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Number of solutions of a linear equation

Source: Polish Math Olympiad 2023 2nd stage P3

February 10, 2023
equationsnumber of solutionsalgebralinear equation

Problem Statement

Given positive integers k,nk,n and a real number ℓ\ell, where k,n≥1k,n \geq 1. Given are also pairwise different positive real numbers a1,a2,…,aka_1,a_2,\ldots, a_k. Let S={a1,a2,…,ak,−a1,−a2,…,−ak}S = \{a_1,a_2,\ldots,a_k, -a_1, -a_2,\ldots, -a_k\}. Let AA be the number of solutions of the equation x1+x2+…+x2n=0,x_1 + x_2 + \ldots + x_{2n} = 0, where x1,x2,…,x2n∈Sx_1,x_2,\ldots, x_{2n} \in S. Let BB be the number of solutions of the equation x1+x2+…+x2n=ℓ,x_1 + x_2 + \ldots + x_{2n} = \ell, where x1,x2,…,x2n∈Sx_1,x_2,\ldots,x_{2n} \in S. Prove that A≥BA \geq B.
Solutions of an equation with only difference in the permutation are different.
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