1997 IMO Problems/Problem 6

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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1997 IMO (Problems) • Resources
Preceded by Problem 5	1 • 2 • 3 • 4 • 5 • 6	Followed by Last Question
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1997 IMO Problems/Problem 6

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