P10398 『STA - R5』Remove and Decrease Game

You are given $n$ piles of stones. The $i$-th pile contains $a_i$ stones, and it is **guaranteed that $\bm{a_i}$ are all distinct**. Alice and Bob take turns performing one of the following two operations, and after the operation they remove any pile whose number of stones becomes $0$. Alice moves first. The player who cannot make a move loses. - Take one stone from every pile. - Remove the pile with the smallest number of stones. Assuming both players play optimally, determine who will win. You need to answer $T$ queries.

**This problem contains multiple queries in a single test case.** The first line contains a positive integer $T$, the number of queries. For each query: the first line contains a positive integer $n$, the number of piles. The second line contains $n$ positive integers, where the $i$-th integer represents $a_i$.

For each query, output one line containing `Alice` or `Bob`, indicating who will win.

**This problem uses bundled tests.** For $100\%$ of the testdata: - $1 \le T \le 2 \times 10^5$； - $1 \le n \le 2 \times 10^5$； - $1 \le a_i \le 10^9$； - $a_i$ are pairwise distinct； - $\sum n \le 2 \times 10^5$。 The detailed subtask distribution is as follows: |Subtask ID|Constraints|Score| |:--------:|:--------:|:--------:| |1|$n \le 2$|$3$| |2|$a_i \le 1000$, $\sum n \le 10^4$|$23$| |3|$\sum n^2 \le 2 \times 10^6$|$23$| |4|$10^8 \le a_i \le 10^9$|$23$| |5|No special constraints|$28$| Translated by ChatGPT 5