 at
B and C. Prove that when I moves on 5, then :
i) the line EF passes through a fixed point,
ii) the line PQ passes through a fixed point,
where P , Q arc the points of intersecti on of MB ,
MC with(O).
t'
lr"
t
*xs =1890)+r2txtt
T81299. Prove the inequality
x1
*x2+...+J
n
l+r(1-x1{1- x)...(l-x,,)
where n > 2 andO a rr, xr, ..., xn S I.
When does equality occur ?
T91299. Prove that for all triangles ABC
:
sinA + sinB + sinC > sin3A + sin3B + sin3c +
sinAsinBsinC
T101299. Let O and 1 be respectively the
of the circumscribed and the inscribed
spheres of a tetrahedrcn AyA2A3Aa. Five spheres
with centers Qr Qz, Qz, Qq, Qt, with equal radii
are arranged inside the tetrahedron tetrahedron
AlAzA3A,4 so that the sphere with center A; (for
every i = I,2, 3, 4) touches the three faces
passing through A; of the tetrahedron and the
sphere with center Q5 touches all four spheres
with centers Qy Qz, Qz, Qa. fuove that the
points O, I, Qt are collinear.
centers
t3
CINI BNI Hi TRUOC
Bni T1/293. 'l'irn btiy
s6' rLgLryitt to' sao cho
t{ch ct)a cluing bring ti'ng ctic lfiy
thia
bt)c sdu
ctia bav s6'd6.
Ldi giii. Goi biy s6nguyOn td llLp1, p2, ..., Pt
ta c6
THaS Thdi Nguyon Nha Trang ; Tp. Hd Chi Minh:
Nguydn Kidu Long,8A, TI{CS Hdng Bdng, Q.5 ; Cdn
Thot NguyAn Minh Luan , 8A. TTICS Nguyen Vi0t
Hdne' TP' Cdn
PPzPzPqPsP6Pt =
= pf +p3*p3 +p3+p3*
p3*pl
Ta cdn ding dinh li Phec-ma ntr6 :
Ndu sO nguyOn a kh6ng chia hdt cho
a6
D6ng Da, Vfi Nhdt Minh, 8H , THCS l,c Quf DOn, Cdu
GiAy ; Hmg Yinz DoanThi Kim HuA',8C, THCS Pham
Huy ThOng. An Thi ; Hii Phdng z NguydnVfi Ldn,Vfi
Ngoc Linh, Dinh Khang, SA,Trdn Haong Giang,8B,
Trdn Xrdn Dfing,9A, THPT NK Trdn Phri ; Nam Dinh
l NguydnVdn Dinh,9E, THCS NgO Ddng, Giao Thiry,
Pham Duy Hidn, N guydn Drtc Tdm, Vfi Khdc Ki , 9A'7 ,
THCS Trdn Dang Ninh, Tp. Nam Dinh ; Thanh H6a :
Le Thi Thu, 8A, THCS Nhfr Bd S!, Hoang H6a, Trdn
Manh Tudn,9C, THCS TriOu Thi Trinh, Tri6u Son ;
NghQ An : Qud Anh Le, 98, THCS Dang Thai Mai,
Vinh ; Binh Dinhz Nguy,in Phrtc Th7,9D, THCS NgO
May, Pht C6t ; KhSnh Hda : Trdn Ki€'n Thrlc, Sll4
= I (mod7).
(C6 thd chrlng minh truc tiOp didu
niy
(*)
7 thi
'IET
th6ng
-
*2
7 .7
.7.7 .7 .7.7
th6a mdn
duoc hO
hdt cho 7, cdn vd phii cira (*) chia hdt cho 7
theo dinh li Phec-ma, di6u ndy kh6ng xiy ra.
.Ndu l
chia cho 7 drt k > 1 theo dinh li Phec-ma, didu
niy khong xiy ra.
VAy chi xhy ra biy sd nguyOn t6 trong dd bai
li
7.
Tri
Tt
hq (2) suy ra
:
TUhq (2) cflng d0 thay :
2OOl(x+ y) = *z +
Y2
>
)
Phri Tho : Nguyin )fudnTrudng, Nguy,inTrung Kitn
A, 8A, THCS Gidy, Phong ChAu, Phi Ninh, NinhVdn
San, 8Al, THCS LAm Thao ; V"inh Phric : Bili Thi
ThuHidn, NguydnThiThily Hdng,9B, THCS YOn Lac ;
Hda Binh: Ngb Nhdt Son,9A, THCS B Ycn Thiry ; Hi
NQi: Pham Hodng LA,9B, THCS Nguy6n Trudng TQ,
(3)
O
Ttd6suyrax+y+1999>0
VAy ttr (3) ta c6 y = "r thay vdo (1)
duoc
*2-*=ZOOOx
TU d6 ta c5 2nghiOm
xl = 0 ve x2 = 2001
Nhung xr = O khong ph6i ld nghiCm cira (1).
VAy phuong trinh (i) c6 nghiOm duy nhdt
x =2001.
NhAn x6t. Rdt nhidu ban giii duoc b2ri nhy nhung ldi
gi6i kirong clrrgc ngan
tdt:
=0
(x-Y)(x +Y- I +2000)
gon'
mQt sd
dd n€u bdi to6n tdng qu6t : Tim p sd nguyOn td sao cho
tich cia chring bing tdng cdc lf,y thta bAc p-l cira p sd
d6 v1r p 3. C6c ban sau c6 ldi giii dring, trinh biy
l4
Q)
lY' - Y =}ooox
it ban lAp lu6n dhi hoac chrra
chudn xdc, hdu hdt ciic ban ddu lim dring. MQt sd ban
NhAn x6t.
THCS
:
l*'-*=2000v
1'
= 76+76+76+76+76+76+76
(*)
(1)
O+t Jf *SOOO, +l = 2y. Kdt hqp v6i (1) ta
s6 trcn ddu bang
o Ndu k = 7, nghia ld chbtty sd trcn ddu li s0
nguyOn td khdc 7 thi vd tr6i cfia (*) khOng chia
ddu
- r-1000../i +8ooo, = lo00
Drlc Giang, YOn Df,ng, Bdc Giang).
kh6cTvdi0Sk<7.
biy
*IAI
BniT2l295. Gitii phuong trinh
Ldi giii. (Cia Phan Tudh Anh,8A,
qua viOc bidn ddi o3 17k * r)3 = 7r + 1 vdi moi
r th6a mdn 0 1r <-6, cbn r lh sO nguyOn)
GiA sir trong bAy sd nguyOn t6 trcn c6 k sd
o Ndu k = O, nghia ld cA
7 thi ta c5
Tho'
16 NcuyEN
Bei T3/295. Goi A vd B ldn ltot ld s6' nlrci
xt,x2, ...,
:
xn(n 22). Chrlrng minlt riing
nha't vd so'ldn nhdlt trong n s6'dttortg
A<
(.r,
+x,- +...+
'2
'")
<28
+2x2 +...+ nxtt
Ldi giii. (Theo ban Hu)nh Thi Thiry Lam,
x1
8C1, THCS Phri LAm, TuY Hba,
Phf Y€n).
cAr aAl xi rnu6c
DAt C
=.rr(rr +..- + x,,) + x2(x2+
+ xn-l(xn-t + x) + xf,
... + x,,) +
= 1xl + ,.. + *l) + @p2 t xfi3 * ... * x1x,,
* x2x3 + ,.. + x2x,r* ,.. + x,r-p,r) th\
c < G? +... + rl,) + 2(x62+-rlx-r + ... + xtxn+
*
x2x3
*
d()ng quy.
Ldi giii. Do C2, Cj ldn luot li trung didm ctra
AtA2, A2As vd, v\ A1A2 ll AsA4nan C2C q ll AtA2.
(h. 1) Suv ,u
'
* xr_yxn) =
x2* ...+ xr12 <2C
+ ... + x2x,,
ctiu ciic futan rhdrrq AjA,,, A1Aq, A2A5. Chtng
rninh riing cdc'dLtitrt,g tlnitrg B1Cs,82C2, BJC3
...
(1)
= (x1 *
Vi A la sd nh6 nh6t trong ci{c sd x1, ..., x,,n€t:t
ut"
BrAt=B'C'
BrA,
(l)
A(r, +2x2+...+nxtt)
= A(xt+x2+...*x,r) + A(x2 +...+x,,) +...
+ A(xu_1 + x,,) + Ax,, I
< x1 (.r1 + x2 + ...t xn) + x2Q2 + ... + _r, ) +...
*
xn-t(xn-r * x,) + x? = C
(2)
Vi B lh s6lon nhdt trong c6c s6 x1, ...,x, n6n
B(x1 + 2x2 + ,.. + nx,r) =
- B(xr + ... + x) + B(x2 + ... + x,,) + ... +
+ B(x,,-1 + x,r) + Bx,,
>
xr(rr +
Hinh
+ xr) + x2(x2+ ... + xt) + ... +
+ xn-1(xn-1+ x,,) + xl = C
(3)
Tt
(
1
...
), (2), (3) ra c6
Ti AA2ll
:
. (x,, *.xr4 *... + '-n/
x.- )2
A<
<29(dpcm)
I r--'
xr+2x2+...+ttxn
NhQn x6t. M6t sei ban sir dung phuong phdp quy nap
to,{n !o1 cho b}ri nay nen ldi giii cdn ddi. tat
uai
giri v€ TS ddu dring va trinh bay rudng tu nhu ciich giii
iaiic
!lcn:
qc
ban sau,c6 ldi giAi irg6n gon hon c6 : frga
Wnh Thdi, Nguy€n Xudn Tr*dng, -8A, THCS Gid],,
Phong ChAu, Phn Ninh, Phri Tho ;Vrt Ngoc Linh,gA,
Ddng Ngoc Chi€h, Pham Huy Hodng, gB, Trdn Xuan
Drtng, Pham Anh Minh,9,{, THPT NK Trdn ph[, HAi
llang_, D6 Via Cudng,9A, THCS Nam Cao, Lf
NhAn, Hi Nam ; Pham Duy Hidn, Nguy€n Drtc Tdm,
9A7, TI{CS Trdn Dang Ninh. Nam Dinh;Trdn Mqnh
Tu:fi'n, )9, THCS Tri0u Thi Trinh, Tri0u Son, Thanh
H6a ;_Hodng Thi Minh HuQ,7C, THCS T; Ooong,
vinh
Phric'
u6 queNc vrNH
BdiT4l295. Cho luc gidc l6i AtA2A1A4A1,46
c6 cdc canh do'i di€n song song voi nhau. Goi
Bt, Bz, Bj ldn luqt ld giao didm ctia irng cdp
dudng chdo Afia vd A2A5, A2A5 vd A3A6, A3A6
vd A/a. Goi C 1, Cz, Cs ldn luot ld trung didm
AaA5 suy
,u 1'8..' = 1t4' p,
AtAq
NhAn theo vd ctra ( I )
BrCt
I
vi
AzAs
(2) ra du-o. c
B,C.
-hCz huu
AzAs ArAq ', BrCz-AzAs
A,Ao
:
13)
Tuong tu nhu tr€n ta c6
crB,
=
ArAq
(4)
BtCt Az4
crB,
BzCt=At\
AzAs
Tt
(5)
(3), (4), (5) ta suy ra
BtCz
*CrB, *CrB,
BtCz BtCt BzCt
=|
Ta c6 thd chrmg t6 Ct, Cz, Ctnam trong ciic
doan thing 8382, 8183, B,B2 tuong ung. ThAt
v6y, gii slr tr6i lai, ndu'c6 ft nhdt 1 didm nam
ngohi, ching han C1 nam ngohi doan B3B2 (vi d
trongA6Bj) (h.2).
t5
cAr BA ri rnuoc
l) MBC
ld ddu, eqi hlit duong cao A'ABC.
Ta c6 k.AB = (hr+ h2+ h)AB =
=2(SMes + Suec + Susc) = ZSmc = h'AB (l)
I
Tt d6 : . ndu k h thi (1) luon dring vdi
trong (kd ciL bi6n) LABC
moi didm M ndm =
nen qu! tich cdc d\dm M lh todn b6 midn
trong (kd cA biOn) crta MBC
.
I
neu k
*
h thi qu! tich tren lh tap r6ng
M ndo th6a mdn).
(khOng c6 didm
Ai
I
A1
Hinh 2
r
Ta
Ndu C2 e 81A4, gei M
li
trung didm AaA6.
c6: CtM ll AlAavd'C2M ll4146
c2M.
nOn
CtM
ll
v6li.
. NOu C, e ArB, vdi chrlng minh tuong tq
cf,ng d6n tdi didu vo li.
Theo dinh li X€va thi 81C1, 82C2, B3C3 d6ng
quy. Ndu trong c6c didm 81, Bz, Bz c5 didm li
trung didm crta A2A5, A3A6, AlAa thi 81C1,
82C2, Blc3hidn nhiOn ld ddng quy.
NhAn x6t. Giii tdt bdi niy c6 cdc ban
Phri Tho: Nguy€n XudnTradng, NguyinTrung Ki€n,
8A, THCS GiAy Phong ChAu, Phir Ninh ; Hii Phdng.:
Bii Ngoc Khbi,8A, Pham Huy Hodng, 8B, Brti Tudh
Anh, Pham Anh Minh, Phqm Duy Thdnh,9A, PTNK
Trdn Phri ; Nam Dinh : NguydnThdTiAn,9A2, TtiCS
Le Quf Don, Y Yan,lhamVdn Hdi,gc, THCS Ddo Su
Tich-, fryc Ninh, Hodng L€ Son,Vfi Khdc Ki,, Nguyin
Dtc Tdm,9A7, THCS Trdn Dang Ninh ; Ngh€ An :
Nguyin Cbng Thdnh, 8A, THCS Hd XuAn Htlong,
Qulnh N{i, Qu}'nh Luu ; Ph[ Y€n: Hu]nh Thi Thily
Lam,8Cl, THCS Phti LAm, Tuy Hda ; Kh6nh Hda :
Kiiu Anh,8ra, THCS Thei Nguy€n. Ma Trang,
Tp. Hd' Chi Minh : Nguydn Ki€h Lang,8l, dudng
Pham
H6ne Blng' Q'5'
"'
Bni T51295. Cho ram gidc ABC (AB 3 AC s
IIC).'f im qu13 tich nlritrtg di1in M ndm trong tam
gidc sao cho tdng khodng uich tit didm M d€it
ba canh cia tam gidc ludtt bdng mdt hdng so'.
8B,
THCS Tidn Thiry, Qulnh Luu, Nghd An).
Goi tdng cdc khoing c6ch tt M ddn ba canh
MBC li ft1 + h2 + h3 = ft kh6ng ddi' x6t 2
trudng hqp :
t6
= 2Sms.a + t'h+DE (2)
trong d6 hq= MG II khoing cdch tt M d€n DE,
cdn / = 0 khi M thuoc doan DE, t = 1 khi M nam
trong LCDE, t = -l khi M nam trong tfi grdc
ADEB (n6u D trung vdi C th\ ADEB tr0 thinh
MBC vi ho= h3). Tt d6
=
Z(Szg:rss
+
MG = hq=
vu KIM THrJY
Ldi giii. (Theo ban Trinh Minh Tu€,
2) MBC khOng ddu, giA sit A,B < AC < BC
(hai ding thfc kh6ng xly ra ddng thdi). Tr0n
canh AC ldy didm D vh tren canh BC lziy didm E
sao cho AD = BE = AB thi ci{c didm D, E cd
dinh. Ta c6 k.AB - (h1 + h, + hr).AB = htAB +
hz.AD + h.BE = 2(Suea + Sueo + S7a6s) =
TSMDE)
t(k.AB -25ooea)
(3)
VAy c6c didm M cdn tim nam trcn ductng
thhng dosong song vor DE vh c6ch duong thing
DE mQt doan fto duoc x6c dinh b6i (3) (c6 DE >
0 vi AB < BC).Vi didm M nim trong (kd ci
biOn) cira MBC nOn cdc didm M cdn tim ld
phdn ctra dudng thhng donam trong LABC.
Chf y ring AB < AC < BC e hr2 hu> hn
(duong cao k6 tit c6c dinh C, B, A).Khi h4
bang khoing cdch tt C tdi DE th\ tit (3) c6
e
dr eAr ri rnuoc
= ZSeoB + 2Sro, = 2Saac = hc.AB e
k = hs, hic d6 d* trDng va d, vd M trirng vdi
k.AB
a2 + b2 +
didm C.
Khi h4 bang khoing cdch tt A tdi DE thi tt
(3) c6 k.AB = ZSMB -2Scon =ZSe,, = heAB
o t = ht,litc d6 d1 trilng vdi da vA didm M
tring v6i didm A. Bii tor{n c6 nghiOm khi
Ito
th6a m6n (3) thi bidn ddi tuong duone c6 (2),
suy ra h1 + h2 * h3 = t kh6ng ddi. Vay qu! tich
cia M li phdn dudng thhng donAm trong MBC.
Nhdn x6t. Chi c6 c6c ban sau day grli ldi giii li ban
Vfr Khdc Ki,9A1, THCS Trdn Dang Ninh, Nam Dinh,
NguyclnTrudngTho,gA, THCS Gidy, Phong Ch6u, Phi
Ninh, Phri Tho; Pham Anh Minh,9A, THPT NK Trdn
Phr' Hei
Phdng'
vir D'NH HoA
Bii
T6i295. Clto ba s6'cltong a, b, r'. Chtrng
lttlttll t'it:,q
I
)t rt
2
t (rt:+b)+c)
+ l' +
S tthc(ah
L'
)-i I u
+ b --c
) (a
4
+ c )( b +
c.
-tt ) < 27 u2 bJ {.2
)(ct+b-<)(a4+c)(b+c-tt) S
+ bc + ctt)
.t ) ( tt + h + c ) I
l( ab+
Loi giii. (Cria
hc + ca ) -1 a2 +
l,: * r.) ) I
ban L€ Phuong,Ldp
i
il,
Q,
t
br
THPT
Luong Thd Vinh, Ddng Nai)
l)
Ndu (a + b
- c)(a - b +c)(b + c - a) 7 O
thi BDT dring hidn nhi6n. Ndu
(a + b - c)(a -b + c)(b + c -a) > 0 thi tdn
tai tam gi6c ABC v6i BC = a, CA = b vd AB = c.
Goi S, R, r ldn luot ld di6n rich, b6n kfnh
ducrng trdn ngoai tidp vh nQi tidp c:tia MBC.
Khi d6 (1) tuong duong vdi (a+b+c)'
li
tam dudng rrdn ngoai ti6p ta c6
(a+b+c)2 < 3(a2+b2+c21 =
=
:[for -ocl,
T
I
Theo chrlng minh 6 cAu (1) ta c6
a2 + b2 + 12
.sR2
<
rt( t. +*llt,
(a b ,)'
+ b + c)
3) (cira ban Hodng Ngoc Minh, 1iA, TIIPT
Hr)ng Vucrng, Phri Tho)
Ta c6 3) tuong duong vdi
9(p-a)(p-b)(p-c) < abc e
a
21 lrR <+
sin4si,
B
rin
&t=
p
C
<
aag
1
2 2 28
Day bat dEng thrlc quen thuOc dd biet.
NhAn x6t. Day li m6t bii todn kh6 co bin vd
duoc
rdt nhidu ban tham gia gi6i v6i nhi6u c6ch kh6c nhau.
Nhidu ban c6 ldi giii tdt trong d6 c6 c6c ban Nguydn
Vdn Vinh, 10A1, THPT L0
Nguydn Vd Vinh 1,6c,
llT,
Qui D6n, Di
Nang,
Tx. Sa D6c, D6ng Th6p,
Nguyen Huy Khi€m, 10T1., THPT Lc Khidr, Quing
Ngai ; lA Khdc Tdn,10.41, THPT chuyOn VInh Phric ;
Pham BdoTruns,DHKIlTN - DHQG Hn NOi, Nguy\n
Vdn Hoan, llAl, THPT Y€n Phong, Bdc Ninh ;
Nguy€nThi Qu)nhTrang, 10A2, THPT Phan BQi Chau,
Nghd An ; Bni U Vfr, L0 chuy€n To6n, PT chuyOn
Hoing Vin Thu, Hda B\nh : Trinh )tudn Hi€p, TIIPI
L€ Van Huu, Thicu H6a, Thanh H6a.
DANG HUNG THANG
Bii
T7l295. llc)i ci tti'r cri bao nhiiu da tlt{rc
P,,(x)b{tc il r'han tltr)ci rnrin cdc diiu kien
.
) Cdc hi s6' t-r)a P u(x) thuoc rdp hop
M = i0,-l, li vd P,,(0) *0
I
2)T6n tai da
tht?c Q(.x) cd cdc h0 sd'tlutAc
sao c:lro P,,(.r) u(x2
M
*t )ek)
Ldi giii. Ddt n =2m.
Da
thrtr'c
Q@)
Q(x) c6b1rc n-2:
= b,,*n-2 + b1x"-3 *...* b,r-lx *
bn-z
Do d6
+@e -oA),
_
'r I l\
12. R' I -*=*- l@ + b + c)
[a b ,)'
^_.1
= 319R" - (OA + oB + OC)'
)
+@A-*f)
<27R'
2) Tuong tu nhu tr6n ta chi cdn x6t trudng hgrp
a, b, c h dQ d}Li ba canh tam gi6c ABC. Khi d6
(2) tuong duong vdi
P,(x)
-
1x2-l1g1x1=
= box' + blxn-| + (bz - b)xn-2
+ (bz
+ ... + (bn-z - bn-a)xz
-
-
bn_t.x
*
b1)x"-3
br_z
Nhu v6y s6 cdc da thri'c P,,(x) th6a mdn dd bhi
bang sd cdc ddy sd (bo, b1, ...,
mdn di6u ki€n
b,-.,
bn-2) th6a
:
t7
cAr BA Ki rRuoc
bie M v6imgi0
bn-z*
O
NguydnVdnThdng,l1T, THPT Le Khidt ; Tp. Hd Chi
Minh: Trdn Vd Huy, Nguy,in Ldm H.ung, 11T,
PTNK-DHKHTN - DHQG Hn NOi ; Ddng ThSP:
vh'
Nguy,SnVdVinh Lbc,
(bn-z = -P, (0) )
Bni T8/295. Tim rdt cci cdc hdrn sA'f ; D + D
trong ii6 D = { l, +aa) thia mdn di€u kiett
(
J(x . f(y)) = y.f(x) v
ta du-o. c
Ldi giii. Cho y =/(1), tU
(2)
=fli)f(x)
f(xf(f(r)))
Chox=1, y=1, tt (1) ta dugcflfll)) =fl1) (3)
(1), (2) vA (3) suy ra :flxl(1)) =/(l)flx) Vx
cho y = 1 tu (1) c6 f(f(l)) =f(x)Vx e D-
Tt
Vi = 1, 3,...,2k-3'
Ta c6 xo = 2, xt = 4 vh xl*1 =Zxk + -r1-1 vi
ndu bro * 0 thi bz**z € 10, bzrl,ndlu b21, =0
thi c6 ba c6ch ldy bz**ze M, s6 ddy (bo, bz, ...,
*o=T
1t +
ral(1) = 1
TiI (1), v6i x = I
SuY
f(f(y)) = y YyeD (4)
Nhan x6t ringfly) = 1 khi vd chi k:hi y = 1'
Thit vAyfl1) = 1, cbn khifly) = 1 thi f(flv)) =
/(1) = 1. Do vAy vd D [1 ; +oo) nOn fly) > 1 khi
xp1'
Chung minh bang quy n4P theo
I
duoc
y>
:
tu c5 1lo = 3, Yt =
/ -\
= /o')/l 1l>ftyt khi.r>yll.
\Y/
Do voy, flx) h hdm ddng bidn tron D v6i
f(l) = 1. Gia sir 3xo > 1 sao choflxo) > xo. Khi
7,
1t+..8)t*1 +1t-Ji)k*t
"
d6/(f(x")) >flxo) hayflflx.))
,,
Do d6 s6 c6c ddy (bo, by ..., b2^-3, bz*-z)
th6a mdn (*) bang
(x*t - x._2).!^_z =
=
(rr*er'
)(u.Or
+s-li)")
+
Irrii
2*-r)'-l
fZf, ngt
18
chuYOn
Lc
T
QuY DOn
)
>flx),
1
mAu
dd/(x.) <
VAy/(,r) = xYx e D.
Thit lai, ta thdy f(x) =,r th6a man didu ki6n
x,,.
ra.
:
D -+ D th6a mdn (l) c6 thd dua vt! tim hhm th6a mdn
di6u kien
(Y) Yx,Y e D
[Xol= f lx)f
;
rdn-T i€'n H odng,
; Quing
Cfrng nhu vAy, kh6ng t6n tai ro
f
Minh Mdn,11T, THPT Luong Van Tuy
Quing Tri: Dodn Quang T ri, llT,
ro
NhAn x6t. l) Day li bdi to6n d6 c6 trong mOt sd thi
liOu (ching han trong cudn "Tuydn qp 200 bii thi v0
dich To6n). C6 gdn 2OO tai gi6i dring. 2) Vicc tim him
D6 chinh ld s6 c6c da thrlc P, (x) cdn tim.
NhAn x6t. Day ln bii to6n tinh to6n hay, hoi phtlc
tap. Tba soan nh6'n duoc ldi giii ctra la bq,], m9t sdt4n
d6 cho d6p sd sai. C6c ban sau c6 ldi giii tdt : Ninh
Binh:
=
thuAn.
bii
- $ + Ji)2*-1 + + 2Ji )2'-t
+0'Ji)'-t +(-l)(;-r)
- 0+Ji)'-t 2
(l
(5)
\ (_
\
L
!.yl=
t(ror)
flx) = "[v
rl '] tl
"\Y- '" - ')| =
)t+r=ZYr+Y*-rVk>1vd
!k-t=
0
/
(du dorln cOng thtlc tr6n bang c6ch ding phuong
trinh dic trmg)
HolLn todn tuong
c6
Tu (4), (1), (5) ta c6
-g - Ji)k*l
..8)t*1
THPT Tx. Sa D6c.
NGUYEN MINH DUC
Chf y : b, = I th\ b,*, e {0, 11, bi = -1 thi
bi*z e {-1,0} , bi =0th\ bi+2 e {-1,0, 1}'
Ta kf hiOu : xp li sd ddy (bo, b2, ..., bzk) th6a
mdn b2, e M vdimoi 0 < i < k'bo * 0 vi
bi*z - b, e M Vi = 0, 2, ...,2k-2;
yr-r Id sd ddy (br bt, ..., bzt-r) th6a mdn
bzi-t e M vdi moi 1 < i < k, bi*z-b, e M
b2p) md bzr, = O bang
llT,
Ngai:
lf (f
*
Yx e D
NGUYEN VAN MAU
Bhi T9/295. Goi ,5, R, r ldn ltrtt lii didtt tith'
bdn kinh dudng trbn ngoai ri|p vd bdtt kirrh
drtdng trdn n\i ilAp cia tam gid.c ABC. Dat a =
BC, b
=
AC, c = AB. Cltttttg tnitrlr rtttrg
cAr eAr xi rnuoc
25AB
*
p -< tl, + c a\stn- + (a - b -r c)srn(a+h-c)sin! -.I
2r
+
Mdi ddng
Ldi
thttrc
.+
(l)
xtiy ra khi ndo
?
(Cua ban Ddrtg Th€'Tt)ng, l0T,
ruNK Hin Thuy6n, Bdc Ninh)
+\a+
D
-c)sln-
I
sinC. .or4 *.or4
222
Ta c6 : sinA + sinB + sinC =
sinB + sinC sinC +
sinA
=-
222
. B+C C+A
( g1p-+Sln-+Sln222
ABC
= cos- + cos- + cos222
*.or9
sinA + sinB
A+B
(
Tathay:P<1
r
2(p-cltg!.or!
' "2 2 =,
o
zr( ror4*.or4 *
2
"orC)
2)
Z
2
NhAn x6t. l) Day ln bii to6n d6 ; rAt nhidu ban tham
gia giii ve tat ca ddu gi6i dring. C6 nhtng ban cdn nhAn
xdt rang ti bhi toiin niy ta lai nhAn duoc ket qui quen
thuoc 2r < R
2) C6c ban sau dAy c6 ldi
giii
tot.
Luong Van Ch6nh, Tuy Hda; Quing Nga'i: Dang Qudc
Vi€t, lOT, TIIPT Lc Khidt ; Bac Li6u: Nguydn Danh
Dfing, LlT, THIrI chuyOn Bac Li0u ; Ngh6 An : Trdn
Dinh Trung, l lA Tin, khdi IrIC To6n - Tin, DH Vinh,
Le Thny Duong, THPT Kim Li0n, Nam Din ; D6ng
Nai: Pham Minh Thdnh, 1lT, THft Luong Thd Vinh ;
Phri Tho z Nguydn ThdTiing, l0Al, THPT chuy6n
Hing Vuong ; Vinh Ph[c : Nguyin Nam Cfui'n,10A1,
THPT chuy€n VInh Phric.
NGUYEN MINH HA
lA, B, C, Dl vir d4t OG = d, thd thi 0 < d < R.
D6t : s = g(M) = MAz + MB2 + MCz + MD2,
A B C)
_.A.B.C(
<3 6l(sm-sm-sm-l cos-+cos- +cos- | <
2 2 2\ 2
)J
-/
Rni T101295. Cho rLr diitt ABCD nAi tii'p mr)t
nut cdu larn O bcin kinlt R. Hii;- -ytic clirtit t't tti
didm M tdn mdt cdu sao cho tottg cdc binlr
phaong cdc khodng cdch ir !,[ tri'i ctic dinh ctitt
tft dien dat giit tri lht nhd1, ttho tthd't.
Ldi giii. Goi G lI trong tAm ctra trl didm
- dry4 cost * rr, - uYr! "o, ! *
2
|
)
Hda Binh: Hd Hftu Cao Trinh, THPT NK Hodng
Van Thu; Phf Ydn: Ldrn Bdo Khd Minh,10T2, THI'I
Vay BDT ben tr6i (1) dfng. Ding thric xiy ra
khi v) chi khi AABC ddu.
+
\
Vay BDT bcn phii (1) dring. Ding thrlc x|y ra
khi vi chi khi MBC ddu.
a+b+c(cos-+cos-+cosA
B
C
2R222
c> sinA + sinB +
t:
c) <2.9=J1
.^(
: Zl ccls-A+ cc\s-B+ crxi-
a co
)ABC
< cotp-+cotp-+cots'2
'2
'2
.2,
o 2P'
R222"rr! *2, "or! + 2, "or9
[
ABC
< cotp- + cots- + cots"2 "2 '2
C
,R, ?!.,
2(p
<- ^(ABC)
zl cos-+cos-+cos- I <
[ 2 2 2)
= (b+c-a)sin4 + @-A*4rir!
22
Ta thAv
e
222
^(ABC)ABC
<+ 2l
cos- +cos- +c6-
giii.
Dat P
.ABC
< 4cos-cos-cos-
(VM)
2)
; r6i khai tridn MF' =GP-Gil12=
GP
-2 +GM
-2 -zGP.GM vdi P e lA, B, C, Dl,
R(sinA+sin-B +sinC)
B
C)
A B. C(cos-+cos-+cosA
e Esln-srn-srn-l
2
2)
2 2 2[ 2
I
ta
du-o.c
: s = q(M) = q(G) + 4GM2, (VM) (do
ce + cn+Ge +cD=d)
(1)
Arc M = O,ta duoc :
t9
crAr BA xi
q(O) = q(G) + 4GO2 =
Suy
ra: q(G) = q(fr -
4* + q1C1= 4rt
*)
Chri
Q)
(3)
Mat kh6c, trong tam gi6c OGM ta lai c6 cric
BDT tam gidc sau:
R - d = OM - OG < GM < OM + OG = R + d,
@)
hay ld | @ - a2
:
GM =R (khdng ddl) e d =0 e G = O e
ABCD ld mOt trl di0n gdn ddu.
. GM + Re d * 0 e ABCD kh6ng PhAi ld
m6t iri diQn gAn ddu.
o
VAy ta di ddn kdt
luin
:
1) Ndu ABCD kh6ng Phai lA mot
thi :
tf
dien
gdn ddu,
&) s,,u*
=
4l(R2
e
- *) * G+ A\=
8R(R + d),
M = Mrld, giao didm cria tia GO vormitt
cdu?\O,R) ; khi d6 O thu6cGMy
= aL(tr - *) * @ - A\= 8R(R - 4
M = M2ld giao didm ctra tia OG v6i mdt
b) s.in
e
f
ring
ndu G = O thi duong thing OG bat dinh
vi s = q(M) = 8f khong tldi vdi mqi didm M trcn mat
cdu ngoai tidp tf diOn ABCD dd cho, vd khOng c6 gi6 tri
Tt (1) vd (2), ta thu duo. c h0 thrlc sau :
g(M)=s=4[(R2 -*)*GM27,YM e '6(0,
R), trlc ld OM = R
rnuoc
klri d6 G thu6c OM2.
2) Ndu ABCD ld m6t tr1 diQn gdn ddu (cdc
canh ddi diQn bang nhau) thi G = O a d = 0,
khi d6 :
s = 4[(R2 - t) * R2] = 8d khong ddi,vu e
v(o, R).
cd:l G(O, R) ;
NhAn x6t. 1) Day h mQt biito6n dung hinh g6n v6i
tim cuc tri hinh hoc, & dey didm phAitim lai-gFn v6i
mOt gi6 tri tuong rlng vdi n0 ta tdn nhat !1y nh6 nhit.
oo a"o, uai to6n Ibi tr-6i pirai x6c dinh vi tri hinh hqc ctrl
lon n[dt, c0ng nhu nh6 nhdt. Hdu hdt c6c ban, do vQi
vhng mh kh6ng biOn lu6n vd vi tri tuon-g ddi c[ra trong
tam-G cira tf dien ABCD d6i voi tam O mit cdu ngoai
tiep tf dien.
2) L6i gihi dua ra 6 trcn da triqt dd !t d,llg lluqng
phrip v6cto, vi d6 Id ldi giii ngin gon hon ci' C6c ban
lO ttrd str dung c6ng thrlc vd dudng trung tuy0n-trong
tam si6c t2'ldnl nhune vi€c tinh to6n den kdt quri cuoi
cDng"sE cong k6nh, ph?c tap hon nhidu ! MOt # ban da
neulu6ng ttr6l quai h6a bli to6n drr6i dang x6t mQt hQ
r didm (trong mit phing n > 4), cdn trong khOng gian
n 2 5) lAr, Az, ..., Anl d6ng vien, hodc ddng cdu
gi kh6
'qO, R). Hu6ng kh6i qu6t h6a niy kh0ng c6
tor4n
biri
quen
lu6n
biOn
ddu
niy
ban
khdn, nhung c6c
theo khoing cdch OG = d, thirnh th[r vi€c giAi biri to6n
v5n chua trgn ven I
3) C6c ban sau day c6 ldi giAi tdt, tuong ddi gon ging
vd tron ven hon ci.
Hi NOi z Nguy,Sn Trung Chinh, 12 To6n, TIIPT Ha
NOi - Amsterdam ; Phri Tho : Hodng Ngqc Minh'
11A1, THPT chuyOn Hing Vuong' Vict Tri ; Hda Binh:
Ngo Nhdt Sar, 10T, Hd Hfru CaoTrinh, lLT, NguyEn
Tuydn, 12T, THPT chuyOn Hoang Van Thq-; Hii
Duong i N guydn Ti€h Thdnh, 1 1 Li, THPT Nguy6n Trdi
Iim
; Ninh Binh: lz Minh loi, Trinh Thiy Nhung' llT,
TI{PT Lrrong Van Tuy, Ninh B)nh ; Quing Ngai:
Nguydn
Naii
fz
chuyen Le Khiet ; Diing
Phrong,11T1, THPT chuy€n Luong Thd Vinh,
Huy-Cung,lzi,TlffT
Bi0n Hda ; Ddng Thdp; Nguydn Vd Vinh Lbc, llT,
TIIPT Tx. Sa D6c ; Cdn Tho: Nguydn Minh Ludn,8A,
TFIPT Nguy6n Vi0t Hdng, TP. Cdn Tho.
NGUYEN DANG PHAT
Bni L1/295. Mit 0!eot'on drtng chu';i'n tling
,,,r'ti t,drt t6'C v. = l0'' nil s tlti ltat't'io vitttt tri
rliin trtdng deu vd tir tr*)ttg
E
rliu. ; vecto vdrt tilc ,1, I E
,eB
t,a LE (ttnh t,il.
i
Xdc dinh dd ktn. vritr tAc
didm phAi tim, ddng thoi cflng phii chira du-d.c gi6 tti t'tis €lectrort d thit'i didrn I
.u. aii hay cuc tiiu ring voididm tim duo.9 Khi d6.'
ngtldt lutdng t'rii t,,,. Biit rang F = t',.8 vi htj
didm phAi iim m6i thuciu tdn tai vh dlrng drro,c' Ndu
vi
tri
duoc
ra
mdi ctri ra duoc giii tri cqc tri mh chuachi
cluo tdc tlun.g ctia trong ittc.
hinh hoc cua didm cdn tim thi m6i chi chfng to ring :
Huong d6n giii. Ki hi0u O lI vi tri Olectron
Neu didm phii tim mh tdn tai thi nhftng gi6 tri cuc dai
b6t ddu di vdo virng c6 diOn trudng va tt trudng'
hay cuc tidu tim dugc chi ld khd ndng c6 thd dat drroc
o
hicn
thuc
duoc
m6i
ma *rbl vh bhi to6n dung h)nh
Chon truc Ox c6 huong trirng v6i vo , ttrtc O!
brtdc phdn tich, chi ra duo.c diiu kiin cdn nhrtng chra
dl. Nguoc lai, ndu chi ra dugc vi tri hinh hoc cira di6m hucrng xudng dudi. Ta thdy l1rc dign truong {
ohii ti"m md chua tinh cu thd gi6 iri crrc dai hay cuc tidu
dung len
ing v6i didm tim dugc itri tai :odn chua daoc girii tron luon c6 huong Oy, cdn luc til f, Ac
(c6
cdn
chta
ch{t
dil
ki€n
didu
ciugc
vii v\ m6i chi ra
0lectron IuOn vuOng g6c vdi v6cto vdn tejc i '
thd chua v6t hdt nghiOm cdn tim).
20
cdr eAl rirnuoc
X6tchuydnd0ngtheophuong
-evnB
mQ))
' fttdt'" = --eBdy.
dt,,.
Ox,tac6:-F,r= gid tri tlrc thdi t'tiu u vd i trottg ntqclt l()
*2,.5 (A) ;d ttt;i clii,nt
ri = l00vE ff)t,d
t t 'u tl*
it = ---"'t^t
mdv, )
tn:-J-= -eBv,dt
! =
dt
^
t1=1,-\1,tctcriu2=100(V)t,r)i2=-2,5rlj
I{d1t
rdc tlinh
ro, u,,, {p t,ci
(A)
t 1.
Hudng dAn girii. Ap dung dinh luAt 6m ta c6
: u = U,,sincot (l)
(v6i mach chi c6 C)
=>
i
=.I,,sin(ro/
+ e + ll uOi 1,, =
2'',
>i=CiolUncos(otr+9))
Tu (1) vh (2)
nit
Cr:,L/,,
(2)
+:-
ra, u2 = U3
C"
a'
Do d6 b c|,c rhdi didm tt vd t, ra
,? =ul,
Ldy tich phdn 2 v6 (vdi didu kicn y = 0 rhi
r', =vo) ta cluoc : y.r. = vo - 'U ,. Kf hi€u v lh
Thay
Olectron
khi d6 : y =
ls Jtm. 0). Ap dung
eB
dinh luAt b6o toin ndng luong (chri f luc tt
{
2t
khong thuc hi€n cong) : \*
eEy = !!-p7
2'2
(r)
II Niu-ton
*?
dt
=
eE
+ eli* E1,
2 truc vd gi6i hC phuong rrinh.
"hidu
len
Nhdn x6t. C6c em c6 ldi gi6i vi d6p sd dring.
NghO An : Hd Qudc Huy, ttAt, TlipT phan B6i
9]ril,Ytrt,Ngh6 An; Vinh phric : Kidu NgocThanh,
THPT chuy€n Vi* pt,t. ;
Nguy€nVdnTy, L0 Hfru Hd,'Ii Dtlc Dat. tt
lL.B3, Chu Anh
Qfrng,
Hi finh:
ilA3,
Li,.THPT NK He Inh ; Uai Duong: Nguyin
e:u61
p.1S.VA Xuin Thdng, t 1 Li, THPI NK Nguy6n Trai,
Hii
Duong.
MAI ANH
Bni
LZlzgS. Dqt hi€u diirt tlr6' Lt =
{l,,sin(at + cp) vdo hai bdn cia m6t tL.t dit)rr cri
di€rr cirtrrg C = lA*314r (F). O thrti clidru t1 r.ti{.
*:+oJ-: 4. =u: +-+;
C'a'
C-
c=
#
G), tir d6 tim
duo.
t li:
= - I'' -ii = l00n rad/s vi
c\ul
-ui
Ngoii ra ta cflng c6
c
:
Un
=
Z}OV.
:
Lrtc tr: z, = 196r[ = 200sin(100nr, + g)
i t = -2,5 = 5cos(100rctt + q)
o Lic tz - 1,51, i uz= l0O
= 200sin(l00nt2+g);
o
iz= -2,516 = 5cos(100n1, +
Tt (i) vd (2) tim du-o. c y - 3v,, = 3.105 m/s.
Chi i. CO ttrd iip dung phuong trinh cria dinh
luAt
c6
.2
f . ---i_
m
vAn tdc cria Olectron ; r4i rhdi didm i ngugc
hudng vdi v,, nghla li v, = ), ta c6toa d6 cua
:2
;
Til ci{c phuong trinh tren tim duoc : g
=
n
J
,
1
=3.3.10--3s.
' = -1300
I1
NhAn x6t. Cdc ban c6ldi giii tot :
Hodng NgqcTudn,12A2,TILFT Duong X6, Gia LAm
._Qu.ang Ngei: Bdr Minh Tri, I I Li, TI-Ih chuyOn L0
Khidt ; Nam Dinh: Trdn Vil Di€u,
ll
Todn
I, THpT
chuyOn Le HQng Phong ; euing Binh : Hodng Dirc
Quang_,11A, THPT Dio Duy Ti, Ddng H6i ; phri-Tho:
!9u1,01K1m Nsoc, t1Bl, THpI chu/cn Hirng Vuong;
L€ Vdn Quinh, Chu Anh Dtiig, trAl,
}1[P
