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I rrgsl
L993
TAP CHi RA HANG THAI.{G
M6i ldn md td tpp chi "ToAN HQC vA TUdI
TRlt', toi lai thdm kham phuc Ban bi6n t{p
tap chi d6 chqn phrrong ttinh I * f = * chu
mgc "D6 ra ki niy" ! Cd 16 nhi6u ngudi d6u tin
li 'dd ra" dd sd drrgc tdn t4i vinh vi6n ! Vdy
mA rdt cd thd Tba soan sE phAi tlm mQt "d6
ra' kh6c dd'thay th6, b8i vi ...
Ngay 23 t&.ng 6 nd.m 1993 llch s& To6n
hqc ghi nhQn mQt sU ki€n quan trgng : Nhir
tod.n hq nguii ,Anh Andrew Wiles cdng b6
chrtng rninh Dinh il bn Fennat. Cri 16 trong
llch sr1 to6n hqc chrra ttng c<i dlnh li ndo duoc
'chrlng minhl nhi6u ldn ddn th5. Vfy thi tai
sao trong sd hing trim'chrlng minhn xudt hiQn
hnng ngdy tr6n th6gi6i, chi cci chrlng minh cria
Wiles li dudc tdt cL cdc td b6o l6n ctla c6c nrl6c
drra tin v6i nhitng bni binh luQn ddi ? Dd hidu
dtro.c di6u dd, kh6ng c<i crich gl khAc hon ld tim
hidu nhtng hI tudng ehrlydu trong chrlngminh
crla Wiles. Mrp dich crla bii vidt nh6 niy ld
trinh bty cho nhfrng dQc gi6 kh6ng chuyOn hidu
duo.c phdn niro nhirng hJ tuding dd. Dd cac ban
thong cAm it nhi6u v6i khd khan ctla ngudi
vi6t, xin dgog trlch ldi crla Ribet, nhd tor{n hge
ndi ti6ng ngudi My, khi trA ldi ph6ng vdn'bria
td "Niru ude thdi b6o" : nCd 16 kho6ng mQt phdn
nghin cric nhi to6n hgc tr6n thd gi6i hidu drtoc
.chrlng minh ctla Wiles".
C6rtng minh mi chring ta dinh tim hi6?u lh
str k6t thfc mQt ch4ng dudng ddi crla hon 3
thd kl, vi cd thd cflng li sg md ddu cria nhi6u
thd ki tiSp theo. Ta h6y ddnh it phtit dd nhin
lai lich sit bdi to6n ndi ti-6ng niy.
fflNH Li Iff FERI!,f,{T r},{ nUryC CIIU',I{G M[NH
NA rrW KHOAI
Nhi6u tdi li6u cho ring Fermat c6ngbd dinh
li cria minh n6m 1637. Tuy nhi6n cho ddh nay,
'ngudi ta chi drroc bidt ddn didh li dd qua bAn
di c6o cria Fermatj,ilo con trai 6ng c6ng bd n*m
1670, 5 nem sau khi Fermat qua ddi. Fermat
ph6t bidu dinh li crla minh bing ti6ng l,atinh,
cd thd dlch nhrr sau : 'Bidu di6n mQt lty thta
b$c ba dudi d4ng tdng hai lfly thrla b$c ba, m$t
Ity thta bAe b6n dudi dang tdng hai hiy thrla
bQc b6n, vh crl nhrl thd-ildn vd h4n, di6u dd
kh6ng thd tnm drioc. Tdj tim drro.c mQt chrlng
minh rdt b6t ngd cria sg kiQn ndy, tidc rirlg l€
sich niy khdng drl eh6 dd ghi'. Cudn sr{ch mi
Fermat dinh vidt chrlng minh crla minh b6n 16
ld cu6n "Sd hgc" cria Diophant, ngdy nay da
thdt truy6n.
CAi '16 s6ch b6 nh6" vi "chfng minh bdt
ngd" cta Fermat d6ldm dau ddu 3 th6 kl to6n
hgc. Sau chrlng mQt thd li tnat bai trong viQc
tim lai "chrlng minh b6 qu6nn dri, cric nhi to6n
hgc d6 tin rilng trong thrJc td kh6ng t6n t4i
chrlng minh so cdp vdi nhfrng c6ng cg crla to6n
hqc thdi ki Fermat. Chi nhiing nhd to6n hqc
'nghiQp du" le v6n tidp tgc c6ng cuQc tim kidm
cho ddn tfn hdm nay. Ni6m tin crla hq vdo
Fermat ldn hon cdcnhato6n hoc chuy6n nghiQp,
cd thd mQt phdn vi ben thdn Farmat ciing li mQt
ntri toan hsc nghi€P dtr : 6ng ld mOJ lu$t su d
Toulouse, mQt thlnh phd mi6n Nam nrr6c Phrip'
Khi de hdu nhrr tin ch6c ring kh6ng thd cri
chrtng minh sd cdp cho dinh li Fermat, c6c nhA
1.
B0 GlAo DUC vA DAo rAo * HOt roAN Hqc vte-r' NAM'
II
m
tohn hoc b6t ddu xdy dqng nhirng cdu truc tdng
qudt dd trI dr5 suy ra dinh li Fermat nhu m6t
trtrdng hop ri6ng. Dinh li Fermat dE thltc srr
trd thdnh dQng l1tc, mrlc ti6u cho m6t thdi ki
m6i crla Sd hqc : thdi ki "c6ng nghiQp hda",
trong dd ceckdt qu6 cfl drroc nhin nhan du6i
6nh s6ng mdi, chring khOng cdn li nhitng su
kiQn ri6ng 16, mi ld hQ qu& cria nhirng li thuydt
m6i. Cd thd ndi ring Dinh li Fermat chinh li
"con g& dd trtlng vdng" cta to6n hgc hiQn dai.
Trong sd nhtng quA trdng vAng mi con gd
Fermat d6 ra, trrldc ti6n phAi kd ddn l{ thuydt
cta Kummer. TrI tudng chrl y6u cta li thuydt
d<i li nhu sau. X6t phuong trlnh
{+{=*
Hidn nhi6n ld ta chi cdn xdt trudng hgp n
ld sd nguy6n td. Hon nta, n6u (x, y, z) lA mQt
nghi6m crla phrrong trinh vi 6 la mqt c6n bAc
n cria don vi thi r0 ring ([r, U, ez) cflng sE li
mQt nghi6m. Vt thd, dd drra ra mQt cdu trric
chung trong d<i chtla drlng c6c nghiOm cta
phuong trlnh Fermat, ta c6 thd x6t "md rQng"
Q(6) crla trudng cric sd htru tl, tdc lir tAp hgp
c6c s6 htu ti ctng v6i cdn bflc n ctta don vi.
Vdi nhfrng n kh6c nhau, trudng Q(6) nhin drroc
c<f nhfrng tinh chdt kh6c nhau. Tt dd tim drrqc
nh{rng di6u ki6n ddi vdi ru dd phuong trinh
Fermat vO nghiQm. Nhtng sd nguy€n t6 n th6a
m6n di6u ki€n dd dugc ggi li s6 nguy6n t6loai
1. Chtlng minh dlnh li Fermat drrgc drra vd vi6c
chrtng minh rlng mgi sd nguyOn t6 d6u li s6
nguy6n td loAi 1. Cho ddn nay, bing nhung
m6y tinh hiQn dai, ngUdi ta x6c dlnh drrgc ring
cric sd nguy6n t6 nh6 hon 150.000 li sd nguy6n
t6 loai 1. Tuy nhi6n ngay cAu h6i s6 cric sd
nguy6n t6 loai I ln htu h4n hay v6 han v6n
cdn chrla duoc tr6 lbi.
'QuA trrlng vdng" thrl hai crla con gd Fermat
li Ii thuydt DQ cao (Theory of Heights) cria
Arakelov-Faltinp. Cd thd hinh dung m6t cdch
'ng:6y tho' vd li thuy6t ndy nhrr sau. Ndu nhrr
'dQ phrlc t4p" cta m6t phrrong trinh dai sd drroc
do blng b6e cria phuong trinh thi d6 phrlc tap
cta mQt phuong trinh s6 hqc (h€ s6 nguydn)
phu thudc vio cdu trric crla tflp hqp nghiQm
cria nri. Dd do d6 phtlc t4p niy agUdi ta xdt
tip hqp cric nghiQm nhu tfp hgp c6c didm ctra
m6t drrdng cong. M6t nghiOrn nguy6n cria
phuong trlnh li m6t didm cta drrdng cong vdi
toq dQ nguydn. Tt dd cd thd thdy ring, phuong
trinh bflc nhdt r* ! = z vi phuong trinh bQc
hai rJ * y- = zz c6 vd sd nghi6m nguy.n. "DQ
caon cta m6t phrtong trlnh s6 hgc ld'd0 cao'
crla drrdng cong rlng v6i phuong trinh dri.
"Dudng cong Fennat" thuQc vdo l6p cric drrbng
cong'gr6ng > 2'. N6m 1983, nhi todn hgc Drlc
Faltings ehrlng minh ring, tr6n cric drldng cong
grdng > 2 chl tdn tqi huu h4n di6'm tga dQ hrru
ti. Cd thd thdy ngay m6t hQ quA cria kdt quA
dri : phrrong trinh Fermat, ndu cd nghi6m thi
sd nghiQm c0ng chi li h{tu h4n. V6i dlnh li niy,
Faltings dE nhfn drroc giii thudng Fields (tuong
drrong giAi thudng N6ben). Sau c6ng trinh cta
Faltinp, nhi6u ngudi hi vong ring li thuydt dQ
cao li con drtdng d6n ddn chrlng minh dinh li
Fermat. Cho ddn ngdy 23 th6ng 6 n6m 1993...
Nhung dd hidu drrgc s1t kiQn c{ra ngdy 23,
cdn phAi di ugrrgc thdi gian, ddn hQi ngh! qu6c
td nim 1955 hqp t4i Kyoto. Tai hQi ngh! dd,
Taniyama d& dua ra giA thuydt v6 m6t d6i trtgng
to6n hgc, dugc ggi ld dudng cong elliptic. Dd
li c6c dudng cong kh6 don giAn, duqc cho bdi
phuong trinh :
f=f+ar&*arx*aj,
th6a mdn di6u ki6n "khOng c<i didm ki d!". Cric
dddng cong elliptic d6 cd mdt tt lAu trong to6n
hgc, tuy vAy it ai nghl ring nd li6n quan d6h
bAi torin Fermat. Li do rdt 16 ring dudng cong
elliptic ld dudng cong "gldng 1", trong khi drrdng
cong Fermat cd gidng >- 2. Muc ti6u cria chring
*a li chrlng minh dtrdng cong Fermat kh6ng
cd didm nguy6n, trong khi dudng cong elliptic
lai cd v6 s6 didm nhrr viy ! Cdc "th6ng tin sd
hoc" cta dtrdng cong eiliptic drroc dno nai
"L-him" cria nri. Taniyama dod.n rang c6c
L-him cta dudng cong elliptic phAi th6a mdn
m6t s6 tinh chdt mi sau nAv Weil chinh x5c
hd; du6i t6n Eoi "drldns coni Weil".
Gii thuydt Taniyama{Veil : Mqi duing
cong elliptic dbu lit. dudng cong Weil.
Ngubi ddu ti6n c<i trr tudng g6n vi6c chrlng
minh dinh li Fermat v6i li thuydt cric dudng
cong elliptic ld nhn todn hoe Drlc G.Frey. Di6u
hdt stlc thd vi li tu tudng dd l?i vd ctng don
giAn ! GiA stt dinh li Fermat kh6ng dring, khi
d6 tdn t4i s6 nguyOn t6 n vd cdc sd nguy6n q
b, c l<hdc 0 sao cho
a/t+bn*cn.
x6t dudng cong elliptic cho bdi phuong trinh
Y2:x(x-an\(x+6").
Frey nghi6n cfu rdt ki dtrdng cong nAy vd
dua ra nhQn x6t ring nd rdt khd tdn tai ! Nh0n
x6t dti dtlEc "cg thd hcia" nhu sau trong dinh
li cria Ribet (1986) :
Dinh li Ribet : Duimg cong elliptic Fryy ndu
tbn tqi thi n6 khnng phii ld. dutmg cong Weil.
Nhtr vAy, ndu chrlng minh dtrgc giA thuydt
Taniyama-Weil thi ta ctng chrlng minh ilrrqc
dlnh li Fermat, bdi vi ndu dlnh li Fermat kh6ng
dfng thi se lQp trlc tim drroc m6t drrdng cong
elliptic khdng phAi ld drrbng cong Weil (theo
c.ich x6y drrng cria Frey), ttii vdi gi6 thuydt
Taniyarna-Weil.
Trong hQi nghi qudc td v6 bidu di6n Galois
p-adic, t-him vi sic dang modular hgp t4i
Cambridge (Anh) tt 21 ddn 25 thdng 6 nim
1993, A. Wiles d6:rg ki mQt b6o cio, chia lim
3 budi. Trong budi ddu ti6n, nhrl thudng lQ,
thinh giA ngdi nghe mQt cich lich str. Sang budi
thf hai, ngudi ta bit ddu x6n xao vi nhirng kdt
quA qu6 nm4nh". Brio crlo thtl ba di6n ra trong
mQt hQi trddng chSt nich, vi ngrrdi ta lirrh cAm
thdy mQt cdi gi dd rrit tlqrng dai sdp xdy ra. Vd
Wiles de kh6ng dd thinh giA phAi thdt vsng :
cutii brio c5o ctia minh, Ong c6ng bd chfrng minh
giA thuydt Taniyama-Weil cho cr{c dudng cong
elliptic "nrla dn d!nh". May thay, drrdng cong
elliptic Frey ndu tdn tqi thi n<i lii drrdng cong
nira dn d!nh, nghia l&. Wiles dE chrlng minh
drrgc dinh li Fermat ! Ngay l{p trlc, tin vd chrlng
miirtr ritra Wiles drrgc tEo-ng'6ao *rap tito gioi]
vA hdu hdt cric b6o hing ngiy d6 drra tin dd
vdo ngdy h6m sau. Kh0ng ai ci; thd thd kh&ng
dinh chic ch6n rang chrlng minh c6a Wles lA
hoin todn d[ng, vi rngi nhd todn hqc d6u cd
thd mic sai ldm ! B6t dttu tt thdng I n6m nay,
nhi6u trung tAm todn hgc ldn tr6n th6 gi6i s6
td chtlc hQi thAo dai ngiy dd kidm tra chrlng
minh cria Wiles. Tuy nh-i6n hdu hdt cic chuyEfr
gia l6n nhdt crla linh vltc ndy cho iing ndu Wiles
nham sai ldm ndo d<i trons chrlnE minh thi sai
Iarir ao cflng chi mang tinh"chdt 'T<i thugt". Con
dudng chrlng minh ctia Wiles ddy tinh thuydt
phgc, vi n<i lir kdt quA cria nhung li thuy6t dugc
xay dgng vftng chdc crla to6n hqc hiQn d4i, li
s{ dric kdt nhrrng thinh t{u crla nhi6u nh}r torin
hgc ldn. Dd vrron ddn kdt quA ki diQu ndy, Wiles
dp..:'qf-qs tf"" y"i nhfing ngudi kldng 16:'. |lsudi
vidt bei niy 16' morg crSc b4n dgc '.16 tu6i cua
t4p chi "To6n :rgc vil Tudi tr6" crinl, s6 dtlng l6n
vai nhtrng ngudi khdng 16 dd cci ,hd vuon ddn
nhrrng tdm cao tudng clulng khOirg v6i t6i dugc.
Chi n6n nhd rn6t di6u : dd c<i thd trbo l6n vai
nhttng ngubi khdng 16, c6c b4n cflng cdn cd g6ng
d6' dat ddn tdm cao niro d<i !
Cu6i cirng xin cd vai ldi an tri cho nhitng b4n
lo xa : dinh li Fermat d6 dugc chrlng minh rdi,
cti phAi nhu thd li chri gi d6 trdng ving dE bi
lim '.ir!t ? R5t may ii con gn d<i da klp dp d
trdng eria rninh sudt 3 thd ki, vi bAy gid chfng
ta d5 ctj cA rnQt din g& d€ trtlng ving !
NHIN BAr roAN ou& coN nAr pHEp s{GHtcH DAo
NCUTEN HUU DU
d truang PTTH ta chua duqc hqc nhi6u v6
ph6p nghlch dfro. Dd tang th6m ki n6ng ldm
bdi, chring ta tham khAo mdy vi dg sau :
1. Dinh nghia. phdp nghiclt dd.o :
Trong mgt ph&ng cho m6t didm O cd dinh.
Vdi m6i didm M c(ra mdt ph&ng kh6c 0, ta cho
tudng rlng mQt di6m M', x6c dinh nhrl sau :
' * M' niUg tr6n drrdng t}Lhng OM,
* OM OM' = k ( ft ln hang sd * 0 cho
tru6c) Ph6p bi6n hlnh cta m6t phing bidn m6i
didm M (;c 0) thanh didm M' xric dlnh nhu
tr6n ggi lA ph6p nghich dAo cgc (tAm ) O ;
phrrong tlch ft. M' gi li Anh nghlch dAo crla
didm M vi ngugc l4i cung d6 thdy ring, M ln
6nh nghlch d6o crla didm M' trong ph6p nghlch
d6o. Ntii m6t crich khac, ph6p nghlch dAo cd
phdp bi6n hinh dio ngtlQc trung vdi chinh n<i.
2. Tinh chrtt : - Ph6p nghlch dAo bidn drrong
ttbn di qua t6m (kho6t M tem) ttrann auong ttring
kh6ng di qt" t6m, vudng gric vdi dudng kinh di
qua t6m cria duirngtrbn vi crich tAm mQt khoAng
Id kld (d li dudng kinh cria dudng trbn)
- Ph6p nghlch dAo bidn drrdng thing k*r6ng di
qua tAm thanh dudng tron di qua t6m (loqr M
didm tAm ), ui drrdng klnh nim t€n drrdngvu6ng
gric v6i drtlng thEng vi qua tanr, dtldng kinh ld
nlp (p ld khoAng cach tr) tAm ddn drrdng thing).
Ilinh I
Chtlng minh : Nhin tr6n hinh r1Q-ta thdy
ri4r LOHM', LOMI + OL OH =
= OM.OM'= ft (ndu kle,d s6, O li t6m) thl
M'h arih ct&ra M vir ngrtoc l1i, M ld 6nh cria
M'.
Do dri mqi didm M fi6n dudng trbn kh6c O
d6u c<i Anh tr6n dudng thing vd ngugc lgi mgi
di6'm tr6n dudng thirtg d6u c<i Anh tr€n drrdng
trbn kh6c O, M "WrOn dudng trbn drrdng
klnh O/ = OM ..OM'/OH = k/p . M'nim
qglr dudngJlhang M'H r- OI :
OH=OM.OM'/OI=k/d
Tinh chdt dugc chfng minh.
3. MOt sd ui du :
Vi da I : Cho 3 didm 4 B, C tr6n m6t
dudng th&ng theo thrl ht dy.X6t dudng trbn
tim O bidn thi6n di qua A" B vd dung hai tidp
tuydn CM, CM'. Tim qti tlch didm gita H cr&a
MM'. (Bdi trong SGK l6p 10 cu)
So luqc liti gid.i :
C6c vl du trinh biy 6 ddy, phdn d6o kh6ng
chtlng minh mi dd b4n dgc t1t nghi6n criu.
_,"t. ,
.... ,,:
,]. i
;i
Htnh 2
Nhin hinh vE (h.2) ta thdy AOMC:u6ng d
M vir MH t OC nln CIryP = OH . CO (CMz
khong ddi vi CItP = CA . CB (khong ddi))
VAy n6n I/ B Anh ct&La O qua ph6p nghlch
d6o tAm C ti sd h = CIP.
TAm O cria dtrdng trdn qua A-B n6n 6 tr6n
dubng thing trung tntc doan A-8. Nhu vdy thi
I/ sE d tr€n drrdng trtn qua C, dtrdng kinh Cf =
= CA . CBICN (N ld didm gita AB c6 dinh).
Sau khi chrlng minh dAo xong ta kdt lu{n
qui tich If le dudng tr6n dudng klnh C/ - ta
lo+i b6 didm C.
Yi da 2 : Cho dudng tron brin k{nh E. Tr}
mdt didm C ngoii drrdng trtn ta vE hai tidp
tuydn CM, CM' vd mQt crit tuydn quay quanh
C c6t dudng trdn d A vi B. Gqi gao c:iuLa MM'
yg ABj K. Tim quf tich / tr6n AB sao cho
U.CK=-Clviz
(Dd thi hsc sinh g6i NghQ An, tg73)
So luoc ldi gid,i.
Tr€n hlnh ve (h.3), ndu ggi trung didm cria
A-,8 le .I' trung didm cria MM' ld N th\ MM' t
J- OC t+i /V. Egi6c_qNlfl' nQi ti6'p drlgc n6n :
CK . CI' = CN . CO.X6b LOMC tacringsrf
CIF = CN . m n6n suy ,^dE . Cf = CIII2
kh6ng ddi. Do C c6 diirh, K ch4y tr6n MM'n6n
ta coi nhu K Ii t4o Anh, .I' ld Anh cta ph6p
nghlch dAo tdm C, ti s6 k = Cifr, hic dci f'
chay tr6n drrdng trdn qua C; dudng kinh =
= Cl,PtcN, (b6 didm C).
Sau khi chrlng rninh d6o vd h4n ch6 qui
tich, ta thdy qui tich I' (ld cung MOM'. Do
cdn tlm qgi_ tich f th6a m6n
CK . CI = - CIP n6n ta ldy ddi xtlng crla -I'
qua C s6 cri L Qui tich .I li cung tron
Mprtz
Htnh 3
Qua hai vi du tr6n ta thdy ph6p nghfch d6o
bi6n dubng th&ng thenh dudng tmn. Hai vi dg-
cudi li ph6p nghfch dAo biSn dudng trbn thinh
dudng th6.ng.
V{ dA 3 : Trong drrdng trdn cho tru6c, Idy
m6J didmA cd dlnh kh6ng trtrng v6i tdm.'Qua
A dgng mQt d6y cung ttyf. Tim qui tich giao
didm M eria eic tiSp tuydn vdi dudng trbn, dgng
tai hai ddu cria d8y cung
(Dd bai 110 Tuydn tf;p to:in sd cdp cttaPhan
Drlc Chinh)
Biti gid.i so luqc: Gqi hai ddu mrit cta dAy
cung qua Ale. K, L, KL L OM t?i C (h.4). X6t
LOKM ta cd OC . OIqI = OKz.
Do OK
kh6ng ddi,
A,O cd dinh, C
nhin OA du6i
I g6c vu6ng
n6n C chay
trdn dudng
trbn dudng
'kinhOA, Clt.
tao &nh, M li.
Anh cria ph6p
nghich dAo
tdm O, ti sd
-'\,\.
r' '"1
i ,\
,tl..i
[ ,o"' i' '*-.'-;..' J
\ .,' " "'r ,i
\ri ,;ri r.,, irr'
,.,'- *I' ' --/ fr'
.., - - t-. ,- _/,
,+t,
- --! I
i
I
il
-^)
,.i,---'
ni
h = oI?. Ta kdt lufln dugc M d tr6n dudng
thing t OA 6 B cdeh O mQt khoAng =
= Ot&lOe. Sau khi chtlng minh phdn dAo xong
ta kdt lufn qui tich ld dudng th&ng L OA 6
B c.ich O mdt kho6ng = OtP lOl'.
Vi dtt 4 : Clto drrdng trbn tdm O, b6n kinh
B vd mdt didm P ngodi drldng trbn. Ve mQt
c6t tuydn tt P c6t dudng trbn E A vd B. VE
hai ti6p tuy6n tt A vd B c6t nhau d C. Tim
qui tich tAm dudng trbn ngo4i ti6p MBC.
(D6 thi HS gi6i NS6Q Nnh 1983)
So luoc liti gid.i; Ggi tdm dttdng trbn ngoai
tidp M.BC lir II, giao cilc. AB vd OC la I. Xdt
L OAC cd A vu6ng vd. N t OC n6n cri :
oL oc : oA2 = ,l?2 (kh6ng ddi)
Drrdng trbn qua ABC c{tng qua O n6n -Ff l}
didm gifia OC suy ra :
OL OH = Rzl2
Qua dAy ta thdy O li tdm ph6p nghich dAo,
ti sd A = R2l2,,H ln Anh cria .[.
/ nhin OP drt6i m6t gtic vu6ng n6n I nirn
tr6n drrdng trbn dtrdng kinh OP.
Ta k6t ludn duoe If d tr6n dudng thing l-
OP crrch O mQt kho6ng ld RZ|\OP. Sau khi
chrlng minh phdn dAo vi gi6i han quf tich ta
kdt luQn :
Qui tich H ld hai nrla drrdng thing Rx vit
Rr', J OP c6eho m6t khoAngld,Rzlzpo (Khong
tinh R vi -R')
.i '-ir:.: -'r.t. -
:-
,' ' itr|. .rr .
l
:
,"
,"'
, ','i. i
:
I
j.
!..
Hlnh 5
Qua b6n vi dg tr6n, ta thdy cric bdi c<i nhtng
n6t tudng t{ nhau. Ndu kh6ng rEn luyQn ki
ning nhfln bidt qui tich bing ph6p nghlch d6o,
c6c ban se gAp nhi6u khd khen trong khi gi6i,
nhdt ld giii theo phirong ph6p kh6c.
i: : .:: i.:,'i.i ;
TRAN PHUONG
Thd ki XD( nhi To6n hgc Drlc Canto (Cantor,
23.8.1829-10.4.1920) cha dd ctia "Li thuydt T{p
hgp" d6 ph6t bidu nguy6n li ndi tidng v6 tinh
giao kh6c r6ng cria cac tQp hgp Compac (hoQe
ldi) l6ng nhau md mQt tnrdng hgp d{e biQt ctla
nri Ii: "IVlgi ddy dc doan thing ldng nhau trong
R cd giao kh:ic r6ng tric lh: Ndu c<i R ) Al )
Az ... f An ) An*r f, .., thi O An + <D'
n:l
Ndu cti th6m di6u kiQn thSt nhau trlc Ii
€
tim lAnl = o'thi n An la 1 didm.duy nhdt"
n+m n=l
Vi du : Ndu An = [O,t
= [ 0, 1] ;Ndu O, = [t
OAn={1i
n=l
Nguy6n Ii Canto tuy don giAn song drtgc
srl dung rdt hiQu quA trong crlc bii toSn chrlng
minh sU tdn t4i. Bni vi6t nny gidi thiQu ki
thuit sfi d\rng Nguy6n Ii Canto v6i hy vqng
n<i s€ ln mQt c6ng cg t6t girip cric bqn hqc
sinh trong c6c kl thi hgc sinh gi6i. Dd cho
tiQn d todn bQ bii ta quy udc: "do4n" tfc ld
"do4n thing trong R'
Vi du 1 : Chrlng minh ring 3a € R sao
cho 1/3 < {an} < 213 Yn e N ({o} ld phdn 16
cria sdo : 0 < {a} = a - lal < l)
Giii : NhSn xit : "MQt do4n A cd dQ ddi
) 1 lu6n chrla mQt do4n con A' nio dd sao cho
1/3<{a}<ZfBVaeL"'
"10 13
Ydin = 1. Ldy Ar = [;,;] thi
t2
5 < {a} = EV"cArTas6xdydungArc
9
Arsaodroll3 <{* } <f v"eAr:Ma€Ar=
r 10 13r ^ 100 , 169
: LT, -r_l non ? = o" * s. D0 dai
.il "ll,o'=
-1.1+1lthi
nnl
doan [#, f, ] = S, 1n6n3ArcA,:
.)
{"'l = ; V a c. Lr... Gie sr1 ta d6 xAy
1
5<
