POTWs 4

Every week, we will select three notable or interesting problems, marked with E,M,H (“easy”, “medium”, “hard”) for the relative difficulty. Easy problems will be around TMO or easier than TMO; medium problems will be around Oct Camp / easy IMO, and hard problems will be around medium / hard IMO.

E4 [Classical]

Alice and Bob alternatively choose numbers from among $1,2,...,9$ , without replacement. The first to obtain $3$ numbers which sum to $15$ wins. Does Alice (the first to play) have a winning strategy?

M4 [The Guardian]

Given a $100\times 100$ grid with an arrow pointing to one of four direction (up/down or left/right) in each of $100^2$ cell. Initially you’re in a cell. The goal is to get out of the grid. In each turn, you must move in the neighbour cell, if not out of the grid, that the arrow in your current cell pointed to, and once you leave the old cell, you must also turn the arrow in that cell by $90$ degree clockwise. Is it true that no matter which square you chose to begin, and no matter what directions the arrows are initially pointing at, you will eventually get out of the grid?

H4 [@gausskarl on AoPS]

Let $\mathcal{P} =\{ P_1,P_2,...,P_{2018} \}$ be a set of $2018$ points in the interior of a circle of radius $1$ with $P_1$ be the center of the circle. For each $k=1,2,...,2018$ , let $d_k$ be the distance from $P_k$ to the other point in $\mathcal{P}$ that is closest to $P_k$ . Prove that