Question 3
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:
each pair $A$, $B$ of cities is 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.
Proposed by Carl Schildkraut.
Manual essay; 1 editorial point, not official contest scoring. Automatic answer fields are intentionally empty.
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:
each pair $A$, $B$ of cities is 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.
Proposed by Carl Schildkraut.
Manual essay; 1 editorial point, not official contest scoring. Automatic answer fields are intentionally empty.