MathDB

Let

n \ge 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

\dots

, 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 \le 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 > 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, \dots , k

Problem Statement