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 C++14 (GCC 5.4.1) (judge verdict: Accepted)

                                #include <bits/stdc++.h>
                                #include<iostream>
                                #include<cstdio>
                                #include<vector>
                                #include<queue>
                                #include<map>
                                #include<cstring>
                                #include<string>
                                #include <math.h>
                                #include<algorithm>
                            //    #include <boost/multiprecision/cpp_int.hpp>
                                #include<functional>
                       #define int long long
                                #define inf  1000000007
                                #define pa pair<int,int>
                                #define ll long long
                                #define pal pair<double,pa>
                                #define ppa pair<pa,int>
                                #define ppap pair<int,pa>
                                #define ssa pair<string,int>
                                #define  mp make_pair
                                #define  pb push_back
                                #define EPS (1e-10)
                                #define equals(a,b) (fabs((a)-(b))<EPS)
                         
                                using namespace std;
                      
                                class Point{
                                	public:
                                	double x,y;
                                	Point(double x=0,double y=0):x(x),y(y) {}
                                	Point operator + (Point p) {return Point(x+p.x,y+p.y);}
                                	Point operator - (Point p) {return Point(x-p.x,y-p.y);}
                                	Point operator * (double a) {return Point(x*a,y*a);}
                                	Point operator / (double a) {return Point(x/a,y/a);}
                                	double absv() {return sqrt(norm());}
                                	double norm() {return x*x+y*y;}
                                	bool operator < (const Point &p) const{
                                		return x != p.x ? x<p.x: y<p.y;
                                	}
                                	bool operator == (const Point &p) const{
                                		return fabs(x-p.x)<EPS && fabs(y-p.y)<EPS;
                                	}
                                };
                                typedef Point Vector;
                         
                                struct Segment{
                                Point p1,p2;
                                };
                                 double dot(Vector a,Vector b){
                                	return a.x*b.x+a.y*b.y;
                                }
                                double cross(Vector a,Vector b){
                                	return a.x*b.y-a.y*b.x;
                                }
                            
    bool parareru(Point a,Point b,Point c,Point d){
    //	if(abs(cross(a-b,d-c))<EPS)cout<<"dd "<<cross(a-b,d-c)<<endl;
    	return abs(cross(a-b,d-c))<EPS;
    }
    double distance_ls_p(Point a, Point b, Point c) {
      if ( dot(b-a, c-a) < EPS ) return (c-a).absv();
      if ( dot(a-b, c-b) < EPS ) return (c-b).absv();
      return abs(cross(b-a, c-a)) / (b-a).absv();
    }
    bool is_intersected_ls(Segment a,Segment b) {
    	if(a.p1==b.p1||a.p2==b.p1||a.p1==b.p2||a.p2==b.p2) return false;
    	if(parareru((a.p2),(a.p1),(a.p1),(b.p2))&&parareru((a.p2),(a.p1),(a.p1),(b.p1))){
    //		cout<<"sss"<<endl;
    		if(dot(a.p1-b.p1,a.p1-b.p2)<EPS) return true;
    		if(dot(a.p2-b.p1,a.p2-b.p2)<EPS) return true;
    		if(dot(a.p1-b.p1,a.p2-b.p1)<EPS) return true;
    		if(dot(a.p1-b.p2,a.p2-b.p2)<EPS) return true;
    		return false;
    	}
      else return ( cross(a.p2-a.p1, b.p1-a.p1) * cross(a.p2-a.p1, b.p2-a.p1) < EPS ) && ( cross(b.p2-b.p1, a.p1-b.p1) * cross(b.p2-b.p1, a.p2-b.p1) < EPS );
    }
     
    double segment_dis(Segment a,Segment b){
    	if(is_intersected_ls(a,b))return 0;
    	double r=distance_ls_p(a.p1, a.p2, b.p1);
    	r=min(r,distance_ls_p(a.p1, a.p2, b.p2));
    	r=min(r,distance_ls_p(b.p1, b.p2, a.p2));
    	r=min(r,distance_ls_p(b.p1, b.p2, a.p1));
    	return r;
    }
    Point intersection_ls(Segment a, Segment b) {
      Point ba = b.p2-b.p1;
      double d1 = abs(cross(ba, a.p1-b.p1));
      double d2 = abs(cross(ba, a.p2-b.p1));
      double t = d1 / (d1 + d2);
     
      return a.p1 + (a.p2-a.p1) * t;
    }
     
                    string itos( int i ) {
                    ostringstream s ;
                    s << i ;
                    return s.str() ;
                    }
                     
                    int gcd(int v,int b){
                    	if(v>b) return gcd(b,v);
                    	if(v==b) return b;
                    	if(b%v==0) return v;
                    	return gcd(v,b%v);
                    }
     
                    double distans(double x1,double y1,double x2,double y2){
                    	double rr=(x1-x2)*(x1-x2)+(y1-y2)*(y1-y2);
                    	return sqrt(rr);
                    	
                    }
                    /*
    int pr[100010];
    //int inv[100010];
    */ 
    int beki(int wa,int rr,int warukazu){
    	if(rr==0) return 1ll;
    	if(rr==1) return wa%warukazu;
    	if(rr%2==1) return (beki(wa,rr-1,warukazu)*wa)%warukazu;
    	int zx=beki(wa,rr/2,warukazu);
    	return (zx*zx)%warukazu;
    }
      /*
    void gya(){
    	pr[0]=1;
    	for(int i=1;i<100010;i++){
    		pr[i]=(pr[i-1]*i)%inf;
    	}
    	for(int i=0;i<100010;i++) inv[i]=beki(pr[i],inf-2);
    	
    }
     
    */
     //sort(ve.begin(),ve.end(),greater<int>());
                             //----------------kokomade tenpure------------
    //vector<double> ans(100000000),ans2(100000000);
    /*
    int par[200100],ranks[200100],kosuu[200100];     
     
    void shoki(int n){
    	for(int i=0;i<n;i++){
    		par[i]=i;
    		ranks[i]=0;
    		kosuu[i]=1;
    	}
    }
     
    int root(int x){
    	return par[x]==x ? x : par[x]=root(par[x]);
    }
     
    bool same(int x,int y){
    return root(x)==root(y);
    }
     
    void unite(int x,int y){
     x=root(x);
     y=root(y);
    	int xx=kosuu[x],yy=kosuu[y];
     if(x==y) return;
    	if(ranks[x]<ranks[y]){
    		par[x]=y;
    		kosuu[y]=yy+xx;
    	}
     else {
    	par[y]=x;
    	if(ranks[x]==ranks[y]) ranks[x]=ranks[x]+1;
     	kosuu[x]=yy+xx;
     }
    	return;
    }
    */


vector<pa> G[100200],G2[100020];


void saiki(int r,int oya){
	
	for(int i=0;i<G[r].size();i++){
		if(G[r][i].first==oya) continue;
		G2[r].pb(G[r][i]);
		saiki(G[r][i].first,r);
	}
}

int fu[100020];
void saiki2(int r,int oya,int fuka){
//	cout<<r<<endl;
	fu[r]=fuka;
	
	for(int i=0;i<G2[r].size();i++){
		saiki2(G2[r][i].first,r,fuka+G2[r][i].second);
	}
}

int t[300],v[300];
int n;
    signed main(){
cin>>n;
    	
    	for(int i=1;i<=n;i++) cin>>t[i];
    	for(int i=1;i<=n;i++) cin>>v[i];
    	v[0]=0;
    	v[n+1]=0;
    	
    	int hi[300],mi[300];
    	
    	hi[0]=0,mi[n]=0;
    	for(int i=1;i<=n;i++){
    			hi[i]=min({v[i+1],v[i],hi[i-1]+t[i]});
    	}
    	for(int i=n-1;i>=0;i--){
    			mi[i]=min({v[i],v[i+1],mi[i+1]+t[i+1]});
    		if(i==1){
    //			cout<<"mi1 "<<mi[i]<<endl;
    		}
    	}
    	
    	int r[300];
    	
    	for(int i=0;i<=n;i++) {
    		r[i]=min(hi[i],mi[i]);
    //		cout<<hi[i]<<" "<<mi[i]<<" "<<r[i]<<endl;
    	}
    	
    	double ans=0;
    	
    	for(int i=1;i<=n;i++){
    		int e1=r[i-1],e2=r[i];
    		if(e1>e2) swap(e1,e2);
    		
    		double tt=(e2-e1+t[i]+0.0)/(2.0);
    		
    		if(e1+tt>v[i]+0.0001){
    			ans += v[i]*t[i];
    			ans -= (v[i]- e1)*(v[i]- e1)/2.0;
    			ans -= (v[i]- e2)*(v[i]- e2)/2.0;
    			
    		}
    		else{
    			
    		double h=e1+tt;
    			
    			ans += (h*(t[i])-tt*tt/2.0-(t[i]-tt)*(t[i]-tt)/2.0);
    		}
    //		cout<<ans<<endl;
    		
    	}
    	printf("%.10lf\n",ans);
  //  cout<<ans<<endl;
    	
    	//          UNRESTORABLE
    	return 0;
    }