Question 56
Let $n \geqslant 3$ be an integer. A frog is to jump along the real axis, starting at the point $0$ and making $n$ jumps: one of length $1$, one of length $2, \ldots$, one of length $n$. It may perform these $n$ jumps in any order. If at some point the frog is sitting on a number $a \leqslant 0$, its next jump must be to the right (towards the positive numbers). If at some point the frog is sitting on a number $a\gt 0$, its next jump must be to the left (towards the negative numbers). Find the largest positive integer $k$ for which the frog can perform its jumps in such an order that it never lands on any of the numbers $1,2, \ldots, k$.