07/06/24 01:39:02
>87
〔系〕 a,b,c>0 のとき
(2/3){a^2/(b+c) +b^2/(c+a) +c^2/(a+b)} ≧ (a^2 +b^2 +c^2)/(a+b+c) ≧ √{(a^2 +b^2 +c^2)/3} ≧ (a+b+c)/3 ≧ (abc)^(1/3).
2. (IMO 1995 Canada)
Let a,b,c be positive real numbers such that abc=u. Prove that
1/{a^3･(b+c)} + 1/{b^3･(c+a)} + 1/{c^3･(a+b)} ≧ (3/2)u^(-4/3).
(略証) 上式の a→1/a, b→1/b, c→1/c, u→1/u とおく。