Question 161
Let $A$, $B$, and $F$ be positive integers, and assume $A \lt B \lt 2A$. 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 $\{m+1, m+2, \dots, m+A\}$ 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
$$
\frac{A}{n+1} \leq B - A \lt \frac{A}{n} .
$$
(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
$$
\frac{A}{n+1} \leq B - A \lt \frac{A}{n} .
$$