MathDB

6

Part of 2017 Polish MO Finals

Problems(1)

Inequality on a sequence [Poland 2017, P6]

Source: Polish Mathematical Olympiad Finals, Day 2, Problem 3

4/4/2017
Three sequences (a0,a1,…,an)(a_0, a_1, \ldots, a_n), (b0,b1,…,bn)(b_0, b_1, \ldots, b_{n}), (c0,c1,…,c2n)(c_0, c_1, \ldots, c_{2n}) of non-negative real numbers are given such that for all 0≤i,j≤n0\leq i,j\leq n we have aibj≤(ci+j)2a_ib_j\leq (c_{i+j})^2. Prove that ∑i=0nai⋅∑j=0nbj≤(∑k=02nck)2.\sum_{i=0}^n a_i\cdot\sum_{j=0}^n b_j\leq \left( \sum_{k=0}^{2n} c_k\right)^2.
inequalities