USAMTS 5/2/35

Problem

Let \(m\) and \(n\) be positive integers. Let \(S\) be the set of all points \((x, y)\) with integer coordinates such that \(1 \leq x, y \leq m + n − 1\) and \(m + 1 \leq x + y \leq 2m + n − 1\). Let L be the set of the \(3m + 3n − 3\) lines parallel to one of \(x = 0\), \(y = 0\), or \(x + y = 0\) and passing through at least one point in \(S\). For which pairs \((m, n)\) does there exist a subset \(T\) of \(S\) such that every line in \(L\) intersects an odd number of elements of \(T\)?

Proof

The answer is just all \((m,n)\) such that \( (m+n-1)(m-n) \) is divisible by 4. The sum of all the x and y coordinates should clearly be even, but the sum of all \(x+y \mod 2\) is \( \frac{(m+n-1)(3m+n)}{2} \). For a construction to even be viable, this quantity clearly needs to be even.

Constructor
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