. X6t trudng h Wn > 5 sit dung gi6 thi6t
tich kh6ng 0m kh6ng mat tinh t6ng qu6t,
gii sir (b -l)(c - 1) > 0, suy ra b + c <1+ bc.
Sri dung BDT Minkowski, ta c6:
Nhu vpy ta chi cAn chimg minh
Ji +te* + JN +tAtU + "Y > 3 + 4(a + b + c) . "Y > 3 + 4(a + b + c) . oe 1i ring $Gr@+ry -4@+c) -JT6+@+8 +4(b+c) 36 Jt6+@+W +4{t+bc)
= (3 - b - c + ab + ca) + (ab - c)(ca - b)
>3(3' b - c + ab + ca) =3(a + ab + ca)' Ta chimg minh
= J36 + 16(1 + bc)z - 4(l + bc)
-o('.*)
:o+ro(r+1)'
Binh phuong hai v6 vd rut gQn, ta c6 thi5 vi6t BDT ndy du6i d4ng
Wl4a+5+9
o v \/ 0 +16a2\ , ,v- /\-" {rl *&*4\ , a' arJ-\'-" '" rh+o* s *9j'' aJ
* t1a2 -t1a -12 +Q) 0 e 12(a - l)2 (a +l), o
aa
BDT cuOi hiOn nhi€n dring. Ph6p chimg minh hodn t6t. Ding thric xity ra khi vi chi khi
a=b=c-1.
Bni to{n 7. Cho cac sd thvc &rong a, b, c thda mdn a + b + c = 3. Chung minh ring
+.+.+2az +b2 +c2.D'L D'L
(Romania Team Selection Test 2006\.
TORN HQC
Lirt girtL Theo nguyOn li Dirichlet, trong ba
s6 a- l,b-l,c-l lu6n t6n t4i hai sO c6 tich
kh6ng 6m. Kh6ng m6t tinh t6ng qu6t, gi6 sir (b -l)(c-1) > 0, suy ra b + c < I + bc. Khi d6
b? +c2 =(b+c)2 -2bc<(b+c)2 -2(b+c-l)= (3 - a)z - 2(2 - a) = a2 - 4a + 5. = (3 - a)z - 2(2 - a) = a2 - 4a + 5.
Sri dlrng BDT Cauchy , ta c6'.
1. Cho c6c s6 thgc duong a, b, c thoa mdn
az + b' + c' + abc = 4. Chimg minh ring
abbcc
_*_*4) a2 +b2 +c2.
cab
2. Cho c6c s5 thgc ducrng a, b, c th6a min a+b+c = 3. Chimg minh ring
(a'z - a + l)(b2 - b + l)(c2 - c + 1) > 1. 3. Cho c6c si5 thgc ducrng a, b, c th6a mdn
ab + bc + ca = abc + 2. Chtmg minh ring
-L---J-+-L<J;b+c.^la+2bc,lb+2ca'lc+2ca ^la+2bc,lb+2ca'lc+2ca
4. Cho cric s6 thlrc a, b, c thbamdn abc>0.
Chtmg minh ring
z+abc+l@2 +b2 +cz -ab-bc-ca)> a+b+c.5. Cho c6c si5 thgc ducrng a, b, c thoa mdn 5. Cho c6c si5 thgc ducrng a, b, c thoa mdn
ab + bc + ca = abc + 2. Chrmg minh ring
ab(2-c) bc(2-a) ca(Z-b) /1A +;;;TF- b, + abc + CT--i--:---:--r > t' A +;;;TF- b, + abc + CT--i--:---:--r > t'
6. Cho c6c s6 tfuc duong a, b, c th6a mdn
a2 +b2 + c2 =3. Chimg minh ring
(a2 + b2 + abc)(b2 + c2 + abc)(c2 + a2 + abc)