Let $M_1$ and $M_2$ be finite non-empty multi-sets whose elements are those of $\mathbb{R}$. Let $s \in \mathbb{R}$ and define the following quantity. $$P_s := \lvert \{ (x,y) \in M_1 \times M_2 \mid x+y=s \} \rvert $$ Suppose that $M_1$ and $M_2$ have the same size and are non constant, i. e., $\lvert M_1 \rvert = \lvert M_2 \rvert$ and $\exists x_1, x_2 \in M_1$ and $y_1, y_2 \in M_2$ such that $x_1 \ne x_2$ and $y_1 \ne y_2$. Then is it true that there exists $s, t \in \mathbb{R}$ such that $s \ne t$ and $P_s = P_t \ne 0$?
I think it is true and my attempt was to prove this by induction on the size of the sets. But I'm not sure about the inductive step.
It is true for finite sets, but not for multisets. Let $M_1 = \{\,0, 1, 1, 1\,\}$ and $M_2 = M_1$. Then $P_0 = 1$, $P_1 = 6$, $P_2 = 9$ and $P_x = 0$ for $x \notin \{\,0, 1, 2\,\}$.