1. (10 points) The rules of the game are as follows: from 64 different items, on each turn, the player must form a set of items that has not been mentioned in the game before, with the number of items equal to the player's age in years. Players take turns; either player can start the game; the player who cannot make a move loses. Sets of items are considered different if they differ by at least one item or if they contain a different number of items. In the game, Vasily and Fyodor are participating; each player has the opportunity to make at least one move. It is known that: a) Vasily is 2 years older than Fyodor; b) Fyodor is at least 5 years old; c) Fyodor always wins. What is the minimum number of years Vasily can be?
Answer: 34 years

Solution: The game move is the selection of 64 elements from a certain subset, which contains as many elements as the player's age. Thus, the problem requires comparing the number of ways to make such a selection: the player with a greater number of ways to choose their "own" subset, i.e., $C_{64}^{x}$, will always win. Let $V$ be Basil's age in years, and $F$ be Fyodor's age in years. According to the conditions: $5 \leq F, F+2=V, V \leq 64$ (since each player can make a move, i.e., at least one combination can be chosen). It is also known that $C_{64}^{F}>C_{64}^{V}$ always (in the case of equality, Fyodor would lose if he went first). We need to find the minimum $V$ that satisfies all the listed conditions.

Let Fyodor's age be denoted by $x$, and Basil's age by $x+2$. Consider the integer function $x \mapsto C_{64}^{x}$. It increases from 0 to 32 and decreases from 32 to 64. Additionally, $C_{64}^{31}=C_{64}^{33}$. Therefore, the inequality $C_{64}^{x}>C_{64}^{x+2}$ holds if and only if $x \geq 32$. Thus, the minimum value of $x+2$ is 34.