Max Dot

AtCoder

IDarc128_c

Time2000ms

Memory256MB

Difficulty—

Given are integers $N$, $M$, $S$, and a sequence of $N$ integers $A=(A_1,A_2,\cdots,A_N)$. Consider making a sequence of $N$ non-negative **real** numbers $x=(x_1,x_2,\cdots,x_N)$ satisfying all of the following conditions: * $0 \leq x_1 \leq x_2 \leq \cdots \leq x_N \leq M$, * $\sum_{1 \leq i \leq N} x_i=S$. Now, let us define the score of $x$ as $\sum_{1 \leq i \leq N} A_i \times x_i$. Find the maximum possible score of $x$. ## Constraints * $1 \leq N \leq 5000$ * $1 \leq M \leq 10^6$ * $1 \leq S \leq \min(N \times M,10^6)$ * $1 \leq A_i \leq 10^6$ * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $M$ $S$ $A_1$ $A_2$ $\cdots$ $A_N$ [samples]

Samples

Input #1

3 2 3
1 2 3

Output #1

8.00000000000000000000

The optimal choice is $x=(0,1,2)$.

Input #2

3 3 2
5 1 1

Output #2

4.66666666666666666667

The optimal choice is $x=(2/3,2/3,2/3)$.

Input #3

10 234567 1000000
353239 53676 45485 617014 886590 423581 172670 928532 312338 981241

Output #3

676780145098.25000000000000000000

API Response (JSON)

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  "problem": {
    "name": "Max Dot",
    "description": {
      "content": "Given are integers $N$, $M$, $S$, and a sequence of $N$ integers $A=(A_1,A_2,\\cdots,A_N)$. Consider making a sequence of $N$ non-negative **real** numbers $x=(x_1,x_2,\\cdots,x_N)$ satisfying all of th",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc128_c"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Given are integers $N$, $M$, $S$, and a sequence of $N$ integers $A=(A_1,A_2,\\cdots,A_N)$.\nConsider making a sequence of $N$ non-negative **real** numbers $x=(x_1,x_2,\\cdots,x_N)$ satisfying all of th...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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