CF1451D Circle Game

Utkarsh is forced to play yet another one of Ashish's games. The game progresses turn by turn and as usual, Ashish moves first. Consider the 2D plane. There is a token which is initially at $ (0,0) $ . In one move a player must increase either the $ x $ coordinate or the $ y $ coordinate of the token by exactly $ k $ . In doing so, the player must ensure that the token stays within a (Euclidean) distance $ d $ from $ (0,0) $ . In other words, if after a move the coordinates of the token are $ (p,q) $ , then $ p^2 + q^2 \leq d^2 $ must hold. The game ends when a player is unable to make a move. It can be shown that the game will end in a finite number of moves. If both players play optimally, determine who will win.

N/A

N/A

In the first test case, one possible sequence of moves can be $ (0, 0) \xrightarrow{\text{Ashish }} (0, 1) \xrightarrow{\text{Utkarsh }} (0, 2) $ . Ashish has no moves left, so Utkarsh wins.