National and Regional Contests Poland Contests Poland - Second Round 1988 Poland - Second Round 2 sum x_i > sum y_i if prod x_i >= prod y_i Problem Statement Given real numbers x i x_i x i y i y_i y i i = 1 , 2 , … , n i = 1, 2, \ldots, n i = 1 , 2 , … , n x 1 ≥ x 2 ≥ … ≥ x n ≥ 0 , y 1 > y 2 > … > y n ≥ 0 , \qquad x_1 \geq x_2 \geq \ldots \geq x_n \geq 0, \ \ y_1 > y_2 > \ldots > y_n \geq 0, x 1 ≥ x 2 ≥ … ≥ x n ≥ 0 , y 1 > y 2 > … > y n ≥ 0 , ∏ i = 1 k x i ≥ ∏ i = 1 k y i , for k = 1 , 2 , … , n . \prod_{i=1}^k x_i \geq \prod_{i=1}^k y_i, \ \ \text{ for } \ \ k=1,2,\ldots, n. i = 1 ∏ k x i ≥ i = 1 ∏ k y i , for k = 1 , 2 , … , n . ∑ i = 1 n x i > ∑ i = 1 n y i .
\sum_{i=1}^n x_i > \sum_{i=1}^n y_i. i = 1 ∑ n x i > i = 1 ∑ n y i .