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Spiiat fa
bdi todn -l(n').
to6n .4(n) 116 cd ldi giii khi rt nh6, ch8ng
l.hanBAikhi
rr:2, B, t. D6 la
Uui to{n d6, Tuy
nhion uri'i n btii kfi A0:-) urtn"*"
cdn td bd.i toan thio
co liri q t&i.
nhu dLrng tluoc nhiiiu bii todn kh6 kh6c,
. Cn_ng
ta
thii b&ng phrr
s6t, phAn t'ictr
l".o
Ioigiii eria bdi toin A(2), i;et, atri
,an
't6ng
tdi ldi gi&i e&a bAi tofn t-ong tluongii't:tep
t, jf
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).
^
TEIANTT
BAy gio ta chuy6n sang xet bai to6n
A(B).
Trudc
ltdt
ta
hdy
tana
gAc
c[c
nghidm
-
chrla t.
TM
r 1. Nhibu bdi to6n hay kh6ng phli bao gi}
ctng hinh thAnh ngay m$t hic. Cd khi nd xudt
ph{t tr} rn6t b&,i toAn d6. Bing nhirng
tlgi
chtt it hay m& rqng them nhitng dibu thay
ki€n crja
bdi ro6n a6 tro ta lgi thu ilugc nh&ng bdi
to6n
thri vi.'
Trong bAi niy chtiag ta hdy theo d6i qra .[}int
-.
PHAN DOC
cd
Gil srl k, y, il lA rngt nghiQm nio dIy eia
..
bii
toAn A(S). D6 dnng
tlay
ty
-nhiu
hai s6 trong b$ ba d6 phAi
nguy&rrtd"eng-;rt
Vfy c6 thb xem 2 (i
Tu ttiBu kign cfia bdi to6u tir cd I
\
t*y * 1: k1z
r-kzl
$rr+
z
lA * 7 - k3r
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) $,
t-
-
r
-Fr-
<
1/2"+ 1/3 + 1lr
I
rl2'$' 4:27124
"*z{=1.
Vay (o, U, z) th6a man.
(1)
lla * llY * llz + tlrgzs !"
Biy gid ta chring minh ec:2' Thqe v$"y ndu
ngo-o. i4i, tue Id n6u r 23 rhl g) 4, z)5 ncn
ta cd
| = tlr + llg * rlz * tlrgz( t/e + 114 + 115 +
+ 1/(3.4.5) = 48/60
iE AiCu kh6ng th8 ttugc' VAy chl c6lhb
(1) =+ 1 = tlz + llg * 7lz * ll2gz
4g=2*5i\z-2t*
4z=3 haY z =? +g=l
ft6Y
r=2'
g:3"
Y$y nghiQm kh6ug chua 1 efra bii to6n A(3)
iA (2;3,7). N6u kE cl nghiQm chua 1 ta edn c6
be
b0 nta:
Tuong tg d6i voi bii toan '{fn)' nghi$m
(r1,...,rn) th6a mlu phuong trinh'
lfrt...nrr:l
(,{ lA s6 tB nbi6n)
MQnh tIE ngpgc cfing
(zt
TM
1/rr+...*llrn*
d{ng: N6u rt, "', r. ll
tq uhien th6a md.u phuuug trinh (2) voi
m$t sd ttt nhitn A nio
cdrc sd
VIE
nghi$m e{ra bdi to{n A(u).
Rdt tlAng ti6c ln trong truor.rg ho.'p n b6t k5r
kh0og phAi ta luOn e6 -rt = i. Trong thqc t4 ;l:1
chl ddi vdi nhitng n =(. SS.
3. Bty gio dB d6 quan slt r,'I phln tith ta
hBy ligt k& c6c nghidrn cria bii to6n A(2). A(3)'
A(4) dudi d4ng mQt blng sau:
a(2): (t,1) (1' 2) e,3)
A(3): (1, 1, 1) (t, 1,2) (1,2, 3) (2,3, 7)
A(4): (1,1, 1, 1) (1, i,1,2) a,1, 2,'j') (1'2,3'7)
(t, :l,l , +i)
'fir b&ug tr$n ia llir$.n ihf;y ririg ir*eg rt:Si
c$t ilghiAul & ilai:g t].rrd"i tiirln tlr-ge tri' nghiQrur
& hdng trou b{rg c6ch th€rn v&o sd tr,
.J
aau.
Bb ila 2. I'i6u bg (rr,.... rt) lil nghiQm cira
thi b0 (sl ,.',:uk, rt..'e.r'*1) lA
nghiAm efra bni to6n li(k * 1)'
Thgc vqy tich eria k sd dhu ti€o c$ng th€m
I chia h6t cho rt+t theo ciirih nghia erie s6
rsal. B&y gi& gii s-& rtit c6 m4t trong tlch'
Ta {9i X; lil tlch cira /t-1 sd d}u ti€n trong
do kh0og chita ri.
Ta chn chrlng nriuh r'frng Xi rr+1 * 1 chia
h6t cho r,. DiBu dd suY ra t*.
b&i tofln A(ft)
:
xk+l: Xi.r,1 a 1 "+Xlrsgl +1:,Ii(Iirl +t) - t
: ritriz + (Xi + i)
s6 hAng thu hai chia h61 ctio ri theo diiu
kign cria bA dE 2.
Nhe bb dE 1 va bB aI€ 2 ta nh{n th6v rang
xust phirt t& m$t nght{m efra bli lo&n '\(k)
ia thu ituqc liai ngtri6m *ir* biii toirn A(l+1)'
Bfrng e6eh dy xu{t phet tir nghi6m cria bii toirt
A(2) ta thu cluqc cae nghiQrn ct" bai tnf," A(3)
vir k6 tlo la thu itrugc cac nghi{m cria bii
AT
(1,1,1), (1, 1,2) va (1,2,3),
E6i vdt bili torrn A(4) 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 $,2'3,7)'
trong bb dE
T
.4
Tt d6 ta c6 bb db bi&n nhieu aeu:
Bb de tr . NEu hg (rt, ..., ct) la nghtQm cria
bAi to{n A(h) thi b0 (rr, ..":rl, 1) lI nghiQnt
e&a bAi to6n A(t( + 1). NguEe lai n6u trotg
k*1 ed lh nghiQm clls b&i ts6n A(k+1) cd
ch*a 1 thi saukhi b& 1di ta nh$n {Iuqe nght(m
c&a bii toAn A(k).
Cdn d&y (2. g), (2, 3, 7),... dlsgc ph&n dnh
.NE
.16
HS
T*
toAn
A(.1).
D6 dang nb{n thdy ring }:i.ng c{ch d-v ta
thu duEc tdt .* *a* ogfri6n, ciia bai tain A(2)'
A(3), A(4). Tuy nhi€n diBu dAng.tig0. lh xu[t
pnai tr) nghi0r:r c&a bii toan A(+) blng c6'eh
iy ta thu dug'c lchdng phdi tdt cd e{e nghiAn
eira biri iodn A(5). 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
tue'hhai 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$m crl a bif
toAn A(A)'
D[t /r: s!,,. rt. T* e&
llq * ."" '1 l/rtr * llfv:
(3)
11
FIou nila'gi& str ot+r * tr seo cho trcl' "'' rk'
ttrinh nghiQm cfia bei i'o{n jr(l'+t)'
uu-r-t) t&p
Ta
1b:r
co
* .,. i- 1/rr.i'llr:s+r
-l- l/(/ra:t-pt)
'*
A id'
Ta l6i thu duqe nghigm theo bb dE 2.
Nhu v&y ndu nghigm crla b0.i toAn A(k+t)
nh$n duqc b$ng c6ch th0m mSt s6 vio nghiQm
ctia bdi toAn A(ft) tht sr5 nAy cluoe tgo n6n
ho{e theo bA dE t ho{c theo h& d& p.
T& aI6 te nhsn th6y rfrng vigc th6rn v&o mQt
s6 khOng mang l4i cho ta dilu gi m&i ngoili
nhtng tttEu d[ bi6t 6 tr6n.
5. BAy gid ta gi& thi6t rfrng (rt, ..., er) ld
trghigm crla bti todn -4(&). Ta e6
Bing quy nqp ta chung minir iluqc rfing /2r.
ering b*ng 6 v} /zr+r e6 r{,n etng bing 2.
{ t{n
Tt
d6 suy ra rfing trong ddy eo s& te1 ci eae
sd /it+r * 1 dEu chia h6t cho 5.
T* tinh ch{t, nAy ta uhgn thfy rlng: khi n 16
nh& bb dE 3 tt d6y eo s& ta trich ra duoc mOt
nghiQm, nghigm ndy nhd bA dE Z sinh re dluge
m{t ddy mdi cAc nghigm.
BAy gi0 ta dl theo mgr hu0ng khtc rtB gili
- 6,toAn:
bii
Xu{t ph6t tr} ughigm crla bii toAn
A(n) hdy tim c{eh thu duqc m(t nghi0m kh6c
erja bAi todn d6. Theo hu6ng uAy te c6.
tsinh lf (Br6tsky).
Gii srl (*1,..,, nn-Z, r1-1,.r1) lA nghi0m thu
duoc tu nghi€rn (rt,.., , *t_2, ro_t) theo b6
dE 2 vA (fn-zxn* t)/rrr*r = dtdz.
N6u hQ pbuong trinh
T
vl (4) te suy Ia
llru.+-t * l/(/rrx+t) * tlfu + ,k+r
- li+ i.
3)
.NE
T&
AT
HS
tlrr*... * tlxy* tlfy* A
(5)
Hon nta gi6 s& (rl, ..., ,rk, ttk+1, rr+ a)
I& ngh.igm cta bii to6n A(k+2) trong d6:
*k+r + l. ruaz * t. fr c6 (v6i cting-,A & tr€ri)
d1y
1l*t* ... * Tlna * l/a..r+r * llra+z *
\ f"-z*lt(10)
g * 1: dzr
f"-z
I
(6)
+ll(7vnx+tst+il=A
+ l/ar+r + llxt*+z + tl$vru+tta4fi *. tlfa fi) cd nghigm nguyen thl
{ry , . . , rn-z, r, y) cEng ld nghi{m cris b[i
+ &k+z=fu.+ (fL + 1Jl|ir;yl-l. - ,r1
(s)
to{n.
Do
1t.1-2,,phii nguy€n n€n d: xik+t -- /r IA
Ohtrng- minh. Biing c{ch nbAn cdc phuong
uoc crla /i * 1. VaV
trinh e&a hd (10) ta e6
(rr1r:/r*d
(
{9)
f?-z.r.u * fa-z 6 * g) a 1: &:?'d-L ,,
(*o+r:fu.+(f?+ttld
Nhd clc h6 thrlc (5) (6) (7) (s) ve ts) tc di d€n
de 3. Gi& sri (.r1,..., rk) lA nghiQrn c&a
- Bbto{n
b&i
AG), f y: trl . . : nk v[ d IA u6.o cris
r rni ao rg
. |/rr,...,
-
\
**, ,^fx+ *'.A
a"fu+ Io'+1 ]
d l
f,n_l
1ly * tl\fn:2*g) : - f,-L *
Cht 1i r&ug
fn*z?t * |
fn-Z frn*l
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$n luOn c6 nghiQm
vit
TM
ff+
hry lla +
VIE
l& nghiQm cria bAi to6n A(k + 2).
Nguge la.i ndu ughifm cfia bAi roen A(k+2)
nh{n
cluoc tu rrgt nghiQm nio it{.y cria bii to{n
A(le) bing e{ch them v{o hai s6 kh{; 1 thi
c6o sd 6d th€m v{o th6a mdn rl8ng thric (9).
Ch(l i- r"rog a : I v{ d = fl.* 1 lu6n lu6n lA
u0c crin ff. 1 1. Erii vdi cr4.e u6c nAy. b6 dE 3
klg"S cho ta til6u gi mdi so v6i b6 a6 :. roy
nhi€n n6u ff + I kh6ng phli In s6 nguy&n 16 thi
bb dE 3 cho ta mOt lo4t nghipm *"i..Ct&ng
hau dl6i v6i ughiQm (?, :i, ?) ta cd /32 4 1 :
.- 1765 = 353.5. Xu{t ph[t tu nghidur naj vd. ru
bA dB S ta thu dugc nghidm {2,:1,2, aZ, :gS) ma
ta 116 ldi d6n trong rnqc B" Tuong tlr xuSt ph6t
tt nghigro (2, A,2,13) nhd bb db Bta thu dugc
ughi$m (2, 3, 7, 43, 1829, 193667).
nghia-. DSy nghi(rn thu drrqe tu nghi€m
. Sja,t
(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.
nguyBn.
ta v6n cbua bidt duEc
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
Cho d6n nay chung
bii
.
Chrie c{c bq.u
con Igt.
rhinh c6og lr8n chflng
tludng
1
a"-
1.-
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
U. V.Si b&i
to6n vE btt&c di c&a eon EgEa'
ie con c& di I6t ieo
vdri n6 l4i niy ra
il6i
mb
nhdt tr0n bln c0, v$y
vi'
thir
to6n
uhi0u bti
Ph&i n6i rXng eon ngrlx
.NE
?R0NG hai s6 b6o 101 va 105, bsn l-€ Th6ng
Il Nnat 6e gi6i thiQu vdi c6c b4n mot loat
c6rc bii to6n trOn bhn cd, md cht y6u In
ci.c bili to{n ph& vir chia b&n cd' L[n. niy t6i
xlrr bb sung mQt sd ki6u suy lu{n khr{e tr€n b}n
T
e#
rrfrP TIJC stJY uu.rgru rmfru mAh&
lt t,rr:'r xca
tsai toen 2. TrOn m6i 6 cria m6t ban c$ 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'?
VIE
TM
AT
HS
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
ddmin0.
nh6.-v viro cac 6 trin6. Suy ra s6 ngrla trlng r-i
Suy ra s6 6 biin cd la
Ldi gi&i. Ta th6y birn eo 10CI X 4 khi t1ug,c ngua dlen ih r:hu nhau.gi&
thi6t birn c& cc so 6
v6i
thr:Xn
mau
cien,
m6i
thi
ehia d6li b&i bflt c& rtudng gi6i han ndo
IA. Vay khdng th6 eho rn6i con dbu ili rndt huse
phln ctng ed mQt s6 chXn 6.
alB sau d6 n.i6i 6 l4i c6 dring mgt con'
Gii sft b[n cd duqc phir kin boi cdc ddmia6
Bii toin 3. Coa n$r,ra cd thb di t"r goc niv crii'
khOng ehbng l6n nhau thi & m6i phhn c6 2
bdu eo th0ng ihucrng toi g6c d6i di€n sao cho
lo4i 6:
qua m5i 6 cria birn ed ilung m$t lrin dugc
*loai o b! phrl ir6i domin& khong bi c[t (I) di
khOng ?
{II)
- loai 6 bi pbfr b&i ilomtnd bi cE"t
Ldi gi&i. Ith6ng thE dusc, gi& o& 6 & g6e eou
I'a thdy ngay sd 0 cria logi I ld sd chin, vi nglr& xu6t ph{t l[ 6 den thi sau mQt sd 16 bu0c
m6i domino irhOng b! eit s6 phrl 2 6 eria bin
,g,y" plai toi 6 tr{ng. Do eon ngua ph&i
c&. Do t8ng sd 6 cric 2 toqi & rnSi ph8'n I* s6 "or,
qua ,r',6i 6 cri* bAn ccr dring mQt l$n db cu6i
di
chin, n6n suy rl sd 0 lo+i II & rn6i phhn ctng c,ing tOi 6 al6i dien v&i 6 xu$t phai n6n n6 d6a
lil sd ch6a' Nhu v{y s6 edc d6min0 t}i c5t la $d o u$y t} l:u6c thu 63' 56 hudc di 16 n6n suy ra
ehla. Nhu v$y n$u tdt cir 102 duo'ng gi6i han O ney te d ir[ng. MAu thu6n vt n6u 0 xudt ph{t
tiuong nio ctng cflt dOmind thi tt nhit tr&n b[u la 6 den iht 6 ai g6c ddi diQn ctng phii 11 6
ed phli cd 162 X 2:204 ddmiu6. Nhung bln den.
cd ilrrqc ph& kin b&i 20tl ddmin6. Vny it nh{t
'I'ruo'c khi sai:g bdi toan '{, ta hiBu rnag u:Qt
phfri c$ Eudug gi6i hqn khOng c&L d6tnin6 riro'
birn cd goi ti ddng v&i hirntl trlnh e &a coll DgUa
Soy ru b&n c0 1.00 x 4 hhfrng thE phir chic ch*n
ndu ccn ngtla ct5 tha €i clua nr6i d dring mQt ldn
b&i c6c d6min6.
vd tr6 vb 0 xu{t ph{t.
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
I X 8 xem sao ? Ngudi ta dd ehtrng rninh ilugc : ccn DgUa thi ph&i e0 sd d ch5"n vi d6i vtri rn0t
(Mei bin c& a6.di0n tleh chhn v& m6! chlBu hi.nh triuh kin c{ra con ngqa thi s$ 0 ilen mii con
l6p bon 4 (trti biii ce 6 X 6) dBu e6 thb tlugc nglra di qun sB l:ing s6 6 tr&'trg rn& nd di qua.
ph& ch&c ch&n >r. Cic b4n hfiy t$ chung minh Vi v$y bflt c{r biLn civ niro c6 s6 S tr6ag kir{c
a6 6 den thi ktrdng rlting v6i h&*h triuh e oo :1gr:a'
rngnh tlE tI6"
,l
I
ditdi6&bi&n.
Cde bnn
n.
h5y gi&i
trii lo{n
E&i ro*l1 "{. Con ngra kh&ng th& &i t&i
mSt lln.
E6i v
qu6t ngudi ta
t$t
dt
cfl
tim
n6u
n: I
VIE
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
(mod 4), ho{c n: Z(mod +), hoae
) 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
n == 0 (mod 4) n6u n
T
Lbi gidi. Giit srl cd A con xe xdp nhu diBu
trii todn. iv m6i 0 c6 con xe ilung bHt
ilhu ta ghi s6 0. Sau iI6 ta x6t tr€n c5.c hirug
kiQn cria
ngang
:
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$ng vao khOng qu6 4n" Tuc li
-
TM
sd trr nhidlr
coo xe kh{c. Ch*ng mirlh k(an/3.
N6u
AT
VSy col ngga ph&i irin luot tu bj6n i,iro gifra
rdi ra bi6n, rbi r'$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
khOng
.NE
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$t 0 6 gi&a vl tdt c& c6c 6 5,g.i Ea 116 e6 rngt
bu6e vAo, v0.y bu0c rn r-rAy phii tt 6 & gitra d6
I >< 8 X 8n sao cho
ili qua noi uho quA rn6t lHr)"
3. Vni bii to6n vB con xe
Ea! tc.&n 6, Ir€n bArr ed nXn x$p Ii con xe &
vi tr{ rn6i con khdng & cung hAng vr}i qud 1
6 mat cfia hinh {ep phuong
HS
Ta 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.
Suy ra:
3k
S.-(+n
--={
+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.
cfia bi.n co thn luqrt
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.
Eii toiu ?. Tr0n 64 6
vi6t cric s6 tu
1.
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.
Chi cdn ehrtng lainh g(c) ^2*313 + 12 lz4r,r.lt:
cd gt{ ui uguyen v6l moi c nguy0a,
4a;3*3c2+5n
(:r.t*n)*(3c$+BrB)+6,
. ._:___-: ___
66
16*D(c+1)+ 3r2(r+ 1)*6r
T
-r-.,* --=-,(rcl=
6
o6t moi giti
*
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
Lbt
c{c s6 6a,2b,
i bft v{y
d
+
D
+ c 1I c6c s6
nguyCn.
:
f{r) =
(r-1)r(r*1)
6o
6
+ 2b
r(.r 1)
*':'
-'-'
*(agD1
c)r.
2
a*D* cr *8*bu*b*
c, 8af 4b*9c.
c le sd nguytu
2b:(et + b*
TM
- VOi mgi c $guy0n tht (:r - l)r(r * 1) lu6n
chia hdt eho 6 v& ak, - i lu0a ehia h6t cho 2.
Do d6 AE f(rl nguy€n chi ehn 8a, Zb, a * b * c
lA cAc s6 nguyOn.
* Nguqc lai cho r lh.n lugt e6c giA tri nguyOn l,
* 1, 2 ta duoc c6c sd nguy€n:
Do d6:
* b - c) I,r sO rguyen
6o = (8a t ab+ 2d + 2(* a+ b
- c) - 3.20 lA
sd nguyOn.
*(-
z
VIE
lp
o
b-
c)
dung mgnh ilE vAo bli to6u:
196{it :} 6a :8928 nguvEn,
197912
:E 9870/6
Yfy /(r)
* 2b:1979 nguy$n,
=
e6
bqu
.N-hi6u
164b nguy€o.
gi6
tri
nguy8n
v0i mqi r'nguy€n.
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
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.
AT
16
.NE
Ch&ng minh r&ng
x thi bibu thirc
,32
r +l9zg *-+b
/(r):1961
:1
26
c6 giA tri nguyen.
ngugdn c&a
HS
Bni 11105.
tri
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
Lbi gidi. Phbn dlo chtng minh khong kh6
khdn. Phln thufn c6 nhiBu cdch ehung mioh,
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-\
Al2
- , PRC 12--/-\
BDC
-\, = BDP
-^\ + PDC =A,
PDC
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
>
(1
*
crXl + cz),..(1+ rn)
L I rtxz ,,, ro
"*
t
r
i
?"'*2(0r8z
+ .." + 0r-16o) +..'
+ dtEz...d"
T
I "t ,ut}(t -f"*,e}.",(1 f *s) =,
= i?'+ 61trt * Ds)...(2 "i i)')
:2" +2o-1t<11 + 6z * ..' I 0ol
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$m
duy nh{t l[ r:S, &:0.
T'T.
r"-'l? + (dr * ... -l E,) +
+ (Srdz
+ 6r,-t6r,) + .." I 0r...6o1
+ (1 + 0r) "' (t + 6.)]
Bei 5/105.
Gi&i phrong trinh
log3 (rz - 3r * 13) = logzc
"+ ...
2"-111
\t * t.lrf ... rn).
a
- .1]-l,r
?tt aI6 cuy ra itiEu Ph&i chung minh.
Bai 41105. Cho nhfing s?i try nhtdn n ud m
&) z, m ,z> 2). Hdg tirn tdt cd c(tc nghiQm ngufidn
c&a phttong trinh
*
Ldi gidi cria Pham Eftc Trung (10D Nguy6n
Kbuydi, Ifir Nam Ninh), Trln Hftu Mrni (10 E
Nam LJ,, Lf NhAn), .Traong Quang Yi6m (L!
Nhan, i{a N"* Ninh), :E(ng Eac Ttqng \9 P
Brii [{tu Nghia, T.P. Hb Chi lvlinh)' Yfi ttqnh
Hirng @75 t 5' tnai Tb, T.P' Hb ChI Minh):
SiBu ki$n phuong trinb phii th6a m6n:
HS
{
'l"heo
.NE
{"
l.bt gidi. lrar :ris l +ai li= 1,2,..., n)
'[ls c6
iliA thiFt r'. l. 1 n4rr )1 )> tr.
,
E.-
tl
.-
Y"*V""-.+il,_--,
I ">o
I
AT
E[tlog2 *=3U € r:23Y,Phuongtrinh
- r'
m-i
,1--"
VIE
do x, y nguyOn duong n6n hai vd nguy6n duong'
Ti6p tgc l$p lufn nhu th6 ta th{Y
n
r'
V"+V"
-
vd V,-
lit hai s6 nguv6n duo'ng' Dit
'
te
o
=l*,
h=
{ x+vt,
=ao*n.
M$r- kh6e do n )> 2 nen ta cd
(g -i l)n );, *o -f n -) **
c6
bo
tl
h.y
tr& thanh
logs (26:
o-s.r)a:VFlfo>o.
-
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$m g - i' Oo tl
nghidm duy nhf,t- vl c{e h}m s6 mt & v6 trdi
TM
-*;;";;'*(er'ra nhibu bqt). Tru0c h€t ta c6
Lbt gti/.t.
nhfln x€t lA n6u -phuong . trinh c6 nghiQm flm
(r,'9) (n ls* thi ( I ,, - g) sc la nghiQm duong
c&a phuong trinh.
Dd tfrdy ring r: 8, I :0 lir mSt nghiQm cria
phuong trtuh dl cho. Gii sr't phuong trinb c6
nghigin nguydn r )&, S ) o. t{6 r{ng r ( g,
Bhng cAeh liiy thua b$e n hai vE ta iluq'e
n .---n
1/
I/ *+"'+1ro =go
W%-/
t-
- 3+v61
(+ s>-n-
\,2-3r-13)o
dEu nghfch bi6n.
VAy phuong trlnh c6 nghi$m duY nhf,t:
s=23:8
Hai ban Ngd Dug Ninft (12 C Quant Trung'
(10G Nam
Qui Nhon) vd Ngug\n ,Dlnlt Khi6n
3
dong guan, Th{i Blnh) d5i logs " - 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 Ngtrudngl{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'
"
bqn khA'c):
Vinh Ph{ vir nhiEu
Phuong trlnh chi cd nghia khi
t*rz>.o
Ttl
hay
$acl
{r
vr
r'+
lcl
-1>o'
fi&Y suy ra
o(ecx + lcl -
1
<1 + l'-L:1'
-\
qrq\
Do il,{---!ggz k2 + larl
* rl(
logz
t:0,
m{t
kh{e nT=;2 } 0 nen phuous truh.chl cd
ughi€m khi vd chi khi rogf tr2 +
lrl _rl -
Bdi 8/105. Tim tdt c&. cd.c s6 tg nhi6n m sdo
,:l ygi s?i tu nhi6n n & trons kho&ng
:y
t<.n(Lml2j tht ph0n a6 (m_dln khOng phEi
T
=/i=T;0, nghia td 1-tr'i = o va nx + Wl *1 tdi gifin.
=t.Khio6 lri -thayr:fl.
. !bi_,At&i efia 3{9ugr6a Ddng eaang. 06p 9 chuy€n
Th& lgi, ta th6yr: -F I nghigm driug phuong toAn Eqi hgc Tbng hop HA NSi) vd nt iBo bao
trinh da cho. V{y nghidm eta. phucrng trinh li kh6c:
t. Ndu mt|: Gitt srl m = 2l + t thi theo d5.u
bAj ta ed 1{n(/.
Trong rrudng hqp nhy ta
r&ng
khi
n:t
rhi phen s6'ltzr.-n)ln^
1!6y lery
Bni 7/1-05, Ggi A^*-2g-ryz. Hdg tim gid tri Q +t)ll ln ph0n o6 tdi giAn. Vay tf,Oog t6u tqi
nhd nhdt ctra A uoi $, U, z Id. nhfrng s6 ta nhidn edc sd m 16 th6e mfin dibu kien efra Ohu bei.
th6a mdn d&ng thuc:
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
lzxz + tlgz + B4zz - 4xg - grz * 369r* 16r *
bni tbl t6t cl c6c ph0n e6 sau
&---Lr t t
.NE
T.T.
*89-2lz*i-A.
Lbi gidi. ESng thrtc dI cho c6 thE vi6t lgi
t t+t t+2
I
g
@rz + gz + 76z2
*
*
-
l},jg * 16rz *
gzz
2Go2+ 92 +
+ 4 + 4rg ag - 72il
1,
hey
ggu hon
-
24gz)
llrz -
d6u
6gz
+ Br +
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,
phii
kh6ng
t-2
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 lS s6 16
(1)
I -L,
lIIt khAc:--- kh6ng t6i gi&n c6 ughia li
AT
* l: :
l(-2*'^" -',
- g * B:
t-t
HS
thAnh
m-2
TM
l oou*
t 2r-3s*42-*r
""o""r, l-r.,
_ s * Bz:Z
-2
a) Gtli hQ I : Gi&i g vi : theo * ta duoc
t-2;=sll vir u+2)- (t-21=4 i q.
lt.+z,
V+y S j 2vA nhu th6 /*2 : 2 ttle
74.
8
,:J*o+1, r:Tr+1.Dod6
/ ld s6 ehEn
e)
(1) vn (2) mAu thudrr. tr{6u thuAn niy
chi
A=+-t
(+"+r)+, (+CI+1) :rts*l. dugc losi b6 khi kh6ng t6n rgi c6c phin s6 nim
r
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
A sE ln s6 nguy0u), ,rgUiu rir khi r: 5,
19" 96
Iric it6 A :
AIL2.
lb, z - 9 vi gi{ tri nh6 nhflt eria
VIE
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
nh6 nh6t khi r=1, g:5, z=3 vi A- t;.
T6m l4i, gi6 tri nh6 nh{t cta A lA bing 0
rlng vdi r:7, U=5 vd z:3.
ban Ngd Hfiu LiCm (l6p 12 truong BrIi
_ CAc
Xuen,
thnnh ph6 H6 chi l,tinr,), Dodn-Qug€t
llf
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.
gifta
lit
"e+.2 Vgy n€u m th6e mEntliEu
crie tlhu bli thi m ph6i th6a min :
ho[c l-1-1:m-2
tucm.=6,
holc
l: m - 2 tric m:. 4.
kj0n
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
+ 15 *.r= 2*s+ (*z _ 1)z ltr] + hl
trong dd [r]: phhn ngugin cfi.a *,
LnJ,
- *[rl
lrl: pthn ti
Tim tdt cd nhfing gid
d.dng thnc trdn.
cfia
tr!
r.
thryc cfia
r
th6a mdn
LOt gl&t crla ban Dtong Dtrc Long (9E Htag
Vuong, Vinh phd):
I
Ding thuc & dlu bai tuong &ucrng v&i:
'
ek4 I zr2 + 1) * (rz -- r)? fr!
]x
- [,
x(r-lrlr - l"l:o
i
i
P
I*
L. H.
- fr]i - ic] x
X(r*[r]r"*[r1:s
++ lr{ (rs4 -- zr2 *
l*]t _ o.
12
f@i: f(tfl :
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] ); -
vdi
f"j =-- 1 thi rla -2*2 + I:0,ia
(loai
a:
^
m (q* rz- 1) c(rr- t)6.t
(t)
1A
t
Chalt..miyh rdno fie) e6 it nidt-1):0.
i1t
ngm€m
g3 nhi6n. Tim nght(m d6.
Lbr gr&i. NhiBu b4n d[ giii theo c6eh sau:
gii-lhi6t f(ai : f6) -
_",r::
ciing
lt nghiQm cria
t.
c6
r:
_1
= I)
thi ua- 2r2:a, te c6 r=rr
hoai r :+ t/z l.
[*j:t tli 14 * 2*z - 1:0, ra cdr
[*]=
Bdi LOl105 . Cho fk) arz { b r{ c (*) rr,
;
=
ld hai sd khdc nhau od.
khtic kh6ng, th6a md"n
T
i
+1) (r
.NE
I
2.r2
vd *= Vt+1/2.'
o
(-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
m
=1/+tl6
(toqi-yt+1/r)
m n&n , 1, 12
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
HS
€) (r4 -
Tdm lqi nghi0m cria bAi to6n lA ff nguytD
rghi€m
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
n]t+n thdy tbng cic h( s6 c&a phuong'trinh
1ri
trf c 2 mdi nAy <1ring bf ng {i), ner,
a,ro,"
k6t qr,A chn tim.
"tug-tf,i,
AT
s6 tr.t
TM
["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).
I
ir
L.H.
Brni 2/1S8. Cho n til m$t sd tu nhi€n
ldn
hon 1. Chung nriilh b6t dlng thuc:
VIE
r
Bei ll10s. c6a s6 [guy€n a, b, tn, n th6a
cAc
cb{t sau dfly: phuong trinh
Tf i ar tinh
n
+ * D :0 . c6 nghi0rn nguy€n, m$ n va
(ln'+am*b)lti.zgan1D)=r.
I
Bai 3/108. Chrtng minh r.Eng r{r c6 ca.e
22cosr+61g4-r
iIEu th6a mtn
1964
a*4b=(m*n)(man-1).
euang S#u (DHSp Vinh)
!
--
Chrtng rninh r*ng
Vfi
(2\
/
2
- D <.
1 \n
t
('
2,4...2'*l
,, ' n.2"-l I
Trhn Xudn Dt*ng (EIITH I,Iud)
3.7 ...
Hh1,
tinb sd
< rtfi {1g7g
nghiQm crla phuong
trinh
tr.6n.
LO Thdng Nhdf (EHSp Vinh)
Bnt 4il0{i. {tho 0 "( a <1 ?[/2. {]irr}ri$ min}r
(lhirrrg rnir:h rln S v,',r /r la sd tu nfritn lt! thr
--
tlsin -1--
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 (EHSP Yinh)
EAi 6/108. Trong taur gi ric lfrC
ABC
cotg --*. eotg_-, cotg_- l-gp th*nh c{p
222
c6
o6
c0ng.
a. Chung minb r}.ng d0 dii c{c cqnh cile tarn
gi6d lap thdnh cdp s6 cgag.
b. Tim
cotga-, eotg-4, cotg:
222
li0n
gi{c bi6l r}ng
la ba s6 Bguyen
g6e ldn nh6t cira tam
ABf
tiSp.
-
Ngrtgi-r't Cdng Qttil
,{ 1 _ 1 , t \
"\;-i';i
=
B
a-yb*c
{ o - o
_:_\
\1,'' ,n--,tl
VIE
-._--.
TM
B/LOB. Cho ba s6 a, b, c khdc 0 vir khl{c
nhrru. th6a mdn
,,
l\
b'"c
\o -rt t
i
i
i.'
[/t*.
,"i{-n
j
i.\
\
;
/
t'--\", i-"-i.i''l
\ o*h !
( -& i
.
1
tr )'* ( -''.0 i "J =
Nom
ljd (BEITH
,
Hu{)
Eei 9ltO8. 'trlin t5t ei e6o ed IiQt kh6ng gi6.m
rt, x9,.,.,:t:7979 g6m 19?9 a6 hqng nguyerr duong,
rls79 < 196i vA
,r* + *E -l :., * r?not
l.q i x'z*.., +rtszs)t
dat giA tri lfn nh{it.
1/ii Qriang trdm {.Hl Nam Ninh)
r{QC SrN}r
riM tdl
B&t l0/lOS. cuo tu dion n&a d6u (cac cqnh
nheu) dBCn. DS ddi c6c cpnh ctng
xust ph{t ttr rrrQt dlnh ld a, b, c cd a*lrtc:S
ddi bing
AT
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$e Sz). Chtng minii rlng cdc hinh t:hi6u cfia
c{c dAy ,{rBt vir !2fi2 l"rdn itudng ndi tim hai
m{t chu l[ nh&ng rIo4u thEng blng nhau.
BAi
i.
,i
ig.
.NE
+
srne
HS
(soan
T
r0 ug
kh6ng d8i.
a. Chirng minh ring n6u qua tlitsm gita m6i
eanh dr-t:rg cAc duung tir&ng song oong v0i cgnb
d5i cfra c.1nh dd thi ta dua'c u:ot tri diQn mdi
.4,8'C,D' .
b. Tt mdt dinh c,ia t::drt trong hai tu di6n
c6e iiinh cira tr.? di$n cdn I4l va $9i
d lA kho&ng c6ch ton ni:dt trong e{e khoing
cAch tr&n. Tim hQ thri'c gita a. b, c 06 a Cat
gi6 tri nh6 nhfrt.
trat n6i v6i
*ilA
ff; tfft,f
iuX"fl*,u,
Tirn hibu sdu thcm to6n hga pktr thOng
.,
HINH PII0NG LANG TRU VA CONG TffiIJC TETE TICH CUA. N0
fiTROHG m0u hlnh hgc
kh0nggtenotrulng
QUdC TRINH
ph8
hi$n uey, c{c bgn iluqe hgc vb e6e
hinh l[ng trp, hlnh eh6p, htnh eh$p cqt
r{ c6eh tinh thl {ich tung hirh {ld. Nay ta
h[y nghi0u cfra mSt lo*i khdi {[* diBn 10i nta
hiuh ph6ng ldng tru mi tinh eb{t e}ia nd vd
I
tUOng
:
1{)
do d6 e0ng thuc thb ttch cira n6 rAt tbng qu6tr
gifip ta euy lei iluqre c&ng th*e thB tieh circ hlnh
l[ng trr,r, hini: ehop, hinh cirdp eqt vil tim ti$p
th8 tleh m$t s$ toEi hiah, kh6i hay g{p troag
thqc td: hinh e&i n4ns, htnh cdi ntrn cut. hlnh
ddng cdt..
t
lc*
$ f . Dinh nghra. Itintr ph6ng taug rlB trr\ hinh
d* dign I6i mi tdt *& e6e &Inh e,&a r3 nfi,rr
T
trorrg bai mAt pbfrug rong cong
$3. Tfnh rhe ticb csa hinh ph6ng limg
tr$.
Lfly tliBm 0 h&t ky n[m trnng thi6t diQn
truug hlnh LMiYPQR. vi n$i Ll vdi tf,t c& eAc
dllnh cf,a htnh phdng lSng trg ta sE tgo fiugc
e&c hinh chdp c6. dinh chung l[ Ci vi cdc dAy
IA hai ilAy vA c&c ffi{t b6u e&r htnh phdng
IIng trg:
I i I :i,;",')'ln,
, t , ',
I 1--{!.i /'r''-.
F,\i"i.'i
l"i.,
I )rI',.-;3J-,';,o{
_.l.r:'.a"*l
i
tr
rI
:r
l
I/..
*? l '* ' '/.:*1-,'
. r-'* -l- 'tr I
i r, \l
]
i
t!.
, I I ,; ^
..t
- -'-- i 'r I
-'' -;.1i
'---'.1
, '.-.-
it ,' : '..i,"' l"'".i
li
"."
/" i;
At'
lLt
'.,
;'
#
1
'Irong blnb I ta c6 htuh' ph6ug leug tr.r.r
ABCD,EFGIT K.
Tr! rlfnh nghia ta suy ra :
1. CAc m$t b6n cta bintr ph6ng l{ng trg ln
hinh tam gl6c hay hinh tri gi6c.
2. Miit b8n t* gi.{c cria hinh ph6ng l6ng trg
ld htnh thong hay ilfle bigt le hinh binh tianl.
3. Einh lEng trg, hlnh ch6p, htrh ehtip crlt
_
ld nh&ng dang ddc 6i{{ crla logi hiah ph6ng
lSng trg.
ie 2. Y&i y6ru td
".
lEng rru.
ktrede
TM
,r
i
li
trlng rz tr/6.
Nay ta hey tinh th6 tich cfia binb eb6p O.DHK
c6 dlnh O vh dAy lt mgt ra[t b€n cfia hinb
ph6ng l5.ng rru (m[t b€n DH,x (h.D.
K6o ddi DO mgt tloan bIng DO tbi c6 D' n&m
trong m{t ph3ng 0 ) chrta diy du6i (vi O
n{m trong tbi6t di6n trung binh). NEu dit diQn
tich tam giitc O.Lltt ni.m trong thi$t dien truDg
binh I& s thi diQnrttch hinh tarn gi{c D'IlJi
blng 1s (vi D'GIX lA hinh v! tg cria O.L,LI trong
ph6p vi tq tim D tf s6 ld 2)"
r.'\\
c&a htnh phong
\
t
\
VIE
t.
"'.1
,
\- ,l
;: ,. ./'
g\-r---1,'--jzHlnh
t-
,
I
AT
i
Tbng th& tteh t6t c!. c{c htnh ch6p nf,y b&ng
tha tioh hinh ph6ng l[ng trp.
Ndu ggi di6n tieh ddy tr€n, dien tich i16y dudi
vi &ulng cao cria hinh ph6ng i[ng trg lhn lugt
It cr, sz vi ft thl th6. tich hinh ch6p O..ABCD
b[n6 sr l/6 va thB tich hinh chdp o.EFGHK
HS
I
.NE
O,AIJCD, O.EEGHK, O.AEK, O,ADK, O,DKE,..
I{oi m{.t oria hinh ph6ng ilng trg nim trong
hai m!.t ph&ng song song (S) vA e) ehria c{e
dinh cria hinh ph6ng IEng tru lA hai itiy cria
binh ph6ng l6ng tru (h.1).
Kh-oing c6ch gifta hai mdt ph&ng song song
tl6 b&ng dudng cao cirs nd.
M{t ph&og (Ir) song *ong v&. e6ch iIEu hai m{t
ph6ng song song chua hai d{y cria binh ph6ng
lilng trq aE tgo ra thi6t di€u trung hinh crla
hinh ph6ng IEng irg.
Trong h.1 thi6t di$n rrung blnh c&a hinh
e!-6ls ^l[ng tru AB$D.EaGgr lA hinh da gi6e
LMNPQII.
\
\
\
\
Einft
2
tich cira llnh cb6p D.D'IIR IA:
a1,o,nr :4sh/B
Th& ttch cria hlnh obbp O.D'HK li:
Th&
Po'D'Htr
-
2shl3'
ll
V{y th& tich cfra hinh ch6p 0.DHK cln tirr Ii :
l-lo.DirK :4sil3 * 2shl3:2shl8.
ln dd suy re ring: tbng lhi tictr r5t ci c6e
hinh ch6p c6 chung aliuh O vd c6 c6c d6y lir
c6e rnflt hen cria hinh ph6ng l[ng trq s6 h[n6
213 di€n ileh thi6t dign trur:g ijtDh xn.
Ndu tlfit di6r tich thi€t d.i6n trung blntr e&a
-hlnh
phdr,g i5:rg tru I.i so rhi cudi crrng th& tich
u cte htnh pb6ng llng tig bhng:
$ 5 Tirn them eong th{rc rhb rieh met
vdi hlnh, lch6i hay g{gr.
l) li&u rnQttrong hri d6y erla hinhph6ngl{ng
trg suy bi6n thA;:h mOt doqn th&r:g, d{y cr}n lai
[& hlnlr tha*g
thl ta eti.ftin,h cci nin {h. 3).
Trong hir:h 3, rloqu ihirng { IiL} sr}rxg soirg
vdi hinh thang d6y ABCD.
.NE
T
:
D:s1ft/6+rzni9*2sohll
I
(s1*
,=
;O
h, )'
4s6 -f.
(l)
sz)
$
HS
trong d6 h li iludn8 c&or slr s6, s2 I[n luqt II
dif n tich d5.y tr6n, di€n tich thi6t di0n trung
binh vA di6n tich ct{y du6i c&a hinh ph6ng
l[ng lr+.
a. Dflo bigt h6a hinh ph6ng lErrg trg.
trtn cfra nd suy bi6n thiruh mgt
1) Khi tldry
ili8m, ta eo hinh cpip" Khi 6y sl=6, ,o
--L
,Z
4
ta c6 ilrb tich cria hinh ch6p
uch6p = !o{n,'
'l Lo,r.r,
' 6 \ r ,r+sz\:
6
u"66o
I
-J
:
,hS2
(lz
li
Ta hay
6
tr, - ftsl
TM
--
,,'(sl+4si-l-st):1.h.6rt
,,
VIE
Dlang
1
(ft : dulng cao, sl : rii0n tich
d6.y lhng trg).
3) Khi hai ddy cria hinh ph6ng l{ng iru yi rW
v6i nhau (cdc cgnh b€n cfi*
AOr"g quy tai
t0m vi tq) ta s6 c6 hinft chop"a
cnt.
i(rri dy
,"
0ch6p
s1, sz
cut =
ll
I
-1 i
3
I
p=--l-ft($+.iso*s2)
6
( NIfNPQ
u:: '6\
ld thi6t di6n trung binh)
o (n.
a)-c
22
b*c
G,E'
tro)l g dbGE,vvudc g gdc vcri dD,
tron
.BC
a1-b
+.-
2l
\
GE
ndrr ia duong
caoo erria hinlht tha ng r15y ABCD : G' E' iA dudng
cao
cgco eria lhi6ltr di 0r
cao
Q{tr trung binh (c{1 ng ln hinh tha:rg)
ndnn 2G,8,: GE,
vay:
u: --I (u * b *
32
t
c)
h.GE
-::----
itc
/rr"t + ,z)
dign tich hai dAy i ft; du&ng cao cira
hinh ch6p cqt).
12
(sr -1-
n€m.
Dung mdt ph8ng vuong g6c v6,i c{c cqnh ,4D,
BC, I{L cfin hirh cdi n6rn ta c6 tiridt diOn th&ng
Ei-G va goi dl€n tich crla rhidt di0n thfrng nny
ln s, dudng eao hinhc{in6m Fil : h.D\t AD a,
BC: b, KL:c.
VAn dung cdng thuc tha rich hlnh ph6ng ling
tru (l) thi tha iich hlnh c{i n6m b[ng:
: (G+G)' ,u, rhay viro (1),
ta e6:
v{n dur:g t 6ng thuc th6 tich hin h ph6ng
llng trq tl6 tintr th6 tich hinh crii
:
: duong cao,s2 : di€n tich ddy hinh e hdp)
2) Khi hai d6ry cta hinh phdng l[ng trq bing
nhau vir cdc cqnh bOn cria n6 song song v6i
nhau ta s€ cd ftini ldng tru.
Khi {y sI = s2 = so ndn thav yAo (1) ta c6 thB
tlch eria hinh ling trq IA :
-"'tang lru
-1
AT
nen thoy vdo (1)
tli*h
I
D= --(a*D*c)..s
.3
ta tim th6 tich cria hinh phdrig l5ng
tru mi bai tl6y Id hai hinh eht nhfl.t cd c6e canh
2) Nay
]
I
I*
I
tuong tiog song song m& ta goi ia hinh nlm ctll
bi€t ho{c }rinh ddng cdt (h. 1). Hinh ABCD.
A'B'C'D' c6 iAc il{y ABCD, A'B'C'D' li hai hlnb
ohtr nh[t vi.
dac.
ADIIA,D" ABIIA,B,.
chiEu cao cria ,4BCD.
A'B'C'D' le
A'B'
: b'
vir
h.
T
B?t AD : a, AB = b, A'D' - a',
u=
1
6
:
h
.
(a'D +
4--a*a
' J_+_y_
4
2
J
t h
Q'b' *ab* a'b* ab' *
a'b'
*
.NE
V{n dqng c6ng ttruc thA tich hlnh ph6ng lflng
tru (1) ta c6 th6 tich cria hinh n6m eqt hodc
cria hinh il6ng c6t lir :
ab)
ub)
ijic,4
6
tuc u-
i- h \2n'b' *2ab + ab' +a'bt.
I
C0ng thuc th6 ltch criu hinh ph6ng
HS
6
Nhu vgy li. eau khi dE nghi0n cuu th& tich
cria hlntr ph6ng l6.rig trg (mi c6c hinh ldng tru.
hinh ch6p, hinh eh6p cgt ln nh&ng dq.ng d1c
biQt cria n6) ta suy 14i iluqc cOng th&c thF tich
eflc hinh ling tru. hinh ch6p, htnh ch6p cqrt va
tim th6m dugc th6 tleh cia hinh cdi n6m, hinh
d6ng c6.i (efi.ng Ia nliirng d1ng dfc bi$t cira hinh
ph6ng ldng Lrg)
D:
1
h
(sr
*
4so
+
llng trg:
sz)
6
duoc goi lir e6ng thuc Torndt Simsorr, 16n nhA
toiu hqc ngudi Anh (1710 - 1761).
C0ng thrlc. t16 qul c6 tinh chf,t tbng qu6t vi
AT
pffi*$'ffi'ffiffi Yffi#ffiffi'
4
thuc t6.
S# W& #'*ruC UltgilS
cefiffi'ffi& ffi*ffiffi ffi*T s# r& w& Tt
I
t.
I i'
l
t:
t
v
bqn d5 duqc gidi thi6u phuong phAp
chung minh sci VXn rA v0 ti trong chuong
trinh phb thOtrg. Ap dqng ki&u I{p lufn
s6 hoc d6. e{c b4n cfing sB chr?ng rninh duoc
rngt loat cdrc s6, eh&ng ha.,n ntr" /a /0, r/r re
c6e s6 v6 ti.
dv fai lao niy tdi xin gini thiQu vdi cdc bari
rngt phuong phap d& c6 th6 IAm dug'c viQc ddr,
thdng qua vigc xet ngbiQm htu ti efra mQl phuong
trinh d4i s6.
Nguoi ta gqi phucrng trinh d4i sd b$c n lh
phuo'ng trinh cd dq.ng:
A.C
VIE
l_
TM
I
#&N
(1)
€n;rn+ rn-lfu*I*...ctr *ro = 0
trong iI6 hn sd lir e;. edn cnt Qt^-Lt,,., 41, c" iir cfc
rB rHdn{c NHAI
sd nfro dd (ricng co+0) gSi li c{c hQ s6 cria
phuong trlnh (1)
& trudng phb thdng crie ban til xet phuong
trtnh tlai s6 b{c 2, bie 4 trring phuong khA ky.
() dey ta x6i cde pbu'ong trluh d4i sd !r{c n m}
cde h€ sd cria n6 la cAc s6 nguy6n (ndu cfc hQ
s6 16, s6 hiru ti, e6c b4n ctng e6 thA iri6n tt6i
tu
Dinh lf . N6u phro'ng trioh dai sd bAc ir voi
e:ia hQ sri nguy6n
,.** i.o-rr'-.1 + ... + ctr -l- co: o
c6nghiQm hftu tl ld alb, v6i rr, b lir cac s6 nguy€n
nguy&n t6 cilng nhatt, thl o lir u&c ctra cr, eOn
l, li u&c cria co.
t3
Chtng rninh. Eo a/& l& nghi0m cria (1) n6n thay
,:alb ta cd
:
crfulb)\ 4' c,*r{a/r)"-r+ ...*cr(a/b) +
NhAn ci hai vd viri bn ta c6:
co
0
=
(2)
(3)
,n-1,-t.
- rD_ , --CnLln-I
CoD
-rl(
-,..-.CLD
Binh lf
td
cnug nhou nCn
tlugc chtng minh.
a Ii
b lA hai
udc cria co.
vA.
Tu d6 te rirt ra, khi cn:1 thl:
ac*aco
triuh d4i sd irqc n v6i
+ c1r* co:B
cd nghigm h&u ti thl nghi€m S6 phli ]{ nghigm
li u0c crla ca.
chrttrs minn
X'a
thf,y
}/ilta
/ite
TM
Sau df,y ta xet rn6t sd thi dg dp drlng tlb chung
minh m6l sd ld mgt ed v0 ti.
'Lbi gi&i.
trinh
s6 v6 rl.
nghipm c&a phuong
*2-7=a
t} dny n=2, c2:1, eo:*7, ct:$
(4)
hQ
- tsei tgp l.
VIE
quA thi ndu phuong trlnh
tl tht nqhiem iI6 lA s6 nguyGu
v& Ii u0e efie -7. Ta thrl t{t cS c6c rr'Oc cria
-7 lir 1, *1,7, -7 nhung hh6ng c6 sti nfro trong
eirrlng IA nghi*m cfra (4). Yg.y phtong trtnh (4]
kh6ng c6 nghigm hftu ti, suy ra lZ W.s6 vd rl.
C6c bqn th& Ap dgng chr?ng minh 16 ViS f a
e6 vO tl vir ehung minh c6c kdt gui sau:
il,I[t kh6.c theo
(4) c6 nghiQm hiru
?; v6i
ho[e ld sd v6 ti
Ch&ug minh r&.ng -6
i$. cde s6 nguy6n
tr& sd nguySn"
s6 v0 ti.
CAc b4n Ap dr,rng lAm c[e bAi tgp
duong,
cr v&
ho{c
:
- fi s6 v6 ti.
Bet t$p 4. ch&ng minh s6 Vr - /s ra s6 v6 ti.
tsAy
-rir'-l+..,
14
BAi rap 3. Chtns minh s6 Vi
uguyOn vd
Tht dq l.
- 2r2 -l: grz
(E)
=+,r,n - 10 ru a 1 : o.
"
Nhrr vay li+lS
ld nghiQm cria (5).
Ap d$ng h€ qu& ra rh6y ngay (b) khOng c6
ughiQm htu ti (c6c bqn 16r tdy) neo /f+ riS-n
ra.
gi0 ta xdt tdi e&c thi dq dqng lugng giAc:
AT
HQ qu&. N6u phuong
cAc hd e6 nguy€n:
*
HS
D
Tu d6 a ln u6c c,fa cob', do a
.NE
*+ *2
I),
- co-lan*l.*...* cldln-z-coln -
s6 nguyOn
*lz-li.
-z*!z+z:s
- 1:z*lz
*2
l& u&c c&a crcn, nhung c vd D IA hai
s6 nguy€n t6 cfing nhau udn b phiri lfr udc cfie co,
Ta Iqi vi$t (3) dudi dqng:
Do dd
t4
vd ti.
:
D(
15
T
rao.n: - 6n-ldo - 1, -. '. * ctab'- l- coD'
cnan =
ta x6t ti6p:
la e6 vo ti.
-3V
x,bi sidi. Gi& srl
":t/{+/s<+"
Binh phuong hai vE ta c6:
Ering thuc {3) co thE vi6t:
n
BAy gi&
16
Thi du 2. chtDg minh s6 y'z + r/r ta ro
cnan* cn-lcn- ID+ ...1c1ab*- l *crDo=0
hay
Bei t$p 2. cbung minh .-3-
Tht dil 3. Chring minh co*20" la s6 vo ti
LSi giei. Gi* st"r ;rr = oos 20., ta c6:
cos 3c =
n€n cos
liay 1/2 =
Suy
60s
=
4
"o.3
4 cos3 20"
ra t/2 :
4o$
hay 8r3
-1
-
-
r-
eos3
20"
-
B cos
,r
B cos2()q
3 eos2'o.
- 3r
6r
-
1= o,
(6)
ygy cos2Oa lA ttghi0m erir (6). Nhung theo
dlnh lj' &. tr6n thi (6) n6u c6 nghign hru tI thi
chi Ii trong e6c s6:
+ 1,+ rlz, t114, t
tls.
Nhung th& crlc gti tri do vto (6) dEu kh6ng
th6a mdu, v4y (6) kh0ng cd nghigm bru ti, tu-c
li cos2oo ld s6 v0 ti.
Crlc bqn hny l6rm e 6e hiri
t[p:
BSt tgp 5. Chilng minh sir 10o lir s6 v0 tl;
ItA{ tap 6. f+6u sos Er t& sd v0 ti rhi cac s6
*iR r, eog e vil tgr cflug lA e{e cS v0 tl,
\ tir. 7 rrr.,i?",
I
,
----.F,,: rOAN HQC
T
Mt5-y,v**-,{F{ ch-3ay
[-Y T]-{u
cua s6 vO TV
o, [aJ, lzaf ,..., liq
-
ue.l
Sau d0y ehung t6i xin gi0i thiQu rdi cAc baa
cdch chung minh tinh ch6t n0u tr6n c,ja sd vd
ti. BE cho gon ta gqi tii:l-i chdt d6 li iiah chit
T. Truoe hdt ta ehuag rninh m6t mQoh dB:
' M$nh db. Cho a lA ffrgi sd v0 tj-. V&i mgt
-
-
-
[z,t-of [(il +.t)/r*al = ...
li1t.a| - [tzir- lr-a[.
D[i rr - I : d thi rl&ng thr]c trea tr& thdnk
Ir-c"f - [tr+a)i.-dl [tr+a)r*al
[(t * 2d)/c*dl.
Nh$n x6t tuo'ng tq nhu vSy. ta co d{y
[r*aJ, frr+a*-orf , [{t+za),t*al, ltr+:alr.ol,...
ia m.Qt c{p ed cSng lni_(d6i ciia cOng sai bX.ng
[t-"]- firr-af ( 112*+t vi b[ng hi€u cria hsi
sS cung n$rn trong khofing (1/2m+r, 112-il. Vi v[y
ph:E! qd It uh6t mQt phhn tu cfia ddy nh6 hon
1,/?-+r: gi& sri rt6 la phln tir lti + ualf-ef .
ta l5y km-Fl : (1 + udU{-.
\'/{.v_\qo gi} cflng tim dugc k6ft sao cbo lk*aral
-:1/!"+', inQnh dii dung vdi n+m* 1(dpcmt.
Chirng ninh tinh chdt T. B&y gid giA s& cho
trudc t ln mSt s6 duong nh6 tty ;i, bao gid c$ng
tim i[uge s6 tr3nhionn aao cho l/ln<8. VOi s6
n nhu vAy, ta chon sd ttr uhi6n k, sao cho
l .t
I q ilz" (chen rluvc nh0 kdt qui cria mQnh
tlE"a
trcn). Chio hiioAug (0, 1t thdnh 2n hhoirg bing
nhau tht trong rn6i khoing dEu c6 it nh{t mgt
phAn tri crla dfly llrro 1, i:1,2,.., V6i r li dibno
b{t ky thu$e i0, ll thi kbofl$g (f - e, r* E) bao
gio cfing bao uong n6 it nhf,t mgt kho&ng ndi
tr€n, tue li bao gio, cflng c6 trong nd it nh6t
p,'t$r pilen tft efi.a ddy f/r,,o
I
[, gi& o& dd l& phln
t t} i i.-lcoe. l.
T6m iai ld bao gid ctug tlm tlLroc ed tu nhi6n
l,rt-rf
.NE
l*-o"f
:
:
-
HS
ac
D6 uhfln s6t ih6y r5ng:
AT
it
s6 tg nhi€o n tiy j, beo gi& cfing tina 6ugc
sd tu nhien l'n Bao ctro [t"af '(:/2".
, Chftag minli. HiEn nhi6n rn[nh dE diing v&i
n : 0 (ko lA m$t sd tq nhi6n l6y ttiy ,). GiA
*s, srl
mQnh dA aIA dung vd'i n:m, ta ch*ng miuh
rq$nh db cirng sd .driog v&i n = ni * 1.
M$nlr dE clring voi nz=m trlc li tbn tgi I'm $no
eho [t*a] {,tlz^.
N6u [,k,,d]< 1/]*+1 thi chl tigc Idy km+t:kn
ta e6lkmlroi < r/r-o'
Ta x6t trudng irqp [l<*al ] 1/2'"*]. xdt tlly
IiL*cx], i= tr,2,3,,.. Chir i: le lik*o.l: {rlt-oJf,
TM
,,
P
VIE
,,
tlA qua cbdng t6i r{e grp mQt bai io&n nhu
sau : Cho a la rn$t s6 vS tj'. H6i riug tgp
hqp c{c phBn thip ph*n
Itol : ta- ftrj: k = !, 2,3,...
c6 trt m&t tr0n doan [0, 1] hay kh6ng ? Ndi c6eh
khdc, c6 phii r&i b6t ky di6m c nao thuqc [0, r]
va b'6t kj' s6 elO n.io cdag tlbu tim tiuqe s6
tr+ nhitn k sao cho *e { [O"l < n+s hay li]rong ?
Chung tdi dA giii ttugc bhi toin,_cq.thb ln de
chung minh riusc tflp hp'p cic s6 [ra] rru m4t
trtn tloan [0, t]. Xet qu& d6 lA mSt tinh ch6t
If tht cria s6 vd t$. Tinh cbdt nAy kh6ng c6
& sd h&u tf, vi ndu o: plqvbi p, g In cfic s6
a trr nhi6n thi vdi b{t k-1} sd tu nhitn ft nho, lkel
etng chi nh$n m$t tron6^ q gi6. tri :
T
}IGUYEN TI'IANTI
n$n ddy niry chia duqc thdnh t&ng doqn sao cho
trong m6i ilo4n thi cic phBn t& Inp thdnh cf,p
s6 cgng ti6n e6ng sat ia {,t-cif . Fh&ra t& tn" ,r
th6a
e
o
m5.n
(ir*l) [r*o]{ 1{ ir lk*af
;rk-a] = tr [,r.a]
[
*
l
{ir
i'-af - t ir I
r)
[,t*a ] = lk*oil
tuc la lirt*al ( {rc*af .
wdu l;ric-a!{tr/2''+1 thi is3r lrnl-t = ftkm.
Hdu lillr-dl > 112'-i-1 thi ti6p tqrr: xdt diy tr&n"
K- IoAn SAO
CnO
:r-e(llel(r+t.
ii vdi vi dr1 rlng dqng tinh ch$t f.
Bdi to&n t. Chrrng minh ring vdi b6t kj. r
vi b6[ ky e ] 0 fiEu tds tqi ud sd sd tp nhi€n n
Sau dAy
eao cho
srr:isinr-sinn * eosr-cos n (e.
Ldi giLi. vi * va e)0co thB thay d6i tily j, nen
bAi to[n 1 tuong duong vdi vi6c ch*ng uninh
r$ug v6'i rn6i r v& e)0 cho lrrr&c (b6t kj') bao
gi& eiing t8n {ai it nhdJ nr$f sd l4r nhiCn n seo
eho s;a,. {e (e{c bat trr I}i gi&i t4i saol).
, [.{y 2r la ? don vi m&i. Khi d6 1 radian s€
lil sd v6 ti' (theo ilon vi rn6i). Ap dqng tinh ch*t
l5
--n- U
they rdng vdi m6i c va 6;0 eho trudc, vi in6i sE ld mgt s6 vrblo,,SiO,"{rg tbn tqi a6 tg nhi6n n sao cho ' n: i,2,..., 8r+0 khi :
^h ehst T,
ln'l (6, trong d6 lal'ld phln thfp phan vdi m6i n ta co od tqr n*.r= -.,,n ,re! .1"
iL"rii;
'i"l'-s6 a tinh theo ilon vi mdi (toc Zct).
eira
{0" (lol' l& phin thgp phan
a theo don
",ia-s6
trlc_la tdn tqi sd tq nhi€n
o{-eo:
In
Mlt khac ta c6 rx.,'= [oin lrl'* eie lrl'l +
;i II';
.^,!l 'IB
l,l-- ini"- l',ro
cos l,l,
+ | si,
+r cos
t"
Z
I
ilo mu6n c6 ssn ( e thi chi vt{e ehon 0
sin l'l'- l"l' |<.,a
"to Ilzl
yi sin a-r0 khi a-+0.
l:r:r-#l:f-ti'--{'?;
i !t-X;r';j-;
khi n --> 6 nOn / (:rrr) -g (a',.):i
(r, + In 13) -
I
g
sao
U+r
Ldi giAi. I'ru&e hdt x6r irurlng hqp
/ vi
g co
cung chu h,- r. Rd ring khi d6 f &): g (r) v$i
moi r vi ndu tdn tai dibm 16 mA & d6 / (ro)
tr& lqi tru&ng hEp tlhu.
TM
Gii sil 11 1rr:ot, l$ inQt s6 v6 tj. yi tbu tar
ro 6 fO, tg1 ma & d6 /(r") - g {r"): c1='0.
[.{y rs Id 1 dun vi mdi. Khi aI6 ri tinh theo cton
dti6m
g(ro)
:
= f(r6-srr)-g(r")
e
o.
n__->E
tr)o ilo ta c6l (r)-.g (r) voi mgi r e lO,r*].
Vi rhtng lfrp luAn tr0n v6n dirng ndu ldy di8m
0 id bdt k5> tli6m nio c6 dqng irg, I ln sd tg
nhi6n, nen ta kdt iu&n fG)= g(r) voi mqi er €
(-*, * *).
C6c b4n tliSn mda l
TrBn dAy Id. m0t vii ung d+ng cfia tinh ch{t ,
T. Chtlug tdi hy vpng r&ng tinh ch6r I s6 girip
cic bgn tiryr ra nhiBu diBu li' thf khee . EE k6!
thrlc bii r.i6t ndy, chring t6i xia luu I'e6c ban
lh kh6ng e6 m6t tioh ch6.t ndo (rn4nh > hon tinh.
cAc e h5.t T, vi dq khdng th6 ehring rnirrh ring i4p I(,
:5lli"{ phi dny t&p t(t5t e& c6c s6 v6 tj.trEn do4n
IO,l I rtuc ld bdt kj. rli6n; r v6 tj. nio iho6" douo
[u,1] crng iim duoe sd tr1 nhien t: A6 lA:al-.t)
B&i vi, udu b4n nio 6h lAnr quen voi i1f thuy€t
tfp hgrp, ch&e c5a ban ihfly li6s cti hIc luong €6m
duoc, m) li tiii e6 lgc lucrng ccntiuurn iioa hon
llrc luoag ttdrn iiuo.e):-hai tAp hgp c6 lge luqng
khde nhau sE khdag thd trirng nhau. C6e ban ctng
co rhE rh6;, th6m ring llsa U f u * i ttri va .i,l ,
khi a/p Ih htru tj'. Ngodi ra L].K, fi.
d€r -
TITAT'IH TiEru CUA MSAN fl$C srHffi
Tlrr TOAN
VIE
TRSNG KY
U hoe sirih c.&a ta tharn dfr ki.thi v0 dich
ffiO.f
w-u toin qudc t4 iln lhu 2l t6 chuc ini
Lu0.n
DOn thang 7 nim 197$ ate dat duo.e thinh
tich ru6t si*. C* b5n bqu dir ihi C6tr durlc giAi.
Mgt gi&i utrrdr:
- ll+n Le lld Khdnh Trinh - hee sir:h lop I0
chuy6n to6n truorg Qudc hgc I{u6.
tsa gi6i nhi;
- B?n Pftam *ru 'l'i|p * hoc siuh lcrp 10 chuy&n
io{u tlu'#ng cffp 3 Chu VXn An }IA N6i.
,r.f
fn
ri nghiQlt iil75
Chi sd: 12884
L5'40Ae.kti
,
AT
g(co): c=tO thi Idy r.r:x,o*2T, n:1,2,...,ta s€ c6 ,t. + * o. khi n ---> E vi m{.c du vay
f (r,) - g (c,r) - c vdi rnpi n, tr&i v6i gii thi€t.
Gil s& / vi g c6 ehu ki' tuong ung tA 11, rg v&
t$' s6 rslrg : plq ld mQt sd h&u tj' (p, q ttr nhi6n).
Khi dd pxg:qx1:7 sE li ehu kj. chung cua c[ /
lhn g vir ta
frn Tf -sr)*g(ru-o)
HS
rnoi :"-. (HAm s6 ip(c) goi lA 1i0n tuc n6u vcri moi
r ta iliiu co lim cO(g) = rp (r).
* / (cl. +
-+/ (ro) -
f 0. khi ., o -. i
fli6u nAy rr.&1i v6'i giJr thi6r lim li (r)* g (c)l -
(diEu uiry idm drrs,c
tsAi to6n 2. Ctro f(o) g (r in hai hdm s6 ti6n
tr.!c vA tuhn hodn tr6n (- m, * e). Chunq minh
r&ng n€u iim ff (r) - S (vr1= 0 rhi f k): g(r) y6i
x,-'F*
{aro
T
j(
lrl'-
.NE
?'.
Hdnq Bb Hd.l{qii, Sd
QU6A
vn*T r{AM
Tg tAr.{ TE{tr ?}
- B?n Pirttm Ngpc AnhCcnt'tq*hqc sinh fip f O
ehu3,4n todri trudug ilqi hgc Tbr:g hqrp IIn NQi.
- fiau }'ri ftj Lattg*;or siuh telp l0 chclEn
to6n trutlng Egi hoc Su pbar* 1 }in Ndi.
Mqr gi&i dSo &igr: Bsn Ld Bd {ittil;,it T,rinh
iA hqc sinh duy nhflt ti"*ng kj'ti:i r.i'.rpc tgng mqt
gi6i d[.e Olet vb io'i giAi hoy mgi bai to6n.
in liti ?9.Iri rottg
t5-S-79. {}f.;.i ttru c?,ibuthing 8-75
Gtd O,IGd
fir
