Problem

Source: Polish Math Olympiad 2023 2nd stage P3

Tags: equations, number of solutions, algebra, linear equation



Given positive integers $k,n$ and a real number $\ell$, where $k,n \geq 1$. Given are also pairwise different positive real numbers $a_1,a_2,\ldots, a_k$. Let $S = \{a_1,a_2,\ldots,a_k, -a_1, -a_2,\ldots, -a_k\}$. Let $A$ be the number of solutions of the equation $$x_1 + x_2 + \ldots + x_{2n} = 0,$$where $x_1,x_2,\ldots, x_{2n} \in S$. Let $B$ be the number of solutions of the equation $$x_1 + x_2 + \ldots + x_{2n} = \ell,$$where $x_1,x_2,\ldots,x_{2n} \in S$. Prove that $A \geq B$. Solutions of an equation with only difference in the permutation are different.