Sum-Product Ratio

AtCoder

IDarc158_b

Time2000ms

Memory256MB

Difficulty—

You are given non-zero integers $x_1, \ldots, x_N$. Find the minimum and maximum values of $\dfrac{x_i+x_j+x_k}{x_ix_jx_k}$ for integers $i$, $j$, $k$ such that $1\leq i < j < k\leq N$. ## Constraints * $3\leq N\leq 2\times 10^5$ * $-10^6\leq x_i \leq 10^6$ * $x_i\neq 0$ ## Input The input is given from Standard Input in the following format: $N$ $x_1$ $\ldots$ $x_N$ [samples]

Samples

Input #1

4
-2 -4 4 5

Output #1

\-0.175000000000000
-0.025000000000000

$\dfrac{x_i+x_j+x_k}{x_ix_jx_k}$ can take the following four values.

*   $(i,j,k) = (1,2,3)$: $\dfrac{(-2) + (-4) + 4}{(-2)\cdot (-4)\cdot 4} = -\dfrac{1}{16}$.
*   $(i,j,k) = (1,2,4)$: $\dfrac{(-2) + (-4) + 5}{(-2)\cdot (-4)\cdot 5} = -\dfrac{1}{40}$．
*   $(i,j,k) = (1,3,4)$: $\dfrac{(-2) + 4 + 5}{(-2)\cdot 4\cdot 5} = -\dfrac{7}{40}$．
*   $(i,j,k) = (2,3,4)$: $\dfrac{(-4) + 4 + 5}{(-4)\cdot 4\cdot 5} = -\dfrac{1}{16}$.

Among them, the minimum is $-\dfrac{7}{40}$, and the maximum is $-\dfrac{1}{40}$.

Input #2

4
1 1 1 1

Output #2

3.000000000000000
3.000000000000000

Input #3

5
1 2 3 4 5

Output #3

0.200000000000000
1.000000000000000

API Response (JSON)

{
  "problem": {
    "name": "Sum-Product Ratio",
    "description": {
      "content": "You are given non-zero integers $x_1, \\ldots, x_N$. Find the minimum and maximum values of $\\dfrac{x_i+x_j+x_k}{x_ix_jx_k}$ for integers $i$, $j$, $k$ such that $1\\leq i < j < k\\leq N$.",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc158_b"
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given non-zero integers $x_1, \\ldots, x_N$. Find the minimum and maximum values of $\\dfrac{x_i+x_j+x_k}{x_ix_jx_k}$ for integers $i$, $j$, $k$ such that $1\\leq i < j < k\\leq N$.\n\n## Constraint...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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