1451D · Circle Game

Problem: 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.

Input Format: The first line contains a single integer $t$ ($1 \leq t \leq 100$) — the number of test cases.

The only line of each test case contains two space separated integers $d$ ($1 \leq d \leq 10^5$) and $k$ ($1 \leq k \leq d$).

Output Format: For each test case, if Ashish wins the game, print "Ashish", otherwise print "Utkarsh" (without the quotes).

Note: 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.