Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.8 AIME Find the answer

Let $f(n)$ be the number of times you have to hit the $\sqrt{ }$ key on a calculator to get a number less than 2 starting from $n$ . For instance, $f(2)=1, f(5)=2$ . For how many $1<m<2008$ is $f(m)$ odd?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

This is $[2^{1}, 2^{2}) \cup [2^{4}, 2^{8}) \cup [2^{16}, 2^{32}) \ldots$ , and $2^{8}<2008<2^{16}$ so we have exactly the first two intervals.

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