USAMO 2006 Problem 5

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 $n$ , then it can jump either to $n+1$ or to $n + 2^{m_n+1}$ where $2^{m_n}$ is the largest power of $2$ that is a factor of $n.$ Show that if $k \geq 2$ is a positive integer and $i$ is a nonnegative integer, then the minimum number of jumps needed to reach $2^ik$ is greater than the minimum number of jumps needed to reach $2^i.$