Polynomial Construction

AtCoder

IDabc137_f

Time2000ms

Memory256MB

Difficulty—

Given are a prime number $p$ and a sequence of $p$ integers $a_0, \ldots, a_{p-1}$ consisting of zeros and ones. Find a polynomial of degree at most $p-1$, $f(x) = b_{p-1} x^{p-1} + b_{p-2} x^{p-2} + \ldots + b_0$, satisfying the following conditions: * For each $i$ $(0 \leq i \leq p-1)$, $b_i$ is an integer such that $0 \leq b_i \leq p-1$. * For each $i$ $(0 \leq i \leq p-1)$, $f(i) \equiv a_i \pmod p$. ## Constraints * $2 \leq p \leq 2999$ * $p$ is a prime number. * $0 \leq a_i \leq 1$ ## Input Input is given from Standard Input in the following format: $p$ $a_0$ $a_1$ $\ldots$ $a_{p-1}$ [samples]

Samples

Input #1

2
1 0

Output #1

1 1

$f(x) = x + 1$ satisfies the conditions, as follows:

*   $f(0) = 0 + 1 = 1 \equiv 1 \pmod 2$
*   $f(1) = 1 + 1 = 2 \equiv 0 \pmod 2$

Input #2

3
0 0 0

Output #2

0 0 0

$f(x) = 0$ is also valid.

Input #3

5
0 1 0 1 0

Output #3

0 2 0 1 3

API Response (JSON)

{
  "problem": {
    "name": "Polynomial Construction",
    "description": {
      "content": "Given are a prime number $p$ and a sequence of $p$ integers $a_0, \\ldots, a_{p-1}$ consisting of zeros and ones. Find a polynomial of degree at most $p-1$, $f(x) = b_{p-1} x^{p-1} + b_{p-2} x^{p-2} + ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc137_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Given are a prime number $p$ and a sequence of $p$ integers $a_0, \\ldots, a_{p-1}$ consisting of zeros and ones.\nFind a polynomial of degree at most $p-1$, $f(x) = b_{p-1} x^{p-1} + b_{p-2} x^{p-2} + ...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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