MathDB

2

Part of 1988 Poland - Second Round

Problems(1)

sum x_i > sum y_i if prod x_i >= prod y_i

Source: Polish MO Recond Round 1988 p2

9/9/2024
Given real numbers xi x_i , yi y_i (i=1,2,…,n i = 1, 2, \ldots, n ) such that x1≥x2≥…≥xn≥0,  y1>y2>…>yn≥0, \qquad x_1 \geq x_2 \geq \ldots \geq x_n \geq 0, \ \ y_1 > y_2 > \ldots > y_n \geq 0, and ∏i=1kxi≥∏i=1kyi,   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. Prove that ∑i=1nxi>∑i=1nyi. \sum_{i=1}^n x_i > \sum_{i=1}^n y_i.
algebrainequalities