5. All integers from 1 to 100 are written on the board. Mitya and Dima take turns erasing one number at a time until only two numbers remain. If their sum is divisible by 7, Mitya wins; if not, Dima wins. Mitya goes first. Who wins if the players do not make mistakes?
Answer: Dima wins.

Solution. Dima can ensure his victory by adhering to the following strategy. First, he mentally divides all the numbers into pairs: 1 and 100, 2 and 99, 3 and 98, ..., 50 and 51. Next, no matter which number Mitya erases, Dima erases the second number from the same pair. In the end, one of the pairs listed above will remain on the board. Since the sum of the numbers in each pair is 101, and 101 is not divisible by 7, Dima wins.

Grading criteria. 1 point for the correct answer without specifying the strategy. Any other correct strategy for Dima (there are many such strategies) - 7 points. Considering any number of examples without specifying the strategy does not count as a solution, but points can be added for substantial ideas (for example, for the idea of dividing the numbers into pairs).