{"problem":{"name":"|A + A|","description":{"content":"You are given positive integers $M,K$. Determine whether there exist a positive integer $N$ and an integer sequence $A=(A_1,A_2,\\ldots, A_N)$ that satisfy all the following conditions, and if they exi","description_type":"Markdown"},"platform":"AtCoder","limit":{"time_limit":2000,"memory_limit":262144},"difficulty":"None","is_remote":true,"is_sync":true,"sync_url":null,"sign":"arc200_d"},"statements":[{"statement_type":"Markdown","content":"You are given positive integers $M,K$.\nDetermine whether there exist a positive integer $N$ and an integer sequence $A=(A_1,A_2,\\ldots, A_N)$ that satisfy all the following conditions, and if they exist, find one.\n\n*   $1\\le N\\le M$\n*   $0\\le A_i < M$ $(1\\le i\\le N)$\n*   There are exactly $K$ integers $0\\le k < M$ such that there exists a pair of indices $(i,j)$ satisfying $k\\equiv A_i+A_j \\pmod M$.\n\nYou are given $T$ test cases, so find the answer for each.\n\n## Constraints\n\n*   $1\\le T\\le 2\\times 10^5$\n*   $1\\le K\\le M\\le 2\\times 10^5$\n*   The sum of $M$ over all test cases is at most $2\\times 10^5$.\n*   All input values are integers.\n\n## Input\n\nThe input is given from Standard Input in the following format:\n\n$T$\n$\\text{case}_1$\n$\\text{case}_2$\n$\\vdots$\n$\\text{case}_T$\n\nEach test case is given in the following format:\n\n$M$ $K$\n\n[samples]","is_translate":false,"language":"English"}],"meta":{"iden":"arc200_d","tags":[],"sample_group":[["3\n6 5\n3 2\n8 3","Yes\n3\n3 1 4\nNo\nYes\n2\n0 1\n\nFor the first test case, if we set $A=(3,1,4)$, then\n\n*   $k=0$: Setting $(i,j)=(1,1)$, we have $0\\equiv 3+3\\pmod 6$.\n*   $k=1$: Setting $(i,j)=(1,3)$, we have $1\\equiv 3+4\\pmod 6$.\n*   $k=2$: Setting $(i,j)=(3,3)$, we have $2\\equiv 4+4\\pmod 6$.\n*   $k=3$: There is no pair of indices $(i,j)$ that satisfies the condition.\n*   $k=4$: Setting $(i,j)=(1,2)$, we have $4\\equiv 3+1\\pmod 6$.\n*   $k=5$: Setting $(i,j)=(2,3)$, we have $5\\equiv 1+4\\pmod 6$.\n\nThus, there are exactly $5$ values of $0\\le k < 6$ that satisfy the condition."]],"created_at":"2026-03-03 11:01:14"}}