|A + A|

AtCoder

IDarc200_d

Time2000ms

Memory256MB

Difficulty—

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 exist, find one. * $1\le N\le M$ * $0\le A_i < M$ $(1\le i\le N)$ * There are exactly $K$ integers $0\le k < M$ such that there exists a pair of indices $(i,j)$ satisfying $k\equiv A_i+A_j \pmod M$. You are given $T$ test cases, so find the answer for each. ## Constraints * $1\le T\le 2\times 10^5$ * $1\le K\le M\le 2\times 10^5$ * The sum of $M$ over all test cases is at most $2\times 10^5$. * All input values are integers. ## Input The input is given from Standard Input in the following format: $T$ $\text{case}_1$ $\text{case}_2$ $\vdots$ $\text{case}_T$ Each test case is given in the following format: $M$ $K$ [samples]

Samples

Input #1

Output #1

Yes
3
3 1 4
No
Yes
2
0 1

For the first test case, if we set $A=(3,1,4)$, then

*   $k=0$: Setting $(i,j)=(1,1)$, we have $0\equiv 3+3\pmod 6$.
*   $k=1$: Setting $(i,j)=(1,3)$, we have $1\equiv 3+4\pmod 6$.
*   $k=2$: Setting $(i,j)=(3,3)$, we have $2\equiv 4+4\pmod 6$.
*   $k=3$: There is no pair of indices $(i,j)$ that satisfies the condition.
*   $k=4$: Setting $(i,j)=(1,2)$, we have $4\equiv 3+1\pmod 6$.
*   $k=5$: Setting $(i,j)=(2,3)$, we have $5\equiv 1+4\pmod 6$.

Thus, there are exactly $5$ values of $0\le k < 6$ that satisfy the condition.

API Response (JSON)

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    "name": "|A + A|",
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    "platform": "AtCoder",
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    "difficulty": "None",
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      "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 exi...",
      "is_translate": false,
      "language": "English"
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