Insurance

AtCoder

IDarc122_b

Time2000ms

Memory256MB

Difficulty—

Snuke has read his own fortune for tomorrow, and learned that there are $N$ scenarios that can happen, one of which will happen tomorrow with equal probability. The $i$\-th scenario will cost him $A_i$ yen (Japanese currency). Following this, Snuke has decided to get insurance today. If he pays $x$ yen to his insurance company, he will get compensation of $\min(A_i,2x)$ yen when $A_i$ yen is lost. Here, he can choose any non-negative **real number** as $x$. Snuke wants to minimize the expected value of the amount of money he loses, which is $x+A_i-\min(A_i,2x)$. Find the minimized value. ## Constraints * $1 \leq N \leq 10^5$ * $1 \leq A_i \leq 10^9$ * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $A_1$ $A_2$ $\cdots$ $A_N$ [samples]

Samples

Input #1

3
3 1 4

Output #1

1.83333333333333333333

The optimum choice is $x=1.5$. After paying $1.5$ yen, one of the following three scenarios will happen with equal probability:

*   Scenario $1$: Lose $3$ yen and get compensation of $\min(3,2x)=3$ yen. After all, Snuke loses $x+A_1-\min(A_1,2x)=1.5+3-3=1.5$ yen.
    
*   Scenario $2$: Lose $1$ yen and get compensation of $\min(1,2x)=1$ yen. After all, Snuke loses $x+A_2-\min(A_2,2x)=1.5+1-1=1.5$ yen.
    
*   Scenario $3$: Lose $4$ yen and get compensation of $\min(4,2x)=3$ yen. After all, Snuke loses $x+A_3-\min(A_3,2x)=1.5+4-3=2.5$ yen.
    

Thus, the expected amount of money lost is $(1.5+1.5+2.5)/3=1.833333\cdots$ yen.

Input #2

10
866111664 178537096 844917655 218662351 383133839 231371336 353498483 865935868 472381277 579910117

Output #2

362925658.10000000000000000000

API Response (JSON)

{
  "problem": {
    "name": "Insurance",
    "description": {
      "content": "Snuke has read his own fortune for tomorrow, and learned that there are $N$ scenarios that can happen, one of which will happen tomorrow with equal probability. The $i$\\-th scenario will cost him $A_i",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc122_b"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Snuke has read his own fortune for tomorrow, and learned that there are $N$ scenarios that can happen, one of which will happen tomorrow with equal probability. The $i$\\-th scenario will cost him $A_i...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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