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ri0i 'rOAN HOc VIE? ISAII{
i$:#iri:;;i"€.t'Y.Sit
gAO RA
FtAl iliANG M0T f,,Y
j}'rAn
flrr lrg tdo so4n i g{fAfi$ CH0NG D0v uor; 59825
Hung
chit nhiQnz: NGUYftN CiNH T0AN 'Iru s&: ?0 - Hs NSi
YHIT 33
,Hi .THI
TcAH QUSc Tg rax
E;.NG I{HT(C NHA}i
fi//t fi\
&1
.bno
3. Nit cAc tlfly lsol nhirno: sd ii,tt,' rl uoilg thoa
rnin tiuh ,.'h{t sau riiy:
vAl'idn
a) Chring rninh ring vrri m6i dE;'' nhu
tai mdt sd nguv{n x.z} 1 sao r,l-ro
thi to&.n qur5i td lln tfrir 23 duqc th chuc 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$€n teiriri trudng D4i hoe Thng !,1p }{e UQi.
2t2rlr
.
.r:ij.rr r .rijrz t-...* r"-,i.r'",,,:J.0lu.
.\
D.rui dav lti di t,Ai : 1. f(n) ld mgt h{m sd x{c dinh'voi moi sd n{uJ'Bn duong n va l{y nh*ng gi& tr! khdng :iur, th6a rn[n e{c tlah ehiit iau r16f :
l;)'finr m$i dny nhu vAy th6a nrfln :' :1 ,2t')I rif .'r +riJ,z '."+ "i-,/,,'o'-i"l
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.
ld-v gin lri u 'lio'ic I :
vOi mgi n) t.
a) \r{ri rno.i cip (ar, .n), l'(m{n) * f,iar) - f(n)
,r"t'
-!
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$n cirng li mOt truc t16i x*ng cria dutrng tron rdi iidp. \tl viv St f,Z - -'l.t'l't (cu,rg dinii hu'r'r1q).
4. Oih s* n l:) rnOt"sd drong. Cirung minlr rhng ndu phucrrg {rinh r" - 3.'g' -+- glo = n *cr ughi6ni 'nguyOu (nghia id co nr6t cnp sd nguycu (e,g) th6a nrln phuung trirri:) rhi no -'rr rl nh5.l ba r:ghiQm nguy6n. Chirng rninh ring vqi n : 2891 phuoug [r'irrh lr'6n kh6ng co ngiriQm ngul 6n.
i
l
it:,t
l
5. Cilc duirng ch6r.r ..iC 'r.i CE t,i:a m$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$, sao cho d,r//Ail :CNlCIl * L. 'i i,'l: h khi li.,tf Yi ,y ihrine hang.
Sau dAy Iir nrdt trorg r:Ae *i*trr g!6i gon rrhiit
cria cic.bii tr€rr.
I
,r,
i
t
/
,t
a
-
-
Trrong tu, 116i rOi iltrtiirg phsn gi6c cria g(,c
Az:.
'
*"
.7---
--\ 7'zTz = - Sefu (1'z al6i xirog vdi Sz; I: d6i xrlng vOi ?'t). C0ng hai d&ng thire tr0n. v€ theo v5. lr aluoc: 51 7'3 = * SS 7e (nghia lA eic eung nh6 t^'3 Sr, 13 52 v€ gi6 tr! thi bing nhau).
la c6 f (:l; = 1.
Do do: Sz.Sr,! I AtAz. trIe A, Azllthltz, n€n .st St I llh llz. 1'ui;ng trl', ta cd Sr5: ll yltit, .\A 53 ,, .1IZ l/S.i,
Hai tarn gi{c Si 52 5s va ilI 1Mz lifs c6 cdc caah tuong ung song song nAn eirtng dbng dang pbdi cf,nh. Ilo d6 ,4/1 !it, ,1'lzSr vi Ms.!s d6ng qui tai l&rn tI6ng ri4rg.
LOt giat flrii 7 (eUa J.i I Lt Qudc l ltdngt. Yi ,f (nt+,r)-*I (m) * f {il bing 0 hay 1 ndn suy ra { trn*n)2' 1ril + f (n). f (D*f (t)-/ (l)> 0. rnh / (2) = 0. ncn ta suy ra / (1):0, Ta lai cd / (3) * f tU -f (il bing C hriie 1, rna f .Q\ = / (t) = 0, n6r / (3) birng {} hcic 1. Iii; /(3) > 0,n€n . Tacsf @>/(3)+l'$)=2 7$i)'/{i,i+ftt)>t f{12)>r({i)J-ftli>4 /'(li) > /' i :'t; r- /(3) :.>., ' r (eee-!,i >/ (eseri) + r(3) )> :ss:i.
i I
Ta cOn phli ch:lng ruish kh&ng thE xt1- ra trunng hop t$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.
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.
L
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
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.
a-bias,ilu'[r0r.
Do do: 0s0-I (1s8ar(/ ftcs:r < i (lti83)=
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$8.
I tl
- NEu StSz r&l;r cting phie v6i ,{s d6i v4i duung ffrr1fz thi rd iing irsz <1 }t jv{2 1d.p.e.nr; * Ndu I'lSr .n&nr khhc phia vdi As ddi v6i l,i*Iz (nhtr irinh v0) thi r$ 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$Iyhl 3 "* ilp.c.rn.
Ta chting minh l (t$"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
661, tr?c e 60 -< f { 198?) < SSl.
-
Ldi gidt &d, 3. Ta chrlng minh mgi d[y sd
I&i gidi bdi 4 G&a Nguyin Hfta frodn). Gi& sfr (r:, .'J) ll nshidrn e&a
'i ilu c0 :
.
,13 - 3..r !r2 -1. #3 : fl (ie)
--"-- ! n fr."
3
ta c6 : Q
3
.r6lr'r 'F "rll.r:e 4 ,... + r;_ liro > t -l-tl't* I
Ir!
"
t\ ],
,rr a
'l'hAt vi3' :
I
lray (y UiEu dd chung ,.o eAo c+p (g -r . -r) vlr do
2,2,\ .u",, --rlrr*l * ,n_. ,/rr :
2,:,\
t 3 ,,2 " ^ 3 +tl*t-Z t\ tTt
,-J
^r.t n-l
-
t
;,,1/( *1-:,,.,-,)( *"*,1,,-,) l 2*o*, ,rq f,n -:',i #n*2 f Jn*l l' ,r 2
''., -2lrrr*l o
i-
\^."
-!o-.
* 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:$-0 nghia ltr n = 0 thi v0 lj", vi
'- ''n-'3'* n-2 ' 'tn-2 ; 2/?.o-3 : 21*lr!oo-S
gi& thi6t eho n lA s6 ngu-s6n duong.
t,
t :l' ,lit'n*l .,- -r -n__r'n-t A" * 3A'l * 3r"y -. .rc" * 3.r:" - lio'g - ,-: n (g --..)s * Br2(x: u) - 13 :., -,.)B - J(.g - d(*ilz + (-r)t=,.,. do cA (* y, r 717) etng in rrghidm cia (*) . BAy gid ta, chrtrrg rninh r&n6; phLrong rrinh (*) iuOnc6 ba n$hii.m [rh6c rrhau. ndrr.'6 nr0t nghiQm ir. g)"
n_t n - r/oo-E
I
nq iSolxt r ... -r f, . llr .zz
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.
,., T 7ia -
at _
.4O
s r a t/'.) ! t ttJ
1_l_ ilq rl_.tt
"
* q* -a
I
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).
11j*-2
-\B,g00tricrugiluong r.6i
\^iy , vA. y d6u c6 dang 3k + l. 3rg2'kh6ng chla h€t cho I n€n Srgz = -1-3 (mod 9t. ryoo srfz:9du 3 thi oB+9i,ffao;ao *J_ 3rg2 + 93 : Z(mod 9), nhung tB + g$ chi cd eic i=2, 1+8=9,S*8:=16khiehia .ddulAIf cho 9, vd lj !
qn - t
. r r/on*i3 , tlL 4; lqt-Z r/rE-2 t J- ti,)-L - ttar -! - 4.2-tt:
..:
ox*g
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
\t'n7 r, ,3iri+ + rl_r1ro' I
"' -r tlz
ildt Cn:2'
'Ia tinh duirc vCri rr:1.! ,.hi Cn )> S,g!g.
Qrii vil nr-''",
! <--,.a /, ,r.,n,!o-Z \
z\'il/'i'Yvv/
' '
(iii,gsg)- ) fi + 1/{600)"
}, I + 2'-?/.luilo : r * {09614000 > 2.
U D$r1 iro = I, :!rt:1i2, 1 Z= 114,,.,r rx =!*n _;i _f n6 sE,trilng v6i CiJ, do d6 i1 truug vdi .N. Nhu vAy
doqn .t{8, v6i ph6p quay dd, s€ trring vdi doan ND. di6m crla AB cAch Li mdr kholng { flf. DE thdl'
ring : a) Lt liirong ,6ng frt,u, f. \'{y ta ed EIfO = 120-.
I{ ntrirr 8ll ttrr6i l gue l.iing l20a nOn Jf n}.rn trcu eurrg iroo tern C" bdn iiinh bring uO. Z| khOrrg rdng vi clrue B. b) ,Li \J Lr: AB. fl c) f? ia hop Iriru han.ciia nh&ng tlr-,an 1i,1nn { (hcec dec liet ia nhirng diEm). 'i'r} t16 tinh cluoc l, - L'N: CE : (:D: CE * lldi.
Lbi gtdt brii sd 6. Goi do 1A m6r rrong hai itdu
mrit cfia l. Kht M, N G t,, La sE vi€rt M < Ar
n6u dQ ddi phAn dud'ng g{p khirc trt lo di:n }1
b€ hon hay b&ng &S dai phln duong g{p liirirc
tri .4o .ddn N 'fil do su-\'ra ring li uh{t co nrdt dibnr l{ €
L't'i L'2. Goi .X iA dibrn G Lr vi. I' ln dilm G
1-z sao cho M.\ -( t/2, r/1'< 1/2. Khi d6 Ir < I'
Phiiu dudng g$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 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 1 -> {Il' + D'Y } ti0+i}9:198, tsAI TOAN RAMXAY LE 1 i6NC NH.{T s
fl
h 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: ' 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. 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: 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 ' irong # quen nirau il0i rngt thi
xon g. khi neng:
1) 3 ngroi
bili to{r gitri 1) Ciro c6.c dl6i tu 6 4 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- 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. 2)Cho c{c ddi tuon6 ltr '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$n tu ss dS I bang r{cti t1:€rr rl-]i tuopfi ti vl
cho d6i tutynfi niry quan hd v6i 5 d6i trrdng kia
bh rrg qrran he III. c(lho 6 ili6m kh6ng c6 3 ili6ni uiro tiring hang
trBn mdt phiing. Gi&a 2 rlibn: lffit k-r- rIugc n6i
ttli nirnu b&i I doan thlng d6 hoic xenlr. Chrng
minh tbn t4i indt tam gi5.c mir rlinh li trong
cAc ttilm rl[ cho, eo eic cq.nh ilrroc.td b6i cring
I miu>. a :E Cic bqrr hn1, phAi bi6u thdrn e ho nhibri Oq!,g
hh6c" B4y gid t* lihai thae t6: bdi toin. Truoc
hdt ta chu _v tdi nhflng s6 luong: '1'o' oi ? -';:: ----'**-o'"' b
[J <{ Irong nrgt hoi ngh! nguo.i ts cin tim ra 3
nguoi hole li tl6u culg guen nhau rltri rnpt ho[e
It khOng .hD quen nhau, v$.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 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$"t 6 d6i tuqng), thi ta liLp luAn hodu toin
gi6ng nhu bei toen ban dhu vA cd kdt qui dring.
Ta c6 bli toAn: dpng cg thB khf,c cria bf,i to&n vta n€u. 5 { 1) C6 6 ii6i tusng.
?) Cd 2 quan h6.
3)Tim ducrc 3 ddi tunng co cung quan h6.
(ii&a circ c,rn sd h, 2.3 cd li6n qrran gl trong
b{i toAn ? CO thE thay 1 trong 3 con sd d6 nrir
bii toAn ,vin dring hrr-r' kh6ng ? 'f a thfr tha_v cnn
sd 6 biri mdt sd ;r ) 6 tlri brlr to*.n lii6n nhi6n
itting. cdn ndu thay s6 6 b&i s6 n{6
thi bii
tod.n kh6ng ilf ng. 'I'a i:ho phirn vi du khi n =5,
b&i so db I. Ydi n =4:3 chi" vi6c b6t 1i 2 ddi
tuong & so db I la ta c6 phin vi riu. Nhu vAr.
con sd 6 li con sd nh6 nlrdt db bai, to6n {h6a
min. Th6 la tr c6 hhi toln : .a 'Ihe,r e$,l:g ih'Jrr drr clli.1; i1'r1. -.4 {tT*j)-l** i C*, Y y-., Ta ehring miah bi.ng qri nap cbng iir,re rar- That ly thir lihi chri _17 il6i turtng lA t7 nhn
hA,c ho*, *<)n'gual l:0 rang budc li i( trao dbi, 1
trr,rng 3 dB'tAi ), tir e6 bii tcAe ::$ 4 crla kn thi
vO dieh tofn qude tr{ lhu tirrr ii tb urlrr?c, rai t,i6n-
.\6 nlm lS.i{. '. \'4-v con sd i? *6 th& lir r:on *i r;hi ;rXrirt trong
irl:i .ri =.] ra I .= I 1.1', !.ir,l::,1 .,r16.: i::r,, hi.t cj:.,
phin vi dri kj:i n =. 16 ii3 lihir:q Sir:h c{u ilii
trin. 116 d*ag, dE nghi e{e ban hfl1' thr.re hi€n.
, BA3' gio ta lai ili theo niQt hgong l;ha: eiie
bdi toAn mii nhrr phin b bii 1/il7 rIE gr;i i'tuc
ln & phHn b b{i 1/117 ban \ i': l-rir:l: Hda ci.icr n '--18 l
tn :2, f : 4. fiic Il gir't ::gur-en cr']r si ri = t
- c&u hAi toirr Sdrr ttrai a;:i
, i, t. Tlir.r i: "!n:t o
tlA-r' cor sd I = 4 e[n {n c]inll trufc. c,,r 5:h.::i
iruv e on sd n. V&i n:18 bii i,::a:: dr'tr:. rilrrigt"
dAy rln ph&i i& ccn'sd nhii *hirt elt;;,r ' 'iilii gei
Ii1 li sd r16i tuong ehd ithii t:iii .5 1;;,* '.-6'i f
ilb bni todn dirng khi cho m = - c,rrin :, thi *ri
lirB tinh ttugc Y1 qua f y:i Yr--1 ther, ci,;r=, ii::':,:
trur' lrbi nAo '' Nirr giit n*,i.i'.i- , :.: r;'i klr; lirar' ,i,'ij ;n tr
ph6i thay dli rr. G*i U* la s$ rlii irir:ng it rii5.t
r'rng v6i rfi quarl hS d6 b*i tr.r6ir v$i d* 3 dirng.
ta du dr:{n rrdt c5ng thfc {ruy h0i ear:: \ r (ihing h.1n::(,'r:;?; L,'g=:: i3'.. li i 2'={i; Az::t(6-i) l-l=l?. '
Sau cung.eic bgn tLri' thav dii c,. m r u i db
.tin: Qm,t rldi tuqong niii nh$t ci3 i-,i I r,::::; rL:r.g ?
l{ong e6c ban ti6p tuc suy nghi r-ir :li'r:rg i:r
sE ,tr0 l+i g{p nhau irdn til bfrc nii-. &s mmn
*$gt*we ry&m gg.J.'d 7q1 u7,i6, phftn bi€t, iEAi .si:1 titttt; t:u'$t tyu& k. I'iiAt virs, e-ii si? J< ) 1ltii2 i*r: l;: igil * r.r
(a nguyfii: dtr,:ng). ,'i6i i,:p li{}p gi}in :.}!2 si ra;."iiJ
sau, m6i sd chira vrrtrt qu6. 1r:
"{-{ltgI 7u, 992*q,.,., t9E! i,r. 1{lSl - r,,,'o
198'l +- ,: 'i 2(t)lil '* a),
tirc sri lrin irhit ti:uci,.: -,i 1u5:r nl:il ltrir 2l;.1 sd
nh6 nirnt tiruOc 3. Tu d:i'' S!!.J I'a lp,;;ii .i ':,.'.r. 1
tln tsi rlil s6 nao l5 l;5i sd ciia sii iilia. ihta,{r
.4r trfi vtii v0u chtr cira..11u l:rii. '['orn ]ai sd it]n nh:it Ent ]-i l=S. i,-ir:; r'ri;i l;;;, "{
'tim giit tri iittz nhdt cfirt i; sao cht tr,,nq .A ei ii nhdt ri:it sa, 1i ,ioi; a,i,, sD' ,;:/,ui phi.i tir;1:1 13E2.
t. :1. i-,if Sfrif . il'a xet lLii cir cic ti;: lri:1.- ,lr (i : l.
*,3,..", $St) vdri:li giiur !l*2 sd t'+ uhii;r lir,;r tirtl,
kii 1ri i trrr i€l nhu sau: B*t 3i!*3. T'int !?it ed edc {t4l'r;ti,T ,r,,jr:5d;r I a,:{t 2 S- 0or}
3I * tlrrc=ng t.:x, tt, z) ,:&a hd lthuang trinh- 1 J 1*t.p2+=2:i9Sr
{r +'g *z:
68. .
.n Liti gidi. Binlr phrrong cria so lr.r' nlrii r: r,'. l'lii
chia cho 8.-cho sd du 1'(n$u n 16). ho6c i) (liLr
n c6 dang {lc)" ho{e 4 (ndu n c6 dang.{1t + 4i.
Vi 1982.E6 (mod 8) n€n tt phrong trinir tirit
nhdt, ta $uy.ra rHlig trong ba sd r, g, z, phli
co hai s6 16 vn miit sd d4ng 4lt * ?. M5.t khie, btnh phu.utg ctra s6 tu nhi6n n, kiri
chir cho 3, eho sd du 0 (ndu n chia irdt cho 3), I.19r,.r:S00, irgl. ... 1981 l.
I Ae91:g{il, 992, '.., 1982 i
DE th{y riing trong rn6i tep htrp i.i lrdrr dbu
tbn tai it nhft n":0t sd ]d h$i s6 e&a sd kirf,c
eirng thude ,li. Y+y cl€n diy ia da tim th6y s6 ,lr
th6a mflr diu bdi le 1982. Ta ehung minh ring
1982 ehinh 1A gid tr! ldrn nhit trong e6e sd k
phli tim. 6 \}- r '1. r 7" --,,,iL.r1,1 -i--l) Iq c) r )> 0, y { 0. He dS cho u& thin}r
. fr+!
1,"'=
1,,2--,,*r.
ry
y - hodc 1(ndu n kh6ng chia h6t cho 3). \'/i 1982=2
(mod 3) n€n tt phuong trinh dAu, ta suy ra rilng'
trong ba sd r, 9,'2, ph&i'cd mQt sri"chia hdt cho.
,d vtr hai s6 klrOng ehia hdt cho 3' {uc id hai.
s6 nily c6 dang 3k + I hohc 3k * 2.-Trorrg phuong
Irlnh thri hai, ta c6 68*2 (rnod'3)', y+y hai stJ
5y phii ilbng thdi c6 dang 3lc + 1. nghigin. r thi tii.piirrr:ng trinti 'thu 2 c&a h$ ta c6 NFu lgi -<
r-y2-1-y:y(ga1)--(0. tr'6i voi diEu kiln f,o ' T6m lai, ta dl ch(rng'i6 ring trong ba sii 'r:,.
Qt .z, c6 hai s6 16 r'd hai sti dqng 3 k + 1, lhn
.vAy phii c6 mQt s6 vua i6, vua cd dang 3ft + 1'
tri'c ltr c6 d4ng 6k + 1. C6 thA coi.ring. z= 6l<*1'
vi
-z(rgsz(1s2,
ndn zlci.ti cd tlrb Ifr.v- mdt lrong 8 gii tri: i, 7
l3; 19,2i.:i1.:17, i3. Nhu vA1' ta ph&i tinr nghiirnr ngul'0tt duong o)0.
Nh.lr v{y tr{rtg i:rir}rg h?p n}y hd cirng vd
ngi)iern.
)
d) -r'(0, g ) N, \'i tlofig hi tih chtr vai trd
tru c&a.ht phucrng trlnh. cia c vil ry li nhu lhatt n€ti
cimg gi6ng nhrr truring irqp c)'. : -l6rn la.i h6 phr.rong trinh dli cho. clii c6 hai rr Crine hai phirong trini] ta drgc rt * gz -2r : 0,
Ndu lul . I tiri A- : i -- gs < o, tLrc h0 vo r'd -r = g:2'.L t 92 = lfrl: -- :2
I '''
1.. +y - 68-:,
vdi z 15y c6c giA tri & tr6n. lIu6n vriy, d3 i ,
ring II. (,u - y)2 : 2(r3 + a') - b -F1i2 a ld s'ri'ngugdn ld. Ilai 4/I"93. 't im i?!t c& cdc cip sd ttt nhi6n
dm, il az i(.p + 1)t'o - p"l dsl gia tri nlfi nhd"t,
lrong tlo .p Lbi gi6i cria -\gug6n floiiri Cudng (Qui Nhon)'
1. li€u n chttn. = 2( tgSa - ,i) - itiS -- :)?
: - 3.2 * 136: * 060.
tl)
Bing ph6p th&, cho : lhn h.rgt I gi{ tri d}
n€u, ta thdy r&ng chi vdi gi6 tri z a 19 thi vd
cu6i cung 6 (1) m6i li nt6t sd chinh phuong,
vA,ta di ddn rghiQm cria hQ df, cho: '(10, 19, 39)
(ctng edc ho6n v! cfra h1i sd ni- ). i(p + t)2'" * p,oi : ltp + t)* * po'll x
x l(p + 1)* + .p",Ei) itp + 1)'" + p"r!l )> zp + t
(vi p vd p *"I, mdl sd ltr chin. mQt s6 ld. 16,
n€n higu c6c lriy thrle cfia chiing - li. !rr-lrl + c
I uo ={ul +, .i Lbi gtdl c&a Ngugln Hadnh Cudng ('Irung' VAy n6u chi x6t n chin thi giA tri n!:6 nhdt
iip+ 1)'o .- psl: 2p *1, dat-d;roc khi m = 1 Yuong, Qui Nhon). "&"
n=2, Xdt h$ phuong:trinh qua cic tnrong hop sau;
a) x ( o, s ( o. He de ci:o tr& {l::\nh 2. )l6a n lE.
Chi ein tinr (rn, n) a61ip + 1)'! * p"l ( rp + r.
a) Tru&ng hsp (p + r)'* - p" { 0. T, c6 ('?:-r+U
l*
I y':-u+f,. $gnlem E :'i:0 2pr:t tr(p"*iF+t):"( CSirg hai phuong trirh.ta du.qc l'..2 * y2 ={t
vly ry =A, g:0. Trl diy v{ hQ phucrng trinh
'dd cho, ta lhdy chi chn x6t th€rrr c*e truting hop
ci r vd y dEu kh6c kh6ng' (") + z :( p' - p2t - .". t'2m.p --( zp + z
<*Zlp( p"-1 -- p'*-J - ... - 2m
' < (Pp +.2)rp ( 3. :b) r > g, y > 0. llg de cho tr6' ttri.i:h 2rn thi .4 chi c6 DAt .4 = po-l , p'*-I tht Eang. 1. 2, :i. t2
)r.:r-rg
I 92 - u i t, e tric ta c6'r=g vA rt -2;r. Ta th'Sy rgay h€ cti
nghi+mn=g:2. Ta c6
' P'-(P*1)2'=dP-1' : '* * N6u,{. = 1 +, ,lr" *.(p + l)2*:.&, * 1.Vd trii .r2m ta 'th5.y r5rg dS cAc i5n thuc c6 nghia, vA-do
vd phii lt}n hon r:d trAi, ehn d{t diBu kiin cho
nghi€rrr 16 n€rr p:2, ta co .r vA tharn sd k:
r )2 0, t< + l, )> o. t-r.) o, trrc lA , Vd tr{i lf, b0i sd c&a 4 trt l khi n } 1 n0n
kh0ng thb bing v6 phii; cdn ndu n'*'1 thl vd
trAi ( 0 cirng kh6ng thB beng r'€ phf,i. ir)rlo'. 1 t'Ndu A-2?p"-(p+l)2--2p t (*)
Vdi cf e atiEu ki6n de dit, ta c6 th'6 blnh phu.rng hai v€ vl duoc ' - t
^n - ,
t.
- l- hdt aho p + l. VAy A't-2. k + 1,1 rc <{ k2 * 2A.r + .r2 _ €-} (p" + r) + 2*- (p + 1)'m :'?(p + l).
VI n 16 ndn p' + 1 ip * 1. Nhung 2 lihdng chia *.
":i
:l I' hay . + I = 3. 'l'u (*) ,r-r' ra p = 2, tr& r'E trrrdni
hop l.: l. VAy tru&ng hqp aAy cfing khdng c6
,irghiim. Coi vd phAi ld m6t tain thuc blc hai tldi r'6i b) Trudns hsp (p + tl?* - p" > o.; Ta e6 A', re phAu tich dugc tr((p+1)r*-pn<2p*1. (t) o((o +y', + 1- k)(r--ii *- rl.
Do di6u kign (*), ta c6 * - l;-,k'{ +) 0< pzn' + ... * 2mp* p' < 2p
{+ o < B - pr**1 +... + zm * p"-l ( z.
I aguy6n n6n chi cS th6'bi.ng; 0, 1,,2. Ta c6
I (p *,I)zn -.pn Bp :1. +B:o+(p*1)2"-p4=1 = x+1/ *+1-& o ('*? - (2.r = I)ft + { {3 - fi, I I ha;', voi t = lT> O, a..!-
u, r,a1:
vdi diBu kiQn 6g, b$t phuong ttinh (1) tuong
rluong v0i
' GiA tr! b€ nhdt rria l(2 + 11:tt" * $'i bing t, .
+il(p + t)* - rl[tp +,)m + 1] = pll
+ (p * 1)m -- 1 : p", (p + 1)"r + 1 = pt voi r, s
nguy€gvi r * s = n. o9 pt -,p" =,2, chj xiyirr
,khi e = 2,.r:1, s = 2 '+ n = S.
khim-1,n=3.' qr'+r+(1 -1r){u.
Trrtde 116t. rlE t2) e6 'nghiQnr, ta phii c6
A=1-l(1-ft):44-3>u, h a:- + B: 1-+ (p +1)3y - I,o.= p * I. Yd lrii 16, vd phii chEn. Vqy B + 1. ' (Z) k ):l/a.
lihi 116 r'6 trAi ci'ra (2) c6 hai nghiQm phan
biet ti ( la. vl l6t fhuope trinh (2) e6 nghiQm'
t, ( t( tz. €+ (F + l)-"' - p" + p" : Cp + 1)o. (**)
ra '6 lp' - pzl t.uorg thA la b6-i eira (p *,1)2,
ra lp" - p?l = p'- p?:
vi- nclrr-vsy thi nlt
p2(p"-2 -"1) l nhrrirg pn-?'- I voi n - i rir kho,rg
, chia h6t cho p + 1. Vay kh$ng thB c6 (**). n .t 7 . .'2nr +. B: 2* (p+ 1)2* * po: 2p * l
't T6m liir v6i p-2 rhi l(p + t)2'- pr" i d+t
gii tri nhr', nh{t btrig 1, khi m: I. n = S Yoi
t6. 16 thi I (p + l)"* - po I rl+t
p ln s6 nguy€n
gii tri nh6 nhrdt bing 2p * 1, khi m= 1, n=2. P.Q.G, vl I : X r 2 0, nen dts bflt ph,uong trinh d5.
che c6 nghiQml ta ghn ph&i c6 tz ) O' Theo hQ
thrle Yiet,-tt * 12 = - 1, v{y f1 { 0, thi.nh thfr'
It, tz c6 d{u trii nhau, do.d6 lai theo hQ thue
Yiet 0)> ttt2=1-.&, t*itd suy r'a,k)1' Nguoc
iai, n6u I{ > I, thi.r!{2 = I - k {0, v4}' tt, tz
khic d{u, mA lr ( !s, do d6 tr ( 0 d fr, vd bf,t
phuong trlnlr (2) c6 nghiQm o q v7( rz. B*! 5i1*S. Gii,t bdbf{n Iudn tfteoktr[f phuong frinlr -._--;'\ o('l(k-(1 +l4k-z)l2 , - (3) \ Liri gidi. Vi6t bdt phuong trinh il$ cho du0i ,---_.-:'
\{k-tlr(k-;c. dqng -
' 't6m lai :
-- ndu ,k ( f , Uat phuong trinh v0 nghi$m,
- n6u ft ) 1 : bdt piruong trtnh c6 nghi+m (3). I E hay va hay a b- )0. B*i 6lJ-93. Gli?. s:? a, b, i tti /la srl /,.?idrug diu,
lidft hC'u|i nhau bitng bi?tt thL?c: a + & * c = 1.
Tin tQp hqp t?it cfr c&* gia tri ci th? ei d*gc )0,
3*7kJ-3/{sr'-+1D(A=r)
18 * 12lr D6.th{y o*'h*c=1. c&s 'bi?a ltt*c sott eldy
.a
^\- --1"1-b* s 1dm lai, ta ludn lufrrL tim duor a, D, c kh0ng
f,m, tbng bing 1, db cd S : A'. Dir]'id rliBu phAi
chung milth. {-iti gii"i cria ba* Ngu.i,6n IIoir.rh Cu<'rrig {Qu.,, Nhon). Dol)>1*(a-ilZ r 861 ?iLPl)" Hdg tim cdt t;i. rti€* co tttl- tiin td'n
lranE mQt t;hdi lhp plwong I , ' /r,t
1*c-l:' + --__=-_--
1+9-c nhdt ndm trgn b€n
cho trrdc.
,,,
LOr gtat.
1. Trudc h6t tim tri diQn-c6 tita ttch 16n nhdt
nXm trgn bdn- trong'niQt.mdt cAu clio tru6c. Ta
thdy tir diQn ilo ' DSrr bf ng ehi .tly la k jri e = b: c.
Ta chirng\1ir .S nhin rnci sia 1r'j thuc t,;n ii,rn \
- CAe dlnh ph&i nim tr&n m[t cEu, vi ndu
Itin hcvn trao ld.v n6 ctng ha;, hdng l. khbng thi c6.tu di€n
nim tron trong nit chu. {,dy k fiiv j-. ,k G li. + * ). (:6 : trirc,.nq h(ip :
,-.
1\ tr.)-- r.r 2
Ta ehon e = 0, cca:tliii tiin c, b sao cho a {-b,.- 1 vAsi2r,=b.i u:k, ' D6 dnng sul ra rlurvc I
t^ -t q * tl rl-i- b: o = ..-.__---_,
r qt, ,
J-r4h k * 1+ (?*2
,) t zt{ - Phiri le iir dien.dEu vi rdu t{r diQn ^},4&r"P0
(cn cAc.clinh nim trdn mii ceu) thi
khOng rI€u
c6 it nh{t hai canh kb_ r:hau l:h6ng.bing nhau,
ching han NrP > NQ. Trong {iirong trdlr ngoai
t!6p tam giAc NPQ (itudng lrdn n[m tr6n. mdt
criu), dr;ng tam giSc cfln N'PQ n6i !i6p, ta co
aiQn ticlr rYPQ *r tu di6n
di$n tich N'PO )
MN'PQ cb rhA tich lon hon tnF tich Lf dien
MNPQ + l:INPQ kir6ng ph&i ln t{r di6n c6 thE
tich l6n nhait ifong cdc tu didn n&m tron,b€'n
trong rrrii t ci u. Il. c hhdng inr Voi c5c gi5 tri nA,rr; r0 rA.ng a,
vi a*&+c:1. D€ thdi. r;ing trit cli"cae tu' ttiQLi r16u, c6 cae
,finti tr6n rnSt ciu, d6u bdng uhau ri6lr dEu la
e{c tir di6n e6 thb tieh iori nhdt. 2) I < tt{,\/%
'l'a biron c : 1!$ ; a, b ilnri:* tinh tu he ph,rysng trinh: a+b+213 I, 2. Cho lAp phucrng ABCDEFGH. Duug rndt chu
tO) ngo4i ti6p lAp phuong tlo. Cic tri di6rr ilEu
A.{:FH v{ BEIIG li cac tu didn, 16 t,hb tich i6n i r'o
tlJ
---j____ ,a
\a ,ti. Tu ,,
F
1-1-lili*t, l*a--ll3*'' '- (27 + 3i')rri- 15 '* i0/r = il. Cbon j '. 1-l-b-a
-_*
flc Ia co.
os (zz + tglt)
nghiQrri n I{ I + k-3 /(k*1XsA + 11r
_
18+12A I
't/: -+ r {rlr, vi v5v :Do lt (
.ta c6 *Itr I k2< + s (e+k)z > e(A-r) (ek+r1,', 80 4 : n€,n a/(t * b-a),>,r(l +- u--b). Cnng r-41:, ta e,i
bl1+c-ol )) bil*b-c), ol{t ta*c) )2 ef*c-a}
-+- Sla(1 * a--b) + bil + b-.c) l- c(1 * c*a)
: c *b*c * h2 + bo+d - ab - bc-ea
: r * l.in -il?'+ (r *eiz + (e -a)sl/z) l. rh6 thi dc'96,.: .8Jr- bing gir: &OC n6n suy ra
/r\ BD>,lC nhiit n:'tn trori tr.rrig rrriLt cilrr (O) phli Li c6ic tr?
di6l'cc thb iich !orr nhflt trong cic tu dign ninr.
tron'{rong lAp p}ruung. NgoAi ra. trong Id.p
p}:rrong- b#t k,i tt'r di€u ntro hhic ciing lA tir di-dn
cd rtinh iihOr'rgr ni-rn trdr mdt cliu. huitc ld tri
di{l1 kh6ng tlriu, n6n sd ;6 thA tich nh6 hon ttrb
tlcir hai tr'r di&tt n6i tr*u. 'I'{rit lai chi c6 hai trr did,r lCFfi rd -Bg'nc
la rdi i.rb tirL lur i.rlrit tioris c{c ttr di4n uirrn
!rurt iroitg l:i,, 1,1,u',*,,t AliCIlDFCll. 'I'n (1) vit (2) su,y ra gric ,{E/J nhi, hon g,ic (1t -{CI), rniu,nr;, 3. Kdt trarin: DiEu kiAn-brjn tlibm ..1, B, C, D
phii fir6a min lA.: hod.c bdn dibm uen.r tren mgt
/ciia m6t trl
cludng trdn, holc eh[nq kI tl?nh
di'Fnr trr.r'c t:rnr. hoHc .hting Iap thinlr rn6t lr-r
,lien c6 c[c canh dcii Iring nhari. ,, iliri gll23. Cjrc hinh chop dpu S..tBCDEE. Ti,
m$t dt?m O tr€n drbng e ao SH cia hinh chop ln
d.qng nti.tt ph&ngic, cilt Sl, SB, SC, SD. SE, S.E'
tai A. B f'C f D
thttt: E t,F t . Cht?ng minh cd.c h€ 1) E*! EIE*S. ?'r'r,r:{,, i;ltdn, €i6$ cjro bdn tlilm
plrri:r biiii :L" E,'C, il. Hi;i ldc dia,tr ctfi cio phi?i
tit6a md* rli?u fiii'n tti d';' bdn d.tibng -trdn, ,l6i
tirirrrg rfi g*c ba lrani1 bdn dibm cla, b&ag n.ha.u?
Liri, gi&i. 1. Edn cti?nt,{, B. C. D dlng ph&ng.
a) ,!.. {}, i:, D lQp llidnh fri' gidc l;3i. Gi6 su
triC ii! irr4t ra:rh eir,r t'i giAd. vi ctrr&ng tron di qua
fr, C, B birg d*.:ng troa di qua B, C, A nOn hai
rtrurrag li-on do pliii ti'riing tihau. Ydl trong trrrdng
?:irp nirr", l/-sll + r/So, = 1/.ra1 + lisr.l '+
' = l/^sc, + llst'- |
1' ' D ris-{l + llsc^l i li !81- = lisB. + llSD, + l/tl.',. -''-''l I -',"' 2. Edn *fjE;:l .J, B, C, D khdng dbng plt&r,g. B6n
iluo*ug tron, rlrii duoqg di qua ba tlong bdn
di8m 1,.{3, C, lJ SbLr *&m'tr6n mdt clu ngoai ti6p,
Lt? digc ABCD; ciLe rludng trun d6 bing nhau
ni'ir tnirt pl:in.q rhr'rr c5e du&ng tron d6 phii
rr:5.l cDu neoa.i ti6p tu di0n. VAy
c6ch riDu titl
t{m h:ri nitt clu r.r6ii.irp rA ngo4i ti6p tL't di€n
Allcl.j pi;iri I,r'irg nLarr. jlI \ Lcri gtdi. Nitr l( soilg sorrg \ r'ri ,15-r' hinJr chdp
ttri c6u. h0 lhuc phii ciiriug rnitrh IA. Iritn nhl6n,
n6n ta g{ st'r:l khdng song,soitg v6.i.rl{y hinh
chdp. Trudc hdt ta chring minh bb db: N6u dn
\trhin gi{c c&a gdc A trong lanr g-i6c dBC,
-4 = 2d, lhi , llAB + tlSC - 2cosai-tD. Thit thd, ta c6 didn tich tarn giAe ABC bhttg
di6n tich tam gific AB.D c6ng vrii ditjn tich tam
giile ADC, suy ra {l$c !a:rr v;ifc nrirl crja tu ilicn lthf,rtg llr} co
g6c tir, vi ndu tr6i lai ching han tam g16c,.-tBC
c6 96c tri. ihi dudng vudll; g6c v6i rn[t phXng'chiua
A.EC. k6 tu tinr ilu't}r,q tron ngoai ti6p tu dien
sE c&t mirt phfing dri tai di.6m ngoAi tam gi{c, di6m
d6 hh8ng ph&'i IA. ti€p dibnr crja iludng trdn n6i
ti6p tr! diQrr vdi rn{t ABC, tr{i voi kdt gufr. tr6n.
C6c cluOng trdn ngoai tidp c6c m[t cria tii
cli6n bing ntrzru n6n hai girc cira hai mdt ddi di6n
r..cri culg ingt canh hdt lij' ph&i bling nhau (kh6ng
i:ri rhau r.i c5.c g6c d6.u kh6ng ttr). AB.,lC.si rfla- AB.AD.siu:t * AC'..4,O.sina sin2o =-.- € AB + ^h
AB. AC (1) *+ Ll AC + 1l SB - 1cosa.i,12 (dpcnr). .i8 > CD . I
f^!
t\t AD. sinct 'i rL eirLing rninh eic. canh tl6i rrhau cria lrt di€n-
{ECl) phii bing ntrau. Thnt vAy, gii sir trei
lai ching iran 1,t $qi cr. li.,gdq girta iluengcx+ c{ra hirlh ch6p 1.&i cA* canh b€n, 6p dueg bb dA tren, ta a6 = llSC, + 1/SFi = ?cosa/S0 MSi ph&rig Az(tO c$t'SC t4i F,_ cit .tr1Cr t4i 1,
c&t CzB* t4i J.I n$m tr.On bai mrt ph&ng AtCtEt
vA rlzEto ndn ph&i nim trSn ,giao tuy6n-E10 cira
hai mIl. phing rlo,. suv ra SI li phAn gi6c u1fia
g6c ,{SC, Chring mir:h.tunng tr,r, ta,c6 SJ 1i phAn
giAc cfia g6c CSE. t ip durrg bh dd tri.rr, ta eo 1 1tl 2) Ta iinh lisl I | 11,i, i + 1..5,! i.
Qua O dgng m[i phing- song soxg r.6i titi.v
hiirh chcp; cit,Sl" .5i, SE iunng r'rng t4i A2. C2, Ez.
Iih6ng m1t tflng iir:Ai, ta cd thb .gii s& vi tri
cria e6c rlibr4 Ai., Ci, Ei (i : 1, Z) ahu hinh vE.
S t/sdt, + 1! sc 1= ltsa1-l TlsP
: 1/,SCq .1- lislr;
1/5P -F liSE
+ trlszl, + r1sc. + ]/.s.e. -
1/Sd.) + 1/,SCg + 1lSE"- i.-osc*/SO : =3*rsa/.!O. Et Bing cAch ttrqirg tr.". ia tinlr tluo-c
lrsBi + 1'Sll1&+ i/SF,
'Iit ci6 suv ra rlibu ph&i chl?ng rninh.
Ban l{.qlrg?n Hoanlt Crcbnq (Qrri Nhc:r) ;,r) loi giii t6t' P'e'c" tsai n{}/12&. Gidi phtong trtn.t r Vr- 1 + Vt..r - 1)(r- i) +.
lc * 1 * VfrE:lX"'+rt :/;;E
Ldi girii. Didu k'i0n nghiQrn phuong tlinh phAi
thrSa mdn': r ) A. Ta bieln ilbi v6 trai cii.a p'urrng
trlnh; db cho gon, ta d{t o * I =,.r, r* l:i,
vd t.rdi bing /," -r- tF+ ', I __'l_ 'tl ^ - ^.*-:- _---
,_,_;:^:
1! ab*a\f abo - b Y arb -f ab !ab'* 2iila 2o + 7no2) q r/a+Vor,r:
-- il , r Vo*o+ n+'liF + z /(,+VFb) (, +-ffiT
= V o + i/rrt+ a+ VJr+ :
-l: { a+ tfazb+t,+
: /{ii. .;- f 61s tri lr h ' i3 t-' 3 r-- T16 Jni b;dn,t., i:r plr&i giii phLrung
,*, r,'ii/;i + V'fu' = tlz*,
ei
l\/r - t + ^/.ri1)3 =:.r:3
+*?r *s Vf.. - i,,t* + ri ><
x (V;;f1 i/..n1-1 = 2,.a. , l- '!'e ed 2
1)-r = 0
2) VEf'-0 *ir = 1 (loai nghigrn r =-t)
s) 3 ile--zVG{:lF B o
<+ s(*2 - t)2 - 11.2
I <+ r'- 1 :1/ 27 12 )i htrng rlo i*i : 3/-..-. 3t''-- 3t'- a; s12 - tz rr"/i)lz +,r:/(z r- r 'i/r yr2. Vr-1* Vr*1=1/2. r:, (lo4i nghi0m 0m). phuong trinh tr'& ,tiiinir:
* e\ -i-:, VAv phuong lllnh c6 3 nghidrn i % P.Q.G. llr*,', i/.rz - 1 :l L'x = 2r3
<+, [r i/z ii rz-]i - l(rcz-l)l * o
{*.r fcl*t bVz *z VGe - 1)rl = o i1 / Tarn.gide zlSC c6 ,Jang ntrru lh6 n?ro thi c6 dflu ding lh*c ? b) Churrg mi,:lr rirrg r'6i -urr.:i ium gi6e ABC ta " ludn c6 ' cos,{ * (l/2.)cosB * (lill)cosft'-.- 7/ti yi v0i m6i sd e}0
rndt clang tam gi6c dBC:db cho tt_v w, luOr-r tbn tai'it nhrit vdi BAi 7/f 26, Cho di1" sd ;r1 , r, ,..., .rl r oQ9 / t/ tirrh chdt D tl :'tltrl.iz : l' vrr CAC DE TOAN THI 6IAI IOAT'
1982 - t983
SAi f 1126. Tiil sd dieL!!{)ticira rndt co.quail,
bidt rirng sd rtd c6 dalrg ibrto, vdi cl2( a { c,
vr\ llng cAc cli& sd crra s6 d6 bHtg bO. - rl
'
al I
t: cocA * (l/2)cosB -t (l/il)cosC)> z/6--e.
Triln Vtin I{hrti *'i1 ffi'tllffi,, vdi moi , C 11,3,....,1981 i. BAi 2ll 25, Chring rn jrrh rf,ng ngo{i nghi6rn oig=0thiphuongtrinh ..
|i-r r r-Lll- i
,., r I
i(lgEi-i) 1982 18 l Cri{ . 1c6 t
r i
r.r. +C )=U Ch ung nrinh ri ng .'
a) r, = r19gl, 13 = rlgg0,
b) r<*id1+ti1980. khdng eds gghiQm ngr:vBn (r, g) nAo khfc. (CHI]C DTC) I/fi Sinh ffda I'rln 1-dn Ii/rdi (Qudc hoc LIud) Bei *1L26,:Iim nghitm ,nguvdn cfra phuong trinh *s*rr3=*.3. Pft am Quano Gidm (HA NQi) " Eai B/126, Trdrg mIt phing cho mQt duong
tron vA mQt dny cung AB tty y. Goi I lA diEm
gita cia AB. M vil N 1d hai di6m crla AB ddi .rr?ng
v6i ohau qua f. L5y r1i6m C tny y tr6n du&ng
trorr ; c{e duong thirng Ci}f, CN vA C1 c&t lai ilrrong
trdn tuong irng & P, Q.vd fi. Dudng ttring PQcit
iluirng thing ,18 6 .1. Chung minh r5.ng rluong
th&ng sfi ra ti6p tu.r'€n eria -;;,3;:"r::r
cho. ,oo
d.)J Bei q126.'flrn ili€rr kiQn ciin vA d& dE mOt
s6 !r1 nhi0n n e6 thB bibu di6rr clutrc duoi clang
, : ,'1 -i- .,2 * .r,t - 3rltrz.f3, trong il6 r: , x^..r lA nhrlng s6 ngrry6n khdng -12i Arn Bai 9 L26. Trong mtrt phdng ciro bdn iliErn
phflr1 biQt A, B,C, D. Ilny dung nr$t hii;h ch* nhAt
sao cho c6c ttudng th&ng chr'ra m5i canh cfia
ird iti qua mQt trong cac tliBm dS cho vl sao
" eho ti sai hai kich, thtroc canh hinh ehir nhf,t N96 Dug.Ninlr (Qu1: Nhon) b&ng urOt s6 k cho tru6c' Trong trudng hqp bni tr'rAtt cr'i.vS sd ldi gili, hny tim qpi tich t&m cria,cAc hinh ehit nhAt' Ngugdn Ddng Pftfil {HA NQi) . i I I Baf 5/12$. Ngucri ta r&i 35 hdn bi trOn mgt
m{.t sdn hinh r'u6ng c6 carlh fing 10m. Chring
minh r[ng lF
q
F
Bit 1O/126. 0ho tani giac AEC'. finr trong
khdng gian m$t ditsm D sao cho ABCD ll rr:Qi
tir di*n tiuo tA:n, d6ng th*i tu ba clo4n th&ng
DA, D,B. OC c6 thb dr.tng ilu,cc rn0t tam gi[e
'tuong t[uong voi tam. gi{c ABC'
t
phai\t euang Gidnt
BAi 61L26. a) Chrlng minh riug vdi moi tarn
tCnu ftlch: Tir di4n truc tr'rm lA tu di{n ed
cAe cqnh al6i cliQn vudng g6c v&i irhau, ciing Ii
'tu diQn c6 cAc dudng cao dbng qui).'
giie ABC l.a ludn c6
'"'o01 ^rl,lr'
(i/3) cosA * (114)cosB +'(115)cos0 < 5/12.
\
\ro\{4
Ngng6.n Ddnq Phdt
fim hlBu eeu thcm todn hoc phb r&Ong
VE CAe sS trGUYfrN TS
}{GO DUY NINH
toen theo khia canh kh6c vA tlm cAch gi5.m nhe
cAc y0u chu bi.ng c6ch -xirt cdc khoing c[ch giila
td li€n
ti6p pn+l - pp. D6 th5y
hai a6 ngul'€n
chl c6 pt=Z,'pz=3ld hai sd nguy€n
tri6n ti6p,
ctin lai : pn+l - p" ) 2.
fAC sO ngur'8rr td lA v{n ilE trung iAur, t.ir ltr
cua ngD de buuc vAo lau diri sd Iror'. t.hiilrg
nhitng nhu th6, r:rd" n6 rlon 6nh hrr6ng s6.u
sic tl6n ci. c{c lginh toin iroc hhAc. Vi tAy no
dA thu hrit su'c lirc cira rii nhiEu nhi., to6n hoc
efla.moi thg ki.
Cncb dnv 2200 u5rri, ngucri Hy L+p tli chirng
minlr rhng : s6 cAr- sd nguyCn td 1A vO han. Tuy
nhi0n. ngu&i ta rnudn co rndt c6ng thirc db vidt
ra rndt sd nguv€l
trj l6n tu]' i. Nhung \,iec tim
ki6rn mdi cdng thuc nliu v&y ching mang lai
mot hdt qui gi hcrn vifc rli tim hon d6 thhn kj
aliibi6n moi kim loai tlidnh virng, hoirc rr:dt c6ng
thfrc toAn nfuig lA ioAn r;:r- ri6 giii moi brii toAn;
(c6 nl'r*ng unc mr: liin iao r.-irn chi li mo uoc
rnA thdi )).
Tuy kh0ng iim ra m6t cdng thue c6 ich nhu
viy, song ngudi ta co thE bi6u di6n khi tlep
m6t sd nguyOn t6 iheo mrii sd.nguy6n tri.
EINI{ Li SIERPI\'SIil: G(,i p Ii s6 nguyon t6
thu n, vA
. --o fl
Trong c{c tAi 1i6u khoing 20 n{m gln ddy,
c{c nhi toirn hoc lai nghidn curr vb cAc bQ b5n
s6 nguyOn t6 li6n ti6p ghn nhau nh{t, goi IA. bQ
sinh b6n, cd nghia l{ sAc sti ngu3r€n
td c6 dang:
pk:=n:4,pt+t=n:2,
Pk+.?=a*2,Pk+3:n+4'
Dhu ti6n nguoi tir xet c{c sd nguyin t6 li€n
ti6p
c6 khoing caci; b,ing 2 vi goi ld c{p s6.nguy€n
t6 sinh dbi, cheng han, clp 3,5; 5,2i 10016957,
10016959 ; ho[c : 1.0C0.000 0t]9..6.19; 1.000,000.009.051.
C6 mQt diEu kiin chn va dri dE cho bi6t cAc
s6 nguy0n .t6 sinh dOi nhLrug thne tt! kh6ng
rip dung tluot::
{ USNff EE 1: EiEu kieri edrn vrl d&.c13 rnQi s6
16 n vdi nyd rz l- 2 d0u ld s6 rrguv8n td I[{n - l) !
1ih6ng chia h6t cho n vd n * 2.
thi
F
0t = L .p1llt
n-l
IIai cdp sd cgng thua nhd.t (co cOng sai Icyn
irhdt) chua t$.t ci cAc c[p s6 nguydn 1.6 sinh ddi
(tru 3 r,A 5) In 6n { 5 r.d 6n * 7. Tuong t.U 4
c{p sd cQng thua nhdt chua c6c b0 sinh bdn
(trri 5, ?, 11, l3) ld iiitn * 11, 30n -1- 1.3, 3(;n l- i7,
Srin + 19.
,,8 -.ri-l ^n-l
"v*.
I't e$.c bi6u thr?c nAy suv ra, tij.t c6 e{c bd
rinh b6n dhu ti6n cho ctdn'10.00U ttrt S, Z, r t,
13) tn khi cho n = 0, 3, 6, 27, ,19, 108, 18?, 314.
' Vi citng 6!rv ra cf"c eirit s5 cudi ci)ng e&a b6
sinh ]:dn li 1, 3, 7, 9. NInr 1919" V.A.Goiuber clfi
'tinh trong 10 triQu.scj tu nlri6n rld,u tiin c6 89t)
b0 sinh bdn v[ 15 trigu sd 6hu tiOn co 1209 b0
sinh b6n. BQ sinh bdn lon nhr{t tinh d6n nEur
1965 li b6 r'6i s6 ugriyerr to nhti nhiL 2Ei:j.30r;.7J1,
do I-eriel chi ra,
DJNII L1' Lt.lCAS : trIot s6 p li *gu5,6g td khi
vA ctri l;hi torr tai so ngu-rdu u il6 np-t-- t i p
vi. v&i mlri s6 n l& ui;c cira p - 1, n."- p - i
thl. on -- I kh6rrg r:hia hdt chci p.
ViQc 6p dung dlnh lf l-ucas di! tim mQt s6
lguydn t$ ln.kh6ng thrrc tE vi n6 rdrn lurr hon
cA,rh clrirrg u siitrg > cua .liratosil:itrre.
\{ic rlir dudng r:hrr v ide ghrrng rrrinh r'.11g : t$p
hop sd nguyEn td sinh i16i ctng nhrr sinh b6n
