P11010 "STA - R7" Divide and Merge Game

Given two positive integers $n, k(2 \le k \le n)$, Alice and Bob will play the following game: - Alice needs to provide a sequence $a$ of length $k$ consisting of **positive integers**, such that $\sum\limits_{i = 1}^{k} a_i = n$. - Bob needs to try to give a positive integer $m$ with $m \ge 2$, such that the positive integer sequence $a$ given by Alice can be partitioned into $m$ **non-empty multisets**, and the sums of elements in these multisets are all equal. If Bob can give such an integer $m$, then Bob wins; otherwise, Alice wins. Assuming both players use optimal strategies, determine who will win. You need to answer $T$ queries.

**This problem contains multiple queries in a single test file.** The first line contains a positive integer $T$, representing the number of queries. For each query: one line contains two positive integers $n, k$, with the meaning described in the **Description**.

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

**Sample Explanation** For the first group of testdata, Alice can only provide the positive integer sequence $\left\{1,1,1,1\right\}$. Then Bob gives $m = 4$ and partitions the sequence into $\left\{\left\{1\right\},\left\{1\right\},\left\{1\right\},\left\{1\right\}\right\}$. Bob can also give $m = 2$ and partition the sequence into $\left\{\left\{1, 1\right\}, \left\{1, 1\right\}\right\}$, obtaining two multisets whose element sums are both $2$, which also satisfies the requirement. For the second group of testdata, Alice can provide the positive integer sequence $\left\{3, 2, 3\right\}$. It can be proven that Bob has no valid partition plan at this time, so Alice wins. **Constraints** **This problem uses bundled tests.** For $100\%$ of the data: - $1 \le T \le 2 \times 10^5$; - $2 \le k \le n \le 10^8$. The detailed subtask allocation is as follows: |Subtask ID|Constraints|Score| |:--------:|:--------:|:--------:| |1|$n \le 10$|$16$| |2|$k^2 \le n$|$27$| |3|$2 \nmid n$|$27$| |4|No special restrictions|$30$| Translated by ChatGPT 5