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 Java8 (OpenJDK 1.8.0) (judge verdict: Wrong Answer)

import java.math.BigDecimal;
import java.util.*;
import java.util.stream.Collectors;

public class Main {

    public static void main(String[] args) {
        final Scanner scanner = new Scanner(System.in);
        solve(scanner);
    }

    static void solve(Scanner scanner) {
        int N = Integer.parseInt(scanner.nextLine());
        int[] ts = Arrays.stream(scanner.nextLine().split(" ")).mapToInt(Integer::parseInt).toArray();
        for (int i = 1; i < N; i++) {
            ts[i] += ts[i - 1];
        }
        int[] vs = Arrays.stream(scanner.nextLine().split(" ")).mapToInt(Integer::parseInt).toArray();
        BigDecimal[] dp = new BigDecimal[200 * 100 + 1];
        Arrays.fill(dp, BigDecimal.ZERO);
        int curr = 0;
        for (int i = 1; i < 200 * 100 + 1; i++) {
            if (i >= 2 * ts[N - 1] && dp[i - 1].compareTo(BigDecimal.ZERO) == 0) {
                break;
            }
            BigDecimal vNext = curr == N - 1 ? BigDecimal.ZERO : BigDecimal.valueOf(vs[curr + 1]);
            if (dp[i - 1].subtract(vNext).compareTo(BigDecimal.valueOf(ts[curr]).subtract(BigDecimal.valueOf(i - 1).divide(BigDecimal.valueOf(2)))) == 0) {
                dp[i] =  dp[i - 1].subtract(BigDecimal.valueOf(0.5));
            } else if (BigDecimal.valueOf(vs[curr]).compareTo(dp[i - 1]) > 0) {
                dp[i] =  dp[i - 1].add(BigDecimal.valueOf(0.5));
            } else if (BigDecimal.valueOf(vs[curr]).compareTo(dp[i - 1]) == 0) {
                dp[i] =  dp[i - 1];
            }
            // System.out.printf("%.1f %.1f \n", i / 2.0, dp[i].doubleValue());
            if (curr != N - 1 && 2 * ts[curr] == i) {
                curr++;
            }
        }
        System.out.printf("%.6f\n", Arrays.stream(dp).mapToDouble(BigDecimal::doubleValue).sum() / 2.0);
    }

}