HMMT February

Number theory Difficulty 5.1 AIME, harder Find the answer

Compute the number of even positive integers $n \leq 2024$ such that $1,2, \ldots, n$ can be split into $\frac{n}{2}$ pairs, and the sum of the numbers in each pair is a multiple of 3.

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

There have to be an even number of multiples of 3 at most $n$ , so this means that $n \equiv 0,2 \pmod{6}$ . We claim that all these work. We know there are an even number of multiples of 3, so we can pair them; then we can pair $3k+1$ and $3k+2$ for all $k$ . This means the answer is $\frac{2022}{3}+1=675$ .

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