Toán học và tuổi trẻ: Số 138 (4/1984)

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VIOT NAM

I Ong

Ph6

BAO I1A

bttu"lQp; NguY6.n Cf,nh Todtt

l'bns bi6n r{P; Ngd Bqt Tu'

rirANc MOl Kt

Tru s& : 70 Trhn liung Dqo, I{t NQi

DAY n6i: 52845

N6i. chuY6n rdi c6c bqn tr6 YOU to6n

bf,n

,, Hinlt

R6 ring chring lg va,

cfeh nhin lih6c nhau vB

xa hsn nt'a il6n ntr{ng

i:.,.\....

\'*,ufi 'FR{irs nn*ffi.u TAIYI

'fl RONG toSrr lrqc c6 r5t nhiFu crii lic rn6i diu

I ,rgir" r{.t la,.,hong khi c6 lhdrn hibu bi€lt mdi'

' chch nhin n'rtri thi Iai hdra quen' Ciu chuyQn

(( Vdns t;on nhiEu tirm )) sau dA1' thuQc vho loqi

ci.u- ciu-vin vb nh*ng c5.i ta mA quen d6'

Gi{ bgn cO tOt du6c di tim lihip th6 gian ctng

kirdng bao gid tim duoc rdng lrdn c6 quh TmQt

tenr ;6ri* io*ng trbn ban tim lir' .vdng trdn mi

bqn .dE lrc.rc trqng chuong tr:inh ptrb thOng' Nhung

,-,e, llAtt .a o-rqt::c6eh uhin^rir6i, circh nhin cria

nhir4g sinh vjit < C[rrr]) eh&ng htl thi . ]rqn'c6

nqal'"nhitng "'Oqg ti'dn l/ai tArn vi hai b6n kinh'

i,ffir;-:l;ti 'ai ,icn" )' li: nh&ng sinh r'{t tu6,g

t,rlrng:sOng trBn mQt m:f t chu niio cl6' kh6ng bao

pio riri kh6i."nr[t cdu vir cflng khdng hay bi6t ti gi

il tle gini ir rigoai nr[i chu' D6i lcri Irhirng sinir

vAt, el6 thi r0 rilng in6i' vdog trr.rn 'cllu c6 hai

,tifnr la hui l-,5rr I'irrlt (h'.1 )' Ddi r'6i clr0rrg ta'

Snhirng ngudi cri tlr6 tlirng ngoii rnfrt ciu -nrir

nhin "vdri tlri,ducrlrg cong {C}'cirng td m61 vdng

lrirn nhuns chi c6 rtQt iim le O 'vI rnQt'b61

r.irn or' iunt ,idi r'6i'sin'h v{t ((Clu) thi (c)

- - NGUYEN ci,xs 'ro.tN

Iir mdt 'r'orrg tron cri hai teirr A?-A vI hai '

liinh AM ri ^B,41.

sinh r zit ( Ciu D c6 hai

( Khoing cich'. EE di

thd gi<r) co vong trron

nhibu tilnr, thirn chl vd sd tArn, chring ta h$.'i: suv

nglii thOm r-b n Khoing clch )). 1'r.ong diri sdng

hirng ngiy, trr, ha1' n6r : ( K ho:rng c6ctr ) tu noi

nirv rldn noi kh;ic, ( Khoing circh l vE trinh el6,

(Kho n${ cich r, vB tu' tirr}ng r'.r... trgav < Iitro&ng

cich ) tri noi n:\1, tl6n rrrri lihic"cfrng c6 thb c6

nhiiiu n!\i dung: ( Iiiroing circh )) theo ilud.ng chinr

bay, e Khr:&ng cich )) tireo tlrrorrg bd, ,r Kho6.ng

cich u theo rludng thrir v r-.'" Nlrrr .,.iy < Khoing

cich'r co ihb c6 rdi. nhiiru rrshiLr nhungdbu thdng

uh{t o ch6 noi I0n'nrri,' Llir l<hAc,nhru gifrll i}oi

srr v.i.t yii rn6t ;;htronq didn rrio d(r : (( l(hoing

circh > cing 16n thi hai stt vAl c-itng r* sr,r ( clb*g

nhiit r x(rt vE .llhuorg diin t16.

Vdri <rlcll nhin tr0n, ia cri ilrb xrly rlung n8n

klrS'i nitn < Iihoiipg cach ;r tlirn irri nhfrng rdng

tron cd :ihrdu t:rn:. Stu dAi'l:i mft 't,i clu: trong

rnlt phing thdi;g thuirng, ta chon nl0t du'dng

thiing rz n:lo ilo riii ggi gOc .44'8 (irong di, .1'

L\ il6i xr'rng ci a ,4 q ua dircng thr'r ng a) l::

(Iihoirrg.6.1-,D giira bai ilit.m A"B (cic riibm

tr6ir during thlng a lifr0ng x6t r16n tro1g viQc

tinh t< Iihoing cach rr'r:d1,) (ld,r' g6c 8fi'i eIr:g

ilugc. 8':-lri ddi xu4g cril ll qua {r, vi

.t'-\

AA'lj * .r.ij?":t). Y rgl'ri* c&s < I'.hoing tich > r1

. '['a hfr.'r xet lem t{ Vdng lror: I tilr ,{ rli qua

.8 li clrrcrlg eong gi'.) T:r thi_r. ngav r',irrgrlo lli.

drrong thSng thdng thuung ,l'IJ ri i Lhi .,1 c6

rllnh, B chav tr0n drrd.ng thing ,4'B rlri hai

drr'&ng tiring A'd vi A' B. c6 dinh, do dri g(.rc

(dlnh'hu6'ng) giiia h*li dirring th&ng ndv klrdng

ri6i, nghia li ( khoAng c{ch, lir it ddn I khclag

.fAi. Nlru vA]' rtu&ng thing th6ng thudng ,4.'8

li m.dt <vdogtron> t6.nr.;{. ll*hung ngoii ti.nr A

I'a' ( ',,dng trdti > nil cr)n c6 r'0 sij tAm l<hiq nrii

qui- ticli iir rluong thing .{8'. Qun viv, cir ldv

rndt dibin r11 lihac .{ trdn drroriE ihirg ;18', tt'

su: IJ:zi_r- nguv ring A1 cing ia tirrr ( \'cng ttirn ]

.{ B (lr. ?) '''r'r i cirng t ll:i n l.:inh r nlrrr ,.ir. \rri

tbng qtrIt thi rar.ri iluring, tl*<rg th&ng tiruring

li ir.rritn vorrg.tron r c6 vd sri tilu nhung ctng n16t

bhn kinh ; qu_i, iich cirrr tXm riir chinh lir ilr-rdng

itiirg ttr$ng {iruiing c1<ii xrrng r-rii rlrir-rng di chr>

(rl1 a.

N(u ilrrcrns thing thCrrg thl.rrng ill cho vta)ng

g6c rdi e tlri < v$ng tr0n ) ttronq irng cr.r birn

kinh bing kh0ng; n{lu n6 s.tne song .. r.'i a thi

<t.r'dng Lrdn > trrrrng i'.rig co ban kinh Lnng ff 12.

Ban rloc cri thb chtlrrg nrinh rirng rnoi 1,dng

trc)n thdng thrrong c6 tiim trrir; ei lai Iir lrdt,

a dudng tlr&ng ). t,l tdu ta c6 l;a dilni ^,1, jl, C

trOn r.ndt vdng irdn th6ug tiruung, c6 tanr tr0n

6 thi ( khoiilg crictr ) dC b[ng tbng hai < iiLoing

cach., 3;'J r.i AC (giting nhtr rlrii rcri cic kiroing

c{ch c1ai -q6 tli6rg tirrrr}ng dC, .{8, BC tr€n mr)i

duonq thir:g th6ng lirrrLing).

DLrong th,ing lai l{ tt rong trdn ) cr.r vd sd lim

r-i,L r-ol c tri;n l:ri la ( rltrono thing ). Qui Ia rn6t

str ( tha_r' bac ilbi ng0i > rat la, plr&i lihOng c6.c

b4n, nhuns ciilg di iribu lirm sao.

Yi du ti€n dA1- Ii vi clr.r don giAn nli5t vE

ur6t l6p lih0ng gian 16ng rii co nhitu tinh chdt

rdt phong ptrrt tlo tric gil bli ni-r.ph6t. minh r.u

:iA-v dung thdnh rn6t ld- thuydt r:6 thb goi li. {{ l-i'

thuydt r:./'Lc kh6ng gian Cri6h - Toin D hay a Ii'

thr"rydt cAc kh$ng girn ui tuyQt ddi ilong )) (nhung

xin miEu giii thiclr ti],i sacr lai goi ii ( c6 tuvEt

dii ddng tr)

Uie u ddng niri r'0i c{c i-rqn ll t5.in qtrrn irong

c'ria t(cicir nhin ): trOr-r nh&ng slr t'At cir, thtint

chi rit cii, ndu c6 chch nhin nrrii vAn cc, ttrb

phit rnlnh nhiiu tliirtr rrrdi nrc r':r l.i'lhri.

" /l irlr 3

d6.v lii dd lQch gcc gi&a phrrong :18' (hoirc BA')

va phuong vudng q6c vrii &. tt6u ia dfnh hu6ng

rniit ph&ng vir x6i grlc 11 inh hrrung giila l:ai cludng

thing .,t';t vr\ '{'lJ rl,;a-v B';l vil 5'B) thi ta cri

kho&ng cic.h rlai sd AB (A truiic, B sarr) xirc

ri!nlr i.rrlri /ifi.

ilAU eryE TRt o

D0 n.i F,tt.\\(;

1a irlrili r: I.t yrl,li: ( llrrllg. ululi.n lv t;ur rtap

nE:-r",-0n 1i' Dir iq'iri0t (\'D rrgu.v0n li Dir.ichiOt

f trb xerl tl:0rri bii < Ngrr-r i-ri lic {-ri-r'ich-10

NGUY$N

rit)r {txt iotlli rr riltl

c Ii-i'tiri lrgc sinh giiri,

i lar: tlrt rrg S rrenr-iir

- KFisi

lrudng .ciurq nlr rr d6

'chtng {r llirrring vi

l-i' chi'r'r:g i*inh: t:ti

S noNr;

t aJ (::i

n lri

)

+:-',.-

2. LeTg Ltrc, l6p 12, trudiirg Qudc

Binh Tri ?hi6n.

3. Dd Quan.el *qt, Iivp l2CT, tluoirg

Su pham lld n-rli 1. _

'':

Gini t{trdng r6 nhiru hgc siiih dr3 thi

h9c FIu6,

ts4i ]rpc

.ll

Giei xrrdt eir: , '' '

- :-- -

l. Db. Drtg lihqnh, lup t0C l. trrung Ngrr.r ilrr

\'f,n Tr'0i, \hrr f i':rrrq.

2. ) gugin ?iin Dfrnr1 .\, ir'rp 10CT. Dai hne

r.\

I ong Il0p I la n,)r.

3. 'Irdn Ddg 1Jini, lrip 11C, IrLro-ng 1'rune

Giai nguni c6 nhibu ldi giii ha3 :

1. Ngu,j€n TiAa Dfing 4, 16p I0C1', trrting Dai

h.:e TBng hop IIi nt)i.

2.-'f rln flrirfn. l6pg0 Cl"r Tlulng Phan Chu

flinh. -ni .\in'j. ;

3. fld Qttang Sai. 'l iip l2ii'I', trrrd,ng I)ai hoc

Srr ph+m tta, npi l.

Giei d{c biQ.t: ye r'(tn phoJ, ttip 9E, rruong

Chu;i6p to$n" Qu.v-Nhop, irighla Binh.' ;

Giii truong dqrt t6ng s6 di6rn cao:

N/idI : Khdi C,lLrv€ulToirn truring })ai hqc 'l'bng

Nii: Truring P?'lH PIan Chu ?rinh, Da, NAng.

Ba: Trri'dng PTTH. Ttrung Yuong, euv Nhon,

l'ghia. Binh. -

ftr nnu

#*r@

BAi l/135. C&ri.ng minh'rdnq ndu sit N bdno

tbng binh phu<t ng c{ta l-3st sd noarudn.thi sd rSA+. *

bdig l6ng bini't phuonll cfia gg2. tgSS * I sd

ng.t ge n.

t.di gt&i. (Cira cic harr Tjrdn Qudn. Ldm 7'itttg

Giang,ifguy0n ?!ung Son,1{t CT Phirn Chhu'i'rinh*

Quing nam Ol\ "".i:ing). .." ,

'Ia chiing rninh bdi to6ri tbng qu{t : Ndu, .&'

la tbng birrlr phrron. cia n s.; nguS:0,, thi n. N

In lbng binh phu'orrg..iir (n --,iin,'2*1 sd ngu.v6n.

fa n

,2 2,

= (al t o2 -f .,.

+l(n?---2rr$z +l

+ -l + 2araz-1,..*2on- 1 c,)

-3',* . + (u}'*zorr, t r| ) 1

1.. Trudng PITH Trrrng.'Yuolrg.

?. Tnrong PTTI'J t.0 llbtrg Pliong,

IIb Chi llinh.

Giii khuydn khich: .

1. Ncug,;n Quang /)ing, lr;p 12C['. lru'ung Dai

2' Trinh'Ydn ftifu, l(rp t1C'1, tnrong,Lam Son,

3. Vd Eai ilr:Ai D\c, Ilrp 11CT, tnrring;,L6

iibng Phong. TI). Ilb Clri l\Iiufi.

4. f rang Ldm'.Bdng, l6''p 11CT, trudng 1,0

Itrbng Phorig, 'lP. Hb Chi Miqh.

5. {6 Thanh Ifting, i6p liCT, truorrg Le tidng

6. Ngutlin i'iit Quarig, l,;p 10C1 , tluring Dqi

herc Tbng holr i-Ii')rlgi,-

7. Iigugitt -tlinh \grin, i,.p lriC'f. {r'udng

Ngul'6n Van Ti6i. Nha Trrn!:-

8. NgugEn Gia Bdel,16p l0CT, tru6.ri}-!{ Iii)ng

Phong, TP. Hb Chi I,{inh.

9. n-gugan lluu 't'tidn, 16p {}E, P]'CS I a Lc.}i 2l

Quy, Nhirn, Nghia Binh.

10. lVgrrg6n Binli Ngpc X&ud, lirp 98, tnriin'g

Iliri ?huong 2,.Pliycu, Gia lai':Corrturi-r

"j.1. Pham XudrtTrunG, l6p 8A, trirdng PTCS

'f rrrng Nhi, Hi N0i.

Quf Nhon,

Thdnh'ph6

*t,otr-zozol * alr4-..., (r,1,-ze2a,, + rl, ) I +

... * [ (r: , -]a ,,.o * ,,1 )1 -

It-I li .

.rf

-- t r ! t' .c. I tt -n ,

- 1o1 -l- dr f ..: -i or)" -,- l(rr1*r.rt)' -l-

... -l- (.rt*rra)rl I l{oa*o3)2 + .", -i (,r2--irnl2l* '

...*(a -*,, )z .

' n-I ll'

ld thng biah phuong cria I * {n.-1) * (n--2) -F...

. n(a*l): \ -

* I:1 + " s6 nguy6n (dpcni).

Ap clung n = 1984 ta e6 lgi gifri bii to{n di cho. l

E.B.K."

, .nei 2/ 135 i Tran'g mlil cupc d.du c6l t1l'im

c&* n ddu ttrrl drqc git ,ao b&ng uuang ^2 6, 6 ,

tnlltxj u0t i+j ld dibm c{ia ddu th& i trang,trQn

gdp ddu ihfi j (thdngi 7, hial. 112 ud. thua td 0.

diam). edc 6 tren d.adng chio chinh qht sd 0.

: 'Chting mtnh rtinq ntu cd thb cdt bfrig uabng

Itreo r$l thdnh.lral trinh ekfr nhd?, oo ilung 1lZ.

tbng s6 dtbn cfia,r46t dtu ttti., t/ri s6 n ogiti ao

: ThAt vriv rlIt IV

(ni ngu-t0u). Khi,Jd

nlY : nc -i-

-::.

,.19C.1

\\,-

Lbl gtdt. cria\,frhn Qudn, t6p 10 CT, Di Ning.

"Gii sri dudng\ clt theo cQt chia bfing vu0ng

nz thAnh hai hinh elii nhAt nXk vi n X (n-t).

Ta lgi cit ngang b'ing vu0ng thanh hai hinh chr]

nhit ft X n vi. (.rr -k) X n. BAng vuong rlrroc chia

thsnh 4 minh A, B, C, D.

Gpi tbng sd tliEp-r-r6ng c6e 6 cria ,4, B, C, D

ruqng ung ltr'u'(A), f (q), T (Cl, T (D). Ta tta ;i

ring tbng.s6 aliBm eira 26 iX j vd j >,. r vdii +j

luOn luOn, bfi.ng 1'. YA.y ta ctl tbng sd dibm c&a

bing vu0ng li bEng (nz - nltZ ti

T(A) : tftz-k)12

T(D) = [(n-i.)z - (n- *)]12

Theo giir thi6t thi I (,{) = I (B) rd f (C) :

I(O) n6n

r(d) + r(D)*f(B) + r(c) - (nz-ili1

hay (,k2-'k) lz "r Un* t'f*(n - /i)li2 = (nz - n)ll

nghia lir; n= nz -4nk*4kz=(n-zk)2'. EiEu

phdl chtng minh.

NhQn r(l: Tdt ci cac bai gi6i g&i to.i tr.ra soan

t J{.p.r

Bii 3i 135. Hdg tim tdt c&. ceic sd aguy€n td

p,thda^ mdn |inh clrdl .goa'. ob.i mgi a, b ngugin:

az +bz z. p khi udrhi khi a :. p udb,. p.

Lbi gi&t. Db gi&i bai to{n tru6'c ti6n ta hdy

cht?ng minh bb dE sarr.

. Bb da t&inn t5, Fecma) Ndu p li s6 nguydn td

thi ;6i mgi s6 tu nhidn o ta c6 : (op - a) i p.

' Chung nrinh (Qui ngp theo a)

a = I : hibn nhi0n. Bb rlb dilng.

Gi6 sfr bb dE dnrrg vdi o ta phii chring minh

bb tl6 dirng v6i a + 1. Ap dqng nh! thuc Niu ton :

. ^n-1

' "..-rL' a*7-a*1,

- (oP .a) + c1 oP-1 + ... + CP-1

rrongd6,l :ffi#:kl!ip

(l < 4 ( p). BB db'dugc e hir.lg rr.rinh.

DB giii bA.i totrn..ta chi chn tirn ldt ci chc s6

nguy6n t6 p sao cfo : v6i mQi a, b ngui'61 mi

a'*b" i p thi,r j p r"a b t p.

* p=2 kh6ng th6i mdn bii tofin vi v6i a, b I€

ta c6 a? + Az i 2 nhung a, b, ! 2.

n p2 3: *)p - 4k + 1: Theo dlnh l1!- Yinxoo

(bb dE 3 bei 31119) tac6 (p-1)! + 1= 0 (rnodp)

"+ (4&): *1 = o(modn). MAt khac

.ltr = :1(mod p)

lk-t - -2(mod p)

:k+1--2k(modp)

-9 (4lr)t*1 * 1.'2...2k.(2/r+1).. .1k+1 =[(2r{) l]2+1

= 0 (rnod p)

nhrrng 11 p. VAv tit ci c.6c s6 ngul'0n t6 clrng

4k + 1 iIEu kh6ng th6a mdn diEu liiQn biii toirn.

+) p = 4Ic + 3. 'Ia sE chung rninh bing phin

eh{tng p th6a mHn di6u kiQn cria dD.u bai:

Gia Sri c6 tr,0 $ Z saor qlrb a2 *bz ', p nhung

chirng han"a i p, Iihi d6 b : P' l

tlreo bb dE ta c6

a(aP-l*t) i p .+ oP-l -1 i p

b(bPl-r),p+DP-t-t:p.

va-v oP-i + DP-l * 2 ; pha-v

k] )Zk+I + (oz)2lt *{ - a = ,az +b2\ A-2 : p

ma a2102 ', f 4 2lt , : t-t, Ij' vi p ) :.i'

\'+y c5c s6 ngu-i'0n t6 p c6 dilrg'tr&*i lir nhirng

sti phii tim. '

NhQ.n rit: C6e b4n Ngugln Yi(t Quung (i0 CT

DIITIi), Ngagln ab ann Ilgugdn(10cT' Dn i'iirng)'

Ld ,lnh Son (tOCt truong Phan B6i Cheu - Di

NBng) c6 ldi gi&i t6t.

f 'T. tl

.sai 4trrS , Gi/ s*t*r, er,.l,. Ao Id cac ,tino

mdt da gitic d.?u crin O lrt tlibm t*17 ! ,bltt ltoah

cita nd,- Chang mtnh r&ng cd lt nhdl tn\t goe

ai?ei thda mdnt ./\\ ,

:rt (t - r/n) (.{i0,r; _( :t

Ldi gril (cta nhiEu b4rr), :

. Gia sr? d'1 ld ttinh g5n O nh5t. C{c drrong ch6o

Il:{i chia da giAe dBu thdnh c6c tanr gi6cl{1'4rAx11'

l<-k 7'rfD.'f *

11h&ng ehi6c lilng lr) eic chri th6 < tl[ng tlong

ire,i l'II vir T'I s.i ;1). ilai bAo nAy nrLiSn gioi

thi6u r,6'i c5c ban 1 nguy0n lj. ci:unu rninh thu

tr.r.cri thb sfr drlng kh6 t6t cho ritt nhiEu L.jri toAn

thude c6.c dang hoitn toin khau nhau, il6 lu nguy0n

li ( Iihli rihu cuc tr! u.

NOi dung cfra ngrrycrn lj- phAt bibu nhrr sau I

'[ronE m6t tdp hgp ttftu htrn (khdc ring) cd.e

sd thurc lrrrin'cd f,rrE cftcn drge s6 bi atidt ud s6

tdn nhd,t.

\ri c6ch chirng rninh rdt drrl gi6.n n0ri nhrrirng

eho,b4n dcc tn thgc hiOn Lir.. Tic giii rh1 xin

ph6p minh'liga r.iQe 5p dung nguy0n lf urav bing

cAc vi du cr.r thb:

Vi d.tt 1, n bau hoc sinh ttri ddu bi,ng bln theo

nguyEn tic ddu vr)ng t{'dn. Chilng mir-ll.i ring

ludn c.6 thb xdp cl n b4n theo hing clgc sno cho

ngud'i rlirng trudc thing lguni thing liE sru.

Gi&i: XOt tdf ci eic clch x6p 1 sd ban thrirh

hing doc sao eho ngLrdi iiuno tru6c th6ng ngrcii

rlrirng liE sau. \ri clch xdp nhu vdy bao giri ,.irng

tDn tq.i vi" s6 cSch xdp chi li hiru han nOn theo

ngu;rOn IJ. kh&i tiiiu cnc tri ta.c6 thb chon cich

sdp cri nhi6u b4n nhflt. Ta sE cirfrng minh ring

c6ch xdp dd phai c6 ci n ban hoc sinh.

Th{t v{y, gi& sG c6ch xdp ci6 ehi gdui co

Ir ( n li4n; theo thrl tu Ie ,{r, ,42...., :1,(.. Khi d6

xdt bq.n .8 khdng xgp trong hdng. Theo nguy€n

t{c d{u vdng trdn, B ph&i c16u vdi clt At, A2,...

AA. Ta tl:{y I kh6ng thb thirng lr vi ndu thing

thi ta cld" c6 aluoc 1 cieh x6p iihiEu hon A ngroi

li. B, At' A2,... A1; trAi vcri c6cli chon At, A2,...

d.k. Do do B thua ,4t. Nhung khi d6 E khdng

thb thfrng i2 I'i ndu thing ta lai c6 c{ch xdp ,_li,

B, Ar, C_s,.,. .A*. Do d6 B thua .{2. L&p luiin

tucrng tu ta suv ra B thua 16. Nhung khi dd

ta lai cirng c5 cfrch r€p cldng h<yn : ,41, A2,... .Ayr,

B. Trong moi trudag hop ta clbu ttrr{-"- vd }j.

vriy 1c phiri bing n (d,rcm).

Yi da 2. Tr€n m5.t phing cho 2n alibin khOng

ed 3 dibrn nAo: th&ng hdng, trorg d6 ed n dihm

rniu t16 r.{ n ilibm m[u. xanh. Chuug miqh ring

tbn tai cich n6i t6t ci cAe ditsm d6 .rrOi .a" Oib*

xanh brli n ito4n thfrng klr0ng c6 diBm chung.

Ci6i: x6t ret ci e6c cd.ch n6i n cip dibm d6-

xrr rrh bdi n tloan lhing. \'i cAch n6i 1i tdn tai

vi sd c6c c6.ch ndi l:\ h&u llan ta chon c$crh ndi

c6 tbng dO dni e6c doan thing ln ngdn nh6,t"

I'a s0 chirng minh d6 I'i cich n6i phii rinr. Thai

r,f.5' ndu c6 2 doarr d,-Y vi 8I' ci't nhau iai rIiEnr

O ie, I} ruii-r ri6 cdn X, I' miu xanh). i<i,i Ao

ndrr ta thay ! rloan .lX vd BI' b&i 2 rloan .{Y

vi 6X vi giir nguy0n c6c doan kia thi do

,4.1' + rJX < (.4.O + OI ) + {EO + O.Y) -

. : ix -l- 8l. .,,

ta duqe I c{ch ndi rnri'i cd tbng d0 rtii cSi doqn

be hon. DiEu niy lr6i vdi c{ch chon ban dAu

crin chring ta, r'ily srUr rft dpem.

Vi'du 3. (Brii'toin Xir{-r'ex-te).

l'r'6n rnirt phing cho n cliBnr (n ) gt. Biet r&ng

rntri itudng thing tli qua 2"clibm bitt lij' dBu di

qua rn6t dibnr thu ba. Chri.ng minlr r*.ng cS n

tli'irrr thang hirrg,

Giii : Gi6 sri' n clibm tihOng thing hdng. I,lag

v*i lrrOi di"bnr tt xit circ lihoilng c6ch tlucrng tri

dibru rli.r t0i t{t ci c6c duong thing di qua 2

tr"ong sd t'ii.c ilinh cdn lai. Vi s6 kho&ng c{ch d6

ii hii'u l.ran nii. s6 dibnr ctng iriru hq.n n€n ta

cri th'6 ciron l<hcirng e6ch b6 nh{t trong tf,t c&

crie iihoiing cich rftroc x6t. Goi ri6 in kho&ng c6ch

tu tlibrn A ddn ilud.ng tli5.ng Af. Ha AIt LBC.

Ihto diirr liit\n diii b:ii, ditd'ng rh&ng BC cdn tli

tiua iii?.;nr thir ba O. Ta thfly hai trong s6 ba

rti6m B, C, D ph:ii nlm rb nr6t plrirr cria dibm

il . Gii si"r C vi D ni.n'r v€ I pliia cria 11. XEt

latr giSc aCC co?tir rr6n ltl,oilig.cach tu C t16n

;iO hd ho'rr lihoing cich tu A dr_1n CD tr-6i r.6i

crich chon A, B, C.l'nong tu ndu rt vi D hay

B vir (.' & rL cirng r:rot plrir criu II, l[ ctng suy

ra kho6.ng c6ch trt d r16n BC khdng ph'ii I:d nh6t.

[{{u thuin cht'rng t6 t{t ci cire tlibm, dbu eirng

nfi.m trOn rn6t ilueng th&ng.

. -t-

:*.--..*. .- ..,***.":l -

A r':

Yi dq. zr.. Chung minh r&ng v.}i mqi sd n

rguy6n > 1, 20 - 1 khdng chia hdt cho n"

GlAi: Gi& sfr c6 s6 n)I seo cho2'* l!n

Khi d6 n 16. Goi p li" udc srl nguyEn td bd

nhdt efra r. Do p etrlg l8 n€n (2, p) = I vir theo

ilinh 11, Phic-ma ta c6 2P '- r, O.

Goi ,lr It sd trr,nhi€n bd nh$t eri tlnh chf,t ld

2k - Ii p. Ro ld.ng ft (p -t < p. Ta. chr?ng

nriuh khi d6 n j A'. ThAt vAv n6u

n'= lig * r vcvi O ( " {i,"tf,i,

2t' * r =(2t' ) q .2I' = (m.p*t){ .2'

-(mp+112r. ' ,

&Ie 2n - r i p vi p l:i u6c cria n ren tn suy ra

2- ip r-Oi 0 ( r { ft triri vtli eSch chgn .k,

lr fi

Toán học và tuổi trẻ Số 138 (4/1984)

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