XXVII Olimpiada Matemática Rioplatense

First, we analyze in which cases three of the considered points form an equilateral triangle. Let $ABC$ be an equilateral triangle with vertices among the points.

Case 1: One vertex $A$ is at a point in the lower part of a hexagon, as in the picture (the case when it is in the upper part of a hexagon is analogous). Assume the vertex $B$ is in the half-plane to the left of the line $AY$ (the other case is similar). If $B=Y$, then $C=X$, and if $B=W$, then $C$ cannot be a vertex of one of the hexagons. So, we may now assume that $B$ and $C$ are both in the region determined by the half-lines $\overrightarrow{AX}$ and $\overrightarrow{AZ}$. If $B\ne X$, since $X\hat{A}Z=60^{\circ }$, we have that $B\hat{A}C<60^{\circ }$, a contradiction. Then $B=X$ and $C=Z$.

Case 2: All vertices are points on the vertical sides of the hexagons. Assume $A$ is as in the following picture. Assume $C$ is in the half-plane to the right of the line $AT$ (the other case is similar). If $C=T$ or $C=U$, the third vertex $B$ of the equilateral triangle does not lie in the vertex of an hexagon. Then, both $C$ and $B$ would be in the region determined by the half-lines $\overrightarrow{AU}$ and $\overrightarrow{AV}$, but then, $B\hat{A}C<60^{\circ }$. Contradiction.

Summarizing, the equilateral triangles with vertices in the $2018$ points are those marked in the figure below:

Now, consider the following numbering of the $2018$ points and the $1009$ pairs: $$\{A_1,A_2\},\{A_3,A_4\},\dots ,\{A_{1007},A_{1008}\},\{B_1,B_2\},\{B_3,B_4\},\dots ,\{B_{1007},B_{1008}\},\{X,Y\}.$$ The beetle has a winning strategy. It wins the game by playing as follows: every time the bee chooses and paints a point in one of the above pairs, the beetle chooses and paints the other point in the same pair. Note that every equilateral triangle with vertices in the $2018$ points has two vertices in the same pair; so, if its three vertices were painted the same color, there should be one pair with the two points painted the same color, which cannot happen if the beetle plays according to the strategy.