下面是@丿孤僻男青年灬提供的题目
1.[一股闷骚的柯西味]已知a+b+c=m.(a,b,c>0),求证∑((6a^3)/(m-a))≥m^2
LHS=6∑((a^4)/(ab+ac))≥6(∑a²)²/(2ab+2bc+2ca)≥3(∑a²)≥(∑a)²=m²
2.[再来一水]已知a,b,c>0且a+b+c=1,试证明(a+1/a)²+(b+1/b)²+(c+1/c)²≥100/3
(a+1/a)²+(b+1/b)²+(c+1/c)²)(1+1+1)≥(a+1/a+b+1/b+c+1/c)²=(1+1/a+1/b+1/c)²=(1+(1/a+1/b+1/c)(a+b+c))²≥(1+(1+1+1)²)² =100,故原不等式成立。
1.[一股闷骚的柯西味]已知a+b+c=m.(a,b,c>0),求证∑((6a^3)/(m-a))≥m^2
LHS=6∑((a^4)/(ab+ac))≥6(∑a²)²/(2ab+2bc+2ca)≥3(∑a²)≥(∑a)²=m²
2.[再来一水]已知a,b,c>0且a+b+c=1,试证明(a+1/a)²+(b+1/b)²+(c+1/c)²≥100/3
(a+1/a)²+(b+1/b)²+(c+1/c)²)(1+1+1)≥(a+1/a+b+1/b+c+1/c)²=(1+1/a+1/b+1/c)²=(1+(1/a+1/b+1/c)(a+b+c))²≥(1+(1+1+1)²)² =100,故原不等式成立。