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Chinese Girls Mathematical Olympiad 2016, Problem 4

Source: China Beijing ,12 Aug 2016

August 12, 2016
algebrainequalities

Problem Statement

Let nn is a positive integers ,a1,a2,⋯ ,an∈{0,1,⋯ ,n}a_1,a_2,\cdots,a_n\in\{0,1,\cdots,n\} . For the integer jj (1≤j≤n)(1\le j\le n) ,define bjb_j is the number of elements in the set {i∣i∈{1,⋯ ,n},ai≥j}\{i|i\in\{1,\cdots,n\},a_i\ge j\} .For example ：When n=3n=3 ,if a1=1,a2=2,a3=1a_1=1,a_2=2,a_3=1 ,then b1=3,b2=1,b3=0b_1=3,b_2=1,b_3=0 . (1)(1) Prove that ∑i=1n(i+ai)2≥∑i=1n(i+bi)2.\sum_{i=1}^{n}(i+a_i)^2\ge \sum_{i=1}^{n}(i+b_i)^2. (2)(2) Prove that ∑i=1n(i+ai)k≥∑i=1n(i+bi)k,\sum_{i=1}^{n}(i+a_i)^k\ge \sum_{i=1}^{n}(i+b_i)^k, for the integer k≥3.k\ge 3.
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