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Brazilian Math Olympiad

Number theory Difficulty 5.0 AIME Prove it Brazil

Determine if there exist positive integers $n, a_1, a_2, \dots, a_{2012}$ all greater than 1, such that
$n^2 = a_1^2 + a_2^3 + a_3^5 + \dots + a_i^{p_i} + \dots + a_{2012}^{p_{2012}},$
where $p_i$ is the i-th prime.

Solution

Just pick any odd $a_2$ , any even $a_3, \dots, a_{2012}$ . Then $a_2^3 + \dots + a_{2012}^{p_{2012}} = 2N+1$ is odd. Then pick $a_1 = N$ and $n = N+1$ , and $n^2 - a_1^2 = (N+1)^2 - N^2 = 2N+1$ .

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