07/06/18 00:46:37
>51 上
[B.3987]
Let n≧4 be an integer, and let a_1,a_2,…,a_n denote non-negative real numbers.
Prove that
Π[k=1,n] (a_k + a_{k+1} + a_{k+2})^2 ≧ (2^n)Π[k=1,n] (a_k + a_{k+1})^2,
where a_{n+1}=a_1, a_{n+2}=a_2.
｛略解(in Hungarian)｝
(a+t)(t+d) = t(a+t+d) + ad ≧ t(a+t+d),
に t=b+c を入れて
(a+b+c)(b+c+d) ≧ (b+c){(a+b)+(c+d)} ≧ (b+c)・2√{(a+b)(c+d)}. (←相加･相乗平均)
循環的に掛ける。
URLﾘﾝｸ(www.komal.hu)
ｺﾞﾎｺﾞﾎ