PUMaC 2009 · 代数（A 组） · 第 3 题

PUMaC 2009 — Algebra (Division A) — Problem 3

Let x , x , ...x be non-negative real numbers such that + + ... + ≤ 9. Find the maxi- 1 2 10 1 2 10 2 2 2 x x x 1 2 10 mum possible value of + + ... + . 1 2 10

解析

Let x , x , ...x be non-negative real numbers such that + + ... + ≤ 9. Find the maxi- 1 2 10 1 2 10 2 2 2 x x x 1 2 10 mum possible value of + + ... + . 1 2 10 ∑ 2 Solution. 810. If we denote x /i by a , we must maximize ia subject to the condition i i i ∑ a ≤ 9. Intuitively, the sum is large if the a -s with large weights are themselves large. Indeed i i ∑ ∑ ∑ 2 2 2 the sum is largest when all weight is placed on a : ia ≤ 10 a ≤ 10( a ) ≤ 810. We 10 i i i have equality if a = 9, and all other numbers are 0. 10 1