1988 CMO 第 4 题

(1) Let $a,b,c$ be positive real numbers satisfying $(a^2+b^2+c^2)^2>2(a^4+b^4+c^4)$ . Prove that $a,b,c$ can be the lengths of three sides of a triangle respectively.

(2) Let $a_1,a_2,\dots ,a_n$ be $n$ ( $n>3$ ) positive real numbers satisfying $(a_1^2+a_2^2+\dots +a_n^2)^2>(n-1)(a_1^4+ a_2^4+\dots +a_n^4)$ . Prove that any three of $a_1,a_2,\dots ,a_n$ can be the lengths of three sides of a triangle respectively.

(1) 设 $a,b,c$ 为满足 $(a^2+b^2+c^2)^2>2(a^4+b^4+c^4)$ 的正实数。证明$a,b,c$可以分别是三角形三边的长度。

(2) 设 $a_1,a_2,\dots ,a_n$ 为 $n$ ( $n>3$ ) 满足 $(a_1^2+a_2^2+\dots +a_n^2)^2>(n-1)(a_1^4+ a_2^4+\dots +a_n^4)$ 的正实数。证明$a_1,a_2,\dots,a_n$中的任意三个可以分别是三角形三边的长度。