Bing nhirng thay tlgi rqng them nhitng dibu ki€n crja lgi thu ilugc nh&ng bdi to6n -.. Cn_ng nhu dLrng tluoc nhiiiu bii todn kh6 kh6c, ta thii b&ng phrr... Ta l6i thu duqe nghigm theo bb
Trang 1\ €y;f:rtt,{l_,
r
ttdr roAN noe
YIST N.{M
ch& nhi€m, neUyfiN C.f,nU foAN
Tru s&: 70 TrD.n Hung E4o - Ha N$i
HAt THANG ruQr Kt
't'hu k{ tda soan: HOANG CHtINe
Ddy n6i: S2SZS
vE r& y#AF{ Ac-r{6N
to6n hay kh6ng phli bao gi}
ngay m$t hic Cd khi nd xudt toAn d6 Bing nhirng thay tlgi
rqng them nhitng dibu ki€n crja lgi thu ilugc nh&ng bdi to6n
- Trong bAi niy chtiag ta hdy theo d6i qra [}int
l.: s:3r:til;i1'*':; iir ris:i"ffi
cho tleb crla b6t kj n ; 1 s6 trong "O**'16"g
th6m 1 chia h6t cho c6 cdn lai r -f,
Spiiat fa bdi todn -l(n').
l BAi to6n .4(n) 116 cd ldi giii khi
han khi rr:2, B, t D6 la
"*" Uui nhion uri'i n btii kfi A0:-) urtn cdn td
co liri q t&i.
Cn_ng nhu dLrng tluoc nhiiiu bii todn kh6 kh6c,
ta thii b&ng phrr<rng ph6p crla < nha -ogtien
l".o r,^ xu{.t ph6t tu viQc quan s6t, phAn t'ictr
Ioigiii eria bdi toin A(2), i;et, atri ii't:tep ,an
tdi ldi gi&i e&a bAi tofn t-ong tluong t, jf 't6ng
q uat.
2 Ldi gi&t erla bAl ro6n A (n) khi n_2, g,4
._ A(2) IA mgt bii to{n khe d6 N6 c6 nghignr Ii
t7, lt {1,2) vi r2, B).
BAy gio ta chuy6n sang xet bai to6n A(B).
- Trudc ltdt ta hdy tana gAc c[c nghidm cd chrla t
Gil srl k, y, il lA rngt nghiQm nio dIy eia
bii toAn A(S) D6 dnng
-nhiu tlay
"eng-;rt ty
hai s6 trong b$ ba d6 phAi nguy&rrtd
",:1Jnfr",r Vfy c6 thb xem 2 (i<y<r.
Tu ttiBu kign cfia bdi to6u tir cd I \
trong (Id kt, kz, /<3 l[ c6c s6 tqt nhi6n
Nhin c{c h6 thrtc fiy theo v6 ta c6
tgz bgz * t*g*) * *y * gz * x.z { 1
- kl Kz kS xllz
*)cg*gz*rz*lirg:
+rA*y:*r:11=Arg: (l lasd tu nhi0n) + llo * llg + tlz * tlry: = A.
Gi{ tri l6n ntrdt cria A dqt duqc khi r, g,:
b6 nh{t, tte lA khi t=2, y = B, z = 4 (tron thue
td z) $,
PHAN DOC TEIANTT
r 1 Nhibu bdi
ph{t tr} rn6t b&,i
chtt it hay m&
bdi ro6n a6 tro ta
thri vi.'
t*y * 1: k1z
$rr+ r-kzl
lA z * 7 - k3r
,1
&
!
I
i
^
rt nh6, ch8ng
to{n d6, Tuy bd.i toan thio
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Trang 2t-T* .16
.4 < 1/2"+ 1/3 + 1lr I rl2'$' 4:27124
"*z{=1.
Vay (o, U, z) th6a man.
lla * llY * llz + tlrgzs !" (1)
Biy gid ta chring minh ec:2' Thqe v$"y ndu
ngo-ọ i4i, tue Id n6u r 23 rhl g) 4, z)5 ncn
ta cd
| = tlr + llg * rlz * tlrgz( t/e + 114 + 115 +
+ 1/(3.4.5) = 48/60
iE AiCu kh6ng th8 ttugc' VAy chl c6lhb r=2'
(1) =+ 1 = tlz + llg * 7lz * ll2gz
4g=2*5i\z-2t*
4z=3 haY z =? +g=l ft6Y g:3"
Y$y nghiQm kh6ug chua 1 efra bii to6n Ă3)
iA (2;3,7) N6u kE cl nghiQm chua 1 ta edn c6
be b0 nta:
(1,1,1), (1, 1,2) va (1,2,3),
E6i vdt bili torrn Ă4) cflng gtli tuang trp'
D6ng vai trd crh phuong trinh (1) Il phuong
trinh
11rc * llg * llz * Tltt * llrlizu=l'
Tu phuong triuh d6 ta thu dugc nghiQm kh6ng
chua i cria lai to{n l& (2,3,7,43)' NEu kE c&
nghi$m chrta 1 ta edn e6 th8m (1, 1' 1' 1)'
(1,1, 1,2) (1, 1, 2, 3) vt $,2'3,7)'
(r1, ,rn) th6a mlu phuong trinh'
1/rr+ *llrn* lfrt nrr:l (zt
(,{ lA s6 tB nbi6n)
MQnh tIE ngpgc cfing d{ng: N6u rt, "', r ll
cdrc sd tq uhien th6a md.u phuuug trinh (2) voi
m$t sd ttt nhitn A nio <tdy thi bO (r1,"',orr) l&
nghi$m e{ra bdi to{n Ău)
Rdt tlAng ti6c ln trong truor.rg họ'p n b6t k5r
kh0og phAi ta luOn e6 -rt = ị Trong thqc t4 ;l:1
chl đi vdi nhitng n =( SS.
3 Bty gio dB d6 quan slt r,'I phln tith ta
hBy ligt k& c6c nghidrn cria bii to6n Ă2) Ă3)'
Ă4) dudi d4ng mQt blng sau:
ă2): (t,1) (1' 2) e,3)
Ă3): (1, 1, 1) (t, 1,2) (1,2, 3) (2,3, 7)
Ă4): (1,1, 1, 1) (1, i,1,2) a,1, 2,'j') (1'2,3'7)
(t, :l,l , +i)
'fir b&ug tr$n ia llir$.n ihf;y ririg ir*eg rt:Si
c$t ilghiAul & ilai:g t].rrd"i tiirln tlr-ge trí nghiQrur
& hdng trou b{rg c6ch th€rn v&o sd tr,
.J
Tt d6 ta c6 bb db bi&n nhieu aeu:
Bb de tr NEu hg (rt, , ct) la nghtQm cria bAi to{n Ăh) thi b0 (rr, ":rl, 1) lI nghiQnt e&a bAi to6n Ăt( + 1) NguEe lai n6u trotg
k*1 ed lh nghiQm clls b&i ts6n Ăk+1) cd
ch*a 1 thi saukhi b& 1di ta nh$n {Iuqe nght(m
c&a bii toAn Ăk).
Cdn d&y (2 g), (2, 3, 7), dlsgc ph&n dnh
trong bb dE aaụ
Bb ila 2 Íi6u bg (rr, rt) lil nghiQm cira b&i tofln Ăft) thi b0 (sl ,.',:uk, rt 'ẹr'*1) lA
Thgc vqy tich eria k sd dhu ti€o c$ng th€m
I chia h6t cho rt+t theo ciirih nghia erie s6 rsal B&y gi& gii s-& rtit c6 m4t trong tlch'
Ta {9i X; lil tlch cira /t-1 sd d}u ti€n trong
do kh0og chita rị
Ta chn chrlng nriuh r'frng Xi rr+1 * 1 chia
h6t cho r, DiBu đ suY ra t*.
xk+l: Xịr,1 a 1 "+Xlrsgl +1:,Ii(Iirl +t) - t : ritriz + (Xi + i)
s6 hAng thu hai chia h61 ctio ri theo diiu
kign cria bA dE 2.
Nhe bb dE 1 va bB aI€ 2 ta nh{n th6v rang
xust phirt t& m$t nght{m efra bli lo&n '\(k)
ia thu ituqc liai ngtri6m *ir* biii toirn Ăl+1)' Bfrng e6eh dy xu{t phet tir nghi6m cria bii toirt
Ă2) ta thu cluqc cae nghiQrn ct" bai tnf," Ă3)
vir k6 tlo la thu itrugc cac nghi{m cria bii
toAn Ặ1).
D6 dang nb{n thdy ring }:ịng c{ch d-v ta
thu duEc tdt * *a* ogfri6n, ciia bai tain Ă2)' Ă3), Ă4) Tuy nhi€n diBu dAng.tig0 lh xu[t
pnai tr) nghi0r:r c&a bii toan Ă+) blng c6'eh
iy ta thu dug'c lchdng phdi tdt cd e{e nghiAn
eira biri iodn Ă5) Ch&ng hqn ngo{i nghi€m
tz, {'l , 43, 1807, thu tluqc theo cach cirr bb
dE 2 lai edn c6 nghitm (2 3, 7, 47,39il' Chinb lf do d6 tr&t Uuqc chrlug ta PbAi ti6P tuéhhai iha on dudng x&y tlgng nghif m cira
bli to6n daog t6t khi n bflt ki"
4" Gi:l sit {rt, ', rk) t$ nghi$m
toAn ĂA)'
D[t /r: s!,, rt T* e&
llq * "" '1 l/rtr * llfv: 11
crl a bif
-
-Fr-(3) FIou nilági& str ot+r * tr seo cho trcl' "'' rk' uu-r-t) t&p ttrinh nghiQm cfia bei ío{n jr(l'+t)'
Ta co
1b:r * , i- 1/rr.íllr:s+r -l- l/(/ra:t-pt) '* A id'
VIETMATHS.NET
Trang 3T& 3) vl (4) te suy Ia
llru.+-t * l/(/rrx+t) * tlfu + ,k+r
- li+ i.
Ta l6i thu duqe nghigm theo bb dE 2.
Nhu v&y ndu nghigm crla b0.i toAn A(k+t)
nh$n duqc b$ng c6ch th0m mSt s6 vio nghiQm
ctia bdi toAn A(ft) tht sr5 nAy cluoe tgo n6n
ho{e theo bA dE t ho{c theo h& d& p.
T& aI6 te nhsn th6y rfrng vigc th6rn v&o mQt
s6 khOng mang l4i cho ta dilu gi m&i ngoili
nhtng tttEu d[ bi6t 6 tr6n
5 BAy gid ta gi& thi6t rfrng (rt, , er) ld
trghigm crla bti todn -4(&) Ta e6
tlrr* * tlxy* tlfy* A (5)
Hon nta gi6 s& (rl, ., ,rk, ttk+1, rr+ a)
I& ngh.igm cta bii to6n A(k+2) trong d6:
*k+r + l ruaz * t fr c6 (v6i cting-,A & tr€ri)
1l*t* * Tlna * l/a r+r * llra+z *
+ l/ar+r + llxt*+z + tl$vru+tta4fi * tlfa fi)
+ &k+z=fu.+ (fL + 1Jl|ir;yl-l - ,r1 (s)
Do 1t.1-2,,phii nguy€n n€n d: xik+t /r IA
uoc crla /i * 1 VaV
(rr1r:/r*d
(*o+r:fu.+(f?+ttld
Nhd clc h6 thrlc (5) (6) (7) (s) ve ts) tc di d€n
- Bb de 3 Gi& sri (.r1, , rk) lA nghiQrn c&a
b&i to{n AG), f y: trl : nk v[ d IA u6.o cris
ff+ r rni ao rg |/rr, , **, fx+ a"fu+ Io'+1 ]
\ - ,^ *'.A d l
l& nghiQm cria bAi to6n A(k + 2).
- Nguge la.i ndu ughifm cfia bAi roen A(k+2)
nh{n cluoc tu rrgt nghiQm nio it{.y cria bii to{n
A(le) bing e{ch them v{o hai s6 kh{; 1 thi
c6o sd 6d th€m v{o th6a mdn rl8ng thric (9).
Ch(l i- r"rog a : I v{ d = fl.* 1 lu6n lu6n lA
u0c crin ff 1 1 Erii vdi cr4.e u6c nAy b6 dE 3
klg"S cho ta til6u gi mdi so v6i b6 a6 : roy
nhi€n n6u ff + I kh6ng phli In s6 nguy&n 16 thi
bb dE 3 cho ta mOt lo4t nghipm *"i Ct&ng
hau dl6i v6i ughiQm (?, :i, ?) ta cd /32 4 1 :
.- 1765 = 353.5 Xu{t ph[t tu nghidur naj vd ru
bA dB S ta thu dugc nghidm {2,:1,2, aZ, :gS) ma
ta 116 ldi d6n trong rnqc B" Tuong tlr xuSt ph6t
tt nghigro (2, A,2,13) nhd bb db Bta thu dugc
ughi$m (2, 3, 7, 43, 1829, 193667).
Sja,t nghia- DSy nghi(rn thu drrqe tu nghi€m
(2, 3) theo bA d0 duoe goi lil nghi(m co s&.
N6ri c6 fT+t i& hdo uJ,u, ,nul b6 dE B ta
t{ch ra dugc,it nhdt Id mgt nh6nh nghi6m
Bing quy nqp ta chung minir iluqc rfing /2r.
{ t{n ering b*ng 6 v} /zr+r e6 r{,n etng bing 2.
Tt d6 suy ra rfing trong ddy eo s& te1 ci eae
sd /it+r * 1 dEu chia h6t cho 5.
T* tinh ch{t, nAy ta uhgn thfy rlng: khi n 16
nh& bb dE 3 tt d6y eo s& ta trich ra duoc mOt
nghiQm, nghigm ndy nhd bA dE Z sinh re dluge m{t ddy mdi cAc nghigm.
- 6, BAy gi0 ta dl theo mgr hu0ng khtc rtB gili
bii toAn: Xu{t ph6t tr} ughigm crla bii toAn
A(n) hdy tim c{eh thu duqc m(t nghi0m kh6c erja bAi todn d6 Theo hu6ng uAy te c6. tsinh lf (Br6tsky)
Gii srl (*1, ,, nn-Z, r1-1,.r1) lA nghi0m thu duoc tu nghi€rn (rt, , , *t_2, ro_t) theo b6
dE 2 vA (fn-zxn* t)/rrr*r = dtdz.
N6u hQ pbuong trinh
\ f"-z*lt- d1y
I f"-z g * 1: dzr (10)
cd nghigm nguyen thl
{ry , , rn-z, r, y) cEng ld nghi{m cris b[i
to{n
Ohtrng- minh Biing c{ch nbAn cdc phuong trinh e&a hd (10) ta e6
f?-z.r.u * fa-z 6 * g) a 1: &:?'d-L ,,
f,n_l
hry lla + 1ly * tl\fn:2*g) : - f,-L * fn*z?t * |
fn-Z frn*l
Cht 1i r&ug
vit
a'r-/o"-1 *1 fn-l = fr-* *n*l
vI nhd h€ rhue tlar"+- * lf rr_1 * tf fn-1 : ,1
ta di ddn h0 thtrc
ltfi = , * Tfrn-z * 1lc' * llg * l\ff,-zrg)=A , Pii" d6 chung t6 ring kt, ,nn-2, r, y) th
nghi{m crla bti to{n A(n)
Nho tlinh lf trtn ruit ph{r tu nghiQm (2, 3,
7, 47, 395, WSTZ|) ta thu duqo mgt ngl,ipm fnac (2,3,7,47,415,8lt1) cria bAi toAn aO Ngfrigm
niy khOng tao ra rluqc bing c6ch drlng b8tb'3.
116 deng chrlng minh rluge r&ng ildi v0i c6,c
nghidm tu d6y eo s& hC (10) lu$n luOn c6 nghiQm
nguyBn.
Cho d6n nay chung ta v6n cbua bidt duEc
bii todu A(n) e6 bao nhi€u nghigm vdi n cho tru6_c (tuy f) Chring ta chl md,i di duge rnQt vii bu6.c tr€n con iludng cdn kh6 dAi iI6 t6i ldi gifii e&a bi.i todn Tr6n budc d.udrig ng*n ngrii
dd de nhy einh ntrtng vdn 6E m6i" dri-i h6i ta phli suy nghi lgi cAc phrong ph6p eii vd ilm
Chrie c{c bq.u rhinh c6og lr8n chflng tludng
con Igt
VIETMATHS.NET
Trang 4rrfrP TIJC stJY uu.rgru rmfru mAh& e#
lt t,rr:'r xca
?R0NG hai s6 b6o 101 va 105, bsn l-€ Th6ng
Il Nnat 6e gi6i thiQu vdi c6c b4n mot loat
c6rc bii to6n trOn bhn cd, md cht y6u In
ci.c bili to{n ph& vir chia b&n cd' L[n niy t6i
xlrr bb sung mQt sd ki6u suy lu{n khr{e tr€n b}n
co mil qua d6 mong edc bqn dugc lirm quen v0i
nhiEu bli to6n thri vi
L Phfi eh*e bin c& b&i c6c domino.
Tnroc h6t ta gqi cac dudng k6 trdn birn c&
li c6c dudng gi6i h4n' Biin cir m X n goi lir duoc
phri ch&e cnan nai cdc dOminO ndu n6 dugc phri
tin r&i c{e dOmino khOng chbng l8n nhau vd
n6u chia birn cd theo bf,t cit ttuong gi6i hq'n ndo
ciiug sE e&t it nhdt mQt hinh tl0rnin6'
Ta r6t bAi tof,n trdn biln cb 100 x 4:
Bai toin 1 Chrlng miuh ring birn cd 100 X 4
id bin cd khOng thb phi eh[c ch&n bivi e6c
ddmin0
Ldi gi&i Ta th6y birn eo 10CI X 4 khi t1ug,c
ehia d6li b&i bflt c& rtudng gi6i han ndo thi m6i
phln ctng ed mQt s6 chXn 6.
Gii sft b[n cd duqc phir kin boi cdc ddmia6
khOng ehbng l6n nhau thi & m6i phhn c6 2
lo4i 6:
*loai o b! phrl ir6i domin& khong bi c[t (I)
- loai 6 bi pbfr b&i ilomtnd bi cE"t {II)
I'a thdy ngay sd 0 cria logi I ld sd chin, vi
m6i domino irhOng b! eit s6 phrl 2 6 eria bin
c& Do t8ng sd 6 cric 2 toqi & rnSi ph8'n I* s6
chin, n6n suy rl sd 0 lo+i II & rn6i phhn ctng
lil sd ch6a' Nhu v{y s6 edc d6min0 t}i c5t la $d
ehla Nhu v$y n$u tdt cir 102 duo'ng gi6i han
tiuong nio ctng cflt dOmind thi tt nhit tr&n b[u
ed phli cd 162 X 2:204 ddmiu6 Nhung bln
cd ilrrqc ph& kin b&i 20tl ddmin6 Vny it nh{t
phfri c$ Eudug gi6i hqn khOng c&L d6tnin6 riro'
Soy ru b&n c0 1.00 x 4 hhfrng thE phir chic ch*n
b&i c6c d6min6
Cric ban hey 6p dgng e&ch lflp lu{n tr€n cho
bin c& 6 X 6, sau al6 ctc bgn thir x6t bhu c<v
I X 8 xem sao ? Ngudi ta dd ehtrng rninh ilugc :
(Mei bin c& a6.di0n tleh chhn v& m6! chlBu
l6p bon 4 (trti biii ce 6 X 6) dBu e6 thb tlugc
ph& ch&c ch&n >r Cic b4n hfiy t$ chung minh
rngnh tlE tI6"
U V.Si b&i to6n vE btt&c di c&a eon EgEa' Ph&i n6i rXng eon ngrlx ie con c& di I6t ieo
nhdt tr0n bln c0, v$y mb il6i vdri n6 l4i niy ra
uhi0u bti to6n thir vi'
tsai toen 2 TrOn m6i 6 cria m6t ban c$ cd s6
6 tn mQt s6 16, ngutxi ta d4t m$t con ngga' Li{u
ed thB cho m6i eou ngqa cung di mQt biroe sao
cho m6i 6 l4i c6 dring nr$t con ngqa hay khdng'?
Lai gi/Li.'l'r8n bda cd dd ta e f,ng co e6'c 0
trfing vA cdLc 6 den xen kE nhau sao cho kh6ng e6 2 6 tr8ng ho[c 2 6 den nio chung c4nh \hu
vfy ta sE g;i cdc con nglte dung &-6 trAng In ngua trXngl cou c6c eon Dgga dung & 6 tien 1A
ngua tten N6u'rn6i con ngua dEu dr mOt b':irc
't e A' *5; 6 lqi e6 6uug mQt con nglta, thi ''1o
cdch di eria con rlgqa nen c{c con ngua tring
s€ nhiy viro cr(c 0 den vd cic con ngta den sE
nh6.-v viro cac 6 trin6 Suy ra s6 ngrla trlng r-i
ngua dlen ih r:hu nhau Suy ra s6 6 biin cd la
cien, mau thr:Xn v6i gi& thi6t birn c& cc so 6
IA Vay khdng th6 eho rn6i con dbu ili rndt huse
alB sau d6 n.i6i 6 l4i c6 dring mgt con'
Bii toin 3 Coa n$r,ra cd thb di t"r goc niv crii'
bdu eo th0ng ihucrng toi g6c d6i di€n sao cho
di qua m5i 6 cria birn ed ilung m$t lrin dugc khOng ?
Ldi gi&i Ith6ng thE dusc, gi& o& 6 & g6e eou nglr& xu6t ph{t l[ 6 den thi sau mQt sd 16 bu0c
"or, ,g,y" plai toi 6 tr{ng Do eon ngua ph&i
di qua ,r',6i 6 cri* bAn ccr dring mQt l$n db cu6i
c,ing tOi 6 al6i dien v&i 6 xu$t phai n6n n6 d6a
o u$y t} l:u6c thu 63' 56 hudc di 16 n6n suy ra
O ney te d ir[ng MAu thu6n vt n6u 0 xudt ph{t
la 6 den iht 6 ai g6c ddi diQn ctng phii 11 6
den.
'I'ruo'c khi sai:g bdi toan '{, ta hiBu rnag u:Qt birn cd goi ti ddng v&i hirntl trlnh e &a coll DgUa
ndu ccn ngtla ct5 tha €i clua nr6i d dring mQt ldn
vd tr6 vb 0 xu{t ph{t
Ta thSy ngay n6u birn co d6*g voi hdnb irinh
ccn DgUa thi ph&i e0 sd d ch5"n vi d6i vtri rn0t
hi.nh triuh kin c{ra con ngqa thi s$ 0 ilen mii con
nglra di qun sB l:ing s6 6 tr&'trg rn& nd di qua.
Vi v$y bflt c{r biLn civ niro c6 s6 S tr6ag kir{c a6 6 den thi ktrdng rlting v6i h&*h triuh e oo :1gr:a'
1
a"-VIETMATHS.NET
Trang 5Ta x€t bAi toiin;
BAi Eo;ilr n" {.Lir*g uiitli r}.dg v{ri rloi s6 tU
rihi8n n iiri hd,a q:ir n X 4 tihorrg d6ug v6i hfruh
{rinh r::n.-il rif r:ri.
Lbt giii Gi6 s* vfyi ss tr nhisn n nir-r &*, ban
ed n X I i-ting v#i hinh trlnh co* ngna igilu
y&y t4i ;n6i S ph&i ed rngt ?rrri;c vSo v&' mgi
hudc ra cira con ngna, tirc iA ph&i cr1 mgt budc
con ngua di vdo 6 5y I'i mgt buoc con llgua
tr't 0 {y dli la ila hinh dur:g ban cd' ndy gdm
4 d&i 6, hai d&i & ngoli ta goi li bi€n, cdn hai
d&i & trong ta goi ti gifta
Ta thfly trl 6 & hion, con ngva chi cd thB di
v&o 6 & gilia Ilo c6 2n 6 6 bi6n n6n phAi e6
2n br.r0c bu6c vAo 6 giira Do m6i 0 di qua
iluag rndt lin n€n & m6i 6 giira chi cdn e6 mgt
bu&c ra Nhung budc ra ndy hh6ng thb di tdi
m$t 0 6 gi&a vl tdt c& c6c 6 5,g.i Ea 116 e6 rngt
bu6e vAo, v0.y bu0c rn r-rAy phii tt 6 & gitra d6
ditdi6&bi&n.
VSy col ngga ph&i irin luot tu bj6n i,iro gifra
rdi ra bi6n, rbi r'$o giira, r.v,., cr! :ten k6 nhau
nhu v&y trong surli hanb trirh Do m6i huoc
tlti con ngqa dr tir (r rniu niy ddn 6 mhu khAe,
n6n suy ra t6t ci cic d & gi&a cung miu vi
c{c 6 & bi4o eirng rndu, v6 li Viy bin co n X 4
kh6ng ddng v&i h&nh trinh con ngqa \,6i moi
sd trr nhidlr n.
Cde bnn h5y gi&i trii lo{n
E&i ro*l1 "{ Con ngra kh&ng th& &i t&i t$t cfl
mSt lln.
E6i v<yi bii to6n t6ng qu6t ngudi ta dt tim
cl'trqc cdc k€li gu& iray:
Vo'i birn cLy n >< rr con ngua c6 thE di t6i tflt
ci c6c S c&a bdu cd' mir qua ede 6 dtng m6t thn
n6u n: I (mod 4), ho{c n: Z(mod +), hoae
n == 0 (mod 4) n6u n ) 4 FIai truirng hqp sau
do N, Nhextvetaev li 1116i hoc sinh & L€ningrat
Li6u x0) tim ra NgoAi ra Nhextvetaev cdri dua
ra mdt hAuh trlnh t&ng qu{t cho con ngua v6i
b)n ed n X n khi n) lS
Ngudi ta etng tint cluoc bdn cr) 7x7, \!xll
cGng th6a mdn d& con ngqa di h6t c&c d cria
bitn c& nii qua m5i 6 dring mQt l&n.
I(h0rg uhirng th€ ngudi ta cOn x6t cdic bdi
toitn d6i vdi cAc biru c& trong khOng gian, ch&ng
hgn nhu con ngga di hgt cdc 6 }gp phuong cira
bAn cb ,1 'X .4, X 4 vd di h6t cric d vu6ng tr6n
6 mat cfia hinh {ep phuong I >< 8 X 8n sao cho khOng ili qua noi uho quA rn6t lHr)"
3 Vni bii to6n vB con xe Ea! tc.&n 6, Ir€n bArr ed nXn x$p Ii con xe &
vi tr{ rn6i con khdng & cung hAng vr}i qud 1
coo xe kh{c Ch*ng mirlh k(an/3.
Lbi gidi Giit srl cd A con xe xdp nhu diBu kiQn cria trii todn iv m6i 0 c6 con xe ilung bHt
ilhu ta ghi s6 0 Sau iI6 ta x6t tr€n c5.c hirug ngang :
- N6u hing nio cO 1 con xe thl 6 c6 con xe
ta c6ng v0i 2.
- Ndu hlng nAo c6 2 con xe thi 6 c6 con xe
ta c6ng v&i l.
Ti6p dd ta lqi lirrn vdi cdc hAng dqe tuong tu
nhu v{y Cu6i eirng & c{c 0 c6 con xe drlng chi c6 th6 iluqc ghi s6 3 hay s6 4.
V{y t8ng S c{c s6 ghi 6' c{c 6 c6 con xe dung
ph&i th6o m[n: 3k =-( S.
ilI[t khdc : & m5i hirng ngaug ta cdng khong
qu{ 2 V6y sau khi lim cdc hAug ngang ta cQng
rAo kh6ng qu6 2n
Tuerng tg v.6i c6e hirng d(.rc Do d6 toin b0
bin cd ta c$ng vao khOng qu6 4n" Tuc li
S.-(+n Suy ra: 3k ={ +n <+ ft ( ani 3.
Khi n:8 thi s6 .k l&ri nh6t lA 10; c{c bqn hdy
thfi x6p 10 con xe th6s mfln bii to6n trEn birn
cd 8X8
C6e ban hdy Idrn hai bii torin sau dAy.
Eii toiu ? Tr0n 64 6 cfia bi.n co thn luqrt
vi6t cric s6 tu 1 d6n 64, theo thit t{ tir trCn
xu6ng dn0i vd mdi hdng ngang tht tir trri sang phii.
FISy tim t6ng c{c s6 ghi & 8 6 nrd 8 con xe dirng & vi tri khdng th8 6n duqe l5n nhau.
Bai to6n 8 Einh sd c5"e hAng ugang tu trdn xudng du6i vA ddnh s6 cic hdng doc tn trAi
sang phii 6 m5i 6 bAn cd ta ghi cou s6 li tich cria chi s6 hing ngaug vd chl e6 hdng doe. FIBy tirn gi6 trf l6n nhdt crla tbng cf,e s6 & n
0 mi n con xe drlng & vi tri kh6ng [n duoc
ldn shau tr€n bdn ed nXn,
Tdi tam dung bti vi6t 6 d0y,.rnong c{c bq, c6 nhibu tim tdi rnoi
VIETMATHS.NET
Trang 6Bni 11105 Ch&ng minh r&ng o6t moi giti
tri ngugdn c&a x thi bibu thirc
,32
/(r):1961 r +l9zg *-+b *
c6 giA tri nguyen
Lbt gidi.'fa chtng minh m€nh d& rdng gutt:
DiEu kifn cln vA itir ilE ham s6 /(r) - ar3 +
br' * ca c6 gi6 tri oguyen vdi mgi r DguyOE
16 c{c s6 6a,2b, d + D + c 1I c6c s6 nguyCn.
i bft v{y :
f{r) = 6o (r-1)r(r*1)
6
r(.r - 1)
+ 2b *':' -'-' *(agD1 c)r
2
- VOi mgi c $guy0n tht (:r - l)r(r * 1) lu6n
chia hdt eho 6 v& ak, - i lu0a ehia h6t cho 2.
Do d6 AE f(rl nguy€n chi ehn 8a, Zb, a * b * c
lA cAc s6 nguyOn.
* Nguqc lai cho r lh.n lugt e6c giA tri nguyOn l,
* 1, 2 ta duoc c6c sd nguy€n:
a*D* cr *8*b- c, 8af 4b*9c.
Do d6:
u*b* c le sd nguytu
2b:(et + b* c) *(- z * b - c) I,r sO rguyen
6o = (8a t ab+ 2d + 2(* a+ b - c) - 3.20 lA
sd nguyOn.
lp dung mgnh ilE vAo bli to6u:
o - 196{it :} 6a :8928 nguvEn,
b- 197912 * 2b:1979 nguy$n,
:E 9870/6 = 164b nguy€o.
Yfy /(r) e6 gi6 tri nguy8n v0i mqi r'nguy€n
.N-hi6u bqu tII gili b&i toan bing cAch chring
minh trge ti6p, ch&ng hqn chtng minh nhu sau:
f(fl = 1964J-3 13 t 197912 t2 + 5r16
= 65413 * 38912 + Zr313 * n2lZ + 5{lQ
6
Chi cdn ehrtng lainh g(c) ^2*313 + 12 lz4r,r.lt:
cd gt{ ui uguyen v6l moi c nguy0a,
-r-.,* 4a;3*3c2+5n (:r.t*n)*(3c$+BrB)+6,
._: _ 66 16*D(c+1)+ 3r2(r+ 1)*6r
(1 * crXl + cz), (1+ rn)
6
CAo e6 hqng crlr t& sd d&u chic h6t eho 6 n€n
g(r) nguyen (tlpem)
Bfl 2/105 T* tti,€m p di itgng tr€n cqnh
BC cita mdt tam gidc cdn ABC nga:Oi la kE 'cdc dabng song song u6i cdc cqnh kia, cdt AB tai q,
AC tai R Tlm qui tich cdc tti?m D ddi ritng udi
P qua QR.
Lbi gidi Phbn dlo chtng minh khong kh6
xin trinh trty mQt c6ch don giln:
C6c tam gl6,c QDP QBF e|rr nOn D, g,'rp nim
trgn ttudng lrdn tAm Q Tuong tu ta ctng c6
D, P, C nim tren dudng trdn t&m .R Vl g6c ir t&m g6p ddi gQc nQi ti6p ch6n cung mot eung n6n
^ BDP = BQp lz^At2
-/"\ /z-\
PDC - PRC 12- Al2
-\, , -^\ -/-\
nghia li D nhtn BC du6i m6t g6c b&ng.4 kh0ng il6i D kh6c phia vdi p ttc crklg phla vOi a aOi
v&i Q.R, do d6 D oirng phta vdi d tl6i vdi,AC
VSy , nim tr0n cung B,AC eria dudng trdn ngoqi
tiGp tam giAc ABC,
Bei 3/105: Cho cdc s6 rr, 12,, , xr th6a
mfrn dibu -ki(n *r)t (i:1, 2, , d ch*ng minh bdt d\ng thae
"*
?"*l >
L I rtxz ,,, ro
VIETMATHS.NET
Trang 7{"
l.bt gidi lrar :ris l +ai li= 1,2, , n) 'l"heo
iliA thiFt r' l 1 n4rr )1 )> tr '[ls c6
{ I "t ,ut}(t -f"*,e}.",(1 f *s) =,
= i?'+ 61trt * Ds) (2 "i i)')
:2" +2o-1t<11 + 6z * ' I 0ol
r ?"'*2(0r8z + " + 0r-16o) + '
+ dtEz d"
i r"-'l? + (dr * -l E,) +
+ (Srdz "+ + 6r,-t6r,) + " I 0r 6o1 2"-111 + (1 + 0r) "' (t + 6.)]
- 1]-l,r * t.lrf rn)
* a \t
?tt aI6 cuy ra itiEu Ph&i chung minh
Bai 41105 Cho nhfing s?i try nhtdn n ud m
&) z, m ,z> 2) Hdg tirn tdt cd c(tc nghiQm ngufidn
c&a phttong trinh
tl
E.-Y"*V""-.+il,_ ,
-*;;";;'*-Lbt gti/.t (er'ra nhibu bqt) Tru0c h€t ta c6
nhfln x€t lA n6u -phuong trinh c6 nghiQm flm
(r,'9) (n ls* thi ( I ,, - g) sc la nghiQm duong
c&a phuong trinh
Dd tfrdy ring r: 8, I :0 lir mSt nghiQm cria
phuong trtuh dl cho Gii sr't phuong trinb c6
nghigin nguydn r )&, S ) o t{6 r{ng r ( g,
Bhng cAeh liiy thua b$e n hai vE ta iluq'e
n -
I/ *+"'+1ro =go - r'
W%-/
m-i
do x, y nguyOn duong n6n hai vd nguy6n duong'
Ti6p tgc l$p lufn nhu th6 ta th{Y
n
,1 "
-
r'
-V"+V" vd V, lit hai s6 nguv6n duo'ng' Dit
Dilu ndy khdng thb riy ra vdi l, b lA eac s6 nguy€n duong" V$y phucng trtuh kh0ng cd nghiEm nguy0n duong
frl nh$n r{.t {fhu ti€n ouy rr phuong trlnh cflng khOng th& e6 nghiQm nguylu Am.
Tdm lgt phuong trisir chl c6 mQt e{p nghi$m
duy nh{t l[ r:S, &:0.
T'T Bei 5/105 Gi&i phrong trinh
log3 (rz - 3r * 13) = logzc
Ldi gidi cria Pham Eftc Trung (10D Nguy6n Kbuydi, Ifir Nam Ninh), Trln Hftu Mrni (10 E
Nam LJ,, Lf NhAn), Traong Quang Yi6m (L! Nhan, i{a N"* Ninh), :E(ng Eac Ttqng \9 P Brii [{tu Nghia, T.P Hb Chi lvlinh)' Yfi ttqnh
Hirng @75 t 5' tnai Tb, T.P' Hb ChI Minh):
SiBu ki$n phuong trinb phii th6a m6n:
I ">o (+
s>-n-I
E[tlog2 *=3U € r:23Y,Phuongtrinh
tr& thanh
logs (26: - 3.23Y * 13) - 39 {+ 33Y',=26Y-3.ZsY-13
e Q7164)t + 3(8/64)Y + 13(t,s4)Y - t
D6 thSy phuong trlnh c6 nghi$m g - i' Oo tl
nghidm duy nhf,t- vl c{e h}m s6 mt & v6 trdi
dEu nghfch bi6n
VAy phuong trlnh c6 nghi$m duY nhf,t:
s=23:8 Hai ban Ngd Dug Ninft (12 C Quant Trung'
Qui Nhon) vd Ngug\n ,Dlnlt Khi6n (10G Nam
dong guan, Th{i Blnh) d5i logs
" - 3 t I vt ila .O *ac bgn kh{c d{t log2t:g, gi&i tuong
tu uhu cAch giii tr€n'
Bdi 6/105 Gi&i phwong trinh
log2 (r*lrl - i=lr:7'
Lbi gidi (cria b4u Ng<i Dug-Ninh; l6p 13C
trudngl{p 3 Quang Trung, Trhn khdc Chdu*
l6p 12-C trlOng c*p"S frhn Qu6e Tudn' thiah
ph6 Hb Chi Mi;h
" D6 DLrc M6n*911 Hrlng Vucrng'
Vinh Ph{ vir nhiEu bqn khA'c):
Phuong trlnh chi cd nghia khi
t*rz>.o hay $acl {r vr r'+ lcl -1>o'
Ttl fi&Y suy ra
o(ecx + lcl - 1 <1 + l'-L:1'
x+vt,
=ao*n
{
bo
' o =l*, h=
te c6
M$r- kh6e do n )> 2 nen ta cd
(g -i l)n );, *o -f n -) **
tl
o-s.r)a:VFlfo>o.
h.y
VIETMATHS.NET
Trang 8qrq\ - \
Do il,{ -!ggz k2 + larl * rl( logz t:0, m{t
kh{e nT=;2 } 0 nen phuous truh.chl cd
ughi€m =/i=T;0, khi vd chi khi rogf tr2 + lrl _rl
-nghia td 1-tr'i = o va nx + Wl *1
=t.Khio6 lri -thayr:fl.
Th& lgi, ta th6yr: -F I nghigm driug phuong
trinh da cho V{y nghidm eta phucrng trinh li
-Lr
&- t t
T.T.
Bni 7/1-05, Ggi A^*-2g-ryz Hdg tim gid tri
nhd nhdt ctra A uoi $, U, z Id nhfrng s6 ta nhidn
th6a mdn d&ng thuc:
lzxz + tlgz + B4zz - 4xg - grz * 369r* 16r *
*89-2lz*i-A.
Lbi gidi ESng thrtc dI cho c6 thE vi6t lgi
thAnh
@rz + g gz + 76z2 - l},jg * 16r z * 24g z)
* 2Go2+ 92 + gzz + 4 + 4rg - llrz - 6gz + Br +
* ag - 72il - 1, hey ggu hon
Qr - 39 * 4)z *2(-2* - g * Zz-2\2 : l.
Yt-c, g, z ln ghtng s6 nguyOn duong nOn dB
c6 d&ng thric tr6n thl phii"ci, '
r l '^" -', * l: : l
(-2* - g * B: -2 ""o"" l-r., _ s * Bz:Z
a) Gtli hQ I : Gi&i g vi : theo * ta duoc
,:J*o+1, r:Tr+1.Dod6
Bdi 8/105 Tim tdt c& cd.c s6 tg nhi6n m sdo :y ,:l ygi s?i tu nhi6n n & trons kho&ng t<.n(Lml2j tht ph0n a6 (m_dln khOng phEi
tdi gifin
!bi_,At&i efia 3{9ugr6a Ddng eaang 06p 9 chuy€n
toAn Eqi hgc Tbng hop HA NSi) vd nt iBo bao kh6c:
t Ndu mt|: Gitt srl m = 2l + t thi theo d5.u
1!6y lery r&ng khi n:t rhi phen s6'ltzr.-n)ln^
Q +t)ll ln ph0n o6 tdi giAn Vay tf,Oog t6u tqi
edc sd m 16 th6e mfin dibu kien efra Ohu bei
2 ;,Cu at chln: Gi& sfr m:21 va f ( il( Z.
NEU c6 s6 m n&o tl6y th6a mdn diEu ki€n d[u
bni tbl t6t cl c6c ph0n e6 sau
t t+t t+2
d6u phii kh6ng t6i gila Nhung ll1 kh6ng
l-1 tdi giin c6 oghia ld (l+1, t-1):dlt va Q+1)-(t-D:2 i d.
V&y d-2 vi ta rney
I -L,
lIIt khAc: - kh6ng t6i gi&n c6 ughia li lt.+z, t-2;=sll vir u+2)- (t-21=4 i q.
V+y S j 2vA nhu th6 /*2 : 2 ttle
(1) vn (2) mAu thudrr tr{6u thuAn niy chi dugc losi b6 khi kh6ng t6n rgi c6c phin s6 nim
gifta lit
"e+ Vgy n€u m th6e mEntliEu kj0n
2
crie tlhu bli thi m ph6i th6a min :
holc l: m - 2 tric m: 4.
T6m l4i c5.c sd m th6a m[n tli6u kigu cfra dhu
bdi In cie sd m = 4 vA ttr: 6.
L H
BAi 9/10.5 Cho d.&ng thac
LnJ, - *[rl + 15 *.r= 2*s+ (*z _ 1)z ltr] + hl trong dd [r]: phhn ngugin cfi.a *,
lrl: pthn ti cfia r.
Tim tdt cd nhfing gid tr! thryc cfia r th6a mdn
d.dng thnc trdn
LOt gl&t crla ban Dtong Dtrc Long (9E Htag Vuong, Vinh phd):
Tu ddy cho thdy ring trong trudng hgp n&y _4
s€ i14t giA tr! nhd r,rh{t khi r Ia s6 t1t nhien
nh6 nh{t sao cho r.r vd z cfi.ng lil s6 tu nhi6n
19" 96 A sE ln s6 nguy0u), ,rgUiu rir khi r: 5,
Iric it6 A : lb, z - 9 vi gi{ tri nh6 nhflt eria
AIL2.
b) Giei h€ II : Gifii E vd z theo r ta ituoc
g,=(l4r + 11)/5, z-(Br+D15 Do rld A:o-_
zIr!* ! 1t)i5 + 3 (8r * 7)t5- (r _ 1)/5 D6
tfd;r rS,nS troug trudng hSp nly 4 dat gi{ tri
T6m l4i, gi6 tri nh6 nh{t cta A lA bing 0
_ CAc ban Ngd Hfiu LiCm (l6p 12 truong BrIi
llf Xuen, thnnh ph6 H6 chi l,tinr,), Dodn-Qug€t
Th&ng (3E khoa ToAn Dqi hgc su phqm Ha Ndi I)
vA Ngug6n VdnTi(n (y6u phong,-Hd B6c) tld c6
nh&ng ldi gi6i dring.
T.T
VIETMATHS.NET
Trang 9I
i
i
i
P
I
*
Ding thuc & dlu bai tuong &ucrng v&i:
'
ek4 I zr2 + 1) * (rz r)? fr! - [, ]x
x(r-lrlr - l"l:o
€) (r4 - 2.r2 +1) (r - fr]i - ic] x
X(r*[r]r"*[r1:s
++ lr{ (rs4 zr2 * l*]t _ o.
Y{y c6 e{c tru&ng hEp aau:
a) l"l:o tue c uguyen
b) ra - zr2 - l*l- o Khi d(r
l'4*z*t+r-r*fr]
tuc 1 * l*l) a hay [r] ); - t
vdi
f"j = 1 thi rla -2*2 + I:0,ia c6 r: _1
(loai a: = I)
[*]= o thi ua- 2r2:a, te c6 r=rr
hoai r :+ t/z l.
[*j:t tli 14 * 2*z- 1:0, ra cdr =1/+tl6
Tdm lqi nghi0m cria bAi to6n lA ff nguytD
vd *= Vt+1/2.'
L H.
Bdi LOl105 Cho fk) = arz { b r{ c (*) ; rr,
12 ld hai sd khdc nhau od khtic kh6ng, th6a md"n
f@i: f(tfl : ^ 1A m (q* rz- 1) t c(rr- t)6.t -1):0 (t)
g3 nhi6n Tim nght(m d6.
Lbr gr&i NhiBu b4n d[ giii theo c6eh sau:
_",r:: gii-lhi6t f(ai : f6) - m n&n , 1, 12
ciing lt nghiQm cria phuong trinh l@) _ m:0.
Phuong trtnh c6 hai aghi€ir nln a'lO uJ ttreo
dinh lf Viet ta c6:
rlr-Fxz = * bla', 0t *.2 = (e
- m)la,
Thay cic kdt qui nly vio (I) ta dugc
m (-bla
- t)tc Lk-m)la*t*bla]_a (+(a-lb*d (e-m)la=0.
Do rr vi *z=FO n6n c - m j=0
J,6f o {-D * o:0, suy la/(c) c6 it nh6t mgt rghi€m s6 tr.t nhi6n v& nghiim d6 chinh bfing i.
Nhdn rtt: Ctng e6 ban nguoc lai dl tr! I(gt
qui cria Viet suy ra hf sd a,"b rbi il"r;e; axj
1ri n]t+n thdy tbng cic h( s6 c&a phuong'trinh
trf c 2 mdi nAy <1ring bf ng {i), ner,
"tug-tf,i, a,ro," k6t qr,A chn tim
(toqi-yt+1/r)
["1-*]z: rni dd r.2]4 hay 14 )4x2.
Nhu th{ ra -2r2 l*J) 4r% _2*2 _ l,rj }
:'+ T.- ["] ] o, trlc litroris ih6a iifln truong
hq.p b).
L.H
I
ir
3.7 (2\ - D < / 2,4 2'*l t ('
m$t sd tu nhi€n ldn
dlng thuc:
I !
,, ' n.2"-l I
Bei ll10s c6a s6 [guy€n a, b, tn, n th6a
Tf i cAc tinh cb{t sau dfly: phuong trinh
n + ar * D :0 c6 nghi0rn nguy€n, m$ n va
(ln'+am*b)lti.zgan1D)=r.
Chrtng rninh r*ng
a*4b=(m*n)(man-1).
Vfi euang S#u (DHSp Vinh)
- Trhn Xudn Dt*ng (EIITH I,Iud) Bai 3/108 Chrtng minh r.Eng r{r c6 ca.e
22cosr+61g4-r iIEu th6a mtn
1964 < rtfi {1g7g
Hh1, tinb sd nghiQm crla phuong trinh tr.6n.
LO Thdng Nhdf (EHSp Vinh)
VIETMATHS.NET
Trang 10Bnt 4il0{i {tho 0 "( a <1 ?[/2 {]irr}ri$ min}r
r0 ug
(soan + srne tlsin -1 <i f{l ig.
2
l',rj van ?',1?dnjt (11[.r Ph0rrg]
Btt 5/108 Gi& er! (i lt t0m vt r lil bfin
ktnh crla du&ng trdn u$i ti6p tam gil,e ABC.
Gol R, Rt, RL, lls tuong uug li br{n ktrrh c$e
ilu&ng trdn ugogi tl6p c6c tam gi6c ABC, OBC,
OCA, OAB ehring minh hg thirc :
-&EeRa = 3fi2r' L6 Qadc Hdn
EAi 6/108 Trong taur
ABC
cotg * eotg_-, cotg_- l-gp
222
c0ng.
(EHSP Yinh)
gi ric lfrC c6 th*nh c{p o6
a Chung minb r}.ng d0 dii c{c cqnh cile tarn
gi6d lap thdnh cdp s6 cgag.
b Tim g6e ldn nh6t cira tam gi{c bi6l r}ng
ABf
cotga-, eotg-4, cotg: la ba s6 Bguyen
222
li0n tiSp.
- Ngag|n CSng Qu!
(DHSP 1'.P Hb Chl Minh) Bdi 7/108 I{ai m[r cBu Sr, Sz e6 c6.c tidp
tuy6u chung AtAz, BtBz (dr, Bt thuilc St, Az, Bz
thu$e Sz) Chtng minii rlng cdc hinh t:hi6u cfia
c{c dAy ,{rBt vir !2fi2 l"rdn itudng ndi tim hai
m{t chu l[ nh&ng rIo4u thEng blng nhau
Ngrtgi-r't Cdng Qttil
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CONG TffiIJC TETE TICH CUA N0
QUdC TRINH
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