MathDB

Let

A, B

, and

F

be positive integers, and assume

A < B < 2A

. A flea is at the number

0

on the number line. The flea can move by jumping to the right by

A

or by

B

. Before the flea starts jumping, Lavaman chooses finitely many intervals

\{m+1, m+2, \ldots, m+A\}

consisting of

A

consecutive positive integers, and places lava at all of the integers in the intervals. The intervals must be chosen so that:
(i) any two distinct intervals are disjoint and not adjacent; (ii) there are at least

F

positive integers with no lava between any two intervals; and (iii) no lava is placed at any integer less than

F

.
Prove that the smallest

F

for which the flea can jump over all the intervals and avoid all the lava, regardless of what Lavaman does, is

F = (n-1)A + B

, where

n

is the positive integer such that

\frac{A}{n+1} \le B-A < \frac{A}{n}

Problem Statement