HMMT 二月 2017 · ALGNT 赛 · 第 7 题

英文解析

7. Determine the largest real number c such that for any 2017 real numbers x , x , . . . , x , the inequality
   1 2 2017
   2016
   ∑
   x ( x + x ) ≥ c · x 2
   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
   p = .1
   k +1
   4(1 − p )
   Note that, for every i ≥ 1,k
   ( )
   x 2
   √
   2 2 i
   (1 − p ) · x + x x + p x = + p x ≥ 0 .
   √
   i i i +1 i +1 i +1 i +1
   i i +1
   2 pi +1
   Summing from i = 1 to n givesn
   ∑
   x ( x + x ) ≥ − p x .2
   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