xyly-x) s lg.
Yi a+b+c=1, n€n 3a+x+!=I=x+Y<l
hay y<l-x. Tri diAu kiQn xSY, suy ra
I0<x<i.Dod6 0<x<i.Dod6 xy(y - x)< x(l- x)(1- 2x) = 2*t -3x2 + x ' X6t hdm s6 f (*):2x3 -3x2 + x,x e[t,]] . ra c6 .f '(x)=6x'-6x+1, f '(x)=0o*=+ (ntran) ho{c x =r*(loai). "rL) -n ,(3-Ji)- J3 Khi d6 "F(0) = 0,f l' '.7J-"''[ 6 ) 18' vi /(x) li6ntuc.* [o']] ncn /(x).*
Ding thtic xdy ra khi ,=L*,r=+
hay a=o,b=+,"='*{ .
Nhu vQy, kh6ng chi gi6i dugc bdi toSn dd cho
md ta cdn chimg minh <lugc BDT "k6P": tln
-#. @-b)(b-c)(c-").#.
Q Bdi totin 4. Cho a,b,c elt,Z]. Chttng minh
,dng ot +b3 +c3 <5abc.
Ldi gi,fii. Vi vai trd cira a,b,c ld nhu nhau n6n c6 th6 gii sir a<b 3c. Khi tl6 t6n t4i y2 x> 0
sao cho b=a*x,c:a*!. Yl a,b,celt,Z),
ndn x,ye [o,t]. BDT cin chimg minh dugc
vii5t lai a3 +(a+x)3 +(a+ y)3 35a(a+x)(a+ y)
e 2a3 +2a2x-3ax2 -x'
+2a2Y-3aY2 -Y3 +5ary >0 (*)'
BDT cu6i ching vi n6 tuong duong v6i
2(a3 - y3) + x(a2 - x2) +3ax(Y - x)
+a2 x + 2axY + Y(a - Y)(Za - Y) > 0'
Binh luQn.Trong bdi toSn tr}n,tadlr <lo6n tting
thirc xiy ra khi mQt s6 bing} vd hai sO bing 1'
Tt d6, uOi uie. sap x6p thir tU ci.r- chc bilin, ta
nhfn ttugc a=l vd x=O,y =1. Do d6, trong
BDT (*) ta ph6i d6nh gi6 d4i luqng a3 vd y3,
tuc ld cdn phAn tich -Y' = -2Y' +Y' ' COng
viQc ki5 tii5p ld ldm th6 nio d6 c6 th6 nh6m c5c
hpng trJr ua r5, ttim b6o rting thric xiy ra ddng
thdi nhu c6ch trinh bdY tr6n.
A Bdi todn 5. Cho cac s6 thrc a,b,c thoa mdn
cac diiu ki€n a> b> c vd a2 +b2 +c2 =5' Ck'img minh ring (a - b)(b - c)(c - a)(ab + bc + ca) 2 4'
Ldigirti.Vi a> b>c n€tt6nt4i x>y>0 sao
cho a:c+x,b=c+-Y. Tt a'+bz +c2 =5, ta
c6 3c2 +Zc(x+ Y)+ x' * Yz =5' BDT cinchimg minh tro thAnh chimg minh tro thAnh
(x' y)xy(3cz + Zc(x + Y) + xY) < 4
a xy(x - y)(5 - *' - Y' + xY) < 4.
D$t m:x-! Yd n=x!, t/t)O,n) 0' BDTtrd thdnh m'n+mn' +4> 5mn . trd thdnh m'n+mn' +4> 5mn . BDT cu6i dring vi mtn+mn' +4 )2 t ntn' *n' +Z+2>5{AA =5mn. =m"n* z * z
D[ng thirc xiry rakhi vd chi khi
, Fm2 ^ l*'r=2 fm=l , -- l*=2
m'n=f --2e\;,
=4e\r=2 haY {y = r
hay a =2,b =1,c : 0 hoic a =0,b = -l,c = -2'
A Bdi todn 6. (Vietnam 2008). Cho a,b,c ld
cac thryc kh6ng dm phdn bi€t. ChwnS minh rdng \a-b)' @- c12 (c-a\' ab+bc+ca
Ldi gidi.Vi BDT ddng bfc n€n c6 thi5 chuAn
h6a ab+bc+ca:4.GiA str a>b>c> 0' DAt
b=c+x vd a-c*!, vbi Y>x>0. Ta c6
4=3cz +Zc(x+y)+xy>xy. BET cdn chimgminh trd thdnh I minh trd thdnh I
- *{*{=r
(x- Y)' x- Y'
e r'y' + 1x2 + y2)(x - y)' > *' y' (* - y)' .
D{t m-y-x,n=xy thi m>0,4>n>0. Ta
c6n chirng minh b6t ding thric
n2 +(mz +2n)m2 ) m2n2 e n' +mo +2nm2 > m'n' . Ap dpng BDT Cauchy,tac6 n' +mo +nm2 +nm' > 4{tr7 :4m2n) m2n2 . Dlng thrc xiry ra o {* =I o{. = f-l n, ln=4 Ly=JS+l o = J5 +1, S =.1$ -1, c =0.
Q Bdi torin 7, Cho a,b,c lii cac sd thrc clt)i
m6t khitc nhau. Cht?ng minh rdng
{o- -fi: -c:-uh-tr-ro{J.- -
\. \a - o1'
I l)21
{a; +;-; )tz
Ldi gi,rti. Gi6 su a =min{a,b,c\ .
Ddt b = a+ x)c - a+ y . Y\ a,b,c ttdi mdt phdn
biQt n6n x>0,!>0 vd x* y .BDT cAn chimgminh tro thdnh (t-xy.the+*.1'l-Z' '(l minh tro thdnh (t-xy.the+*.1'l-Z' '(l
v' (*-v)')- 4
, ) ,- (x' -xv+ v'\2 27
<> (x- -xY+ Y')--;-> xy'(x-y)'- -4'
/ t " r3
I x'-xv+v" 1
^l , r I
-tt trvl / ' ,'-ry+y' ,
xy Dat r-*'-xY+Y'=L*!-r.xyyx
n6n I > I . Ta vi6t BDT tr6n du6i dpng
4lt -n- ^
/{ > 21 e(2t-3)21r+3)>0.
B6t ding thirc cu6i hi6n nhi6n <hing n6n ta c6
diAu phni chimg minh.
C5c bii toSn d6p theo sd minh hoa th0m cho
tinh uu vi6t cria k! thuQt niy.
& Bni to6n B. (Arkady Alt, Sqn .Iose, USA).
Cho tam giac vcti d0 ddi cac canh ld a,b.c.
Cltu'ng minh: {ct + h + c).min{a,h,c}
! 2ab -r 2bc + 2ca * rt2 - b' - ci .
Ldi gidi. Gi6 sir ring a=min{a,b,c} . O6t
b=a*x,c-a+y,thl x,y>0. B6t d[ng thtic
trO thdnh (a+b+c)a<2ab+2bc+?ra-a' -b' -c' e2a2 +b2 +c2 <ab+ac+2bc
e2a2 +(a+x)2 +(a+y)2
< a(a + x) + a(a + y) + 2(a + x)(a + y)
e x2 + y2 < x(a+ y)+ y(a+ x).
BET cu6i lu6n dring, v\ a+b>c vd a+c>b
hay a+x>y vil a+y>x.
E Bili tor{n 9. Cho a,b eRl thda mdn diiu hiin
2"t2 +3ab+2b2 <7 . Chwg rtinh rring
Inax {2nr - h.21, * ,rl '. 4.
Vi x>0,y>0
Ldi gi,fii. Vi vai trd cria a vd b ld nhu nhau n6n ta c6 th6 gi6 sri a> b . Khi d6
max{2a+b,Zb+a\ =2a*b.
Ta sri dgng phucrng ph6p phin chimg d6 chtmg
minh BDT. Gia sir ta c6 2a+b>4 huy
b>4-2a. Khi <16 tdn t4i r>0 sao chob=4-2a+x. Khi d6 2a2 +3ab+2b2 <7 b=4-2a+x. Khi d6 2a2 +3ab+2b2 <7
e 2o2 +3a(4-2a+ x)+2(+-2a+ *)' <7
e 4a2 -20a+25+2x2 -5xa+16x<0
*(u-s-1.)' .*o +l*<o(v6 ri vi r > o ).
8 Bdi todn 18. (Mctrian Tetit,a, Ilowania,
CR{,X problem #3246).
Cho a.h,t.,d>A, r/=n_rin!ra.b.c,dl . Chrl:ng
minh rctn,q: a' + lr'+, I +r/-' --4uhcd
> ult o - 4' + (b * (r' + ( c - c{13 -\" - cX b - a}{c *,t };)
Ldi gidi. Y\ d =min{a,b,c,d} n6n ta c6 thti
d(t a = x+ d,b = y * d,c : z * d,vfi x,y,z) 0 .
n6t ding thric cAn chtmg minh tro thdnh
(x+d)a +(y+d)a +(z+d)a
+d4 - 4d(x + d)(y + d)(z + d)
> 4d(x3 + y' + zt -3*yr) .
Tac6 (x+d)a +(y+d)a +(z+d)a +da
- xo + yo + za +4da +4d(x3 + y' + z'1
,!
- 4'
+ 6d2(x2 +y2 +22)+4d3(x+y+z);
4d(x + d)(y + d)(z + d) = 4da + 4d3 (x + y + z)
+4d2 (xy + yz + zx) + 4ry2d. Do tl6 BET tr6n trO thenh
*o + yo + zo +6d2 (x2 + y' + z')
-4dz (ry + yz + zx) +\xyzd > 0
e*o +yo +ro *Adlf +f +11-1xy+y+gf
+zdz (xz + y' + z'1+8xyzd 20.
BDT cu6i cirng lu6n tlung vi
*'+y'+z'> xy+yz+zx vd x,y,z,d)0.
Ding thric xhy ra khi vd chi khi x- y- z-0hay a-b- c- d . hay a-b- c- d .
Q Bdi torin ll. Cho a,b,c ld cac sd thaclchong dtn. Chri'ng minh rcing lchong dtn. Chri'ng minh rcing
4(a+b+c)t > 27(abl +bc'+ca2 +abc).
Ldi gidi. Kh6ng m6t tinh tdng qu6t, ta c6 thti gi6 str a=min{a,b,c}. O6t b = a* x,c = a! !, trong d6 x) 0,y > 0. BDT cdn chimg minh tro
thdnh 9(x2 * xy + y')a + (2x - y)2 (x + 4y) > 0.
BDT cu6i dring. Deng thirc xdy ra khi vi chi khi x=!=0 holc a=0,2x=y, tuc h
a=b=c hof,c a=0,c=2b vd c6c ho6nvi. Q Bni tudn 12. (Gabriel Dospinescu). Cho circ' sd thtrc: durrng a.b,c vd thoa rudn diiu ki€n
n'wx{a,b,cl - rcin\a,b,c\ 31. Chting tninh ring
l+ a3 +b3 +c3 +6qbc>3a2b+3b'c+3c2a.
Lnrt gidi. Gii sri a:min{a,b,c} . Opt
b = a4 x,c = a+ y,x,y e [0,1]. ra c6
a3 +b3 +ct -3abc:3a(x2 -xy+y')+x'+y'vd a2b + b2 c + c2 a-3abc = a(x2 - ry + y') + *' y. vd a2b + b2 c + c2 a-3abc = a(x2 - ry + y') + *' y.
BDT c6n chimg minh tro thdnh 1+l +f >Zf y.
BDT cudi thing vi l+x3 +y3 23xy>-3x2y, do
0<x,y<1.
BAI TAP
l. (Spain 1996) Cho a,b,c> 0. Chimg minh ring a' +b' +c' *ab-bc-ca>3(b-c)(a-b). 2. (Bosnia 2008) Cho x,y,z ld c6c sl5 thgc.
Chtmg minh rlng *' + y' + zz - ry - yz - zx
,*u*[3@ .Y)' .3(Y-z)'z .3(z ;x)'?\.
14'4'4)
3. Cho a,b,c ld c6c s6 thgc kh6ng 6m. Chimg minh ring a' + b3 + c' -3abc.r(ry- ")
4. Cho a,b,c ld c6c si5 thgc kh6ng 5m. Chimg minh ring (a + b + c)3 > 6J1@ * b)(b - c)(c - a).
5. Cho a,b,c ld c5c si5 th1rc. Chimg minh ring
fld +t +t +*1+1a+b+c+d)2 >12(ab+tu+cd).
6. Cho a,b,c litc6c sli thgc. Chimg minh reng
21a2 + b2 17b2 + c' 11c' + a' 1 > 7a - b1' 1b - c1' 1c - a12 .
7. Cho a,b,c ld c6c si5 thgc kh6ng 6m thoa mdn tli6u kiQn a+b+c = 1 . Tim gi6 tri lon nh6t
cira biiSu thirc P : (a -b)3 + (b - c)3 + (c - a)3 .
8. Cho x,y,ze[O,t]. fim gi6 tri l6n nh6t cria bi6u thfc P = (x * y)(y - z)(z - x)(x + y + z).
g. (BDT Schur) Cho a,b,c,r ld cilc sO ttrlrc
kh6ng 6m. Chimg minh
d (a-b)(a-c)+U (b-c)(b-a)+c' (c-a\c-b) > 0.
10. Cho a,b,c>0. Chung minh rdng
o-b-cb+c' c+a' a+b b+c' c+a' a+b -3 7 'r- 16'',u* {1, - a)',1b - c1',7c - ")'l 11. Cho c6c s5 thgc J mmn rang ab+bc+ca phdn biQt a,b,c. Chmg -( r l r ) q
(a2 +b2 +c'11---: - '[{o-r)' "a ' (b-")' - +---:- ' (c-a1'z . )- l)-2'
12. (Vasile Cirtoaje). Cho a,b,c >0. Chimg minh rlng (a2 +b2 +c')' >3(a3b+b3c+c3a). 13. Cho a,b,c ld c6c si5 thgc khdng 6m ph0n biet. Tim gi6trinh6 nh6t cira biiiu thric
r =l1a + b)2 + (b + c)2 + (c + a)'f
^t t TO6N HOC
J4 t dtrOiga S eet ot-zoto