China Girls' Mathematical Olympiad

Question

Let n 2 be an integer. Find the largest real number such that the inequality a_n^2 (a_1 + a_2 + + a_n-1) + 2a_n holds for any positive integers a_1, a_2, , a_n satisfying a_1 < a_2 < < a_n.

Accepted Answer

For a_i = i, i = 1, 2, , n, we have (n-2) n-12 = 2n-4n-1. Since a_k a_n - (n-k), k = 1, 2, , n-1, a_n n, then we have aligned 2n-4n-1 _i=1^n-1 a_i & 2n-4n-1 ( (n-1)a_n - n(n-1)2 ) &= (2n-4)a_n - n(n-2) &= (n-2)(2a_n - n) (a_n - 2)a_n. aligned That is, a_n^2 2n-4n-1(a_1 + a_2 + + a_n-1) + 2a_n. So the largest value of is 2n-4n-1.