l' +r 't tn$
H*r 'roAfi fItlC
ylEr xru
Chri ni:ifni, nOUYtX
Trr,r r&: 7{) 'Irlrr IIung
8ACI RA
cimm roln
tlgo - Hra N6i
,1, rn,{Nc MOr xy
yho kg tdta loan HoANc cn0nc
[)5y rror : 52825
I
t
I
I
i
I
,H# B$n;G 8.t-{Ar NIFM OUGNG TRON CLE
YA SLTSNS THANC OLI CHO DA GIAC NOI rfiP
Cle ban thil mdn I
- Kh{i nignr du0ng trr)n C_Ile vA.itudng lhf,ng
Ote dA etugc xAy drng d6i 'roi tarn giue. Droi
gdc tlQ cfr* verto, & bAi ndy t6i xin gi&i thi6u
m0t cdeh nr,l rdng cA,: khAi ni0m niy d6i v6i
da gi{c nOi ti6p. Tru6c h6t ta hiy nhlu lri
lrudng hgp iam gi6c ir g6c ilQ ndy.
D6i vdi tarn gi{c -lr^4e.{s, t.{c barr d& dang
shung minh iluocrngay hai tinh ch{t sau tliyl
'linh ch?it 1; EiE:r ki€n c&n vd drl dB r', lf
trong tAm eria tam giAc :ltdzCs li : t
+-.-4
rlitl = (l/3i(o.{r + oAz * ii-13) il}
vdi S la {6m d*dng trdn sgoai ti6p tam gi&e
1tt.{?As.
Tinh ehdt 2: $i6u"kien ciin v{ tlrl i[€ J{ tt
trr:e lAm cria tam gi{c .{t;{rrlt l$ :
+ .<
Olt*OAr*0Az*0.{g, (2)
r&i 0 l*m tAm iludng trdn ngoli lilp eia tam
gi{c.41r13d3.
-r\.oAi ra ndu gcri //r. ltt, Ita l6n lugt l& c{e
tli0m il6i xrlng cria t) qLra rlrAl, .ls.{.r, .4r,{3 thi
ta s€ c6 tinh ehdt:
l
' f inh chflt .1 : C6c ilulng trdn co tArn lin lugt
l*- tl t, llz, H3, vi bin ktnlr b{lg bin kinh
cfia itudog trr\n ngo4i ti6p cirer tunr-giec..{rA.l,le
sE clt rrhau lai m6t fii$m chung H chi.h IA
tr-gc tflm cria ta nr gilc AtAzAi.
(Ctc tran hfly ttr chring mintr ba tiull ch{t trCn)
Biy gio ggi E tAdilm Bt0a cua OH, thd tbi "
0- = (tl?) (Otr+ 0Az + OAil (t)
CAc bq.n de bi6t rludng trdn tim E bin kinh
hfrng ltia bAn kinh cria clu&ng trdn ngoai ti6p
trm gi6c rl1.{rds di qua cAc trur,g di}m Mr, }Is,
'i113 cria cic eanh .,12.43, ,.11.131 At.4z ; cAc tr,rng di6m
(.r, Cz, /i3 crle cAc doao t'hing l/.tr, uaz) tt .lt;
c{c ehla ilrrdng vu6og g6c Br, Bz, BJ n+ tu
ele dinb dr, .1t, ,{s xudng cic larih a6i A;en.
During trdn ni.r. duoc gpi Id drbng trin g dilm,
lray cdn ggi li dubng trdn Ote Ifra tarn gi{c
.{til2.l s.
Ngo&i ra, do (r), (Z); (B) n€n 4 (li0m o, M, H, E
In th&ng hlng vA ilrrdng th&ng ili gua 4 rtigrn
nny ggt lA iludng thing Ole crla taur giitc AtAzAs.
Btroe ([Au ta thii m& rOng cjte khAr ni{nr tr6n
cho m$t tu girc nQi ti{p.
-
--/
r
Gi& sr} I a cd ttt gifc nQi i i 6p '41 A2 As At'
tludng trdn S c6 t6.rn O, li6n kinir rT lA dutrng
tr&n ngoai ti6P tti' glrio niY'
Biug nr$t cielt rri-'in tuorrg tLr circ tliir:g th(c
(i), (2), (3)' ta dinh nghia:
Trrong tu ndu goi lit, Ez, Es !A tirn dudng
lrdn OIL cfra cdrc tam giAc ;t2,431i+ ' '4sAaAt ;
,44.t1.{r ta r," :
EE1-*: liE2 -' ;''1;3 :::: ;)f;!' .
Nhu vAy cac <1i6m ll,t. Ez, Ez, E4 nirm lrdn
duong tron. CJle eira tir giAc Jl"1z'leAa, hay cAr
rludng trdn Olc cira brie iaur giec {::'13'14' Agrll3i'
.4+.ttiiz, JrAzAs cJit tri::rrr tai iirlrn l:r'
Cu6i Cirng, e6c l;*r i:li.,; ir'; 1!i1' 11;. i:ri:rir iinh
elrJt sat, ddr':
Tinh chdt 5: Goi Mt, XXz, tr13' lla l:]n lttgt
ih trong tArn cria e lLe tam giiic. AzAele, '{s'4aAi'
,ladlAz, ;ltAzAt tht 4 doan thlrng 11!'11' 'l'M2'
.13i'tf3, d.lMa sE ciit nhau lai rnQt ili6m '11 chinl:
tA tLi7ng tArn cira liL giae LrSi 1i6p 3t"1:ls4'r vir:
.\;Irtti 3l tli: 3 (i: 1, 2, 3, "l)
Iihdng nhtng thij, cilc ban cr)t; cri ihi: i'iiii:lg
minh dusc ba tiuh rr,rtit r'iii ti:t ';] nir* :
Tinh chdt 6' Goi ifil ld '.rttng di6m rloan A;3;
(i*j) t ri ,4f Iir gioo clilrn chring cfr:r cat: doan
itrirqg ^!Ii;Msr ,-Ol (r, /, ii. ir li'rnrii irr-'ilr -"i eira
(1, !, 3, {),
Tinh chdt 7 : Ci,rc itoan tlrrrng 't1Ili, 'lzllz'
A3ttg, AaHt cit nhau tai mdt cliSrr' E Id tdnr
A"ong trdn O-ie cfia tu giirc nQi ti6p lr'4zAsAa'
.f inh ctrdt 8 : Siu tluong vudtn{ goc ha tu
trung tlibm c&a m6t canh (ha1'n'rot rtrucrng ch6o)
toi c-anh d6i di6ir {hay cludng r:heo eirlr iai) cirng
cit nhau iai lAm slrlrirrg 1r'tn {'1i* ciirr trri giir:
ndi lidp Jt.1z.1:i.{4.
Nhu virv cuc batr se hinir dttt'ig ril cr'ieh m&
rArrg lrl,ii niQnr drio'n11 irdn Ole '''i rlurilg llirlg
(jie cno rn$t tla giac rriii lit^'i) b'i1 kj-' Chting
ta dinh nghirt bilg qui nap: giri siL i:11 c khf,i
ni€m tr€n cll dinlr ughir rho {'dL tit circ rla giic
n6i tiFp c6 sd canh nh6 hrrn n: thd thi:
Fsinh nghiLt 1' : Truc tirn li eia -rr
* gidr
,11J2.13...J6 l$i ti6p li-cng drrdr:g tit)u S ld giao
Ai6rn cnung ciia n drrt}rtg lr.on bingl (tuolr[i trdn
S vi tri t.im tai "6" ii1;:', lit:l i:i:a rti*'1 )'- giec
iao b&i (n'- 1l <?inh eri:r iz - rrii:c '1t'4'r'4:1""'1t'
0ll - 0'1,t -r- {}lt i "" + (l'{n
r'6i 0 in tArn efia drlr'rnq trriir i:gi:rli ti$1r n ^
giac;1tda...dn'
:Binh rtghitt 2' : 'l'rr.rng lArn' ]/ tr&a n - gi6c
AtAz,.'An ndi ti6p duirtg trdn ^S la-grao t1i8rn
*troig cia circ iloan thrrng n6i inSi dinh critt n-
gi,ec vtii trong tAItl o&n (n-1)-giac iao boi ciic
Iinh' *on lai. Nltu v.ly ta 's0 cc ll{ chi:t irong
.il' oonr, ti&ng nev theo t? sd (n- l) : 1 hb tri
dinh cria n*giAc vA :
AM = (O,\ + n.1? '... I QAntrn,
-Ei rth ngl,io 't : Tr u c t fr nr tt c irr
{t Ar .13 Aa nOi ti6p troug iltrong trt'rti
bun kinir Ii 1A diBm lhila mdn:
@ 4 4 d
O lt :0Ar * OAz + OAs i- tlA+
iri gidrc
S. tem Lt,
(2')
s
{
Oinh nghia 2: Trong t0m '\y' cirrr..tl- Cie"
Ar,qf es.la"noi ti6p tlong tlutrng tron S' tAr;r O'
Lin kinh ll ld di6m ttr6a mdn:
4 s 4
ail:(tl4) (oa\+ ij|, + tiat + OL4i (1')
Dinh nghia na-v hohn loln phri lro'p v6i
khti niQm"trqng t6'ar nra ta bi6t tir trutlc ddn
rr*',' (c{c bqn trr ehirpg rntnh)'
Bitth ngltLtt 3; Ndu ff Ii trrc t6rn eira tii gif,c
Irezasan"oOi ti6p trong ilulng tron-'S'.tini O'
bln kr,rli /? thi duong trdn v6i tim E li irung
ililrn ctia OH, Lrh4 kinh tl,'2 iluoc goi lb' dudng
trdn Ole cria tu gi6e nOi ti6p '4t rlz ;1g ;1+' Nhu
v iv 4 diErl O, /,l: E, fJ sE riinr trdn mrjt^fluong
;il?r;,;;;,,g',r.,ing tln:" sqi li drrdrns th8.ns ak
c&a tu giiic n6i tidP 1r '4xA3 Aa'
. BAy gid ta hd5', I gihi0n cilu mOt viri tinh clrdt
,'i,, clc khai niQm vira ni"'r r$ng dcii r'#i tir
gihc n0i ti6P .At.4zA3Aa.
Tinh chdt 3': Gr-,i tt1, t{2. tt3, Ha lin luqt ln
rrt," tarn citn c6c tam $iic AzAlAq' - Ag'{a'4i'
.lnir.fr. ,1142Al; th6 thi cSc duong tlr)n tiiur lA
it,-Hz, H3, ll 4 co cr\ng bin kinh fi (bin hintr
J,roni t.an nsoni ti6p) sE t'il lha.u 1:ri nr{'1'
diim tr/ cirinh it iruc t$m rira tr1 gifrc'
Chtl'nq mitth:.
- t+
fIfi4:lotl.
-- ffi''r-fffi: +
1'rrr,rng ttr. ta
V{y ,t <ir&'in
trda 1im i:
cirirng m ini:.
'la c6 :
+r 1,.+ ,..@ + +
Otla\= l(O,Itl {.}'1 ? + {i-i3 ' t)14\ -
& r ,+
rr*)i-ltl,lai -R.
cti : li i?1 - llLlz * IiIJs n Ii'
{"tt, IIt, Ils. lia liini ii'$n drOng
5*r irir:h Ji. ra elr rliErr r'llii
'ltnlt cttdt 4 : iliiii ilrr!;:g tlon Ole cria c&c
,r-"gia* ,ta" iA4, arAqAl, AqAtAz, AtAzlg ciit
ntr*r, tol urQi rh6m .ll ehinh li tArn dudng tron
Clle cita t.u giic n6i"ti6p SlAeasCa'
j
t_htrng minh: N6u ggi Ea ltr tlL[r aTaU ',S
Ole ciia tarn giAc .lt.ltA3 rbi : [Ea =\OU-OEtl
- ,.,".,6 4 -4 ,+ 4 .4
= ,(r/2)iA;i + Oiz, O $'()ALt ( ti':t O'{t*O'42 +
. €,
* oAs) l=filz)l loJ4l = fli'].
,l
t
I
,B
$
$
"i'ru<}ng hop n:f, tr'gng t.im ri/. oiia doa,,
th{ng ,{1,42 lir truns diErn crja do4n thing d6.
'Eitth nqhiu 3' : Dudng lrr)n O'le ciia n* giAc
-{1d2.13.'..4n ndi tidp rltrr}ng trirn S ti clucrng trdn
ili gua tfit ci cic tr^r ln eira circ clrrdng trln Ole
crin cric (n* l)*siAc lao hilj (n-1, <linh ciit n-
gi6t.5!irr thE lirrr E cria tlrrring tlt'rn nirv thiru
n:ili :
()E: t.r"{r -r OAz -r...+ ()ArJi2.
'I'rrang lrqrp n:2. duorre 1r'ou Ole cria diL-v
cung lttAs rria iluring. lrdn S ban kinh ll lA
d*ring tron birn kintr l?i2 c(r tfirn tai lrung dibm
cfra dir euug.t1.{2.
B6n di8m O, M,11, li nlnr tr'6n m{t duo'ng
thing goi lS dairng thfrng Ute ciiir rr - giAc
.'1 I.t:....in
Yi€c ehung rriinli tinh torr tu i rria e Ac dinh
nqhia 1'.2',3'li rirc biri toirn tlinh slro c.1 c tran
tn giiii'
Ngcii ra. cic ha:r co thl thdv thdrn ciic tinh
ch{t :
'{inh cl'fit 6': Tit ci eac tloarr thitrg n6i trong
tdm cfr..i l * gi6c lao tr&i lr tlinh cira n - gi{c
vlri trgng. tim i:ria (n-lc)-gilic tro b&i (n--rt)
tlinlr rclrr lai $iiu cli qua trong tAnr,V cfra n-
Tinh chii.t 7' : Circ dorrrr thiug n6i moi tlinh
*fra n*{iirc vcri trrJc tAm cira (n-l)*giic lao
b&i (n- l) tlinti cdn lai rii cill rrlrau tai t{m dirdng
trdn Ole cira n -giirc, r'd bi lirrr rrlv chiri thlrrh
? doan h:lrng nharr.
[iorr nfra: 'l'6t ci cric do4n tlrJrrrg n0i tluc
t*.nr cria i.-giric tao biri & dinh criLr n'-giAc vdi
irqc 1Ailr cirr (n-1r)-giAc lrro b&i tn-t) einn
con Iqi cirng rlEu di qua tiim ([unng trdrr Ole
cria n-giac vA bi tr+in uiy chia thAnh 2 doa n
htrng nhru. (O'rli1 t:r hihu [r'uc tdnt //;1 cria doarr
,'{iJ; [n dr]m thrla n,an, ?ili,i =d, +[il
7'itth chdl 8'I n(n.- l)/2 drrdng vudng g6c ba
i* tiinr rlurlng ilon Ole cia (n-2)-gi*e tao b&i
(n*2) dinh ct'ra n-giic tdi drrting thing n6i 2
dinh con lai cit nhau tai mdt di6m chinh lA
t$m duong trdn Ole cfra n*gi{c.
(ludi ciirrg iOi xin gcri' ra m6t hrrdng t6r:g qtia,
hon nira kh{i nidm dudng trdn Ole vi ilurYn
th:ing Ole eho i1a gi{c rr(,i ti6p, cac b4n h;i1--
klrai plra thdrrr.
Gi:i si'r lho da gi{c ndi ti6p AqJ243...Ao, ta
diuh nghia rlrdng lron {.lle tftri' k cira dt giac lir
rlrrong tron tsnt li diiirr E{k) th6a rnen :
-
+
-
oE'(6) -- ({)-{t + uAz ..,.- ,,s,,tf 1;
vA b{n kirrh bing ltik i trong d6 O Ii tinr
ftudng llt)n ngoai ti6p n-giric -,1t,42...-{n, eou 1l
ia brin liinh cria drdrrg trr)n ngoai ii6p nA1..
Nhu vty tiirn cira tludng tr.r)n (Jtc thu I li
trlrc tdrn, t0.m cira iludng trirn 01e tht ; l*
trong trtrm cira n-giAc, cdn it,rrtrng trdn Ole tlrLr
2 e hinli lA ilu<ilg trdn ()le ta tI5. goi iir llrr6,t'.
Cic brrr hny chrlng rninh cirrr tinh ctrdt:
7'inh chttt 9: ( {c tim e ,ir, ,.Ac rlrriirrg trdn
Ole thir l cia (n-lr*gi{r-: tao bi}i (n-1) dinh
nio il6 cri:r rnrit rr - giac rr0i ti6p sE cring nlm
trtln cluting trdn O'le tltri /i ciia n--giac nilv.
7'inh chlt 10: CAc tI,ran thing n6i tinr cria
drrring ildn ()le tht'r i c&a &-gi{c tao b&i I rlinh
nAo il6 cta ntrit a-gi6c ndi ti6p ldri tinr cira
dudng trdn Ule thri ,1 cria (rr*k)-giAc.tao b&i
(n-lr) dinh ct)n lai s€ di qua tim -E('*i) cira
ttutlag trdn LIte tht (i+j) *ir', y1-giic ndi ti6p
116 vA lii E(i+i) "n'* theo ti s6 j: I ka tri tinr
itrr0ng tloir (IIe ctia k-giirc.
T'inh ch(tt /1 : C{c tludng vu0rrg goc ha t& tim
iluting trdn CIle thri f cia cac (n-2r--giuc tao
b6i (n*2) ilinh lrao d6 cril mgt n*gi6c ndi trEp
xudttg <1oau thi.ng n6i 2 ilinh con lr.i sd c6t
nhau tiri tdm Ji (t) cia clrdng- tron Ole thu i ciia
n * gi{c d6.
- CAc ban thi_v ngav lA tJt c;'r circ tam dnulg
trdn Ole nim tr6n nrOt dudng thing vir drir':rrg
th&ng nny gei li drrting thirng Ole.
Cudi
hign bb
cdrng chirc cic
ich. hoo n&a.
l:an cd thOm nhilu phnt
r"E rH6i,iri NHdT
(Thomkhdo cudn <Qug nap trong hinh hoct
cfia I. .!{, {cr.-Gl6ar padtt b&.n n&m t963 tgi l.lut-
l{cq-ua\ '
L
'Bli t/lf 9. HfrA tixt tdt c&. adc s6 ehinh
phuong cd sdu chti sd md tdn eing eia nd ld ba
cir? sd khtic khdng gtdng nhau.
Lbi gidt (c'le Ygug6n liug lftrii Srn, 808 NgrryEn
LluF, llud|.
Trrrdc h6t ta cd v:ii nhin xdt:
t) Sd chinh phuong c6 ttn'eung li i,$ s0 cd
chri sd hrlng chuc la s6 chAn. ThAt v$y. sd c:hinh
irlruulg 5v si' cri dang:
(10a -l- t )2 : t00a2,-! 20a -t I ,
(lkz ;[3)z =100a2 :i:6Oa +{} ; 6- 0, i,2,...
'1'* hai d&ng thuc nA-r suv'ra dpcrn
2) D6 'drtng lhf,y ri.ng s6 chinh phuong c6 t{n
eing lir o sE c6 clrir sei hAng chuc li 2.
3) 56 chinh phunng c6 tAn er)ng li 6, sE cd
ehir sd hing chuc tn sd 16. ThAt v{y, s6 chinh
phunrrg dv e6 dang:
(ttla:L 4)2 : l00al -}' 30a * td: a:0, 1,2....
'l'u dSr tld dAng syy ra rlpcnl.
'l'r& Iai bli toln : Gi6 sfr sd ph&i rim, vj6t
du6i dangrlhAp phfin la ,{z =;[};r o,. n,},..r
t ) tr, t. 2.....9 I vir rn r * {t.
\ri Az li s6 chinh phuor;g vi do r =7- 0 nen
r ehi cir th0 nhan cic gi5. .tri, I,4. b.6,0. Lai do
3 .rrhAn -rdt tr6n rrdrr {:l chi c6 r - {. I)o vflr'
sr1 irhii lilir cc drrro.
.42 = rnnp444,
Suy ra "4. = lOa - 2 hodr: .4 * 1{.}n : Z.
Xdt hai tnrong hqp:
l) .,1 =-10n J* I. I{hi ci6: ,{2 = 100a2 -1-{0a + 4,
Suv ra .lLla *,tr ==- {,1 (p1od I00) .+ {0 (a * t):--
(l (rrrod I00) "+ 2 to - 1):= g (rnr.rd5) =* c :5b -+-1.
VAy : d : 10a * 2 = 50b -l 12. Suy ra ,{P:250062+
12000 + 144 j= 4i14 {mod 1€00) + r00b (So +Z):
300 (mod 10{.}0) + b r5b 12) ,= B (mod 10).
l)o clti D phrii ln sn 16. Da t b = 21,. + I, klri tlo
(1.['+l) (10k+7):.3 (rnod l0)+4/rf Z:3
(mod I0) ".) 4k.= 6 (mod l0) -+ t : 4 (mod 10)
ho{c 1t:- I (nrod l0). Ci hai lruring hop d6u
din d6n b:2k* t =9 (mod 10), tri'c b cri dang
D: l0c'- I.
Suy ra d, = 60b * 12 : 500c * 38.
Ta c6 t05(a2 = (500c - 38)2 ( to6 + t00VIo
. {soOc - 3s < 103 "+ c = I ho+c c::,t..
{
'\:di c =,1, 'tim ra A = 462! = 213444.
Vdi c rr2, tim ra ,{ = 962r = 925444.
2) d =. 10a * 2, Bing ,c{c phep bi6n t18i tuong
lu nhu tr€n, te s€ cd
A-500cr38.
'la cd 105 (.42 = (S00c * 3s)z < 106
.+ loo/;l { iooc + 3s < to3 ,+ c ,,- 1.
Khi do dr - JitB' = 28g.lq l.
Vfly cd 3 s6 th6a rlEn dE ra Ii: 2131.1{. 2891+{,
vn 92544+.
NhQn cdl: [Jan, 8fin91 ltotng An \l0T Ngt, Si
:Li€n, B{c Giang. Hi Bfre} clng c6 ldi giii dhng,
tuy tihi8n trinh biy cdr rusm rir.
N. ti . lt.
Bni 2/If 9.'fim bdn sd Is nhiin cd tinh r,hlt
saa; Binh..phtrongl cfia mdi sd lrong chitng egng
uoi ba sd .<:dn lqi lri sd chiuh phnong.
Liri gidi (cria N.qugdn Dtug 7'hal .\on).
Gii sfr b6n sd lrr nhi€n a, b, c, d cirr tinr th6a
nri n
u)> bb t)> d ({)
,{ = u2 * b*c +d, B:o* bz + c -l d,
L -r.r* b.yc2 +d, D=:6* }i- c*d2
li c{c s6 chlnh phrrtrrrg,
.Tr! cliBu kign (*) ta c6
,2 ( .r ( u2 +'ro ( (a t z)2.
v4y r{ :ot + t * c * d:(a * "t)z
ha.v d-(0*e+d-1112. (lt
Tir (1) vi {'t) suy ra a (30/:, va
tr { g < ia,/2+ bz + ztt : bz + ?blz<.rD + 2'l2.
\,*y I : a *b2 * c*ri ={} + l)2.
iray b: (u * c * d -- 1),2 (:i
lru (l) clrtr (2) r'E voi vd ta tlLrqu
o=0vAa=c*d--l (3)
.Vdn dB dlt ra ld tinr c.,.{c sd tqr nhidrr a, e, d
l,t)?c >d) dA cac s6 o:2o *cz* d. D:2u*
. +l rtl ia "ninn plruong.
X6t hai lrudng hop:
l)d=1. Ttt (r) ta c6 a=c vi do dl6 C=
(a * t)2 lA chinh phtrong.
-Db In sd chinh plruong l-hi D==lto-{- t - (3kt
l)2. Khi ilo o:b:c:If'2:LZ.
2).i > 1. Tri (3r ta c6 c ( a (2c. r'rr (c * 1)?
*zc +-cz + l( () < 4c + c2 +c ( rc + t)2. yA]'
c = {c1 2)2,trlc ln2a } ca I d =(c 4 2)2 lrrry
2a*d=4e*4' ("1 )
-it
i
Gifii he(iB), (4) ta ctugc
' e:itdl"L - i,
Iu day la thfly ;=;:'; '
o = 5m - { } 1=t- "- 3> d =.-!m. Suv 'a mb 1.
D : 2o * c -]- d?-=4mz + +Jnt -- 11>> (2m ' 2)! tdo
m > rt, vL. 4mz * l3m - tt ( (2m + +.r2.
Ve.v chi c6 tlrF:
a) I]= 4*2 + IBm-- tt:(z'm +2)?(+m =8.
4 = 6=5m- 4 : II, c=Sm * S =6, rJ = 2m:6.
h) D - (2nr -t- 3)2 +* m = 20.
a = D = 5m - 4:96, c = l}n - B = 17. rt - Zm
=,10.
'l'6m lni cAc sfi tu nhi6n tl.nr rlucrc lA :
rl=l)=c:3Jt1 +2k (t. I{ s6 tu,,tien), rl:1;
11 = g: i = 3*1 - 2A (& I* sd trl nhi6n), rl = I I
a:b= II. c:d=6:
(1 =b=96, c=51. ri=40.
BAi 3iIl9. fry hiQu ao tds6
nhdt. ildg tinh tbng
3
nguq€n san li
Vidc linh toarr sd tr& n6n nhe nhang vi ilon giin _
hon rrhi6u. n6u dE { rdng.
[(r + rtz)Bl - |* * r/:)3J :
'- { 3lr2 n6u {.- i (mod 4) I i=t .2,8
* I r*' * r n6u r.: o (mod 4)
Khi d6 ta c6 ngn-t
Ir
s - X. (g*?rr.) + t/{ -F llti -+- rlt,J + 2g/tl
r
-i
a-r
1t
=$ E k+tl4+ t/s+l/l?+2&/13
k:1
= :ao I9,1
312
N.K.t.
Bet 4lll-9. .Chnng mtnh rdng u6i moi ctich
chialQp hop 1,2. ...9 (gbm g sd tV nhittr ddtt tient
thgnh ra hai tQp hgp cirt lu6n lu6n c6 mgt tSp
hyp coh chnq ba sd llp thanh mgt cdp s6 cgng.
l.di gilti (cria .Vgd Duu Ninil. ,
- Gii sr"r ugrrqc lqi : t6n tai m6t cAch ehia tip
di cho thanh hai t{p con L vA B sao cho trong
{ cfrng nhu trong B kh6ng c6 3 o6 ndc, Iep
thtrrh mQt c{p s6 c6ng.
xct 3 c{p s6 .c6ng (1, s,trt; {3, +.5) va (3,5, 7),
1'a th{-v cdp sd i3,b) klrd,ng thE nAm tror)g I
tap hqp, vi n6u ngugc lai thi cde s6 1,4,7 iap
thAnh rnOt c(p s6 c6ng vi crlng thuOc m6t, t$p
hop, m0l lhu6n vdi gii,thi6t.
Trrrrng tu, .ret edc cdp s6 cgng (S. S, Z) ; (5,6,7)
vA (5, 7.9), Q,4.6) I (4, S, 6) vA (4, 6,8) ta c6 :
"c{p (q,7) cting nhu crp (+,ot ttrong th} nim
trorrg mdt tAp hop.
.Do ctrp (3,5) ctng rrhu ctrp (5, ?) khOng th&
ndrn trong mOt ttp hop ndn B,T crtng thu6e nr6t
tAp hgp vA 5 thudc tAp hop kia.
Kh6ng miit t6rrg qu6t, gii s,i I,7 G i{ va b G R.
Xet hai trudng hcp:
1)4t,.{. Vi 3, 4 G C fi6n 2 G A. Vi 4.7 C/
n6n 1 C B. \ri 2,8GB n0n S €,4. Vi Z,S t ri
ndr. I G B. Nhung khi d6 1,5,9 (:E vd IripthAnh
mgt cf,p s6 cdng mAu thudrr v0i gii rhi6t.
2) 4 c a, khi .td 6 c d. vi J;6 G rl r,ofi g L"
E, Vi 5,0 A,ir€n t c r4. \ri l,g q A rr€n t
B; vA vi 2,5 G B l$n I G .4. Nhrrdg khi d6
ti,7, 8 C A vA IAp th{nh rnQt c{p s6 eQng, mAu
thudn vcyi giA thtdt.
C[c mtu thu6u tr*n chAng t6 kdi lu{n crla
bai toin Ii t*ing:
N.K.M.
t,*
S -- tlst * tlaz + ... + t/arsar.
_ Lb!-!idi_ (cria Ngd Dug Ninh - b? Trtn phri,
Qu-v Nhorr).
Do Vn li nrgt sd tr nhiCrr ho6c mOt s6 r,6 tj.
n6n crieh x{e dinh oo c{ri bAi ra ll dur nhdt.l
Ta nhAn x6t moin6 tu nhi&n n th6erndn,**
tl.2),3.> n,)(r. - t!z)3 ae"u "J
';,;;.'ilu o,u,
so.(k f t!z)" ve & _ rlzt" t.6 ri-r. ci ta (t + i/zrg
- (L' - ll2\' sd ttr nhi€n.
. r'a lai..cd (tS + l/il3 ) rlisi ) ,.. ) lgr4 >
( 13 - tr!)ts:12 + l/:)3) t95:t, rrJn
"_ ,$rr t t_. * ,+ lrt + r,zts] _ ltt .* trys
tlul' L
i:I k:l A.
+ (rsrr * 195{-t- 1)it3
- 3 + l2l2 | 2713 + 4gt4 .-r ji,i + t0st-6 + 14717
+ lgriS + :43i9 + 300110 + 363irl + 43$/t2
+ 28/13 r
IL
=J --E ,nr * {ir{) (1+ 1 ! + tl3) + 28/tB
arl: I
- 236-l!L
312
--.S.ed" rcdt,:-Cteh giii tr€n cria ban Ng6 Duy
Ninh g{p phli mOr sg rinh to{n }h{ c6ui tfntr,