7
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#
*
B
YrEN IiH0A. EOC
yrEr NAlr
EOI TOAN EQC
YIET NAE
n:7-
tong Dten top l
Phd T'bns biAn tQp:
Nguy6n C6nh Toin
NgO D4t Tir
Trg s&: 70 'Trln Hung. Dao
Df,.v n6i : .52825
- Hn \ioi
ffiffiffi
TBING
x6r rsAun TicH zo TrIM EoaT oOnc
H0r E0Nc NHA NrI0
TaXC THrr0NG
BLo roAN Eqc vA'rubr
H UAN CHUONC tAO
HANG NHi
c
TRI
OQttc
r
t
s
THU C0A CHo T|CH
HOr obNc B0 TRuoNG.
PHAM VAN OONC
Gr?i bgn ilgc
nhen dip Bao To6n Hg.
vA Tu'bi T16 trdn 20 tubi
\-
)
Hd Nfi, ncdy 31 thdnrl I ndm 1984
C{e ban thAn mOn,
'I'rong 2U nIm rJua, b5o ( Torin hoc r u
Tubi tri, ) ill cr-r nhfrng dong 96p quan trong
<ldi vivi phong JrAo hoc toin cfia tubi tr6.
Cii dgp trong s6ng vS tf.c dung kj' diQu cfia
to{n hgc khii!n ngudi xua coi to{n hoc nhu
c Bi Hoing crja c6c khoa hoc >. Ngdy na-v,
truoc nhii'ng tng dgng ngd.v cdng to l6n
va sAu rong cia toAn hoe. ngrroi ta coi
toAn hoc li ( Nguoi girip vigc 116c luc >.
Hoc toirn vi lAm to/rn'hhOng chi ld rli tim
c6.i hav, c6.i rlep cfia to6n hoc md. phli rlcm
to6n hgc r?ng dqng c6 hiOu qui vio moi
iinh v1rc ph{t tliEn kinh td vi xE h6i. Thi.nh
tich cfia.ddi tu-vbn hoc sinh nuoc la trong
r:Ae ki' thi toAn qu6c td chung t6 tu6i tri;
\,-iQt Nam. nguoi Vi6t ft-arr ta c6 nhfrng kh&
neng r6t Aeng cr"ri trqrrrg r i phrit huv vE
to6n hgc. 'I'u duy va ki6n thric toArr hgc
la cOng cu ngdy cAng ciin rhi6t tl6i voi
nhrtng ngu'di ldnr khoa hoc k! thuAt, nhirng
ngudi lim kirh t€ vd n6i chung ddi r,6i
nroi ngrrdi trong linh r-uc hoaI tldng c6r
urinh db tinlr'toArr dring, lua chon nliing
phuong 6n t6t trong cAc qu1'6t djnl "a
trong r.i$c'ldri-r.
Chric t.[c barr ire ;rhan cl{u db tien bo
khdng ng*ng trong viQc hoc to6n, lirrn to6n
i'a ung dqng toin, Ch0c b[o < To6n .hoc
vi tubi,tr6> c6 th€m. nhi0u nguoi doc r:i
ngi-v- cing c6 tic dung thrdt thr5c.
FhSm Yau Dang
*r.-...:--l--
I
,j
't
tAc. b6o T'oAn hoc vi Tubi iri rlir nhu i:hy nrlng
vunn l6n giira mira ddng bio. r-urrt qua rnoi ]ilio
Iiht'rn gian khb gA-r. nOn b6i nhr-rirq brriic leo thang
chi6n lranir ngA_r. cing irc iir.t cira qutn thir,
Jriruc vu ban doc drroc llirrriltg ru.r-0n vi li6n
Iuc. Trong lric rn53' ba1' M1- nriil ilinr a'ieo r[c
dau thtru^ng vii it6 nAt tr0n circ pher phrrirng lang
mac, bAo Toi.n lioc..r.:\ Tu.Oi tri rr=in vrro.t qua
bom iian d6n r,6i ban iloc l;0n miinq hiinr tlii
dn hav du6i rn[i trrrdng so tirn, rrrili theo circ
hdrn thrt tha-v cl6i cria cAr ciridn sl pl:r)nc lihOng
r.lijn b€n rnilrlr ph6o, canir trlnr ra clu, qrirritlr diu
[6n lirri hal iip fr trong Lra 16 ciir r-i, unh h0
aliri Cu I{I dang < x6 cloc 'I'rrtting sr-;u rl i cL'ru
rt iruc u,
Cudc lilriug ehi0n rircing II i gi:\nL tl,rng nirirt
<iril nucrc kdt thirc thdng .lcri nirrn lgT5 dir nr6
drrring cho b5o TH vi 1'T bav yao cAc tinh thinh
nti6n Nam,. MOt phong trdo cloc b6o r-:i giii bni
tr0n b6o TH vri TT Iai di-t' 10n trong circ nhi
tnrong cria c{c tinh, thAnh nr<ii r.luoc giiLi 1;hdng.
Sd luong phiit irinh biro t[ng lin r0 rit vi ngrrcri
ta thdy xurlt hion trdn b{o TH vd TT nhirng ldi
giii hal cfra circ thanlr thieu ni0n. nhirng bii
r i6t cira cAc tiriv, c6 giiro cria i:irc irrrong
thridc cric tintr phia \anr, llot so giii thrrirng
iiong cic cu6c tiri giii to6n cfra biro ili clugc
ting cho circ hoc sinh r'0u to[n r-:i gi6i toin A
c5c tinh, thAnh phia Nam. Tur- nhi0n, trong vong
nriroi nim tr6 iai d$r. nhirng kh6 khhn mdi d6n
r'<ii b6o Tll vir T1'khong phii ln it. Song, nhti
c6 su cirng tlic nhiQt linh \,,i to l/rn c&a d6ng
dao ban cdng t{c, ban bi6n lip r-r\ cihc biQt li
c6 sB irng'hd tich cuc cfra Vi0n Khoa hqc Yiit
nanr, b6o I'II va TT de dap bing :;tine gir, c1t
irirric rg ban clgc ngiil- m6t t6t lrcrn.
Nliin iai qu5 tlinh 20 niim lthur- vu bqn ilgc
nerrrvi ta c6 thb th{y nhii'r:g tltin}r lich n6i birt
cua Bao TH vi 'I1 :
.-*l - Nh&m drlng muc tlich vLi cldi tuong cia
nrinh, hAo Tll v:i 1'T di,. thuc sr.r gfl1, cluoc rn6t
Iilidng khi sdi nbi r-a irhong tiio say m0 hoc
toAn vi giii toen iOng r.ii trong c6c thanh thi6ri
niOn hrrn hoc, rliic biQt t:i tro::g hoc sinh cac
trrrdng ph6 ttr0ng trung hoc.
2 - Dring g6p ciirng kb i-Ao \.!0c nnug cao chdt
luong hoc tip vi trinh tld toin hoc ciia hgc sinh
c{c trutrng phb th6ng tr.Lrng hoc. Dr.rng gop qrran
tlo4g vdo vi0c nflng cao ch:it luo-ng bii giing,
tlinh d0 r,r\ ldirg 1'8Lr nghir cfia c6c tlriv giAc,, t:o
gi:io lr'ong cfic trrrcing phb ihOng.
'.?
HII ftIIT.OI NAM HAS YSAffi E{OC YA Tff&f, rRE
f)O sAng Iiidn eirr Ban !in rlong thirrir lrilr i{6i
B ToAn IIoc Viit Nam viL su frng ird tich ctc
, cria S;- ban I(hoa hc.rc Nh:'L n,,-c i,,gil u"o-
li 'Ii.r ban Khoa l,oc vd K! ihuit Nhi rru'Lr:) []ho
1'oip }roc vi TLibi tr6 ra rnit brn doc vio liriing
[4udi nhrrr 1964 i'iri rririe tiich ( gd17 l;hdng ktti
sdi n6i, ltito hftnq hac loan I ron q i lrunh tht(a nl€n,
rtQc hi(l ld .hoc sl.nh cric trtitng phtt lhing cdp i}
(ngdg nttg Li cnc trtrd'ng pt{6 th|ng trung h1te) l
Nh&nr rnuc dich di db I'a, ndi dung cfra biro
lip trung vio nhirng cl6 mr.rc: ^\6i ciru-r.0n viri
bg.n l.rt3 -1'0u toirr; 'I"im .hibu sirti toin iroc ph6
th6ng; DE ra lii na-r' va giiii b)i k\' trrr,rc (11i-r-
la 1lflq ,'lr',r .viu ri Ilrrrrirrg xu.vi n cii:r rriui ,-;d
LrA'ot: Bir0,' rlSrr tiiii hitri lorin lro,. lrirrr dai:
f'oiln hoc vi rldi sdng; Liclr srl-. ijtr .", cAu
chu_vQn vi gili lhoai toin hoc ; Ciii rt6 p thic
m5c; Giii tr'1 to{ri hoc. 'llong riui'trinh pirit
tribn cr"ra bAo tiir hi:rh ttrrinh th0m mr)t .srj rlI rnr-ic;
I3an c6 bi0t; Hoc sinh har. ban cloc tim tr)i ; -i-i-rro
A6'i vOl ban doc hay ban doc rlro r1bi.
. Vrli rnlrc dich vi ndi durig nhr r-a.r-, ngu-v [r):
lihi nrdi ra cloi bno TI{ r.ir l'.I dl (luoc cic ban
tre khnp nrri lrosn ngiri,rrlr nlriit li(,1 S',i Irrr,, tl rirr
ti6n dE phfri in lrai 1iln d'6 tiing s6 lLr.crng l0n
g{p dOi rnA vin kh0ng dirp riirg clir riiru i:ilrr ciin
ban iloc. '['ri lihli11 noi trOn rnii]n Birr bai.l cioc
tdii tdp giri thu v0 toa soan hoan ng-i)Onh, l)r\v
t6"nilnr:hin hoan sau lilii dc-rc sci blo dlr tiOir
vii'n6i I6n nlrftn thirc ciin mii:h r-I j' ngiria, r:ii
trt)'vir tic dr.rlg cira hlio. Sd luong vi n6i drrng
nht'rlg birc thu r:&a ban rlrrr: liir0ng nliirng noi
lOn ldng thi6t tha 1,0u ihitrh torin I:oc ciia rl6ng
rJrto c6c bnn tr'6 tlong circ nhir trrring. xi nghlip,
coi r{uan, c6ng tlirgng, ndirlt lru,dng va trong
tluAn cl6i mtL cr)n cho th.ir- sir ilAp iLng rlirng Iirc
cia tti l;:io ddi vri'i uhu ciiu lioc tiiir r,i tirn hi6u
Lo'6n hoc ciia nhirng thrnh lIiii;rr rrii,n hlrn irr.ic
cfra ehirn,'ta. Do cirng la nhfrng loi dfrrg r.iOn,
t'6 r I nlrii't lhjrnlr crjrr harr'rloc riiri r.,ii Irrit .or,,,
bqn bi6n tip, c6c ban cdng tirc r-i. nhirng ai di.
rl6ng g6p cdng siic viio su ra ildi cira tc b6o.
PhIn tirn trong i0 nlnr diiu hoat ilerng cira .
b*o in ihoi gian quAn r1frn nribn Bic nLr6c ta
ptiiLi ilrrong cliri vrii cudc chidn h-rrnh phi hoai
vd cung ire ligi cira tiO qrrdc 1i:-. Tu-v r-iy, nhir'
cd BrI quan tAm clilc bi+t ciia Ur- ltan Klror. lroc
Silri uuue, nho co srr gi(r.Jr tlo r:)ri-rl tinlr r. lia r.iit.
cd; quan xudL birn. in lcii, phirt lrinli ttuo ciri,.
biru r1iQn, cc)ng r'6i nh irng cd gd ng b6n bj. tie h
crrc cr'ta han bi0n trip. Ir-rlr slln vi c.6c barr,:r)ng'
r)
wP:'6r ! --'L
3 * C6 tAc rlr.rng s{tt xa vi. tft2irg ydu ddi vtri
viec alito tao, bbi du0ng vA phit hiQn nhtng tii
ntrng trir 1:\nr nt)ng cdt cho vi0c nhanh c1.r6ng
tlho tao vt xirv dung {0i ngil cAn bd khoa hgc
ki thurit n6i chung vl to5.n hqc n6i riOng.'
Dinh- gi6 thi'nh lich cria bf,o TH v:\ TT trong
20 ndm qua, tl6i dbng NhI nudc Nusc Cdng hoa
Xe bdi Chti nghia Vi0t Nam d[ t{ng thrr&ng b6o
TH vn TT I!uCrrt t:huong lao dQng hgrtg nli'
E)A-v ln phin thu&ng crto qul' vri vinh du i6n
lao mir Eing vi Nhi'L rruric tli\nh cho t:.1 p thb iOn
nhfrng ngudi'tii. g6p ct)ng 96p sirc rA1' dtlng vit
lirm nAn th)nir tieh cho bAo i'll vi T'f .
Ngny 10 ttring 10 nx.rn 1967 Chri tich I16i dbng
BO tru6ng Pharn Vfu'r Dbng cln c6 l.hu gii'i ban
clgc cira bio'I.IJ ld T-l nrii l€n vai trd cfia b6o d6i
vdi tubi tr6 r,i rldng vidn thanh thi0u qi0n ( n€a
cao chit nllhiu anh ltting rhch mqng ttong uiec
tidn qud.n uao l;hort hltc, k! lhu!'l t. Nlrrin cl!p blo
Ttt vi TT t|dn 2tt tubi ( hfi tlch l?i vi6t thu cho
ban ilqc eria biro r{c nhqrn r 7'rott11 2a ndm qua,
bd"o ?r Torin hoc utl .7-ttdt tri.tt dit cd nhftng ddttg
gdp quan t rgng tI6i L,ot phonq lrdrt hr.tc toan ct)a
tubi tfi ) vd An cin chi biro cho chtng ta c llpc
.l
todn od, Mm tod.n khdng chi td etl tint cdi trcg,cdl
rlgp cila lod.n hgc md phdi ttem todn htlc ang
tlung cd hiQu quit udo mp! lTnh or;c phdt trltn A:t'tih
'E uri .rd hli tt.
Db xr1ng ddng. vOi phhn thu&ng cao qu! crin
Nhir nuoc, Ab lintr hQi s0.t sic vir thrlc hiQn sinh
l6ng nhfrng loi chi bio cira Chri tich tlQi rldng
B0 tru6ng, db chdp t{nh uoc mo cho c[c b4rt
tr6 r0u toin, girip hg nAng cao nhiQt tinh vi
ph{t hu-r. 6c sing t;1o, n}ranir chtlng chi6irr linli
nhftng dlnh cao cfra lri tr0 hay gdnh lric nhiiltg
trgng trSch trong c6ng cuQc dtrng xi1- r'i\ bio vQ
clf,t nr-r5'c, biro TH r-A'IT'nrong cluoc sq cdng t{c
chX.t chd hcrn nii'a cira cAc ban 'cOng tic, t-nong
cluoc'sg gi(rp dO vd irng tr6 nhiEu hon nfra cria c6t:
co quan criq Ding vi. NtrA nudrc, mong drqc str
hrrirng ung, a6p j', phe binh cfia ddng diio ban tlgc.
NlrAn ngir-r' ki' nif6 v6 vang nAv b5o 'II{ r'A'l'T
.r.in bhy ti1 li,ng l.liirt on rldi v6i cAc co quan Ding
vh Nhi nttdc tli hudng d5n vi ginp dd' bdo
hoat rIdng, xin chin tltinh ciim on cic c6 rihin, co
quan, ctodn thb aln cOng tic vtri biro vn dA gop
phiin tirrn rr6n tlr:1nh tictr cia ]:r1o.
Biio TOI,X HOC VA T{.TbI TEG
I[6i xhuyArc t]bi c*c bqn tr& ydu to{rn,
\i-'
d
sO vA LfrNt-t
f ,+C sd thdng ttrut;ng (s6 thrrc) cdn c6 gi rlirrg
v su-v nghi nira nhi ? Nl.rrng chi viQc clbr c6.ch
nhin thi lai lim dibu mrii xuf,t hiE'n, it ra ciing
li moi r16i voi nhitlg ngudi ehi c6 ki6n thr?c
tohn hoc phb th0ng. Ta hlv ldy mQt ph6p nh{n
hai s6 thuc, ch&ng han ll. 4. Ta c6 thB coi (il)
Iil mdt lQnh vd. (4 ) lA vit chiu lQnh: llnh <<3>
tirc dung l€n.'vit < 4 )) li\rn cho vlit niy tr& thdnh
v{.t ( 12 ))..( Lenh ) r.ir <r Vrit > <.r dni dDu ln sri
thttc.
Ndii ta thuc liiQn liQn tidp lQnh { 3 D rbi l€nlr
( 5 )) l;r1 viit (4 )), cu thb ]i.: 5. (ll. '{), thi ta ilugc
vit ( 60 ). Ta cirng cltrgc ktit quL di.r, ndu ta thi
tliqtc hiQn c6 mOt lQnh du-v nhf,t lA' < 15> l6n vAt
<4>. Nhu-th6 ngnoi ta n6i ring lQnh ( 15) li
tich ctia hai lQnlr (( 3 ) y-e ( 5 ' x6t theo thu tt5
d6 (& dAy ti-lu tg kh6ng qrrhn trong uhung shu
ciil', khi m& r'dng khiri -nif m (lQnh ), chring ta
NGUYEN CANH TOiN
sd th{y ri.ng thri'tr.r nriy ld, quan trc.ing)'MQt c{ch
t6ng quAt, ta n6i ring tich ci'ra hai ( lenh ) (ldJ*
theo m6t thu ttI nno d6) Id mQt ( ICnh ) duy- nhf,t
c(r tdc dgng 10n ucn m5i <vit> gi6ng 4hu tdc
drlng cria v-iQc thrlc lriQn liOn tidp hai <lQnh>
ril cho, (theo tht' tu d[ cho) 16n ctrrg < v4t I c16.
R6 ring cich rrliin mdi n[y c]ro phdp ta m&
rdng cAc <lQnh> vr\ c6c'< viit > ra ngoiri pltam
r.-i c6c sd thge. N6i r-:hinh xilc hon thi ir d$y c6
(lOnhr /,, <r'qt chiu l{nh> V vi (,Kdt gua.satl
khi l$nh di thi hinh n Q, t$t cA dbu li. s6 thqc.
Sau ddy, lihi n& rOng, c6 ci.i lihdn,g cdn'li. sd
thrrc n[la, Ta hiy di trl'n"g brr6c:
Buoc rhtr ahdt : tr/ vn Q ln nh&ng ci.p s6 thgc
sip thir tu cdn L vitn IA sd thr.rc. Ta qu;r uoc
.ing, <lQnir)) c tic dQng 10n (r,itr (o, D) vdi
o, i, , tA nhirng sd thuc thi cho ta ( v4tr (ca, clr)
li cIp sd tbr.tc ca vl cb.
BueG tht h'ai: 1.. vA I' dEu IA clp sd thr.rc
s&p thrt tg, cdn Q ln sd thqc'
ta guy u6c r$ng ( Itnh I (p,g) tac rtQng l€n
dvft.r (*,U) ttri eho kdt qui ld s6 thuc pa*qb.
Btr*c th& ba: i, 7, Q dEu ltr cIP sd thtc.
Ta qu1- udc'rBng <tlQnh) (f,g) ta" dQng len
qv4t ' (a,6) tfri cho kdt qui ln ci.rp sd tlrnc'
(.po, qb).
'Budc thtt tu: L ld hai efp sd thuc sXp thdnh
rr$t blng gbm mQt c{p & tfCn 1A mQt.c{p fi
&usi, Y la'*Ot e[p sd thuc vd Q ctng lit mSt
cdp aci thge.
s vqt ) (o.b) tfri ciro ket qul IA cr'rp sd thne (pa'r*
qb, ra * sbl.
-.=i\{r
q€u trong (4) < ISnh >
k€t quA ei v€ sau cfia
rr vS sau efra (3).
Dem P
mQt su kiQn nrdi la khAr : m6t Ph[n
( vl khdc 0 nri. mqi liry thria cria n6
n6. M6t ph[u tr] nhu v{"y gqi }n mQt
tilg d&ng ".
hsi cta -1. Ta thu x6t lgnh i;
Dern i tdc d$ng vAo mdt clp (4. t) nno $.
Ta ctng cbu 1, ring,
(f :) ., ,'u"* (61),,,,
(4) sE gi6ng nhu kdt qu&
T6nr lgi, m6 rQng nhu ir (1) le dA bao trrlinr
edc m6 .9ng & (t) ,i,i (3) \,a'1A viQc thuc hiqn
hai l&n m& rQng & (2). Cho nOn ta tip trung
nghiOn criru rn& rrlng (4) r,i ta th{v ngay nhitsu
diEu nroi, c6 v6 k1' lg. n*a, xu{t hi{n:
/01 \
loy linh: Te ih& -tdt lQnh o= {.0 O/'Dent z
tdc ii<)ng vio rn6t cip (o,li) ndo d6. theo ('l), ta
cd clp (b,0) I ti6p t.heo i[<'.r ta ]q.i cho lz tec dQng
l6n nira vAo c{p_(b,0); cu6i eing ta duoc citp
(0.0). \hu r,f,y az l{e iking viro c{p (a, b) gi6llg
y nhu lQnh <t 0 D. Vi]' ta c6 thB vi6t:
c-2 = A (nhung e f 0. r-i t{c rtong cia l€nh
a klric t{c"ilgng cria lQnh 0)'
Tt d6 su1 ra t t3=a2,a=0,s=0 t
rrii din di n su.r' ra :
a"=0 v6'i moi sd tu irhi0n n.
Ita g{p tndt su ki$n mdi'lgr: mdt phin tir
a7.0 ntrung mqi ltiy tht'ra cia n6 dEu bH.ng
hhdng. MOt phhn tit nhu v+l gqi li mt)t phnn
t& a fu?g linft ;,.
ra quy udc rXng ( l$nh > i:1) tac rlqlg lon
c. (4,&)=(cr:, c'b)
(p,q).(c,D)= Pa*qb
(p,q). (o,o)= (pa, qb)
(|:) (e,b)= (pa*'r1b
< vAt l (a,e) d3 co ra*sl;'
ta l4i chir 1i- rlng, nCu tron.g
(::) .. nn"* (6,1) ".' ri€t qun
(+) *e gi6ng -v nlrrr li6t qui ir s.
(;:) . (rr'rr)=(cer, cD)'
D6n d*y, ch*c b1n d& h{obrc niutjn nr6'rdng
.tli xa'hnn ntn. Nhuug th6i, ta hEy tam dung 6
brrdc thir trr clb di sAu c{i di- Ta chir j rirng
(4) thuc chf,t la. 6p dgng, hai lin (2)' ten thtr
nhdt [p dqng <i6nh, (p'q) vio < viit)) (o.1,) da
c*J pn*qlr, ltn thu hai 6p dgng < LQnh > (r',s) v{o
tnc d+ng vio mdt clp (4, b) nao d6,. theo (4),
ta c(r cep (o, 0); ti6p tuc ttc dong..B vAo eip
(a,0) ta r'6u duoc cr.rp (c,0). T+)' p{ ctng tae
rlgng I6n c{p (o, b) gi6ng nhu p, tirc li 92=?.
Tu d6 lt' :Fz .P= P.i3: (3 .
rbi suy ra po=p vbi moi s6 tu nhi0n n'
Ta sE vi€t lgi bdu qu,v rnc
ti'&n nhrr $su :
V{.y hai ( l$nh r c
qBt, D{c bi9l hai (lenh } 1 v&.
6S& ngul'€n I vA. thai l?nh > 0
t*nh a tri*t ti&u r vi X.(a,b)=
ti l:dn budc n6i
(,1i, ( lenh ))
& vd sau eia
sttu crla (l ) :
Luy iting. Ta thit x€t lQnh lr': j;3)
Ta g[p
i* p khac
dEu bing
phhn t& n
' CEE b4c
/ 0 - 1\
(r o)
'u (;l) .. 'nn coi nhu Ia
,) ==(3 3) . ,-, b) *{0, o)
(; { ) <IEu ri r{nh
/00 \
'n too/ dau lr
(il1 r"'ui=,,,"
d6, theo (+), ta c6 c{p (-a,a); ti6p tgc t&c
d6ng i vao c{p (-0. o), ta duqc c{p (-4, ;b)'
Vdy tdc d6ng efia iz gi6ng -v hQt uhu tAc dQng
cria l$nh << - l r' \'i-r' iz _- - l.
/ u1\
Di niritn -i= { -i 0 I cirng c6 binh phuong
\ -:/
bsng -1. D6 hiBu lim sao. ii,C *e ngiy xua
ngudi ta kh6ng hiEu nbi vi goi i Li 16 ngu
ng6c d{r'l
0"{et,
tiin hge hai cria
cl6u e6 i.iiuh phuong
Ta lrlv xdt l6nh e=
ta cltrgc cap (0, 0). t"1y:
p+0, C+0. q ra p gr.ri
m$t ili6u eirng mdi lq.
chi bl6t *1 vd *l
1. Nhuug dAu di hdt.
R6 rAng d6 khOng
tlch qp = U m{c 4u
li ciLe uri'c c&a lthlng,
l. Ai
b[ng
(?;)
- Tlch hai lsnh adi ehung kh6ng giro boin'
B4n cu thir tinh rliA x!ru. Trin ki-a tp=0 ahung
pq +0.
Rd rAng ta burvc vAo mrjt thd git''i nrCvi, rn&
r6ng thd giiri mi & '116 cdc < lQnh > lA cdc sd
thuc. Ct&c ban hlo huc nru6u bude vio thd giOi
dti. Nhrrng di vlo de;,, phii r{t chn {hAn nhu
lihi burlc vflo mdt bei rnin. t)bHng hair vio dAy',
niu gir.p d&ng thitc ab: ac. Voi o *0, ch6 vqi
rrirc lugc dF c6 b:c, Vi aD=ac cho ta oD-sc=0
hay c(b;c) - 0 nhung c(o-c)rO c6 thb x$y ra
r'6i cr f 0 vl b * c+$ ttc D + c. Hoi.c ph&i rdt
thfln trc2ng voi tht tu c6c thria sd trong rnqt
tich. Dir vAy, ldng h6.o hrlrc ri6i r,6i c6i rndi chfie
sD thuc dhl' b+n sfim mOt nr6r der nrin dt di vtio
bli min do.
Cudi cirng xin n6i llng, t0i gr.ri " l$nii ' eho
n6 nOm na, cori trong to6nl hr.re ngrrdi ta ggi c6c
< l$rh ) Lrlng nrot ilanh tti v5.n i'6 hryn l.ir :
a to* n tr? >.
phii ln lQnh < giit nguy0n r. Nhung ban tu minh
c6 tlrb itrir d'0 th*._v ring c" la. (( lQnh ) < gii!
ngrr,r,6n }). VaJ' 1, -1, e vir -e dEu cri binh phuong
I-ring I
L'&c cta khoug. 'Ia hd1' 6p dgng l6nh:
/ t-z\
,= ( -i iJ "*" c6p (rr, tr). rheo (1), ia c6 c:ip
ta *2h, - 1o -l-4b). Tn lai 6p dung lgnh :
s = (l l) "*" clp (a - 2r5, -.ro +lb)' rheb (4),
ttm hibu sd,u thAm todn hqte phb thding
Hit xet bii io[n: Iim mpt rla th{tc odi hi
sd ngugdn co bdt: nh6 nhdt nhdn irrrgt s$ a
chq lrtdc ldm nghilm; Eir-r' li rndt bli toin
eb clibl c6 li6n quan clr[i cliE ddn r."iQc phit
tribn nrQt sti lihii nidm co s6 cria toAu hqc.
Ngirl' lay nguoi ta Lidt r.ir phii gi6,i quydt biri
toirn nay nhu thi nAo. Tir.r' "nhidn vdi lrinh dqr
phb thf.,ng cI6 vfrn ld rn6t bni to6n kh6.
Cd ,t!ts r;r,rnfr hoa rlibu nity qua trudng hop
z = 11 2 + V3 . Dtv li rudt iIB bii cria ki' thi
hoc sinh girii toun iap tf ioan qudc r rr 'qua.
Hhrl h6t thi sinh cliiu tim thflv:
P(.[):x6-6x1 *6x3 + nxz - 3ot + 1
ln mOt da tlruc r,0i h(! s6 nguyan "fr,1n fiT-+ Va
lirn nghiQrn uhung lihdng ai chring urinh cltrgc
116 li ala thirc c6 bic nh6 nhf,t th6a rnln tliEu
kifn tlEn.
ViQc tim r:r P{{) .hoirn tofn tlou gi&n vr}. dqa
vd.o nhin x6t sau: Tich cfra ni0t da thirc c6
nghignr li .r v6i rnirt tla thuc bflt lii vfin lir m6t
da ttr*c co nghiQnr k\ a. Da thirc ci' h*c nhir
Nco },IfiT TBUi\G
nhdt (r'ni he sd thr,rr:) nh+n /il+ V:f tao,
nghil:rn lA x -1it.- Vf nA lo1i c{n bg.c }ra ta
rrli*n a6 r'rri (x -lTlz + (x *t':=lVil+ ivf*
uir uhAn tlrror. tlu thtrc'
rr - r/ilt3-s : ^\3 -ty'e*x? + 6.I .* ztT *'t.
Db to4i cintqic hai ta lai uliin ti6p.r v0i
-\3 + G.y-: + y'z f rrz * l) r.n nhSn drrr.yc .r
. Pl,Y) = 1x3+sx*B)l *z(.:xe+e)?
Sau lihi tinr tlttgc .P{X) nhibu thi sinh nh6"n
ret li: r.i 2 vd 3 ngult\n t6 t.ung lhau nOn 6:
2.3 phii ln bic nh6 nh{t r'fra c6.c da ttruc veri
hQ sd nguydn nhAn iy + Vf-lar,, nghiCm. Do
d{t P(X) l[ da thuc chn tinr. K€t lrrin iibu nhtrug
rrlt giin r\r th$t. Tai sao khdng ti6p luc n:i6t e5e}l
bigJr thurlng b$.ng r.ifc gii_si c6 nr6t ila lh-rk:r
Q (x) ** *-q1{ * ..' * a5x} r'6i trqr sr5 rrgnyon
nh{n y'J- +Vr ttn, nghif nr ri)i chirng minh
f,o =' . '=oS = 0'l
Dibu ki4n duy nhft r.Ang bu6rr.: c4e sd nguy6sl
e0,.. . , rr5 vdi nhau lir g tVt*+ Vill* 0. V{n dE
)
ilr ol IHSc Yfi rcnltil cm nu0c
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