NGUYiN MINI] HA
*Ilhi'110/393. Tint tat ca cdc s6 thu'c k vd m
s00 L'170
t(.r3 + ),3 + z3) + nxx)12> (;r + y + z)3 t,o'i tttei x, y, z kh1ng Am.
I.rri gitii. (Theo ban DSng Hiru Trong, 1 1,A. I ,
TFIPT Quj,Hqp 2, NghQ An).
c Di€u ki€n cdn. Cho x = ! = z> 0 ta clugc 3k + m227. Cho x = y >0, z = 0 ta ducyc k > 4.
. Diiu ki€n du. Gi6 sir k> 4,3k + m> 27.Khi
d6 k(x3 + y3 + as)+ mxyz
2k(x3 + ), + z3)+(27 -3k)xyz
=4{l +)} +73')+1544.+(k-4)@ +f +23 -3xlz) > 4(x3 + )3 + 7t)+l5xyz, do k > 4 vlr
x3 + y3 + z3 > 3ryz (BDT trung binh cOng -
trung binh nhdn).
Bdi to6n dugc gi6i, n6u ta chung minh dugc
4(x3 + y3 -p a3) +15ry2> (x+ y + z)3 (1)
e) x3 + y3 + z3 +3xyz
> ry@ + y) + xz(x + z) + yz(y + z)
<+ .r(x-1,)(x-z)+fu *zX y - x) + z(z- x\z- y)>0 (2)
D6 chung minh (2), kh6ng gidm t6ng qudt ta
gi6 su x > y > z. Khi d6 (2) tuong tluong v6i
.r(.r- y)(x - y + y - z) + y(y - z)(y - x) +z(x-y+y-z)(y-z)>0
<+ x(x- y)2 +x(x- yXy* z)-y(x- y)(y-z)
+z(x - y)(y - z) + z(y - z)' >- 0
<+ .dx-))2 + z(y - z)2 +(x- yXy -z)(x-y+ z) ) 0
<+ .{x-y)2 + z(y-z)z +(x-1,f,y-zXx*l+z) >0 (3)
Vi x > y > z nOn b6t ding thric (3) ddng. Vay
b6t ding thric (l) dugc chirng minh. R
{Nnqn x6t. l) Da s6 c6c ban khi din d6n b6t Oing
thirc ( I ) diru n6i d6 ld Biit ddng thirc Schtt vd kOt thtic chimg minh o ddy.
Btu ddng tlwc Schur duqc ph6t bi€u dudi d4ng: Vdi ba s6 thlrc kh6ng dm x, y, z vd s6 nguy6n khOng dm n ta c6 b6t ddng thirc:
l'(x - i)(x - z) + )'() - zX.y - x) + ztt (z - x)(z - "v) ) 0 .
2) Cdc b4n sau ddy c5 loi girii t6t:
Vinh Phric: Ddnt Vdn Tti,l}A3, THPT DOi C6n; Phri
Thgt Nguvdn Qudc Hiutg, 10. Torin, THPT chuy0n
Hirng Vuong; Nam Dinht Trdtt Quang Huy, l0Al,
TI-IPT Hai Hau A; fliii Duong: D6ng Xudn Bdch, l0
Torin, THPT chuyOn Nguy6n Trdi; Thdi Ilinh: Dlnlr Th! Nho, 10 Toiin, THPT chuyOn Thdi Binh; NghQ An:
Phan NgLtydn Thanh Son, l0Al,.THPT Di6n Chdu 3; Dir Ning: NguyAtt Htitt Mirtlt Tudrt, I lT, 'I'HPT chuy6n
L0 Qu! D6n; Quiing Ng6i: l,trmg Khdnh IrAc, l0
Torin, THPT chuy0n LO Khi6t; Ci Mau: Nguy€n Phqm
uy, tac6CA.KA CA.KA
=4'-l'L
,//grtut
TORN HOC
ftrutt Artl"t, l2Tl, THPTchuy6n Phan Nggc FIi6n; B6n Tre: Kirtg Hinr HiQp,l I Toiin, TIIIn chuyOn BOn'Ire.
DANG TIUNG THANG
*lili 't'11/393. Cho dd1' s6 thqrc' (r,,) t't)'irt = l, 2. ... tluio nt[irt ln(l+r,2)+nr,,:l v'ci'i rt = l, 2. ... tluio nt[irt ln(l+r,2)+nr,,:l v'ci'i
trroi .sri ngu.\'On dtrtrttg tt.
n(I-rr.r..)
7)lr linr t "'.
/t ),{! .{,,
Lo'i giiii. Voi m6i n € N. ta d6tJ',,(x) :ln(1 + "r2) + nx *1' xe IR'. J',,(x) :ln(1 + "r2) + nx *1' xe IR'.
')y (x+l)2
'l'a co 1",,(x)=---' - +l =l---:+n-l>0.
l+x2 I +r:
J'',,(x) - 0 <+ n:1, x = .'1. Do d6 .f,,(x) lhh)rm sd ting thuc sU. Chti i, f,,(0) = -l ( 0, h)rm sd ting thuc sU. Chti i, f,,(0) = -l ( 0,
/;, f 1) = tnft . +] > o , suy ra c6 duy nhAt
''\u) [ ,,'/
rnQt sO r,, € R thoa mdn l, (-r,, ) = 0 vh
I
0(ir,, <-. Btri vay
n ,. rr(l - rx,, ) ,. rr.ln(l + r,2 ) llnt -- = llln ir ) r xil ,t-)p ,Tr, / r\ : Iinr I nx,,.ln(l +;r,2) r; | : 1. /l ,a@\ ) I Do lirn ln(l+x2)..: = I r;0 Yb, t"tx,, = I -ln(l +x,') + 1 khi n -) +@, vi x,, -+0 khi n -+ +-). ru(l+m,,) u Chir 1i ":' '"-"' - " +n2 -)+€ khi n->pq tr X,, xn
{Nna, x6t. l) I)o so suat de rodn dI bi in nham
rr( l+rn,)
thlnh lirn \ -7'l,
tda soan thinh thlt xin l6i tiic giti n+F )i,
vir ban doc.
2) DAy l) biii to6n gi6i h4n dang.co brln, khOng qu6
kh6. C6 mt)t ban_ hr;c sinh dd viet duoc cd hai gi6i han tr0n ld NguyAn Manh QuAn, I lAl. TI{PT chuy6n
Vinh I'hric.
NGT]YBN MINH DUC
* ltli 'tl2l3g3. 7'im tat ca c'tit: hitm so li\n
rtrt' .f' :liR -+iR thod n'rdn cli\u kiAn
./(,,,+ /(1')):2.r'+./(.r),V.r,1'e 1R (1)
Ldi gi6i (Tlrco da s6 cdc bqn).
Nhdn xdt ring I lh m6t clon 6nh. Thdt vAy,
n5u /(y1)= f 0) thi ung vdi m6i x a c6
J' (x +./(y, )) = f (x + f (yz))
hay 2y1 + f G) = Tyz * J' (x),tuc y1 = y2' Ti6p theo, tir diOu kiQn (l) cua bhi ra, ta c6 tpp gi6
tri cua hirm '/ (n6u t6n tai) li lR., n6n t6n tai
ce R d6 f @)=0.
Tu (l), irng vdi ! = a, ta thu ducYc
f (x + f (a)) = 2a + J'Q) haY
J'G)=2a+f (x), t'lc a=0 vi /(0)=9.
Tu (l ), ung v6i .{ = 0, ta thu dugc
f (/0D=2y+ "f (0) = 2y, hay
f (.1'6D = 2),V)'e R. (2)
Ti6p tuc thay x = f (t) trong (l) vh su dung
(2), ta thu duoc
f (f (r) + l'(v)) = 2v + f (l'(t))
= 2), +Zt = 2O) + t) = f 17 1y + r)),
hay /(x* y) = f (x)+ /(y),V,r,ye R (3)(do tfnh clon inh cuafl. (do tfnh clon inh cuafl.
Tu d6 (3) 1) phuong trinh hlm Cauchy (cQng
tinh vir li6n tUc) n6n c6 nghidm f (.x) = 6* .
Th6 vho (1), ta thu duoc bzx=Zx,Vxe R
n6n b =+Jl.
Thu lai, ta thdy hai hhm sO /(rl = +J2.t thoa
mdn bhi ra. B
{Nha" x6t. 1) Ngodi circh.gitii trOn, mQt s6 bqn cdn
su dpng (2) tl6 dua trUC tiep vG phtrong tfinh hirm
Cauchy dqng 71.r + 2v) = J'(x) + 2J 6,),Vx, _y e R.
cilng c6 th6 chuyOn d6i ve dang (3) duqc. 2) Cdc ban sau diiy c6 loi gidi d{ng:
Th6i Nguy6n: Lutr Si Ting,1lAl, TIIPT Dai Tir; Phri
Tlqr: 7'a Hai Nam, l lTl, Il{l']T chuy6n llirng Vuong;
Bic Giang: Vfi Thdnh Dqt, l0T, TFIPT chuy0n Bic
Giang; Hir Nam: Ddn14 Du\, Ilidn, Ngrty€n Vdrt-Tinlt, 10CT, THPT chuy6n Bi6n Hba; Hi NQi: Ngu,-€n Vdn
Qui, 11A1, KhOi'chuy€n DHKHTN fia N6i, lusrr-yan
Ngpc Srrn, l lTl, TFIPT chuyOn DHSP HA NQi; Hung
Yiat Ltrong l)tic HiArt, 10T1, TIIPT chuy0n Hung
Y0nl Thrii tlinh: Dlnlr Thi Nho, l0T, THPT chuy6n
Thrii Binh, Vii tlul- tlodng, l0Al, lllPT Tiy ThUy
Anh, Ngu1,Q71 Thi Minh Hrong, 11A THPT Lli B6n; I{di Du'ong: Phqm D*c Vurntg, l0Al, T}{PI' Thanh
TORN HO(
24 ';{rli[;