National and Regional Contests China Contests China Team Selection Test 2015 China Team Selection Test 2 China Team Selection Test 2015 TST 2 Day 1 Q2 Problem Statement Let a 1 , a 2 , a 3 , ⋯ , a n a_1,a_2,a_3, \cdots ,a_n a 1 , a 2 , a 3 , ⋯ , a n n ≥ 2 n\ge 2 n ≥ 2 ( ∑ j = 1 n ( ∏ k = 1 j a k ) 1 j ∑ j = 1 n a j ) 1 n + ( ∏ i = 1 n a i ) 1 n ∑ j = 1 n ( ∏ k = 1 j a k ) 1 j ≤ n + 1 n \left (\frac{\sum_{j=1}^{n} \left (\prod_{k=1}^{j}a_k \right )^{\frac{1}{j}}}{\sum_{j=1}^{n}a_j} \right )^{\frac{1}{n}}+\frac{\left (\prod_{i=1}^{n}a_i \right )^{\frac{1}{n}}}{\sum_{j=1}^{n} \left (\prod_{k=1}^{j}a_k \right )^{\frac{1}{j}}}\le \frac{n+1}{n} ∑ j = 1 n a j ∑ j = 1 n ( ∏ k = 1 j a k ) j 1 n 1 + ∑ j = 1 n ( ∏ k = 1 j a k ) j 1 ( ∏ i = 1 n a i ) n 1 ≤ n n + 1