Let
![p,q,n](/media/m/c/3/7/c372bd04bedef34df3d07435a49a7689.png)
be three positive integers with
![p + q < n](/media/m/7/3/c/73cb72438c4ca0a0895942ca3fe6d6c1.png)
. Let
![(x_{0},x_{1},\cdots ,x_{n})](/media/m/2/e/1/2e18289799124dc1e7a0f27fb73ce0d5.png)
be an
![(n + 1)](/media/m/8/a/5/8a5eeda99b96726b7e47d16dcee50440.png)
-tuple of integers satisfying the following conditions :
(a)
![x_{0} = x_{n} = 0](/media/m/0/c/3/0c3f4b997b9ea54c9d55fe4e646876d6.png)
, and
(b) For each
![i](/media/m/3/2/d/32d270270062c6863fe475c6a99da9fc.png)
with
![1\leq i\leq n](/media/m/1/e/e/1ee4f10bd28ea7bb5a88120fbf9e78a1.png)
, either
![x_{i} - x_{i - 1} = p](/media/m/8/3/2/8324bce8e2776f013118a763ad2880c6.png)
or
![x_{i} - x_{i - 1} = - q](/media/m/0/d/c/0dcc3f58b2a157660099f4bc11a1ca72.png)
.
Show that there exist indices
![i < j](/media/m/5/1/e/51ecc47ba337386b413e10d240ce6439.png)
with
![(i,j)\neq (0,n)](/media/m/a/d/f/adfd829ebc128b36d309d049457e51d7.png)
, such that
![x_{i} = x_{j}](/media/m/2/f/f/2ffeec1734d8205caefb7c65f307f715.png)
.
%V0
Let $p,q,n$ be three positive integers with $p + q < n$. Let $(x_{0},x_{1},\cdots ,x_{n})$ be an $(n + 1)$-tuple of integers satisfying the following conditions :
(a) $x_{0} = x_{n} = 0$, and
(b) For each $i$ with $1\leq i\leq n$, either $x_{i} - x_{i - 1} = p$ or $x_{i} - x_{i - 1} = - q$.
Show that there exist indices $i < j$ with $(i,j)\neq (0,n)$, such that $x_{i} = x_{j}$.