Problem:
﻿Score : 400 points

Problem Statement

In the year 2168, AtCoder Inc., which is much larger than now, is starting a limited express train service called AtCoder Express.

In the plan developed by the president Takahashi, the trains will run as follows:

A train will run for (t_1 + t_2 + t_3 + ... + t_N) seconds.

In the first t_1 seconds, a train must run at a speed of at most v_1 m/s (meters per second). Similarly, in the subsequent t_2 seconds, a train must run at a speed of at most v_2 m/s, and so on.

According to the specifications of the trains, the acceleration of a train must be always within ±1m/s^2. Additionally, a train must stop at the beginning and the end of the run.

Find the maximum possible distance that a train can cover in the run.

Constraints

1 \leq N \leq 100

1 \leq t_i \leq 200

1 \leq v_i \leq 100

All input values are integers.

Input

Input is given from Standard Input in the following format:

N
t_1 t_2 t_3 … t_N
v_1 v_2 v_3 … v_N

Output

Print the maximum possible that a train can cover in the run.

Output is considered correct if its absolute difference from the judge's output is at most 10^{-3}.

Sample Input 1

1
100
30

Sample Output 1

2100.000000000000000

The maximum distance is achieved when a train runs as follows:

In the first 30 seconds, it accelerates at a rate of 1m/s^2, covering 450 meters.

In the subsequent 40 seconds, it maintains the velocity of 30m/s, covering 1200 meters.

In the last 30 seconds, it decelerates at the acceleration of -1m/s^2, covering 450 meters.

The total distance covered is 450 + 1200 + 450 = 2100 meters.

Sample Input 2

2
60 50
34 38

Sample Output 2

2632.000000000000000

The maximum distance is achieved when a train runs as follows:

In the first 34 seconds, it accelerates at a rate of 1m/s^2, covering 578 meters.

In the subsequent 26 seconds, it maintains the velocity of 34m/s, covering 884 meters.

In the subsequent 4 seconds, it accelerates at a rate of 1m/s^2, covering 144 meters.

In the subsequent 8 seconds, it maintains the velocity of 38m/s, covering 304 meters.

In the last 38 seconds, it decelerates at the acceleration of -1m/s^2, covering 722 meters.

The total distance covered is 578 + 884 + 144 + 304 + 722 = 2632 meters.

Sample Input 3

3
12 14 2
6 2 7

Sample Output 3

76.000000000000000

The maximum distance is achieved when a train runs as follows:

In the first 6 seconds, it accelerates at a rate of 1m/s^2, covering 18 meters.

In the subsequent 2 seconds, it maintains the velocity of 6m/s, covering 12 meters.

In the subsequent 4 seconds, it decelerates at the acceleration of -1m/s^2, covering 16 meters.

In the subsequent 14 seconds, it maintains the velocity of 2m/s, covering 28 meters.

In the last 2 seconds, it decelerates at the acceleration of -1m/s^2, covering 2 meters.

The total distance covered is 18 + 12 + 16 + 28 + 2 = 76 meters.

Sample Input 4

1
9
10

Sample Output 4

20.250000000000000000

The maximum distance is achieved when a train runs as follows:

In the first 4.5 seconds, it accelerates at a rate of 1m/s^2, covering 10.125 meters.

In the last 4.5 seconds, it decelerates at the acceleration of -1m/s^2, covering 10.125 meters.

The total distance covered is 10.125 + 10.125 = 20.25 meters.

Sample Input 5

10
64 55 27 35 76 119 7 18 49 100
29 19 31 39 27 48 41 87 55 70

Sample Output 5

20291.000000000000

Solution in Kotlin (1.0.0) (judge verdict: Runtime Error)

import java.util.*

fun main(args: Array<String>) {
    val N = readLine()!!.toInt()
    val t = (listOf(0L) + readLine()!!.split(" ").map{it.toLong()} + listOf(0L)).toLongArray()
    val v = listOf(0L) + readLine()!!.split(" ").map{it.toLong()} + listOf(0L)

    for (i in 1 until t.size) {
        t[i] += t[i - 1]
    }

    var res = 0.0

    for (i in 1..N) {
        var l = 0L
        for (j in 1 .. i) {
            l = Math.min(l, v[j] - t[j - 1])
        }

        var r = 1e18.toLong()
        for (j in N + 1 downTo i) {
            r = Math.min(r, v[j] + t[j])
        }

        val x1 = (r - l) / 2.0
        val y1 = (r + l) / 2.0

        val list = ArrayList<Pair<Double, Double>>()

        list.add(Pair(t[i - 1].toDouble(), Math.min(l + t[i - 1], r - t[i - 1]).toDouble()))

        val x0 = v[i] - l
        if (t[i - 1] <= x0 && x0 <= t[i]) {
            list.add(Pair(x0.toDouble(), v[i].toDouble()))
        }

        if (t[i - 1] <= x1 && x1 <= t[i] && y1 <= v[i]) {
            list.add(Pair(x1, y1))
        }

        val x2 = r - v[i]
        if (t[i - 1] <= x2 && x2 <= t[i]) {
            list.add(Pair(x2.toDouble(), v[i].toDouble()))
        }

        list.add(Pair(t[i].toDouble(), Math.min(l + t[i], r - t[i]).toDouble()))

        for (j in 1 until list.size) {
            res += (list[j].second + list[j - 1].second) * (list[j].first - list[j - 1].first) / 2.0
        }
    }

    System.out.printf("%1.10f¥n")
}