Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 7.9 National olympiad, round 2 Find the answer

Is there an infinite sequence of real numbers $a_1, a_2, a_3, \dots$ such that
$a_1^m + a_2^m + a_3^m + \cdots = m$
for every positive integer $m$ ?

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

No such sequence exists. If it did, then the Cauchy-Schwartz inequality would imply
$\begin{align*} 8 &= (a_1^2 + a_2^2 + \cdots)(a_1^4 + a_2^4 + \cdots) \\ &\geq (a_1^3 + a_2^3 + \cdots)^2 = 9, \end{align*}$
contradiction.

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