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Chfi nhi€m; NGUYEH cAag roAn
BAo PA tlAl ruAue MQT KY
rtu ks fda sosn, :lj::i:rrr::
Tru s&: 70 Tr5.n Hu,ng D4o' Hi N|i
ldm suen mOf ch0t v1i
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LIEN PHAN 56
I i'
f. 6 tOp dlu c'6p III. ban iI5. hoc vD cln c6
vi bi6t ri.ng cEn l.ric hai cria cdc s6 kh6ng chtnh
phuong ld sd vd ti vA c6 thb bibu di6n blng
mOt sd th$p phdn u6 hqn kh,ong tuhn hodn. T6.l
nhi€n, d6i.v6i sd th$p ph0,n v6 hqn kh0ng tuhn
hoAn ta chi e6 th8 bi€t tlugc mgt s6 chir s6
ttrrd.p phsn cria n6 mir thdi. Thi dq:
y 2 :1,414')t... rt:3,141592...
Ban ch6.o c6 lrio t\r illt vdn dE: llm th5 niro
dD tirn c{c chQ s6 th{p phin tI6 vdi s6 lugng
mir te mu6n' ndi c6.ch kh6.c: lim thd uio tlm
mQt phAn s6 hay mdt sd thap phf,n hftu hgn
biBu di6n gln drlngi s6 v0 ti tl6, v6'i tIQ chlnh
xic cho trudc ?
2. Trulng hgp tton giin nhfit l& v0i c6.c c[n
b{c hai cfra c6c sd, ngutri ta dd tim re mQt thu{t
to6n &8 gili quy6t v{n tlb tr6n (thufl.t todn khai
phuong cA.c s6). Tuy nhi6u nhiEu bgn c6 th6
khOng bi€t thuSt to{n niy, mQt s6 ban khdc c6
thb bi€t, nhung khOng hiBu eo s& lS lu{n cire
thu$t totrt vi vi kh0ng dirng thudng xuy0n u6n
c6 th6 kh6rg cdn uh0.
lBey gid, ndu c6 ngu&i ohd bg.n: e vi6t girlp
{3 ghn thing du0i dqng mQt phAn a6 thu0ng
HOANG, CHING
hay m$t sO thip phAn, v6i sai sd khdng quri
11105D, b?n'o6 thb srl dgug ki6n thrtc erla lOp
tlhu cdp III thdi dE tri ldi rturyc khong?
Sau d&y li m$t c{ch rfit don gi&n giirp bgo
trA ldi cAu h6i d6.
Chri f rilng
, r< /5:<e , (1)
Vr - t .{-llci. (a1 nguyon duong)
do d6 . et:11(/t- t) = (/ s + r)/z
Do (1), ntn 'c6 th6 d{t:
or:'(y'f+ t)lz = | * tlaz (a2 nguyen ducng)
bay li
a2=21(o13-l)=Ve+r
Do (1), n€n c6 thA d[t:
a2:l3f 1=2."*,lios (as nguyen. duong)
hay lt o3 = rltf;T- 1):(/t+.1)/2
Ta thfly ds = at, do d6 c6 th8 vidt a3 = 1 ! tlat
(v&i ae=azl, a4=2*llal (v0i airyaa)v.v.'.
',+t. .'
,,t '
::.'
i ,,,
\.
,},:
ry
t
{
.l
TOU lgl lr o6:
y'T= r + l/or
dl = 1 +l'laz
d2= 2 * llat
as: I * llat
at=2 f 1/45
a5=tr*lloa
trong d6 tilt cA *.'o, ua, ln s6 ngu-vtn duong'
Tri cic k6t qui tr0n dAy. ta c6 th& tim phln
cd biEu di6n ghn ,tilng Vf v6i d$ chln\ xAc
ngiry cirng cro. Ch&ng hgn n6u ta l{y
ar:1 thi y'TE 1 + tlt:2'
NEu I{y
az=2n,iVret+ -j-=+'
,* ,
d3:1 ttrt /T:, * --= = +
, + ---T
,* i
(tinh tn du&i l€n: 2* t--3, 1+ -l = n
,
r33
I + -q- --r" ).
14
NEu ldy s4 x 2 lhl dutTc
t- 1 rg
V3Ei+-- '-=--'
r + -!-, ii
2+
-- I
r+T
05:I thi iluoc
,- I 26
V3:l-:- = -'--
1". 1*- 15
2+i-- 1
'1
_1
Cdc b4n e6 thB th6y rlng c6.c phin e6 tr€n
d[y cho gtd tri ngily cing "lrioh *{" "&u
y' a ,
2, 513, 714, tlltt,26/15, v.v...
Thuc vfy, binh phrrong cic s6 d6 ta c6 c6.c
girir tri ghn {Iung bing 3, v0i sai sO ngly clttg b0:
22 :4, (s/g)z : zblg - 2,77...
(t l*12 - 4sl t8 - s,ob...
(rs/tr)2 = g61
I t2t- 2,98...
ea l rc)z : 67 6 I zz5 = :i,oot...
,}
3. Clc bilu thtc nhu
1l
1+-'.1*- ,v.v...
1
1+-
2
1
1+-
2+1
,l
ri
LI
f.
i
I
ti.
f
dugc ggi ll'e6c liin phdn s6. M0t c{eh t0ng quit'
1i0n ph0n s6 bfc k liL biEu thue dgng
1
oo*T
sr+ I
a3*1
i
I
*;
tnong d6 o6 lI ngugdn, cdn dL a2;-'., ak ll s6
ngagd,, duong. Ll0r: phln s6 b$c k duqc tf bif u
ggn ll
[ao, at, dz,.'., dk...]
Qua cAc tht dq & trOn, ta d[ thfly c6 thE tl8i
mQt 1i0n phtn sd th')nh m$t phAn s6 thudng'
Nsuli ta chung minh ring nguge lai, m5i s6
hiu tl dEu c6 thF bi8u di6n duy nh{t dudi dqng
mgt li€n phln rd b[c k niro dd' Thi dr.r:
.rn'tll.
::--s+:-=5+3=5+' :[s.a.2]
7 7 7 s+j-
22
2s6-2+22 -l+ I -2+j-
7r7 tr7 _1I 6 + t
21 22
=2q -t ,2+ ' r =[Z,S,S,Z.]
ll
' 22 s+l
77
37 0 1 -. I
-J---= -6+ _- =-A1--:-6r
7 7 7 ' 1+ 1
,58
- [- o, t, oJ.
4. Ndu ta k6o dni m6i c{ch llm cln bSc hei
"&"
y' s nhu dd lnrn & tr€u tht te di ddn mgt
titn phdn sd u6 hqnt
1
.l-
r-r I
l+.----_-
I
2+
-.
I
lJ- I
2+-
l+-
: [], l, z, l, ?n 1,",,.1,
J
1
{
Nguli ta d[ chrtng minh rlng mdi
td ud han bi?u diEn mit sd ud ti, od
nrdi sd oG ti ttaqc bil"u diln Ddi mft
sd o6 hgit, 5. Li6n ph$.n e6 16 !B$t cOng cp e6 hl€u lpo
trong nhi6u v$n tl& toin hge. N6i ri0ag. n6 girip
trjiu di6n x{p ri cic s6 thlrc rilt tl$n lgi.
Cbo s6 thuc a bi6u di6n b&i li0n phtn e6
d : loo, ot, az, ,..)
Xet lien phon s6
Lioo phin s6 v6 hen bi8u di6n t/ g lr *Ot tleo .,
vidt:
/s=tr,t,2l.
C{c ban dE ding tim th[y rlug:.
',lT*j,21, r/F= lz,7), /o=tz, r, r l. j co=[oo, at, "', a,]'
' Sau khi thqc hi&n c6.c ph6p tinh trong do
Ntrung kh0ng ph&i bao B!& ta cfing cd kdt qui i?(trl Cu6i l€n), ta tlugc rr6i ph$n o6 p,/0,,1 fi
c(Igp> nhuv{y.TugiiiathdkiXvlli,nhitoi.u r'. goirln=prlenli gidn phin'bdc ncriadinfoa
hgc vi dai 0-ie dd chung miuh rXng nrgi s6 v0 ;j':'c5 btc l& A' thi n { d.).
ti deng ta+t-/i)la {a, b, c, d ngu5'€n ", j. I _, Trong tht du tr€a fi{y v& tinr gia rri
d * Ol c6 th6 vi€t du&i dgng lien phin s6 vd . ,-- .
han tutn hoin, vi ngrrEc lqi m6i li6n phdu s6 ' cria V 3:.
o6 hqn tu!.n.troiru niAu atAn rngr s6 iu.,t.:o , le-: [t, 1,2, l,2, t,2,1,...J.
dsng (o + b'l;)/,t. i{hrr v3y, Ii6n phAn s6 giup " - - - L:: " :' '' z' t' z'
ta il{y mQt net bin ctr{t eia cdc ra un tt iang !a rla tinh cie giln phan
uiy. NOi ii6ng, lidn phin s6 vd hsn tuhn dr:[t, f]:flt, a2:lt, 1,21=714,
hoin .:, ds = h, 1, 2, lJ _tsltt ,....
1 1
-1-
: I r, 1] * Ggi da: pn/Qn vi dn*t = pn+rlQo+tlr cAc gilu
phdu b{c n vt bec " a i .&a'"ilOn phan sO a biEu
I + I di6o s6 a. Nguli ta ch*ng minhl {luqc rlng n€u
I + ... dnng PuiQn bi6u di6n s6 ct thi sai s5 mfr.c phii e8
bieu dr6n s6 e - (t + .l;)n.s6 rp xu{t hiQn rrong aT$ 'J;?";3;J'}r:}'it1'o1,':;:'; iYil
a ph6p cbia v3rng ); phep chiar mdt doan thSng d4:26115 (0a - 15), ta m*e sci s6 nh6 hon
thirnb hai ph5.n r, y iltrocgoi lir mdt( ph6pchia 1/(11.i5) ( 0,007, tuc li nh6 hon i/100. Tuong
virng' (tuo ln mQt ph6p chia r{t dgp,l, n6u nhu . tt1 nhu v&y, e{c bqn c6 th$ ki$m tra lqi !'iing
a + g * n€u ta bi&u di6n .o / z lXng giin phan
= -. ft, z, z, 2, 2, 2,2] : 239/18S, rhi sai s6 se nh6
G y hon 1/10i; do iI6 ndu dbi seng s6 thSp ph&n
L{y y=1s1,1 r+l= i 0., *2 -*-1=0. }ili,;,?':l-;:if};,,_;ln',1"'uurl,ot"ol',1
Nghl€m duong cria phuong trinh ndy chinh Il -01-.
s6 cp. Tu rhli thrrqng e6. ;guoi t" aa'tney ,ang D6t v&i s6 0i, tr c6 c!sfe z, t;, 1,292, 11,...J
m$t hinh ch& nh&t (hinh d4ng quyEn s6ch, m{t nOn dr: [S, Z] *2217,
Llt i,lruil:xi;: };:;lt"'1J,1J*l;i,"tlX d2 : rB, r, 151 :,Bs/,06,
dep mft ,rhat, do do (phep chia vingr;;;: as- [3, 7, r5, 1] = 365/.113,
*ri dgng rong rdi trong ddi 06n3. da= [3, 7, 15, 1,2g2)- 103tig3/33102.
Ddi vdi s6 e (co s6 cta to-ga-rir tu nht€n), Do d6, n6u lfly x o 22h tht sai a6 uh6 hon
nguli ta tim thdy rhng 1/(2.106) < 0,002; vt 22lz =8,1428..., n€n c6 th6
e:12, 1,2, t, 1,1, r, r, 6, r, r, B, r, 1, r0, l, r,...1 li:lr";ij{rf:_It $:J,irltifl, :llt..:;
Lieu phfln s6 vs han niy kh6ng tuln hoAn, nhung nh6 hon 1/(11S.J3102) < 0,0000008, vi vl B5f/ilg*
ciing duqc la.p theo m6t quy IuAt: l, 2,.hai l[n :3,141bg2$... non c6 thE vi6t 3f.=:8.14160 (ci
1,4, hei l&n 1, 6, hri l[n ,,8,., E6i vrri e6 cr, ugu&i o6r,! sbi] 16 thgp ph&n tl]u chtnh xAc).
liln phdn ",I.ii,ta chuo ttm thr.y mgr quy luet gl trong biBu di6r
nguqo tqi n6 b&ng iiOo phAn sfi: rr:13, V, 15, l, 8l2,
ti€n Phdn 11'"'l'
*
t
s
f
B4N EOC rlA ril:
nA mgt n5.m qua, tOi tham gia sfrn xu[.t &
v hop t6c xil, nbung nh&ng trli to6.n mir
tOi g{p khi odn ngbi tr€n gh6 nhh truong
v{n sdng trong tOi nhu m$t kj'niQm kh6ng qudn.
Vi thg, ngoili nhfl'ng gio lao i10ng, tOi lg.i d6u
v6i oic bAi to6n.
NEm trudc, khi hoc bhi: ( Dung htnh biug
phuong ph6p tlqi ad r, thhy gido tld rs cho chring
t6i bii todrn ssu:
Bdi to6n !,. Cho tru6'o 6oqn th&ng a. II0y
dgng bHrg thude vi com pa doab thins aVZ.
T6i cdn nh6 l& khi d6, ci ldp ehring tOi dBu
chiu bd tay. T4i sao I4i d+ng tluo'c clogn
ttr8ng oV2 ? Vt ch"Jrng tOi m0i chi hqc c6ch dung
iloqu th&ng oy'Z tf.ni bidt doan th&ng a) rh0i
mir... Thdi gian trdi ali, md b&i to6n d6 vEn
khi la, khi hi$n trong tri nh0 oia tOi.
E{n nay, sau m$t th0i gian loay hoay, t6i d[
kh6m ph6 r'a su bi mff.t efia bli todn 1 vfr th6y
rf,.ng cdoh dung deiqn ttr&ng aViti dqra tr6n viQo
c6 do4n thfrng ay'il
Tru0c h6t t6i cf,ng mu6n n6i thOrn diEu n&y:
Trong lrlc duug tloan th&ug ol 2, tIi tim thdy
mQt tliBu_thri vi lA: dna trtn crich dr.rng clo?n
th8ng oVi, tr sE dung duo'c rloau tninla/r..,
drra trtn cdeh dgng doan ih&ng o/ilr, tn re
dr;ng duge iloqn th&ng n-fin ti,int f),
Nhu v&y, ttr vigc glli bni to6n don gl6a li
dyng tto4n thang aVe t6i tld gili dugc bii to6n
sau :
Bdi toln 2. Cho trudc fioan thf,ng a. Hny
dgng blng thu&c vi eom pa Aoan th&ug ohr,
vOi n lA s6 tu nhi6!.
Biy gi& t6i tr& lai bii to6n 1.
E& f ring tol,lilz - ulili - "."{f,-
ta thfiy oflZ fA trung binh nhAn cria hai ilo4n e
,a of/2. (CAch dung ohrlng ta dErr de bi6t).
T6i di ddn bLi to6n t6ng quit sau:
trudc fioan thEng a. HEy
k
com pir tloqn th8ng ary'I.
Bdi toen 3. Cho
dqng bing ihu0e vlr
k k-l
DE i, ring trztilz - ,.r't/{
ta s€ c0 kli giai cira biri tolu 3 (hiah 2).
.t.
::
Tr\ circ hii to6n u6i tr6n tOt tin ra bii todn
t6ng ttrult ho'n:
_) tsSi to6n 4. Cho tru0c do4n tb8ng a' Hfly
k
dltng bXng thu&c vir compa tlogn thf,ng azy'i;
vOi n,'k lir tirc sd llt nhiOn'
NGUYf,N VIf,T TII}'NH
(Odng lfd, Phric ThP,
Nghi Ldc. Nghf Tintl
i
6
.,i
1
J
Btt I/f0f. Cho tam gidc ABC cd g6c t uafngl
dabng cao AH cd d6 ildi z. &4t IIB=t. HC-C.
Chrhng minh rdng nlu ta cd ** g + z = {g.
thi z2li.
,Lbi gidi. Vi tam qitc ABC vu6ug ntn ta co
z' : xA. Tt do zo: tgz,
Tbeo gii thi|t, *gz- r * g * z nOn ta c6
*r ,3=n*g*z (l)
Ndu goi Il ln bin kinh fiulng trdD ngoal ti6p
tam gi*.c ABC thi 16 riug \
:
P * 3 : AArlOh * BBtioBr I CCllOCl
; :S/Sr*SlSz*SiSe
> 3s/Vsrsrsa
= gsisVs]ilsg ) es/(sr + sz * sa)
= 9S/S = s.
- Vfly P tlat gia tri nh6 uh6t bing 6
Sr=Sz:Ss, ttc ln khi O li trgag t[m crir
giitc ABC.
. Bfi 3/101. Chttng minh r&ng u&i mgt tuc gidc
l6i bdt bdt ki thl bqo giir cilng c6 thb tim ilwqc
m$t dinh efi"a nb sao cho ba d.adng .chio *udt
phitt tir dinh d6 c6 th? tiig tdm ba cqnh crt"a mgt
tam gidc. Cd thb tim droc hai dinh (cfia l4c gitic
lbi bdt k!) th6a mdn d.i?u kiQn b6n nag iningt
Lbi gi&i. X6t duong ch6o dii nhdt cria'lgo
gtic. C6 hai trudng hgp.
7. Tradng hqp drb'ng chio ddi nhdt chia lue
gitic thdnh m$t tam gidc-u.d. m(t ngil gidc.
Ch&ng hqn A1 ,{e li tluong ch6o dti nh{t.
Ndu gii sri 1.1 khOng th6o min tinh ch{t q{o
dudng eh6o xu$t ph{t r* n6 c6 th8 ldy l&m be
e_4nh cria nrgt tam giac, la chring minh As pbll
th6e mEn tinh ch6.t d6. Tuc li u€u
4
. A1A5* AlAt ( Arls (c)
thi phii cb Aals * .{s,40 },{rAs
Th{t vey. Tu gi6c ArAsAsAo lbi n0n cic tlu<rng
ch6o crig a6 phii cit nhau t4i.[ nim ttong tu
t\
EH ffiffiH
*$ {r*wc
{
khi
trm
Vi v{y
Trl (1) vi. (2) suy ra
's)32
,2>l
, >.1{
r+s-Illzz,
t+g+2232; (gl
(it.p.c.m).
*
.:*
Bni 2/l0l.tCha tant gridc ABC, O ld mgt di?m
trong tam gidc, Cac itadng th&ng AO) BO, CO c&l
cdc cqnh cfra tom gidc taong trng tai At, Bt, Ct.
Tim o! tri crt,a O d.? bibu thirc
F - OAIOAT 4 }BlOh + OCIOq ;
dat gid tri nh6 nh?it.
Lbi gi&i. (cria nhiBu ban). Goi S li dign rich
tam giSc , BC; Sr, Sz, 53, tuong ung ln diQn tieb
c6c tam gi{c OBC, OCA, 0AB. Ta c6
S-51fSz*Ss.
Db tim gi6 tri nh6 uh{t cfia P, ta tim gi6 tr!
nh6 uhf,t cria P f 3.
A1