51st Ukrainian National Mathematical Olympiad, 4th Round

Enumerate the burrows from left to right with numbers from $1$ to $n$ and define the "distance" between the burrows $i$ and $j$ by $i-j$ (it can be negative).

First Tom checks all the burrows from $1$ to $n$ one after another. If at the moment when Tom checks the first burrow Jerry is in a burrow with an odd number, then Tom will necessarily catch Jerry during this check.

Indeed, if at the moment when Tom checks the first burrow there is no Jerry, then at this moment the "distance" between them is a positive even number. If he still did not catch Jerry when he checks the last burrow, this would mean that the "distance" between them at this moment is a negative even number. Note, that after every Jerry's run and Tom's move the "distance" between them either does not change (if they move in the same direction) or changes by $2$ (if they move in different directions). So if we start with a positive even number and end up with a negative even number, there should be a moment when the "distance" between them is zero, which means that Tom catches Jerry.

If at the beginning Jerry is in a burrow with an even number, then at the moment when Tom checks the $n$-th burrow, Jerry will be in a burrow with a number of the different parity than $n$. After this Jerry moves to the burrow with a number of the same parity as $n$. So, if now Tom checks all the burrows from $n$ to $1$ one after another, he will necessarily catch Jerry by the above arguments because the "distance" between them is now even.