HMMT 二月 2017 · ALGNT 赛 · 第 7 题
HMMT February 2017 — ALGNT Round — Problem 7
题目详情
- Determine the largest real number c such that for any 2017 real numbers x , x , . . . , x , the inequality 1 2 2017 2016 ∑ 2 x ( x + x ) ≥ c · x i i i +1 2017 i =1 holds.
解析
- Determine the largest real number c such that for any 2017 real numbers x , x , . . . , x , the inequality 1 2 2017 2016 ∑ 2 x ( x + x ) ≥ c · x i i i +1 2017 i =1 holds. Proposed by: Pakawut Jiradilok 1008 Answer: − 2017 Let n = 2016 . Define a sequence of real numbers { p } by p = 0, and for all k ≥ 1, k 1 1 p = . k +1 4(1 − p ) k Note that, for every i ≥ 1, ( ) 2 x √ i 2 2 (1 − p ) · x + x x + p x = + p x ≥ 0 . √ i i i +1 i +1 i +1 i +1 i i +1 2 p i +1 Summing from i = 1 to n gives n ∑ 2 x ( x + x ) ≥ − p x . i i i +1 n +1 n +1 i =1 k − 1 1008 One can show by induction that p = . Therefore, our answer is − p = − . k 2017 2 k 2017