2005 OIM Problems/Problem 2

Problem

A flea jumps over integer points on the number line. In its first movement it jumps from point 0 and ands at point 1. Then, if in one movement the flea jumped from point $a$ and fell at point $b$ , in the next movement it jumps from point $b$ and fall at one of the points $b + (b - a) - 1,\; b + (b - a),\; b + (b - a) + 1$ . Show that if the flea has landed on point $n$ twice, for a positive integer $n$ , then it must have made at least $t$ moves, where $t$ is the smallest positive integer greater than or equal to $2\sqrt{n}$ .

~translated into English by Tomas Diaz. ~orders@tomasdiaz.com

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2005 OIM Problems/Problem 2

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