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Harvard-MIT Mathematics Tournament

Number theory Difficulty 4.6 AIME Prove it United States

Problem:
How many of the integers $1, 2, \ldots, 2004$ can be represented as $\frac{mn+1}{m+n}$ for positive integers $m$ and $n$ ?

Solution

Solution:
For any positive integer $a$ , we can let $m = a^{2} + a - 1$ , $n = a + 1$ to see that every positive integer has this property, so the answer is $2004$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.