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Tru, r.i: 'ii;'1'1trr, Huag Oao - Hf HOi :
HAr THANG' MoT rY
'{hx,ki* rri€ roe$: SSANe CE{Ui\G
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Wffi ffi**r. FKE#ffNG TftiNH HAM
ft'r,t soan bi,o 'foAn huc tubi trE e6 nhii.n ilugc
- ,rrdt sti lt:ri gi{i bAi td&.n s6 1 tiong ki thi
, 'l +irrr & L{rc-xern'-bua :
Bii ioin i l lffy rtnr mgi /rAm sd f(o) crtc
dinlt rr*i prr{ e hitu tij, Itirja, nrfin cdc drtu &ign
lil) * ?, /(rg) = f(.r) Ttgt * f(r f g) + I
u&l tt,p! " y triin fij.
, .CAc trii gr-aigoi tlClir h& ra ciirrg dip t'rrrg dugc
1,0u cAu dr)i hiii, vl bil.-i to6n n6i ring: htrs tlri.
irr0t hAn,.,. 5o ddr chhng tril d&i df,ne c6 th!
,rlrra ngay;,:: hi.i:n *6 f(r)c*c*- l, rec dinh r6i
Irloi si lrirt, tt,
Nhrrng rl,1 khC.'ng ph&i it ib6m .? ctia bg1 lo*n !
Thuc,.clrrrl lrii ..of,n ilui h6i tirn t{f cI eic hAm
tfi tlroa nrftn uac diErr ki6n de nQu. i'a ifrtr.-il
l* /{"1 = r I I (:r irti{ ti): te ng}i9*';"".'nlrf-
cir-a b*i lo{x. flet ti6o trr,irs-{.d. r-
khdng e.ti Larr nAo c
nhd t rS1-" , ., .,
-. iliirrg chi Fhqrn
so4rr ird hdi !Si :
. PH.AN }T,C CHINH'
hAm 1r(r) xAc iliuh vlyi r hf,"u t-i, lidu c.i rni.
rpng tluqr cho ugi sd thrro a. ching- ? ' .
B&i toAn 1- tbu6c loai c phtrrrng trlcli hArr'r e,
{rlc-In &n l{ hAnr sd vd ph{i tim rdr cir e{,c h*i*
s6 ngiriQm bAi toflrr. ilh6p giii m6t phuang trinii
thOng thudng ndi chung dt lA viQo klrSrig rla*
gt&n, lO di nhi€n ph;p giii rngt phu,rr-ig trinh
him lai eAng phril: tap hoa" .
. 'Irong lrgt bni toAn vE phrrong tlinh IiE:m.
h{rn 16.phli tir\ buOc p}r&i th6a uran rngt rhrrl-
nhiiirr) hf tbrlc dgi sdi co bin. \'ir nui chung,
ndu khdng burlc thdm.,urgt v$i di0u kigr: phir,
thl c6 v0 s6 hdm sd 116 da:;g rSt kir6e l,hau,
nghigm bAi to&n. Ch$ng lran :-
Bil torfln 2. T!.m tdt ett ,:ric hdrri frr)" ;i.iic
nhh u(ti- rnpi sd il*tc x, pd lJirie arfip diEr l:fl;
f(.. rgi=/r.r)+/(9)
,T0Ag.
" dur s6 nghi6ro h;t i -l;.
li,'
i.j':
l
i
i
t "*"
[ --:... ,- : - *-*'*- "
-
|
\-?.*--k\\,**.*-**
-->'?-''
l' , i
i I trong do cIe rn0t lrlngo6:{tr1lryie6 ttiair. }rihung.
,'..,.,kh6rrgthak{tluln,r:l1'g.:"..
:. ' -,, , _./(r) = C.r vui nrc,i sd flipc..- l'', (.1)
t.trLi rrg hrn. rdi 4 vA C La hri lrlng ^"fi tir.vlJ,
' ) :c6 diith, h{rtl 'i'- ...''. -. ' - ';)r''
f(: g) =;(.r.)/tgt - f(r + tJ) * t(2)
.,/
i) t'rong (ll): cl,n rr e g -: {r lIt rl6r,c
,
q(0) =9r({)) ji:er(0)- ivrrJ},
t {_r g(0) - g.
2) T'rtng (3), ,'1,u ., = L t:i drr,;t
. g(r) = 9(.t') g(1) "+ g(r, * p(r) - gi y + !1,
\'0 r
tg(**11:.g({)I.}frr+11. t1)
il !'r
N{u g{ t)=r, thl g(.rr t ii=r} vii.i r.,l,,i .t.. tue'
i3. S(, t dbng nirit birrg r), ,rr r r v,',i qii rlii6t.
'flrnSh thir 9(1) g (1. 'lirplg (4), rko .r = *. lr
t hi dluoc
it=g(0):=g(- l* t):.gr i)Lgi .. trr :i,
niA g(l) * u, nen s(-li - .. i'
13) Trong (3).ehn g,= * 1 tlli .iltrirc
.\
gi* r):= 9{* t) g{a) +ig{j t) .1 gr,.) - ,(.s-- 1)
" : --. 1 ^- g(.r: '.''i).
Nhan cA hai v6 ;ri' virj *r.gil), rhi ttiir: e)
- g(l)g(- *) - 6 tr):l.i '+ S( * * r]i : rjtr; (5)
ireng lrg ihrlc rrrr r', citrl .r.. = r- i, lhi'drrrrc
_ _ .' a .-
-g2,1)=gr*l)=.-1. r
r'4I gtr tl = t. N6rr 9(rt * . r. tiri (4) ir& rlrsnh
. g(.r':"lJ=- I .lir),
'-/
I t-
\ .{r ! Cs pdu r cd rlang ..,'= rV2 -r s,
. /t-u): ( ! vtYi i' r h&'u :ti'
I it nf,rr s kh0ri.g biilrr diin drdi dang
'1i m$l nghi{m cira lili loAn.
' ilE irti tp{u 2 chi cCr nghigm (i), ehu,:Ait th*,:,
' di&u kien php. llirdng thrtirng_, dl,li toi C* sd
:.. cac.phunrig trinhlh&nr, dO ld?{itu hi*rri-fdt)
, '. - K-hiri tritrn lrdn tuc htt s,tiE rlu a p tl'ong irr;ng
tqln.h,..rc. lihung khAi ni{m. Oy' rtii linh-.vi,,ntl
phii.dr.ra ti6ir quan nif,nt cl,it che vii sd thrra.
Vi t'{y, & Irlnir {d, ph6 .th0ng., neu cO thl cf,r:g
cbi cd tha dE edp c16n kh&i nigrn-', lian tu\- rlrgt
cdch. str lugc.
toAn lr chi u6i ddn hArn /(r) xrilc dirrh vrli r h&u
'tt. Dt sno eAu hiri cfia ttAng chi'Giirr,r df tlrOi
th6t' 1$i; mir r.6ng bAir tr.rAn .1 clia eie .s6 th.rte,-
kh6ng phii r&'ding khti li{m tiCn trrc. N6i
r cich khte, co the li hf t^hrlc dai s6 trorrg bil
I ta{n. 1 dfl rgxnr bao giinr kh*i iiem liea trio ?
. - Itghi.sat hon, tOi ih{}' r$ag r{ring l& nhrr r.ly,
Dilng thdi c6 thb giri nr nhq gii tlr i€q, r,ri ia
Biil to6n 3.Gi,i sri /(.u) ld m!.t hdm s6 stic.
:dfnft r.rdi rnpi *6.{ftirc tr; ttd.'th6a rr&n diYa, *{dn
?l
.f,"
_{
t?
:
\
d r clil
'glb
:i t
:
l
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t
*
;1
f"
titi{ mpi: e: ud s. Tht ihi f(r") "p}rrii ld rl,lt'irsng
.llal hdm sutt d.dg :
t' -fioec/('r:);=1,&t"mqiri t ',t' 1,,
.
I . - lti{c'f(:r) = it t- i, ucri nrpt c. -t: ''^ '
, .' ild' tilfrc (2) tuy cS.r"6 da{ giirn. nirrog hoi
klr6 lAm vi{e. lr[r 9(r) * 1'G) * I, {hi !ai ro($
, Bai.to6ra 3'.'{iiei,,sr? rr(r)r ld nt6t'hdm rd *la,
' dlch uot tnp.i sd liqrc a:, ui lhf.a rr:dn rlt'Ia A1{n
17(r:g) =,gt c) 9(q) +g(,r)+.g(g),:- g(r- 1 ., ,u)
:
**i rnQi..r'r,d Y. ?'fid l-lri :" . , '
'',: 9(a) dbig nhfu I'irtg $;
,r.\ = .r t'6'l met t.
.t1flctii$t:g121-":g{a)*{t.'}ihtrilg,lrhidd
,,,, * 1 * g(t) * g\Z.tia,\ =.-g(:;gltil) i
+ 9{i) i1 gtl,rfi ,- s(2 1 1121 - str':l) - gil/31 i: g,
rnAu:.thuf,u. \'{_t ta ph&i i:6 gil) = li. 1-& i4), {S)
\ - .. .l
g(.r. r.li = f(r) -.5 i; gi* r,) -:, ',r1,{r).
;':-: ,4) l;ai, ttreo. (3), tts. ,lheo irtl. 9{,r * :ili :=
E g($ + l),+ l rer g{trI l- X vAtrt(.z} =. r: g,
' 0{2sE}"* g(ligt-'i 4., g{?.i 1 S(,r'} + g{i: *r ?r
.-, = 2g(r', + ? r- s,(,r: * y(r't " 2=9d(r).
5) Ul-j' r'&lg
*9(r'g)=g(-'rg)
' ., , I ;;(rr)g{,.i.vri,-hatiJ.}-.q( --,gi,:9(r-.-1;J
'.1619'l -i* $i;r:) -- .liyl * 9i:. -,Ui.',
;i (3), arr.r !.:i
, :*.-t
- 2g(.r') r= g{lir) ^
thi ta e&.
clilng vdi' rnpi r,'a, hay vi€.t lgi
g(x) * g(fi r= g(c + g). i,
${cc) -=ig{,)9Q)
- Trim lai, hdm g(.r) c6 ci,c tlrrh cii$t:
i) s(1) - 1., '
ii) g(.r * L,) : C(.r) * giy) v6i mqi-a, g,
iii) 9(::y) ",, g(.r) g(g) v0i msi s, g, *
'fr'r ii) bLng plrip euy nrp, ta durvc g{rr) =
irg(d r'6i nroi r: irgrrr0l. Do dd vdi'n nguy*n,
m nguy04 dubng \ -.
gQtf rn):; ng(1ir*), i : 9(n.t'rii) = mg(Ilm),'
Yiy /
r g(.nlrt) - nim.
l'u iii) db " ring adu r > 0 tht g{.r} } u
}]oI vl KtI.r (o
. 9{r) =. s[y' r .i "l - szt',71 ," o,
rbnnh thr! u6u * ft t1 tl'.i . '
' o< g(.lr'* y) * pinr).1- s(-s) * s(;s) =g(yr,
ha1'g (a) ? sU).
(iiA fhn r Ii rndl sd tlrrrc,
tf, ta c6 r,gev q(,r)'J .r- N{u
brirrg e{cli xiip xi tiiisrr vA
dlucrc lrai dRy sd h&tr tjr
, rt>r,rl)..'}r" }..
st{sz<...<: so{.'.
Yi m6i mqi rr
trly y. Ndu r hitu
r,vd t!, ,:l.rrerng l.ran
ihri4 sd .r', t*'-tim
! iirr s1 ix !r,
t,
rn.
-"r r, v*I g{r)el,
n*a
haY
, ':t" { g(')
Cliro n + €o thi $n .."* {i:, -\
GL&r cAfr BAr To,AN THI budc rfi NX,,v{ 1980
\
' ,,; TAI PH
'l'16rr b{o Tofu:. hoe vi ttir}i tr:6 16 2 n[m 1983
dE girli thi&u: vd dSrg i?E tofrr cria e6c k-] thi
to{n q.udc td l[m tr980 t+i ilhEn Lan vir Lric-xem
biia. 'i'da s*3.n rih6n du.gc hh6ng nhi6p Ldi gill c&a
eAc bg.rr gili clCn. 1ii6ng ddi vdi edc bAi'toBn thi
& Ptirn Lron, chi cd bqu Ngugln Y,frn nf inh {ir}p
I2ll trrtlng .l'han Lhdu Trinh. I)& Nlng) giri l*i
gi&i nbiBu h&i, tror:g dd *iaii tdt c{c.bill s6 2, 3
v 4,6,, . I
Du<}i d{y 1A d*Li d-e vA }&i gi&i eft }*&i to6n
thi,&: Ph&n Lan. .. ' '
B&i l. Trong lant gttlc .lEC. cit; tl.udng lrung
trLrc cfia t'tit cqnh AB uit eC tdt dudng thdng Bi
r$i X, I. fjJrringraiiir,l rdrg rlB 8C =,IY, mp{ diPu
.tifn dfi li {gpfge = 3. tldll ch*ng,I6 r:rtrrg di}t
Aifn nrig ktdnu ph[tt ld c&n ut" !EtL*m,di?u ki€n
cln rrr.' ari r{} Bt; = x}t.
: j.::.j'-:
Icri girii. Vi X, I Ia n
th&ng SrJ, n&n dE I{p {uJ
hiuh v6, ta iriy itinh h
A tl6n (.' nhrr ti'€B hlu
AN LAN
' -'i' -
Dbng thdi'ta e&ag dlnh hu6ng.rri6e goc I vi
G nku t"r0n hinh 6y. 'khi dd dti goc ^8 huy e lri
nbpu hay t,i. ta lu&n luOn eo
I nr * c/i2cosB), Yrl = ll{2cosC),
v*v
8{ = ex + 'Y}' l- Yr''
= cv f.c-lQg.i+8] * Dllt ' "
r
:..lf-::.:
t)N{u BC i-q XY, thi iu (11 vf rri dinh lf h*m
sd sir, suY rn. - -: --
. {.) _= ecrs^ll casC
sin 8er,sB -i- sinCcosC
r: a r:r Ee r) s{,'
. sirr2F * slrrlo
: -^"-_'-_'.--
- '..lcr.rs 8r oqtl :
sirr 1E 1 (,') cos (B *"(j)
- -:!L1l:,l. {6 --c)
Yi sin { =}i 0. neil
.eoii (8* CL
_, {j= 'eoii (8* CX $cs-,9cost..L--l- .cin,AsittC
.'.
nghi€rn trgnj (do thtdu cfr& ( hoic rt; t:hi cr1 .thb
li8u o6eh . r'i6t tIO ll hai ding thr'rc phii ddng
ihdi. th6r men.
. B&rl Z. l]&i 6or El,.(r1r,.., aB . d.ar,st tat rllnh
tLht ruu:
, ' '6u+1/!; .
(:h*qg mtnh.rring 1,- l/n ( a' <- 1.
\
L$i gid,. Dt i rlng l
' ll2=0o (.ui ( a2 { . -.- *o.
(16 th6 vi6t qu3' luAt x6c dinh d.Iy sd drror dar,g
- sl+l = dr '1. oli r, = u* 1n"r-: a1)/n,
.1o dd '.#
1n.11
-=
-:---="*-- (1)
ok+l at {n* a]r) .- a k n -! air
l) Do 11)0, nen t* (1) suy, ra
1l{1v+r} l/ri. * 1ln.
ViFt cie bllt drtng i'hr'rc nilr'vdi A'= 0, l, ?,...,
n-1,. r0i c6ng lai thi dtrcrc
1ia" } tlao -'n'ln o 2 - I = l,
va1' an ( l.
..2)veiodr(n,
(l) suy lrL
coslit'osf
. :1 * igBtgr
hey
r'o s #cos{','
.' cos Bcos{'j
,. I
tgEtg{ = - 1
,'f,
,(Di"nhi6u k6t rlul ti1' cr)n 'tLtong dgohg voi :.
la.--c.l -nlt).r .
2) fi6u sa : ) J o d, tiri tu (t) su] ra
tb
'r,,- ' L **=-
.&tt -2ctrs8 2cosC
do dd thco Ofah,lf .lr*m cd ein, vi ril dungl phdp .
bi{n dtii& tr0n
sin[.' sin8 sin,4.eos (B *C) .
4sin A :.*--. -l- -- = ---:.--..*- '.':-----*.*-i I
\tl sin A + o, n$n siry r.:r I l
cor (b -t,) cosBcos(l I sirrRainC
{ .- -------.- :. -.-*:=_.--,*-,-
' ae 1.* tgf.)tg(, i
hl.v ' ..
i tgt8gc = 3' '
'lhAnh thir'rli"u ki6n cDn vA d& dt BCs.it lA
ho:le tgdtgC: - 1 hc[c tgiLigC -3'
Nhrr r"5i, tgEtgC:,3 ciii lir mOt ttritsu ki$n tI{
db BC =,Yl', nhung hhOng pirrit ii. di&u ki(p ciin..
,\Ig.t vni bgnps.csch gi&i khAd, nhring ldi giAi
"&u sai s6t. Cd lian vi8t k6t,IuAn
I
',' L\ t-rgBtgc -;;
'- )' r, :a,ttn
l/ctat { 1/rrr,- li(n i- 1)
\iist ca.c trlt rl:irllg thric ,nLy vOi k:= 0,'1 t,..,,fi. l,
r0i,cgng lai thl drrrc ' ,
l/qr..( I/ao * n/inl+t)'= i - n/(n -1" 1)
' = (n + ?)/(n-t-, 1),
Y+y .
. . a,r)(nf'l)/(n+2) * 1*1 (n'J-2)'
vA di nhi0rr qn ) I * lln..
t,
,T/t"<rn r d{. Ban .\:girrTin ycin ,1/intr c6 lr"ri giAi
ii;1, bAng c6,:h dttrl rrr hdt di iig thire kelr
2n+l-/i 2n.-k
Db r:hr'rng .tir ring b{l dting thr?c k6p niy. .d*rig,
ben \Iiuh u|{rr .r*t.fi6 itung c}ro isa l, sau dd
dur.rg plr0p. qu3' ri+p theo'k, o'
-; ' 'x ,qit plrrrorrg phirp, r'ifc ilua ra
i:6. phB.ri e kh0ng tr,tnhitn D,
ria rndt : qui Iriinh i!o{n
ta. c6 a; { 1, nen. kl tir
'€n elirrg ld rngt phutt.n$
'rtng kidu lo+i" Eey lA
p
-6
i.,. ,
mqt:bAi-to6n do'fhr;y p;!6 rlE nghi ltm.{B'rhi
troog ki. thi ToAn th5 gidl Ifii 16 (1974) nhuns
l&n hy, IlQi {t6rig, gi{rp .k}i&o cho, r*.rig"bAi n'1-
guri khd I) : . ." ,.
r Cho ct ld mdt td thtr.i, 0 1 a { I,'t,d.'n.ld mdt
..rd tr nhtin tiy {}. Vdi mdi n lip ddy s8
Aoi (II, A2,,,,, dn ,
0o = o. rrk+1= tts + t1f,17(0 < k ( n)'
.r'.,i'I * ,, :
Jr.4i+t//d"+1 di-p1 1-t f tfi f - J, 2;i,.1 n -- I ; ta chring
rrrinh .4 1.41o, /Ao -.to+1. (ii.ri tlcr ,lAi.currg 31.{i41 ld
n; ,y(1i i * 1,2,..., n * 1 rA rli dni cung .+rr.A[ tf,
,ailo. Try .l,tA.zl lAn+t4n+,1. suy r.a' -'- ' ,
rrz -i ag14... + aD = rln .l:2 r ttn -i *...* 6ga (l)
'l rrong trr. lw.ltAi4t '.{n-}-r rl,,;i r 1, i = 2,.".. n*1,
.:3 t a4 + .... t- {r4g1 ,a on*E * ...,*,112o,* a1 Q)
-.i{ + o5'+..' * rrs+i = oa44*..;*o:n+41}43 (3)
tldq elrittg'tntnh rdng
(phttthugc ud..a a) grro cha
lbn /ci ntf t s6. thr.rr ,l
(/(1(A*rr.)<(jra.
nii 3: -Ye.ra,pfirryrn g t-rini
n , , - ..n+1
I + l:Y'''
tronrg rt6'.;gi*td ntll c6 tv il:.eil l6n hon 1. Chnng
nritth .rin'g phrung trtnh'khdnq cu tttlhi/,m ngugin
duorg (,r,u) ubt r ud n{'/ ngugin t6 ctitrg,niou.
t-bi gtd:r icira bqn i[.qirydn l:ria .&ffnh). Gii sfr
n gu..re i ai, ph uo.n g tr iirlr c6 n gh i t in,n g uy 6n d irorlg
(ro, go) l4A (.r'q, n+l) E l, ta c6
I l; =(gr*l}(yn + ,X-'n . . +- r) (1)
lln biEu rlidn .{ = r,l} ' n - I
,o r 9., + . " * 1 =
ca * o'u-1 i..,*a2n.l:.o2n * o1 *.,,{an-2 (o-r}
Yiri cAc.rlirug thri,c trdn- l6-v t l) llri i2.r,. (3) ir.r!
tll. ..., (Lr *2i trri (n - 1) r e yCri vd lbi cQng tdt
ci kr\t ilrr:i lli it,l, rlusc* - - I l
01 + q2 +.., -f an-l : rln*l * ... -1- nzr*r
Dlng llrr'rc nll t'lrirng .ti, dt -{en ll Ao An+t.
l.!
, ' 2) Ydri,,n chirr tlrt cr) dla gi6.c 2n e-qnh nQi li6p
c6 dtL-v nlrrri <'Al\ clnlr ClCii klif'rrg sgng song,
Th$t vrr-v, tit urdi rlu giAc I in-. l) c4rrir, it-l
t6. c6..tal ci" cie c,lp cqrrh ildi s,rng song r fi lih6nE
hing ulrirrr tcheng hlrt da riut' tlria rludrtrg tron
ngoai titip thdrlh c{r: crrirg r:6 tld dAi. ttll,ulr-1),
Zit ll(tt-.1) rcn kd rrlritrrl; irr Bgi rrrdt clp canh
'rI,1i lA AB, (.D, l.;.1' l.- r,".u crLrrg .,lB vA F trdn
, .eung CD sao EAB = FDt. :
Khi d6 si c(t AE; lDE" LrO dAng chrlrrg rninh
<1 trgc EB r A Ci: lilr6nr: :ong song r'6i nltau. Da
. giile'2n earh c6 oic clinh 1& cAc diirh efia tla gi{e
rfn-l) cgnh Dily ia'hai. dinh li. .&' r.a F' sE c6
. duy nh{t'o{prcqnh d{i Br,4/r kJi6ng son-q song
'.1 rii nhau.
'/\!_t
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