Tạp chí Toán học và Tuổi trẻ: Số 233 (Tháng 11/1996)

; hay li

u E,_[J

v6i

J

uv

Tt dd P, d4t cuc dai 'khi r = r' "--

32r

b) Gsi ,I' li dbng qua tr, I,o ii ddng qua doan mach AB cci chrla R1, R2, g, R3, R. Ta crj t, 7 - E-U r =t-tz=--T--E;= E,= 8o #vir,= #

NhQr? x€t. Cdc ern crj lbi Biei dfing z Trd.n Sy,ipon Ha, 1l Ct, P?THns.rrgkhidu, ere&aag ffiinlr ; Phan Vdn il*c 1"L"{,, pTTi{ phan gOi Cxr&u, Vimh, F$ghQ s^:e.

h,{Ar A.NFi

48 -r = BZ +r --'r = 32O, v&do dtir = 16

= ACz +AB2 _;@Ap _Be) suy ra AP=ry ^ b2+c2 e.2 ma= 2, - 4' _ Q.dd.t_cfuph trinh ldy I t11 nhi6n vi cho th6y 16 ydu td cdn tim, cdn-c6ch trinh bdv 2 lai sfi minh.mi iac tX gla dung dring ki thya! -chrlng dE dirng trude dri ddi v6i-dinh li ham s6 c6sln. Hon ntta cA 2 edch n6u tr6n dpu cd t6e dung cring cd dinh nghla tich v6 hu6ng crta Z veet6 mA. crle tdc gie dA dt{a ra. ,_ Cu6i ctrngxin cd rnQt d$ r:gh! v6i c6c tr{e giA : N6n cheng vi6c dua dlnh t{ h6m sd sir:" l6n tril6c, g6n vi6c xAy dung dinh Ii hAm ed e6srn v6i vi6e x6y durng cQng thdc tinli dO dei trung tuydn tam gi6c erhtt thd tV d6 trinh bdy trong n}:Ung:rem trutdc d6y"/.

nx-1

NcuvBtl utuu otjc *'*!:

cho

b) lim a, , n+*a

SE RA Ki XAV

thrlc f{x) = xa + 4x3 - c2r2 - tzx + l' H6y tinh tdng v6i n li sd nghi6m vd r, Id

S = ! -:

Bai T fi83 : n 2r?+l - ,!!r(xrt - 1)z nghiQm cfia da thr3cf(r)'

ooan rri6 pHrE.. (Nam Hd)

Ehi T1/233 : Cho daY s6 nguY6n an *z * a, - r = 2 {on n t * on) Yn > 1 Chtlng t6 ring tdn tai sd nguy6n M kh6ng

{ ""} ff = o th6a rndn :

BAti T8/233 : Cho ba sd thrrc a, b, c th6a m6n di6u kiQn az +b2 * c2 = 2. Chrlng minh ring :

M *4an*r.an ld sd chinh phr-tong Yn > 0

DOAN Q{JATG]4ANH

1)lo'rb*c-abcl<2 z)la3 +b3 +c3 -tubcl <2{2

ohu thu6c z sao cho : '

Bhi T21233 : Chrlng rninh ring n6u ")

(a+c){a+b+c) < 0, thi (u - r)i;;+?,["of '

{IIdi Phdng)

Bari T3/233 : Cric sd nguydn kh6ng dm, a, b,

BAii T9/233: Cho tam glecABC cci c6c c4nh A, iiii Artdng cao vi trlng tuydn trrong ring l;-h;,;;,h;,-;;;,*u,rr,, diq"n tich s' clrng ", rninh ring :

c. d th6a min di6u ki6n : '

a)a*b*c>2W.'{S b) tr.,.m.! + hu.nt! * n,".mf, > g {F's2'{s t-ti6lc NcQc vu

f"z + zbz + xcz + 4d2 = 36 (1) lzoz +b2 -2d.2 = 6 (2) ii* gia tri nh6 nhdt cirap= o21162l-"2162

'"fi##r'f"

Bhi T10/233 : Ciro trl diQn ABCD vi mOt didm P sao cho P A2 + BC2 + CD2 + DB2 : i-*+ coz+ DA2+ AC= P G + Dfr + AB2+ + BDz = P D2 + ABZ + BCz + CA2, Ct riog minh ring tdn tai trong khdng gran

m6t didm M th6a m6n

BAi T4l233 : Cho tarn glilc ABC.tlhg+' Csi (0,1) la drrdng trbn tam D, brin kinh bang 1 ngoai tidp tam g16c ABC. - Chfn! minhrang : a *b * c > abc' Ddu "=" xlY ra khi nno ? .

N vAN TR6c

(Qudng Ngai).

AGr= BGz-- CGr= D6o Trong dd G 1, G2, G3, G 4ldn lrrqt ld trgng tdm cta oic trl d iC\tt'' MB c D, *1?*-T?fl"lMAB c'

(Btnh Dinh)'

^ Bili T5/233 : Cho tam girie vudng ABC ( ) = 9V ) drrdng cao AH, trung tpydn BM vir pt ao eia'" CD d"dngguy-tai rnQt didm' Chrlng {5 - 1 minh ring sinB = z

Tt dci suy racdchdtrngm6t tarn gi6c ccj tinh

rnAN xuAN PANc (Nan Hd) n

vr oueic D,NG (Bdc Thdi).

cAc oii vAr li BAi L1l2S3 : Tt 2 didm A vi B c6ch nhau 100m. xe 1 vi xe 2 cilng xudt phiit v6i ctr-ng vfln ;6;;': :,;= L0 m/s' x"e 1 di tLec hrrdnghop v6i eS iAcOOl. niAt ring 2 xe so g?p nhau d C 'H,ey x6c dinh : ---

cAc l6P TrIcE Bei T6/233 : Cho m la s6 thgc duong' Vdi m6i n nguY6n drrong daY sd thgc

- ifuang chuydn dQng cria xe 2? - Th,di didm 2 xe gAP nhau ? - Tga do didrn c ?"

I e,,,i\ f = 6 d,roc x6c dinh nhu sau :

,1

chdt n€u tr.n'

an,o= ! on,i*t= an,i ('*;;o,,,r)

l=0, lr...n-l en,n2fi+fvdimoineZ+

BBi L2l233 : IvtOt diQn tich didm o - *2. 10-s c drlns c6ch tdrn kim loai phing irOt aat *6t Lt oang"o -= 3cm' H4v 16: dinh lgc iJo"iiacsi,r" diQTr tich s. y? t6+ kim loai dd khi dEt chd'ng trong chdn kh6ng i

ltl

-

-z+

NCUYEN DUY TRUY (:fhdt Btnh).

pHau soNc ouYd,r. (Hd N/,i) ooi

i$

2) Gie sil m > tr . Chring minh : uj on.n. * 4 vdi mol rL e' &'

)

1)Provethatifnr.,{r"t+L

r g

tl a ^

fi't

u) on,r, <

!.2x!+t _ ,y ,4,r*r,

",'*:,,*=#' T71233. Consider the poll.nomial f(x) = xa +4F -2x2 - 12x I 1 . Calculate the sum t : Where n is the nurnber of roots and

-

"zi

T'8/233. The real nurnbers o., b, c satisfy a2 +b2 + c2 = 2. Prove that : l)la+b*c-abcl <2, z) I a3 +br + c3 -Sabcl < 2{2 . T97233. Let h.u, h6, hr' and nzo, trl6, nL. be respectively the altitudes and the medians of a triangle with sides @, b, c and let S be its area. Prove that :

@N$Ill{il we" uQlL itst * ";is?#"a7nir*> 1 prove rhat: n? * lfor every n e Z+ FOR LCIWER SECONDARY SChIOCILS - T1/233. Let the sequence of integers { ""} ff :, satisfy an + 2 * an _ I = 2(an + I * an),Vn >- l. _ Pro_v-e that therq exists an integer M, not depending on z, such that is a nerfect =o,ru"J#BJ "h , a. TZl233 Provd that if (a+c)(u+b+c)<0 then (b - c)2 > 4aia +'b'+ c) . T-8/2P3. Thg-non positive integers a, b, c, d )", + 2b2 + Sc2 + 4d2 = 96 (1) lzoz+b2-2d2=6 (2) jz . Find tile least value of P=a2+ 5za T4l28e" The radius of the circumcircle of an acuteangled triangle {BC equals 1. Prove that a*b*c>abc. When does eoualitv occur ? T5/233. The'altitride AH. the median BM and the4ngled-bisector CD of a right triangle ABC (A=90o) are concurrent. Frove that sinB :

a)a*b*c22W{9, b) homf + hrpt + h"mf; > s 1{3 s2 {s. T10/239. Let be given a tetrahedron ABCD

x; (i = 7 , 2 , ... , n) ate the roots of f(x). satisfy the conditions :

',1 5 - 1' -, triangle.

FOR UPPER SECCINDARY $CHOOLS T6/233, Let nr. be a positive real uumber. For every positive iirteger n,let { a,,. il i: o

,1

Deduee from it the construction of such a

be the sequence of real numbers defined by : _1' ^ @n,o - tt an,i: an,i+,(, *;non,i) Q = 0, 1,..', n-L).

ffiA,* dure^gt e6a4t tituib ffiffiffi@mfriilWffiffiffiffiT$Nffiffi

o6 Nnrl NGec *:no'n'":1'

+ +- +

= (IA + IC12 + {IA + IC)z : 2IA2 + 2iC2 }eay bz + c2 = 2nlo +, (b'. su, ra cong

thtlc d6i vbi m " -- "-a ---- i- ---

--> +

and a point P such tlat pA2 +BC2 +CD2 I rtg2 =pg2 +CD2 +DAz I 4C2 = =P0 +DA2 +A824BDz - PD2 + AR2 +BC2 +CA2 . Prove that there exists a point M in space such that AG, = BGz = CGs: DG4, where Gl , G2, Gj, G4 are respectively the centers of gravity of the tetrahedra MBCD, MCDA, MDAB, MABC.

..-t

Trong phdn "H6 thrlc ktone trong Tam si6c" cua srich-girio khoa l'dp 10 Ban l{hoa hbc Tu nfiion, c?D tac gie de xAy dgng c6ng thric'tinn AO aai dt.tdns truns tuv6n tam siric benp nhtronE ohd,o vecto-(nhtr da tlm ddi vdi dinh iffibm s5?tisinl. Day lh qich lArn ggn vi. d5 hidu. Tuy n-hi6n, cdn ccf i:rich trinh bdylhric hop li hon.

Trddc ti6u, dhring ta'iem c6c tdc giA vidt : "Ggi.I l& trung didrrecanh BC. + -+ Tacri: CA=IA-IC -..) -+ BA=IA-IB -t

lry

O dAy, theo t6i vi6c viSt CA, B$lh4lrrra trr ub6a*"i hoc sinh thildng vi6t AC,AB ho4c CA,AB (kidu ho6n vi vbng quanh). Mat khr4c, muc dich li timmocbn b! "khudt". Theo tdi nghi, cd thd trinh ba'y theo 1 fiong2 crich sau dAy :

Qg dci :* CAz +BAz = 4+^++ (rA - rc)2+ (rA - rc|t:

(xem tidp trang 7)

a) Cd.ch I : Goi f lA trung didm canh BC. fheo $S t$1c trung didm ctia doan th&ng :2 AI = AB + AC -r8 * 4AIz : AB2 +ACz +2AB .AC = ABz + ACz +

-

codnfr cfro cdc 6aru cfrudh 0i tfri r.'do dai froa

$ru$ $uru$ E{cm PHAH vfmrx un#r af,qn*

t

h

Brr6c 3 : Kdt lu4n Lint Sn = [ 161ax

a

tt+ @'

o Nhie lai : Dlnh nghia tich phAn (GT12) Cho/(r) xde dinh tr6n [o, b] 1. Chia dogr [o, &] thenh z p]rdn b&ng nhau

-

trudnshqp

1-

,2,

,n,

2(b -a\

bdi (n + 1) didrntti",, 1i =ffi1nhu sJu ' xo:a;*l=a*

.n ls^zlr f \

I

I

ffo= a <.rl < x2 1 ...< xk< ,..4xn = b v6i b -a n ;o2:r*-,,, = u

-_

-

drJ&"jIsJEalfI ll tl#fff,*+p B"r.iAe 1: BidfrddiS,?thinh d?ng s,=;[r(;) *r(;). nr(;) i= ,1. - n.L-t \n l'

b--a

eua"d,=qii ra him /vd chring rninh f li6n

tuc tr6n-[0, L]

=6i=a*i-.Tinnf({,)=f

I

...xn= a *n .* 2. Ldy Ei : xi e Lxi - 1, ri1 i = L,n , \o*i

3. LQp tdng

NcuvEN TI{ANH GIANG (Hdi ltwng)

b-a, " ) s, =) f(6i) (xi - xi - ,l = .Lr (a + i*)

"

l=I

/!

n2+12' n2+22'"'' n2+n2'

\ b*a, n I

n L'\

Tilrh Limsn n+*a

Et b-a b-a. x-=-lflc*-\+ n 2(b-a\. ,

b-a.

*r (" +i::---2) *... *r (" *".7) ).

f@)=i -

ra cd : s.=:t:*.xet ,",=r

1* (;).

tr6n [0. Ll + f(r) li6n tuc tr6n [0. 1l

( S, : Tdng tich phdn (tdng Rieman) eria hnrn / rlng vdi phdn ho4ch d6u [o, b] rlng vdi uich chgn Ei6budc2) 4. Ndt flx) li6n tqc tr6n La,bi tni

fia

\;)'

r

Khj dd : S, lh tdng tich phdn c,5,af\i) tren [0, 1l v6i ph6p chia [0, 1] thnnh z phdn bing nhau bdi cric didm chiar , = f, G = on) raic6ch chgn ;_ €i= oelxi-1,x,i (i = 1z)tfclir: q' = if(6'14' =i__+- := i: tl + i: I , -1( - tL.,/-. , i,; (o, =xi-xi-r=;)

Lim Sn= [ Sg1ax. "l** tim sr6i han- tdls Sn = ur * u, * ... * un (Lim S) phg thuQc vdo n € N trong nhi6u trua#inrlp ta ari thd ddn ddn d4ng tdng tich phdn ) ffeil { rdi tinh tich phAn tuons rtns. gin&;;ch dnh tlch nh6n ta tinh drroc fi6i h"an cdn"tim. Bni torih : Clio S, : uI* u, * ... * zr" Tinh

'n d,x

Lim Sn? n+6

r1 7[ = arctsx l; = ?'

Budc 3 : Kdt lu@n Lim Sn = { Y61ax. o Sau day ta *gti?It i.au'o VD1 : Cho Sr=-! ^+-2+...+-tt-

Ngohi c6ch t{nh tr{c ti6p tdng S, nhd c6ng thdc bidn ddi. (dac biet li c6ng thdc cdp sd cdng (cdp sd nhdn) ia rrit 'gq, S, tEeo rz tii dri suy ra Lim.Sn ta cei thd tinh gi6i h4n nhd tich phhn

ir,'*X bu6c sau :

f ,=,1 * \n) o" uo!:Tj" = Jr u ", VD2: Tinh 11 -1 ,,':T*i#+dry**wffi) Tacri: * '- - n - 1 -r 1_-: * Dn - iTilry' {4:$4r'''' {TF:F

,\

n

&

.

r

-tjl vtl i

, t

s --

"a* h -n.'- zL L,\ .t &-.s,. &*fi 3LJ_1

n.#-"\7t,)

i___-_____* i tn; r -"- I LH - nZ

Bu*c fi : Qli ra rra::r pvhlh{lng minh flii6n

'crirc tr6m [#, &],

ItJ

Hudc i ; Bi6h ddi S,, th&nh dang ; b *a, b*a Sl.. = -"-- :" 1 f {a +1.:-:'r"r f {s +2." * i + ,, n n / / &or, +, "+/"i tt-t.tL.-- ) I = -- 5' I { cr r- j ,'--

g.

. n+**

VD4: TInh

tr6n [0, 1].

i._

;

tdng

:_t, Ta ed ttnl1

L

+ LimS, = J e.osttx * =Y li = ,. **) (, * ?-l , F.*\* ,yT-l(r D4t ,,='l(r.,*) (,n?.1 ?":\: vd s,=rn Pn=i[* (r+) *...nr" (r+)] X€t harn s6 y = f(x) = ln (l +r) li6n tsc tr6n [0: 1] IGi dd S- li tdne tich phdn ctia him f(il trfun [0, 1] coi pH6p chiE [0, 1] thanh z.ph'dn'blng nhau bdi cric didm chia x, = I vd ehgn Ei = i e lxi - p x) (i = Ln) v6i Li = i.

i-1 '1

71+*m

o 11

Chia [0,.1] thdnh'z phdn blng nhau bdi c6e didm ehia *i=* Q=7e. Tr6n m6i doan x)ldy E, = * (i = Ln) ; t;=*;*;-r=1. f*i tich ph6n

=l

OO

'

dx

:1+ 1 _, "?'uS-e)' "" ,':T!.:[#=*i il (;r

1+*o

eari= cosf, (, = [ E,t1) + dx = -%osirtt d.t 1 x : 0 =+ / :

Do dd limsn= [tnql*1ax = rtr(.Itu) l1 - 1, - I # =rn2 - I a* * I # = 2tnz - 1 ,+ lim P, : lim do = , JTJ" - e?tnz - L n+*o Ta x6t rl du *a him l6v t{ch phin x6e dinh

[ ; * = t *

-

.xL o _

zsrn zsrn

,). U Do dti Jt

lt

n

tdn [a, a7 marig lt t0 ; 1] " l- . xt . xt sln- . I Sln- "=+--+

Tinh tim\l ln

-'l

I

n-+&z lL *..#** , **7fu* , fut nsln-

++-fr1

zri','u

- r '" -2sint dt 2- '=;tu# = [o':'1*=Z-X:Z ,a '^t^yT:"=t \lDB : (D6 thi DHQG IIaNOi kh6i D - n6m t99b) Tim

Det s:fr'l

'

n

'

nill 1 *cossp2 o , . I sin{ n + n * "z lt +co*f, r +.oir%t

ln

Det

?n.2n

.nnf Tn-n nlnlnnrun

-h,,T-t=-t._-.=--T--r

2n

xL

0'-n, *"'4 cos: +coff + co#)

lT; (r +cos L +"n T. *.o,4o) ,,,:*(r +""i +.*3+...+"o r**): 7 / n-l =n_\:': ." "

Ltt zxt - -

+...+-

_oLrn$ 1ao.&- =*,2,'(T)

Xdt f(x) = costtx, tr€n Chia [0, 1] thinh z phdn blng nhau bdi crtc (i=o;n-L)

,*"r[l TXtt ..tL1t 4Stn-nn ^llxt 7 + cot- n

I , 0,n

[0, tl

Yi ': ..

1

=+'t'r(o+'+) Kftf(x) =,ffiLi6n tsc tr6n [0, z]

r?.

,r-r1 i a4" "* .; = ; ( g6s ;o)=sn

n-l -)r"G) o, = + cosL+... + "orL-

(xem tidp trang 13)

11

orem cnra xi=i(L=on) chgn €i= n thu6c [r, - 1, xiTvdi i = I7 *,a a, = -L. *

r,$ IT,illllt ilr ffi l'fi [Jl[i iffil

I

il]r1ilI ilrilrli $fiIlfiIft rfilfir; I

NcuYEN THUC HAo @a Nai)

@)

Lx, a7 = -_r1ta,bll [r, b] = Ela,b) l

Ta suy ra

(5)

Ch* !. - Cdihd

62+q2=t minh d6 ding ring "n,frrg OAt OB lh,hai b6n kinh li6n hop, trlc AA', BB' td. tzai dubng h,inh tihn ho.p crta dubrug elip'

Trong hiuh hgc phd th6ng, ngrrdi ta dlnh nghia drrAng elip vd hypebol, vdi 2 ti6u didm F, .F.1 ve fing Ai6u ki6n li tdng ho4c hi6u cta kho&ng cacfr tt didm M ddn F vi .F''khdng ctdi' Ndu ia tdng, qui tich ciaM ld drrdng elip ; ndn l}r hiQu thi quy tich ciraM lA dirdng hypebol' Cbn paratrol thi lb qu! tich didrn M ei'ch ddu mQt Aldm cO dinh.F vA dridngth&ug cd dlnh A' Cflng cbn cci dinh nghia chung cho 3 drrdng, vdi mQt ti6u didm .F' va dribng chudn tudng rXng A' Di6u kiQn det ra trh ti sdp cria khoAng c6ch trJ M ddn F vi A ph6i Id hing s6. Quf tich c&aM li elip,. hypebol hay parabol ld tny theo,u < l, lt > t, hay li pr : l.

II" Dinh nghla dudng h5rPebol Ch.o ba d.idm. A, A, B kh\ng th&ng himg' Ta s€ gqi tit hypebol qui tich didnt' M sao cho hiQu sf-crta nai iirun phuong d,iQru tich cila hai tam' giar MAB, ruOd &tio thf tg) biing binh pt uong d.i\n iich cila tant gid.c AOB' Trlc li (hinh 2)

Nhung trong hinh hoc afin, chring ta kh6ng thd lim nhu vf;y, vi khod'ttg cd,clt. lit' nt|t khdi niQm khdng c6 ! nghia gi trong hinh hqc afin' Thd cho n6n sau dAy, ta sE dirng diQn tich afin (ctra binh hdnh hay tarn gi6c), li mQt bdt bidn afin, dd dinh nghia 3 c6nic.

(7)

oBI42 - OALrt - OAB2 (6) Qcd phrrggg trinh-gecto- bing cdch dat OivI = x, OA: a, OB = b Phrrong trinh (6) s6 vidt drroc li fx,bf2 -fx,a)z - fa,blz

Thay vdo (2), ta duqc

I. Dinh nghia duirng eliP Cho 3 d.idnt. a, A, B khOng th&.ng hdng. Ta sd gqi l&--"lip quy'tich didnt M sqo cho oAr,I2 + oBlli2 = oAB2 (l)

s6 cci drJgc, c6n cri vlLo (4) :

(8)

Ez_rf=t Ta thdy rlng O li nm ctia drrdng hypebol, didm A thudc hypebol.cbn didm B thi khdng' Hai dttdng thing chtra OA vd OB ld hai dildng hinh han hqp. Chil !!6t

trlc lir : Tdng cfia binh phuong diQn t{ch hai tam gid.c MOA, MOB bd,ng binh phuong diQn tich tam gid.c OAB (xem hinh 1)

Phrrong trinh theo tga dQ, vdi co sE {o, b},

(8',)

thaYjigu ki0n (6) bing OAI,I2 - OBIO - OABz thi quf tich cria M s6 cd phuong trinh €2 -'12 = -l

Fllnh 1

R6 rdng le A, B thuoc dudngelip. Cbn O ln tdm (ddi xttng) cta elip. Ta h6y tim phrrong trinh crla drrdng eliP.

(Trlc tarn girnc OAB ln m0 t tant gidc lieghqp) ' Ta h6y d4t x = OiuI, a - OA, b = OB Ldy diQn tich binh hdnh (gdp doi) thay diQn tich tam giric, phrrong trinh (1) vidt dugc li

(6')

(3)

Dd la dudng hypebol liAn hqp criadrrdng (8). Nd chrla didm B mi kh6ng chrla didm A. Trong hinh vE dudnglf l}dudng cd phrrong trinh (6) cdn .H' la dudng hypebol li6n hgp c

x-$a+rp