2017 IMO Shortlist N2

Let $p \geq 2$ be a prime number. Eduardo and Fernando play the following game making moves alternately: in each move, the current player chooses an index $i$ in the set $\{0,1, \ldots, p-1\}$ that was not chosen before by either of the two players and then chooses an element $a_{i}$ of the set $\{0,1,2,3,4,5,6,7,8,9\}$. Eduardo has the first move. The game ends after all the indices $i \in\{0,1, \ldots, p-1\}$ have been chosen. Then the following number is computed: $$M=a_{0}+10 \cdot a_{1}+\cdots+10^{p-1} \cdot a_{p-1}=\sum_{j=0}^{p-1} a_{j} \cdot 10^{j}$$ The goal of Eduardo is to make the number $M$ divisible by $p$, and the goal of Fernando is to prevent this. Prove that Eduardo has a winning strategy. (Morocco)

设 $p \geq 2$ 为素数。 Eduardo 和 Fernando 交替进行以下游戏：在每一步中，当前玩家在集合 $\{0,1, \ldots, p-1\}$ 中选择两个玩家之前未选择的索引 $i$，然后选择集合 $\{0,1,2,3,4,5,6,7,8,9\}$ 中的元素 $a_{i}$。爱德华多率先行动。在选择了所有索引 $i \in\{0,1, \ldots, p-1\}$ 后游戏结束。然后计算以下数字： $$M=a_{0}+10 \cdot a_{1}+\cdots+10^{p-1} \cdot a_{p-1}=\sum_{j=0}^{p-1} a_{j} \cdot 10^{j}$$ Eduardo 的目标是使数字 $M$ 能被 $p$ 整除，而 Fernando 的目标是防止这个。证明爱德华多有制胜策略。 (摩洛哥)