2025 USAMO 第 3 题

Alice the architect and Bob the builder play a game. First, Alice chooses two points $P$ and $Q$ in the plane and a subset $\mathcal{S}$ of the plane, which are announced to Bob. Next, Bob marks infinitely many points in the plane, designating each a city. He may not place two cities within distance at most one unit of each other, and no three cities he places may be collinear. Finally, roads are constructed between the cities as follows: for each pair $A,\,B$ of cities, they are connected with a road along the line segment $AB$ if and only if the following condition holds: For every city $C$ distinct from $A$ and $B$, there exists $R\in\mathcal{S}$ such that $\triangle PQR$ is directly similar to either $\triangle ABC$ or $\triangle BAC$. Alice wins the game if (i) the resulting roads allow for travel between any pair of cities via a finite sequence of roads and (ii) no two roads cross. Otherwise, Bob wins. Determine, with proof, which player has a winning strategy.

Note: $\triangle UVW$ is directly similar to $\triangle XYZ$ if there exists a sequence of rotations, translations, and dilations sending $U$ to $X$, $V$ to $Y$, and $W$ to $Z$.

建筑师爱丽丝和建造者鲍勃玩游戏。首先，Alice 选择平面上的两个点 $P$ 和 $Q$ 以及平面的子集 $\mathcal{S}$，并将其公布给 Bob。接下来，鲍勃在平面上标记了无限多个点，并为每个点指定了一个城市。他不得将两个城市放置在相距最多一单位的距离内，并且他放置的三个城市不得共线。最后，在城市之间构建道路如下：对于每对城市 $A,\,B$，当且仅当满足以下条件时，它们与沿线段 $AB$ 的道路连接：对于不同于 $A$ 和 $B$ 的每个城市 $C$，都存在 $R\in\mathcal{S}$，使得 $\triangle PQR$ 直接类似于 $\triangle ABC$ 或 $\triangle BAC$。如果 (i) 生成的道路允许通过有限的道路序列在任何一对城市之间行驶，并且 (ii) 没有两条道路交叉，则爱丽丝赢得游戏。否则，鲍勃获胜。通过证据确定哪个玩家有获胜策略。

注意：如果存在将 $U$ 发送到 $X$、$V$ 发送到 $Y$、$W$ 发送到 $Z$ 的旋转、平移和膨胀序列，则 $\triangle UVW$ 与 $\triangle XYZ$ 直接相似。