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1997 IMO Problems/Problem 6

Revision as of 15:22, 6 October 2023 by Tomasdiaz (talk | contribs) (Created page with "==Problem== For each positive integer <math>n</math>, let <math>f(n)</math> denote the number of ways of representing <math>n</math> as a sum of powers of <math>2</math> with...")
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Problem

For each positive integer $n$, let $f(n)$ denote the number of ways of representing $n$ as a sum of powers of $2$ with nonnegative integer exponents. Representations which differ only in the ordering of their summands are considered to be the same. For instance, $f(4)=4$, because the number 4 can be represented in the following four ways:

$4;2+2;2+1+1;1+1+1+1$

Prove that, for any integer $n \ge 3$,

$2^{n^{2}/4}<f(2^{n})<2^{n^{2}/2}$.

Solution

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