2006 USAMO Problems/Problem 5

Problem

A mathematical frog jumps along the number line. The frog starts at 1, and jumps according to the following rule: if the frog is at integer $\displaystyle n$ , then it can jump either to $\displaystyle n+1$ or to $n+2^{m_n+1}$ where $2^{m_n}$ is the largest power of 2 that is a factor of $\displaystyle n$ . Show that if $k\ge 2$ is a positive integer and $\displaystyle i$ is a nonnegative integer, then the minimum number of jumps needed to reach $\displaystyle 2^i k$ is greater than the minimum number of jumps needed to reach $\displaystyle 2^i$ .