I have the following problem from USAMO 2006:
"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 $k\ge 2$ is a positive integer and $i$ is a nonnegative integer, then the minimum number of jumps needed to reach $2^i k$ is greater than the minimum number of jumps needed to reach $2^i$."
I found a solution to this here:
https://artofproblemsolving.com/wiki/index.php?title=2006_USAMO_Problems/Problem_5#See_also
However, in this solution, the author makes the following statement:
" The sequence $J_{i_0,k_0}$ cannot contain $2^{t_0}$ because it takes more jumps to reach $2^{t_0} k_0$ than it does to reach $2^{t_0}$." What is $t_0$ here and how has he arrived at the conclusion mentioned?
Given that $t_0$ makes no further appearance in the argument, it seems reasonable to surmise that there are a few typos and this should read:
"The sequence $J_{i_0,k_0}$ cannot contain $2^{i_0}$ because it takes no more jumps to reach $2^{i_0} k_0$ than it does to reach $2^{i_0}$."