Question 6
Let $p,q,n$ be three positive integers with $p+q\lt n$. Let $(x_0,x_1,\ldots,x_n)$ be an $(n+1)$-tuple of integers satisfying the following conditions:
(i) $x_0=x_n=0$.
(ii) For each $i$ with $1\le i\le n$, either $x_i-x_{i-1}=p$ or $x_i-x_{i-1}=-q$.
Show that there exists a pair $(i,j)$ of distinct indices with $(i,j)\ne(0,n)$ such that $x_i=x_j$.
(i) $x_0=x_n=0$.
(ii) For each $i$ with $1\le i\le n$, either $x_i-x_{i-1}=p$ or $x_i-x_{i-1}=-q$.
Show that there exists a pair $(i,j)$ of distinct indices with $(i,j)\ne(0,n)$ such that $x_i=x_j$.