cloudRo wrote: Functions f, g are defined on the set of all integers in the interval [-100, 100] and take integer values. Prove that for some integers k, the number of solutions of the equation f(x)-g(y) = k is odd. ( A. Golovanov) Let $n_k$ be the number of solutions of equation $f(x)-g(y)=k$ Note that $\{f(x)-g(y)$ where $x,y$ are integers $\in[-100,+100]\}$ is bounded and so $\subseteq[u,v]$ for some integers $u,v$ $\sum_{k=u}^vn_k=201^2$ is odd Hence the claim.

FTSOC assume for each $k \in \mathbb{Z},$ there are an even amount of ordered pairs $(x,y)$ that satisfy $f(x)-g(y)=k.$ By the definition of the functions, $f(x)-g(y)$ has a unique integer value. Summing over all $k \in \mathbb{Z},$ we get the total number of ordered pairs of $(x,y)$ is even, which is a contradiction as there are $201^2$ ordered pairs $(x,y),$ an odd number