CAPS 2014 Problem 5

Let all positive integers $n$ satisfy the following condition: for each non-negative integers $k, m$ with $k + m \le n$ , the numbers $\binom{n-k}{m}$ and $\binom{n-m}{k}$ leave the same remainder when divided by $2$ .

(Poland)

PS. The translation was done using Google translate and in case it is not right, there is the original text in Slovak

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