Question 176
Assume $P_1, P_2, \dots, P_n$ ($n \gt 10$) are pairwise different polynomials with coefficients $1$, $0$, or $-1$ and such that do not have integer roots. Additionally, $\forall i = \overline{1,n}$ $|P_i(5)| \le \frac{n^2}{2}$. Prove that $P_i + P_j = P_k + P_l$ for some $1 \le i,j,k,l \le n$ and $\{i, j\} \neq \{k, l\}$. Note that $i$ may be equal $j$, and $k$ may be equal $l$.