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Not all checkers are in the same row

Source: APMO 2006, Problem 3

March 24, 2006
number theorycombinatoricsAPMO

Problem Statement

Let p≥5p\ge5 be a prime and let rr be the number of ways of placing pp checkers on a p×pp\times p checkerboard so that not all checkers are in the same row (but they may all be in the same column). Show that rr is divisible by p5p^5. Here, we assume that all the checkers are identical.
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