| Bài 1 : cho $a,b,c \ge 0. $, thoả mãn $ab+bc+ca=3. $chứng minh rằng:
$\frac{1}{{{a^2} + 1}} + \frac{1}{{{b^2} + 1}} + \frac{1}{{{c^2} + 1}} \ge \frac{3}{2} $
Bài 2: cho $a,b,c >0 $.CMR
$\frac{{{a^2}}}{{b + c}} + \frac{{{b^2}}}{{c + a}} + \frac{{{c^2}}}{{a + b}} \ge \frac{{3({a^3} + {b^3} + {c^3})}}{{2({a^2} + {b^2} + {c^2})}} $
Bài 3 Cho $a,b,c \ge 0 $ thoả mãn $a^2+b^2+c^2=1 $.CMR
$\frac{{bc}}{{{a^2} + 1}} + \frac{{ca}}{{{b^2} + 1}} + \frac{{ab}}{{{c^2} + 1}} \le \frac{3}{4} $
Bài 4 cho $x,y,z \ge 0 $ thoả mãn$ x+y+z=3 $. CMR
a) $\frac{x}{{xy + 1}} + \frac{y}{{yz + 1}} + \frac{z}{{zx + 1}} \ge \frac{3}{2} $
b)$\frac{x}{{{y^2} + 3}} + \frac{y}{{{z^2} + 3}} + \frac{x}{{{y^2} + 3}} \ge \frac{3}{4} $
[RIGHT][I][B]Nguồn: MathScope.ORG[/B][/I][/RIGHT] |