_--7 =-
L
,
ri0i 'rOAN HOc
VIE? ISAII{
i$:#iri:;;i"€.t'Y.Sit
gAO RA
chit nhiQnz: NGUYftN CiNH T0AN
'Iru s&: ?0 j}'rAn Hung .bno - Hs NSi
FtAl iliANG M0T f,,Y
flrr lrg tdo so4n i g{fAfi$ CH0NG
D0v uor; 59825
,
Hi .THI TcAH QUSc Tg rax
fi//t thi to&.n qur5i td lln tfrir 23 duqc th chuc
fi\ tai llungari v&o tblng ? nam 1982 ,.c6 30
nrt0c thum gia. M6i nudc dr19c cir t hoc
sinh du thi. Tbng s6 thi sinh li 119 (Angi0ri chi
'cr] 3 hoc sinh). Hoc sinh ph&i lAm 6"bhi to6n
ilong 2 ngily, ,rn5i ngny lim 3 bAi trong { gi<;
30 phrit. Bdi I vi. bdt 4 crla Anh, bii 2 vd bii
5 cria ltA Lan, biri3 cria i-i0n Xd va bni 6 cria
Yi4t Nam. HQi dbng ginm khlo clA quydt dinh
ring 61 giii, trong cI6 c6 10 giii nhf,t, 20 giiri
nhi rA. 3i giiii ba. Bdn hgc slnh ciia nuoc ta tlf,
. dat 4 gihi : girii nhdt : Lt Tg Qrdc lJidng - hoc
i.,qinh truslg,Ld ildng Phong thdnh phd I{6 Chi
' .\Iinh ; gi&t nhi: Trdn Lltnh - lroc sinh Icrp cbuydn
todir .trudng Dai hoc {Ang hqp I-in i\eii li -\'go
Phi 7'hanh hqc sinh trucrng Qu&c hqc Hu6; girii
ba: NEag6o Hftu IIodn -- hoe sinh lop ehu$€n
teiriri trudng D4i hoe Thng !,1p }{e UQi.
D.rui dav lti di t,Ai : .
.\
1. f(n) ld mgt h{m sd x{c dinh'voi moi sd
n{uJ'Bn duong n va l{y nh*ng gi& tr! khdng :iur,
th6a rn[n e{c tlah ehiit iau r16f :
a) \r{ri rno.i cip (ar, .n), l'(m{n) * f,iar) - f(n)
ld-v gin lri u 'lio'ic I :
YHIT 33
E;.NG I{HT(C NHA}i
b) r (2.1= 0, / (3) > 0 vt / (ir!)!ir) *-- 33i13.
Tinh f (1s82).
2. Cii sir et.ll .4g 1tr 6rit ta.m giAc khOng cdn
co cic canh la al, at vd rr3 (di lA canh ddi tii6n
,4i, I = 1,2,3). t{i ii trung di6m eria canh ai; ii
lA. tli8rn ti6p xuc cire r-ong lrrrn noi tidp t*m
gif,c voi cq"nh ai vA Si 1A di'bm r16i xirng cira Ii\
qua dludng phAn giirc trong cfia g6c & dinh Al.
Chirng rninh riug c{c iluong thiing M1 5r,'S{q.!2
r A .1y'3S3 co mQt ,1i6.nr chung.
3. Nit cAc tlfly lsol nhirno: sd ii,tt,' rl uoilg thoa
rnin tiuh ,.'h{t sau riiy:
a) Chring rninh ring vrri m6i dE;'' nhu
tai mdt sd nguv{n x.z} 1 sao r,l-ro
2t2rlr
.r:ij.rr r .rijrz t-...* r"-,i.r'",,,:J.0lu.
l;)'finr m$i dny nhu vAy th6a nrfln :'
&1
vAl'idn
:1 ,2t')I
rif .'r +riJ,z '."+ "i-,/,,'o'-i"l
vOi mgi n) t.
,r"t'
-!
it:
,t
i
l
l
2
\
4. Oih s* n l:) rnOt"sd drong. Cirung minlr rhng
ndu phucrrg {rinh r" - 3.'g' -+- glo = n *cr ughi6ni
'nguyOu (nghia id co nr6t cnp sd nguycu (e,g)
th6a nrln phuung trirri:) rhi no -'rr rl nh5.l ba
r:ghiQm nguy6n. Chirng rninh ring vqi n : 2891
phuoug [r'irrh lr'6n kh6ng co ngiriQm ngul 6n.
5. Cilc duirng ch6r.r ..iC 'r.i CE t,i:a m$t luc giie
cTEu lSCJIEF dLrgc chia lheo tlrrr tu b&i rihirng
dibm M r il .V nim irong e5"c rToan l.ir&ng d$, sao
cho d,r//Ail :CNlCIl * L. 'i i,'l: h khi li.,tf Yi ,y
ihrine hang.
6. Gii srl C Ii nrrlt hinh vuiing nil. clri0u cl{i
.cria rn6i canh id 100; /-: A"A1..t.4a lA. m6t drrcrsg
g{p khirc k}rflng rlirng (,{o -;.tu} vh kh6ng tLr c{t,
ehri'a trong C, sar. ctro vrii rnr,ri diOin P thudc canh
cria C t6n tai rn0t cirtm cira L n-rh ki'roing cich
lni P khOng qu-a 1/2. Chirng minh rirg tiiu lai
diEm .{ vi }'tiiudc I- sao cho k}io&ng c{cli gi ira
X vir Y b6 iion hoirc b&ng I v5. srao cho bd phan
cria dutlng g{p kirire 1- nHm Siirrl -"{ r':i I'cr} ri:iFlr
drii I6n h<;n hodc b&ng 1$8.
Sau dAy Iir nrdt trorg r:Ae *i*trr g!6i gon rrhiit
cria cic.bii tr€rr.
,r, a ,t - / i - t I
LOt giat flrii 7 (eUa J.i I Lt Qudc l ltdngt.
Yi ,f (nt+,r)-*I (m) * f {il bing 0 hay 1 ndn
suy ra { trn*n)2' 1ril + f (n).
f (D*f (t)-/ (l)> 0. rnh / (2) = 0. ncn ta suy
ra / (1):0,
Ta lai cd / (3) * f tU -f (il bing C hriie 1, rna
f .Q\ = / (t) = 0, n6r / (3) birng {} hcic 1. Iii;
/(3) > 0,n€n la c6 f (:l; = 1.
. Tacsf @>/(3)+l'$)=2
7$i)'/{i,i+ftt)>t
f{12)>r({i)J-ftli>4
/'(li) > /' i :'t; r- /(3) :.>.,
' r (eee-!,i >/ (eseri) + r(3) )> :ss:i.
Vi gi6 thi6l c6 / igSgi]) = 3333, r dn ta phii c6
tl6u E: r 6 crie i;5,t Cirrg thri'c rriri. nghia lA
/(3n): n, I n, : 1, ?,...1Ji13i.
Nhu vil' f i ig!O) * 66i, ; l' ilgBJ) = ti6l.
I!I[t ktrhc, n6u a, ]r lA cie sd tir rrbi&n, o ] 1,, t]rl
/," (d) >/' (l)i+t (a-:ll :>./" (e) (vi s}'$ n6n
a-bias,ilu'[r0r.
Do do: 0s0-I (1s8ar(/ ftcs:r < i (lti83)=
661, tr?c e 60 -< f { 198?) < SSl.
Ta chting minh l (t$"qzi - il$1. 1'hfrt e.{y, ndu
/(i932):601 thi I (3!)6{} = /' (i982 a lgSz) >2.
r (1982) * 1322; I (5940) = l(3961 +r es2) ) 7',$eO+t-1-
. / (1982) = 1322 + 0S1, r*c r (5t46) p trSt. Ua
2
594S:3. 1982 n6n theo lrau / {5916}: 1982, rAy .,
rnflu thu6rr. Liip luAn 1s6111g tu, ta suy ra vdi n -(
:t- 33. rhi 1 r;1r : In/J j.
Loi Eiai bdi 2. 'f z r A 1'r J6i rri ag nhau quu
ph6n gi6c g6c ,4r. Sr vir ft efing d6i xung nhau
qua,.tluong p:hAa gi6c cira 'lt. 11i. rtueng phin
gi*c qua tim {1. n$n cirng li mOt truc t16i x*ng
cria dutrng tron rdi iidp. \tl viv St f,Z - -'l.t'l't
(cu,rg dinii hu'r'r1q).
Trrong tu, 116i rOi iltrtiirg phsn gi6c cria g(,c
Az:.
--\
7'zTz = - Sefu (1'z al6i xirog vdi Sz; I: d6i
xrlng vOi ?'t). C0ng hai d&ng thire tr0n. v€ theo
' *" .7---
v5. lr aluoc: 51 7'3 = * SS 7e (nghia lA eic eung
nh6 t^'3 Sr, 13 52 v€ gi6 tr! thi bing nhau).
Do do: Sz.Sr,! I AtAz. trIe A, Azllthltz, n€n
.st St I llh llz. 1'ui;ng trl', ta cd Sr5: ll yltit,
.\A 53 ,, .1IZ l/S.i,
Hai tarn gi{c Si 52 5s va ilI 1Mz lifs c6 cdc caah
tuong ung song song nAn eirtng dbng dang pbdi
cf,nh. Ilo d6 ,4/1 !it, ,1'lzSr vi Ms.!s d6ng qui tai
l&rn tI6ng ri4rg.
Ta cOn phli ch:lng ruish kh&ng thE xt1- ra
trunng hop t$rn dbng dqng dr v0 eing. trlc lir
phii ehr?ng r.rrinh hai tam gi6c M1 Lt2 Mt vir Sr SZ Ss
khOng thF bing nhau. Cu thB ta chung mitrh tarr
gi6"* St.r:,S:r nh6 hoa tanr gihc iI1 MzMs.
Th$t vfly' cAe di&nr'St,Sa,Ss tllm tren duong
tron n0i ti6p tan giic ,4rlri4: n€n phli nlnr
ir6n trong Iarn giac' (chtr i trrn gi6c At e2 e3
khiing cin,t.
- NEu StSz r&l;r cting phie v6i ,{s d6i v4i
duung ffrr1fz thi rd iing irsz <1 }t jv{2 1d.p.e.nr;
* Ndu I'lSr .n&nr khhc phia vdi As ddi v6i
l,i*Iz (nhtr irinh v0) thi r$ r*ng clurrng cao
dung ti' Ss cta fanr gidc S1S2S3 co (Q dAi ngin
hon dO drii e1u&ng cao tuong irng cria tam gidc
M$Iyhl 3 "* ilp.c.rn.
i
I
L
I
tl
-
3
Ldi gidt &d, 3. Ta chrlng minh mgi d[y sd
'i ilu c0 :
--"-- ! n fr."
.r6lr'r 'F "rll.r:e 4 ,... + r;_ liro >
t -l-tl't* +tl*t-Z
t\ ,rr I Ir! " t\ tTt
], a
'l'hAt vi3' : I
2,2,\
.u",, --rlrr*l * ,n_. ,/rr :
2,:,\
.,- -r ^r.t n-l t :l' ,lit'n*l ,-J
-n__r'n-t
,rq
f,n -:',i #n*2 t ''., -2lrrr*l f Jn*l l' ,r 2
o
\^." i- -!o-.
'- ''n-'3'* n-2 ' 'tn-2
; 2/?.o-3 : 21*lr!oo-S
I&i gidi bdi 4 G&a Nguyin Hfta frodn).
Gi& sfr (r:, .'J) ll nshidrn e&a
. ,13 - 3..r !r2 -1. #3 : fl (ie)
ta c6 :
Q t " 3 ^ 3 ,,2 3
A" * 3A'l * 3r"y -. .rc" * 3.r:" - lio'g - ,-: n
(g --..)s * Br2(x: u) - 13 :.,
lray (y -,.)B - J(.g - d(*ilz + (-r)t=,.,.
UiEu dd chung ,.o eAo c+p (g -r . -r) vlr do
do cA (* y, r 717) etng in rrghidm cia (*)
. BAy gid ta, chrtrrg rninh r&n6; phLrong rrinh (*)
iuOnc6 ba n$hii.m [rh6c rrhau. ndrr.'6 nr0t nghiQm
ir. g)"
* N6u irt + ga * o tni tnong thii x&1- ra
(r, g) : Q-r, - g.) hodc {g-a', *e) - (:-y, r-y)
hodc (iy, r - g) = {r, g). Vi m6i trong 3 d&n$
thuc tr6n dBu dan d6ir * = g:0.: Nhu vAy r.6i
*'t g'+ 0 thi phuong trinh t*). ndu eir 'n:gt
nghi6m ihi c6 3 nghiQnr IihAc nhau : (.r, g), Q-t-,
- r), (- y, r - g,t.
* N6!u.:r:$-0 nghia ltr n = 0 thi v0 lj", vi
gi& thi6t eho n lA s6 ngu-s6n duong.
B:ry gi& ta chung urinh riing voi n:2891 thi
phuong trinh i*) khOng c6 nghiqm nguyeu., Gifi
su ngrrgc lai (r, y) lA nghiQrn cria (*). Voi n= 2891
thi n:9 du 2.
L*p phuong cria mQ! sd ho{c chia hdt cho g,
hoac-c6 d+ng^-(3k + r)3 thi chia eho I ^du f i.
Nohung ngu J3 (rro+c y3i ma ! I thi sryz i slcn
U] (hay r"):9 du 2 do a:2imodg), Vd ly' vi
-o3 : -{- I (mod 9). I
cho 9, vd lj !
Ndu 3r-y2 chia cho I du 6 thi es * gs ehia cho
n-{ooU, vd If I \rfl5' khOng tdn iq.i rndf c{p (p, 9}
dE rr - 3rg' * gr:289l . n
- Ili girii bdi 5. Ndu quay dC quanh trini O
ngirqc chi€u kim dbrg hd trQt g*e bir:g 120'thi
;,,1/( *1-:,,.,-,)( *"*,1,,-,) l 2*o*,
nq t,
iSolxt r ... -r f, . llr .zz
n_t n -
s
r a t/'.) ! - r/oo-E
at I ttJ t ,., T 7ia
_ .4O
. 1_l_ ilq r r/on*i3
* q* " rl_.tt , tlL
-a 4;
t J- ti,)-L I r/rE-2 -! lqt-Z
ildt Cn:2' - ttar "' -r tlz - 4.2-tt:
'Ia tinh duirc vCri rr:1.! ,.hi Cn )> S,g!g. \^iy , vA. y d6u c6 dang 3k + l.
11j*-2 3rg2'kh6ng chla h€t cho I n€n Srgz = -1-3 (mod 9t.
Qrii vil nr-''", -\B,g00tricrugiluong r.6i ryoo srfz:9du 3 thi oB+9i,ffao;ao *J_
! <--,.a /, ,r.,n,!o-Z \ 3rg2 + 93 : Z(mod 9), nhung tB + g$ chi cd eic
z\'il/'i'Yvv/ ' ' .ddulAIf i=2, 1+8=9,S*8:=16khiehia
qn - t ox*g
(iii,gsg)- ) fi + 1/{600)"
}, I + 2'-?/.luilo : r * {09614000 > 2.
..:
U D$r1 iro = I, :!rt:1i2, 1 Z= 114,,.,r rx =!*n
\t'n7 r, ,3iri+ + rl_r1ro'<n.'
_f
_;i
n6 sE,trilng v6i CiJ, do d6 i1 truug vdi .N. Nhu vAy
doqn .t{8, v6i ph6p quay dd, s€ trring vdi doan ND.
\'{y ta ed EIfO = 120-.
I{ ntrirr 8ll ttrr6i l gue l.iing l20a nOn Jf n}.rn
trcu eurrg iroo tern C" bdn iiinh bring uO.
'i'r} t16 tinh cluoc l, - L'N: CE : (:D: CE * lldi.
Lbi gtdt brii sd 6. Goi do 1A m6r rrong hai itdu
mrit cfia l. Kht M, N G t,, La sE vi€rt M < Ar
n6u dQ ddi phAn dud'ng g{p khirc trt lo di:n }1
b€ hon hay b&ng &S dai phln duong g{p liirirc
tri .4o .ddn N
Gqi ABt:il lA hinh vu6irg dl cho. Tr6n. L l{y
cAc tlibm A'. f]',/-". D'sao cho c6c d& dAi .4At,
BB', CC'. /),il' trhring vrr,rt qqi 1/2. Ta ludn
lu0n c6 th6 gii sfr ,r'(.r'< B'. Tip hop eAc
l, ili6m eua.L(Iug(, chia lArn hai tAp hop r.,,r:
' Z1 gdm cAc cliEnr -s{ n', I.2 gdrl c6c di6m } D'.
Tap hgp c6.c cliBur crja fioa,r.r tlrfrng AE r:fing riiroc
chia' lAm hai t iir hop c,:t L' , t,Z: ,1,i gdrn r ir itn g
di6m crla AB cAch Li mdr kholng { flf. DE thdl'
ring :
a) Lt liirong ,6ng frt,u, f.
Z| khOrrg rdng vi clrue B.
b) ,Li \J Lr: AB.
fl
c) f? ia hop Iriru han.ciia nh&ng tlr-,an 1i,1nn
(hcec dec liet ia nhirng diEm).
'fil do su-\'ra ring li uh{t co nrdt dibnr l{ €
L't'i L'2. Goi .X iA dibrn G Lr vi. I' ln dilm G
1-z sao cho M.\ -( t/2, r/1'< 1/2. Khi d6 Ir < I'
Phiiu dudng g$p ktrric gi6i hqn b&i -\. )' pbit
ctrua D'. ri I <1 D'< 1', bgi vii-q, n6u gci lla
dO dAi ,'[:r no lhi
1 -> {Il' + D'Y } ti0+i}9:198,
{
tsAI TOAN RAMXAY
D6 chinh lA ,n$i dung cfia bai toA.n 1/117 clu
a, trdn b{o fo.{n hoc vir tu$i tr6, mA ban
Er? dinh llda tlI suu tlrn, gidi thiQu vdi
ci.c ban. Ta rilric lai br\i io6rr: ( T'ri 6 rrguoi tri-v
y ta cd th3 c:hr2n ra ll ngudi, dOi rnrlt quen bi6t
nhau, ho[c d6i rnQt kirOng quen lri6t nirau >. Bii
to{.n duoc giiii ntro pli6p su-v,luAn hh6 ha1. I{dy
l{y ra 1 uguoi iiy 1f; pri ta gqi l{ A, cdir lq.i 5
ngudi ta chia thanh 2 :r[rf,r.r tlreu tluan hB quen
hofle hhOng quen A, nhLr vur' si, co ni,Jt nh6ni
c6 nhiEu hon 2 ngubi. i: rci nlrotr: niv ln 76 td
ta lfly 3 ngrr,)i ci:'a Zfi . .'',e u nirtltrr cfti q *e n, A thl
ta thdl- co I khi nenq:
1)Bngudi trongT6 ktiOng.hB quen'nhau ihi
bdi to{n giii xorg.
2) 3 ngudi lrong W co ! ngu'ri qirr.rr nhau ihi
2 rgrrdi nAy cring A l,lp th5nh nhdtn 3 ngrrdi
atr6i mOt quen nhau, irAi to&n giLi xong.
),tr6u nhorn ffi i.hOng quen ,4, ta cring cd 2
khi neng: '
LE 1 i6NC NH.{T
2),3 rrgir6i trong 7(3 c6 2 n:gu,ii khdng queir \
nhau thl 2 ngndi niy cung A lap thtnh nh6m 3
nqL;ai hhdcg hE queh nhau, bAi toSn giii xoag.
\I{,r, thuc ch{t c&a }rii tofl nny li cho 6 ddi
fuqng vA, 2 quan /rq al6 ndi 2 ddi trtgng vdi uhau
C{c d6i tuong iv da-r' IA c ngudi > vi 2 quan hG
li n quen l) vi .( kh0ng quer ). Phiit bi6u trlu
trrorrg biti to6n drt#i dqng sau: '<r Cho 6 d6i tuong
vir giiia 2 ddi,tuung dudc r*.ng brrQc b&i 1 trong
2 quan h$ I hoiic IL 'l'h6 thi tlrn ilugc 3 ddi tuoog
mi c10i ni$t lAng hridc nhau b&i crlng m0t
q ua n 'ha. .r.
C{ch phit bi'6u truu tuqng cho ta thfl;" thr;c
chdt cfia v{n dE 6e tu d6 co thB ph{t biEu h{i
toin thluh nhiEu dang blng rich hly eu tbE
c5c ddi tucrrg vA.'cie gurln hQ. Ching han:
1) Ciro c6.c dl6i tu<rqg lA. < di8rn tr€n niit phing )>
vi 2 qrranlr0 lir (ndi ci'i ubau b&i I i[u&ng congD
vi ( khtng ndi v&i nhau b&i I tludrng cong E
ta c6 bii 1651 ; << Cho 6 di6m tr6r rnlt phing vA
tldrr urf t phing c6 r'dt nhiEu tltrdng coug. Chung
'n.inh luOn tim itugc 3 diEm mi ho*c ll chung
s
fl
h
1) 3 ngroi
bili to{r gitri
4
irong # quen nirau il0i rngt thi
xon g.
6
tlugc ndi r16i m{t bfli cAc du&ng eong holc lir
chirng khdng tlrFgc ndi r.di nhau ciOi mQt b&i
ci.c rltdng cong-
2)Cho c{c ddi tuon6 ltr <di'6m trdn mXt phiing
mi khOng cf 3 rlibm n{o th&ng hing )i vi 2 quan
h6 li n rr6i vdi nhau b&i doan.th&ng mriu 116> vi
( ndi vdi nh:lu biri cloan th8ng miu xanh ) ta c6,
bLi to{n:
c(lho 6 ili6m kh6ng c6 3 ili6ni uiro tiring hang
trBn mdt phiing. Gi&a 2 rlibn: lffit k-r- rIugc n6i
ttli nirnu b&i I doan thlng d6 hoic xenlr. Chrng
minh tbn t4i indt tam gi5.c mir rlinh li trong
cAc ttilm rl[ cho, eo eic cq.nh ilrroc.td b6i cring
I miu>.
Cic bqrr hn1, phAi bi6u thdrn e ho nhibri Oq!,g
hh6c" B4y gid t* lihai thae t6: bdi toin. Truoc
hdt ta chu _v tdi nhflng s6 luong:
1) C6 6 ii6i tusng.
?) Cd 2 quan h6.
3)Tim ducrc 3 ddi tunng co cung quan h6.
(ii&a circ c,rn sd h, 2.3 cd li6n qrran gl trong
b{i toAn ? CO thE thay 1 trong 3 con sd d6 nrir
bii toAn ,vin dring hrr-r' kh6ng ? 'f a thfr tha_v cnn
sd 6 biri mdt sd ;r ) 6 tlri brlr to*.n lii6n nhi6n
itting. cdn ndu thay s6 6 b&i s6 n{6 thi bii
tod.n kh6ng ilf ng. 'I'a i:ho phirn vi du khi n =5,
b&i so db I. Ydi n =4:3 chi" vi6c b6t 1i 2 ddi
tuong & so db I la ta c6 phin vi riu. Nhu vAr.
con sd 6 li con sd nh6 nlrdt db bai, to6n {h6a
min. Th6 la tr c6 hhi toln :
I3dy gid ta tim c*ch ttray cou. s6 2 eria bii
to*n. Ndu thn_y s6 2 b&i s6 i thi bai to6n ll tiim
thu"r)'ng. N6u thay sd 2 bori sd nr)2 tht th6 n{o?
'lla r6t trudng hgp m =3, khi d6 bni toAn trtr
thdnh; (Cho 6 d6i tugng vir giira 2 tl6i tu0ng
brit ii1: bi rdng 'budc bni 1 irring. 3 quan h0 L
II, IIL ihi thi tdn tai 3 drli 1o,rnU mn ddi mQt
rdng budc nhau b&i cing mdt .quan hQ D.
'fa th{y bir,i toiLn ni1 sai, biug caeE cho phln
vi du -bing so 116 2" 't'a thi'v su db 2 li phdt
tri$n tu ss dS I bang r{cti t1:€rr rl-]i tuopfi ti vl
cho d6i tutynfi niry quan hd v6i 5 d6i trrdng kia
bh rrg qrran he III.
-';:: ----'**-o'"'
'1'o' oi ?
lihu vly la pirii tha-,- siv ll hoac do 3 trong b{i
toan khi m:3 dB bei toan dirng. Ndu thay sd 3
b6i s6 t)>3 thi bei to5 n eiLng sai khi giiI nguydn sd
6, con.f =2 thi bAi to{n tH.m i.}rudng' V{y ta giA'
r = 3 con thay s6 6 b&i sd n nio cho thich hgp'l
Khi rghi6o criu ldi giai cria trLrdng hop n = 6
m:2, t = 3. ta th{r, khi l5y btrt di rnOt d6i tunr,g
thi sd cdr lai cbia th.{nh 2 nhirm kh6ng bltg
nhau r.6sd luong.-Tuolrg tu nhu vAy khi m:3
tbi sau' khi l{f' 1 d6i tuglg d thl. s6 cdn lai
phi.i chia thdnh 3 nh6m theo guarr h9 ddi r'6i .4
vi s6 ltrgng 3 nh0m niy cilng !ihoug bdng nhau,
hrrn niia pUai .O mrit nhi,rn 7{ a4,* biQt v6i sd
lutrng tirich hcp db lip iuan cho hu,i to6n dl6rrg.
Gifr- sil' tltbm V'6 quan hQ voi A theo guan hQ I, md
trong cac dd i tuong thi|c 76 c6 2 d6i trrgng quan
hS I v6i rrhau thi b'ei to{n dting (uiiung diBu n{y
v[n lA rnal'min), ta phii .rft ilen trong ff khdng
hE c6 quan h0 I, v4y e{c d6i tuong c&aWcbi
quan hQ vdi nhau b&i 2 quan h0 I.I vi III. da
bAi toAn tl{rng v6i V6 nt ffi ph&i c6 6 il6i tuqng
li it nhdt (theo trudng hq.p dd xet) VAy todn
bQ ph&i c6 it.nhdt le 17 ,15i ttrgng^(vl 17 - 1 =
i6 mA 16 chia 3 phii c6 nh6m 76 aa" biet it
nh$"t 6 d6i tuqng), thi ta liLp luAn hodu toin
gi6ng nhu bei toen ban dhu vA cd kdt qui dring.
Ta c6 bli toAn:
a
:E
b
[J
<{ Irong nrgt hoi ngh! nguo.i ts cin tim ra 3
nguoi hole li tl6u culg guen nhau rltri rnpt ho[e
It khOng .hD quen nhau, v$.v chi c}n ldy it.rrh{t
bao nhi6u nguni mdt s6ch hu hr.ra tI& trong s6
uh&ag'ngu&i ni_r. sE chon dugc 3 ngu&i theo
_vdu cdu ? r.
C/ac ban hay phdt biEu dang lriu tuong vA eAc
dpng cg thB khf,c cria bf,i to&n vta n€u.
5
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