2012 USAMO Problems/Problem 3

Problem

Determine which integers $n > 1$ have the property that there exists an infinite sequence $a_1$ , $a_2$ , $a_3$ , $\dots$ of nonzero integers such that the equality $a_k + 2a_{2k} + \dots + na_{nk} = 0$ holds for every positive integer $k$ .

Solution

For $n$ equal to any odd prime $p$ , the sequence $\left\{a_i = \left(\frac{1-n}{2}\right)^{m_p\left(i\right)}\right\}$ , where $p^{m_p\left(i\right)}$ is the greatest power of $p$ that divides $i$ , gives a valid sequence. Therefore, the set of possible values for $n$ is at least the set of odd primes.

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2012 USAMO (Problems • Resources)
Preceded by Problem 2	Followed by Problem 4
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2012 USAMO Problems/Problem 3

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