Problem:
Score : 600 points

Problem Statement

There are N holes in a two-dimensional plane. The coordinates of the i-th hole are (x_i,y_i).

Let R=10^{10^{10^{10}}}. Ringo performs the following operation:

Randomly choose a point from the interior of a circle of radius R centered at the origin, and put Snuke there. Snuke will move to the hole with the smallest Euclidean distance from the point, and fall into that hole. If there are multiple such holes, the hole with the smallest index will be chosen.

For every i (1 \leq i \leq N), find the probability that Snuke falls into the i-th hole.

Here, the operation of randomly choosing a point from the interior of a circle of radius R is defined as follows:

Pick two real numbers x and y independently according to uniform distribution on [-R,R].

If x^2+y^2\leq R^2, the point (x,y) is chosen. Otherwise, repeat picking the real numbers x,y until the condition is met.

Constraints

2 \leq N \leq 100

|x_i|,|y_i| \leq 10^6(1\leq i\leq N)

All given points are pairwise distinct.

All input values are integers.

Input

Input is given from Standard Input in the following format:

N
x_1 y_1
:
x_N y_N

Output

Print N real numbers. The i-th real number must represent the probability that Snuke falls into the i-th hole.

The output will be judged correct when, for all output values, the absolute or relative error is at most 10^{-5}.

Sample Input 1

2
0 0
1 1

Sample Output 1

0.5
0.5

If Ringo put Snuke in the region x+y\leq 1, Snuke will fall into the first hole. The probability of this happening is very close to 0.5.
Otherwise, Snuke will fall into the second hole, the probability of which happening is also very close to 0.5.

Sample Input 2

5
0 0
2 8
4 5
2 6
3 10

Sample Output 2

0.43160120892732328768
0.03480224363653196956
0.13880483535586193855
0.00000000000000000000
0.39479171208028279727

Solution in C++ (GCC 9.2.1) (judge verdict: Accepted)

#include <bits/stdc++.h>
using namespace std;using ll=long long;using uint=unsigned int;using pii=pair<int,int>;using pll=pair<ll,ll>;using ull = unsigned long long;using ld=long double;template<typename T>void _(const char*s,T h){cerr<<s<<" = "<<h<<"\n";}template<typename T,typename...Ts>void _(const char*s,T h,Ts...t){int b=0;while(((b+=*s=='(')-=*s==')')!=0||*s!=',')cerr<<*s++;cerr<<" = "<<h<<",";_(s+1,t...);}// break continue pop_back 998244353
#define int ll
#define pii pll
#define f first
#define s second
#define pb emplace_back
#define forn(i,n) for(int i=0;i<(n);++i)
#define sz(a)((int)(a).size())
#define sqr(x) ((x)*(x))
struct init{init(){cin.tie(0);iostream::sync_with_stdio(0);cout<<fixed<<setprecision(10);cerr<<fixed<<setprecision(5);}~init(){
#ifdef LOCAL
#define dbg(...) _(#__VA_ARGS__,__VA_ARGS__)
cerr<<"Time elapsed: "<<(double)clock()/CLOCKS_PER_SEC<<"s.\n";
#else
#define dbg(...) 
#endif
}}init;template<typename T,typename U>void upx(T&x,U y){if(x<y)x=y;}template<typename T,typename U>void upn(T&x,U y){if(x>y)x=y;}mt19937 rng(chrono::steady_clock::now().time_since_epoch().count());const int D=4,dx[]={+1,0,-1,0},dy[]={0,+1,0,-1};
const int N=2020,OO=1e9;
//дописать точное
/*after all: #define vc vc<int/myf/whatever>*/
const double EPS=1e-9;//long double?
const double PI=3.1415926535897932384626433; //long double?
struct myf{
    double x; //long double?
    myf()=default;
    myf(double _x):x(_x){}
    myf operator-(){return -x;}
    myf operator+(const myf&rhs){return x+rhs.x;}
    myf operator-(const myf&rhs){return x-rhs.x;}
    myf operator*(const myf&rhs){return x*rhs.x;}
    myf operator/(const myf&rhs){return x/rhs.x;}
    myf& operator+=(const myf&rhs){x+=rhs.x;return*this;}
    myf& operator-=(const myf&rhs){x-=rhs.x;return*this;}
    myf& operator*=(const myf&rhs){x*=rhs.x;return*this;}
    myf& operator/=(const myf&rhs){x/=rhs.x;return*this;}
    bool operator<(const myf&rhs)const{return x+EPS<rhs.x;}
    bool operator>(const myf&rhs)const{return x>rhs.x+EPS;}
    bool operator==(const myf&rhs)const{return abs(x-rhs.x)/max(1.,abs(x))<=EPS;}//ld1
    bool operator!=(const myf&rhs)const{return abs(x-rhs.x)/max(1.,abs(x))>EPS;}//ld1
    bool operator<=(const myf&rhs)const{return*this<rhs||*this==rhs;}
    bool operator>=(const myf&rhs)const{return*this>rhs||*this==rhs;}
};
myf sqrt(myf x){return sqrt(x.x);}
myf abs(myf x){if(x.x<0)return -x;return x;}
std::istream&operator>>(std::istream&is,myf&p){return is>>p.x;}
std::ostream&operator<<(std::ostream&os,myf&p){return os<<p.x;}
template<class T>struct vc{
    T x,y;
    int i;
    vc(){}
    vc(T _x,T _y): x(_x),y(_y){}
    bool operator<(const vc&rhs)const{return x==rhs.x?y<rhs.y:x<rhs.x;}
    bool operator==(const vc&rhs)const{return x==rhs.x&&y==rhs.y;}
    vc(vc<T>a,vc<T>b){x=b.x-a.x;y=b.y-a.y;}
    T length(){return sqrt(sqr(x)+sqr(y));}
    T sqlength(){return sqr(x)+sqr(y);}
    vc operator +(vc b){return vc(x+b.x,y+b.y);}
    T operator *(vc b){return x*b.x+y*b.y;}
    T operator %(vc b){return x*b.y-y*b.x;}
    vc operator *(T k){return vc(x*k,y*k);}
    vc rotate(double angle){return vc(x*cos(angle)-y*sin(angle),x*sin(angle)+y*cos(angle));}
};
template<class T>T dist(vc<T>a,vc<T>b){return sqrt(sqr(b.x-a.x)+sqr(b.y-a.y));}
template<class T>T sqdist(vc<T>a,vc<T>b){return sqr(b.x-a.x)+sqr(b.y-a.y);}
template<class T>std::istream&operator>>(std::istream&is,vc<T>&p){return is>>p.x>>p.y;}
template<class T>std::ostream&operator<<(std::ostream&os,vc<T>&p){return os<<p.x<<' '<<p.y;}
template<class T>vc<T>turn_clockwise(vc<T>a){return vc<T>(a.y,-a.x);}
template<class T>vc<T>turn_anticlockwise(vc<T>a){return vc<T>(-a.y,a.x);}
template<class T>struct line
{
    T a,b,c;
    line(){}
    line(T _a,T _b,T _c): a(_a),b(_b),c(_c){
        //normalize();
    }
    line(vc<T>one,vc<T>two){
        a=one.y-two.y;
        b=two.x-one.x;
        c=-(a*one.x+b*one.y);
        //normalize();
    }
    line(vc<T>norm,vc<T>cont,int mycop){
        a=norm.x;
        b=norm.y;
        c=-(a*cont.x+b*cont.y);
        //normalize();
    }
    void normalize(){
        T div=sqrt(sqr(a)+sqr(b));
        a/=div;
        b/=div;
        c/=div;
    }
    bool on(vc<T>p){return a*p.x+b*p.y+c==0;}
};
template<class T>struct circle{
    vc<T>center;T radius;
    circle(){}
    circle(vc<T>_center,T _radius):center(_center),radius(_radius){}
    circle(vc<T>_center,vc<T>_v):center(_center){radius=_v.length();}
    bool inside(vc<T>p){return dist(p,center)<=radius;}
    bool strict_inside(vc<T>p){return dist(p,center)<radius;}
    bool on(vc<T>p){return dist(p,center)==radius;}
};
template<class T>T vc_segment_dist(vc<T>O,vc<T>p1,vc<T>p2){ // vc O,segment [p1,p2]
    return((vc<T>(p1,O)*vc(p1,p2))*(vc(p2,p1)*vc(p2,O))<=0)?min(dist(O,p1),dist(O,p2)):abs((vc(O,p1)%vc(O,p2))/dist(p1,p2));
}
template<class T>T det(T a,T b,T c,T d){return a*d-b*c;}
// 0 if parallel;{1,vc} if OK; 2 if match.
template<class T>int cross(line<T>one,line<T>two,vc<T>&p){ 
    T dx=det(one.c,one.b,two.c,two.b),dy=det(one.a,one.c,two.a,two.c),dd=det(one.a,one.b,two.a,two.b);
    if(dx==0&&dy==0&&dd==0)return 2;
    else if(dd==0)         return 0;
    p=vc(-dx/dd,-dy/dd);   return 1;
}
// 0 if A in C;{1,vc} if A on C;{2,vc,vc} if OK
template<class T>int kasat(vc<T>A,circle<T>C,vc<T>&p1,vc<T>&p2){ 
    vc OA(C.center,A);
    T l=OA.length();
    if(l<C.radius)return 0;
    T x=sqr(C.radius)/l;
    vc OP(OA.x*x/l,OA.y*x/l);
    vc P=C.center+OP;
    T hor=sqrt(sqr(C.radius)-OP.sqlength());
    p1=P+turn_clockwise(OP*(hor/x));
    p2=P+turn_anticlockwise(OP*(hor/x));
    return l==C.radius?1:2;
}
// 0 no intersection; 1/2=OK(1/2 vcs)
template<class T>int cross(line<T>l,circle<T>c,vc<T>&p1,vc<T>&p2){ 
    if(l.b==0){
        p1.x=-l.c/l.a;
        if(sqr(c.radius)>=sqr(p1.x-c.center.x)){
            p2.x=p1.x;
            p1.y=c.center.y+sqrt(max(T(0),sqr(c.radius)-sqr(c.center.x-p1.x)));
            p2.y=c.center.y-sqrt(max(T(0),sqr(c.radius)-sqr(c.center.x-p1.x)));
            return sqr(c.radius)==sqr(c.center.x-p1.x)?1:2;
        }
        return 0;
    }
    T tt=(l.c+l.b*c.center.y)/l.b;
    T ta=sqr(l.a/l.b)+1,tb=(tt*l.a/l.b-c.center.x)*2,tc=sqr(tt)-sqr(c.radius)+sqr(c.center.x);
    T td=sqr(tb)-ta*tc*4;
    if(td<0)return 0;
    p1.x=(-tb+sqrt(max(T(0),td)))/(ta+ta); 
    p2.x=(-tb-sqrt(max(T(0),td)))/(ta+ta);
    p1.y=(-l.c-l.a*p1.x)/l.b;
    p2.y=(-l.c-l.a*p2.x)/l.b;
    return td==0?1:2;
}
// 0 no intersection; 1/2=OK(1/2 vcs); 3=match
template<class T>int cross(circle<T>one,circle<T>two,vc<T>&p1,vc<T>&p2){ 
    if(one.center.x==two.center.x&&one.center.y==two.center.y)return one.radius==two.radius?3:0;
    line l((two.center.x-one.center.x)*2,(two.center.y-one.center.y)*2,sqr(one.center.x)-sqr(two.center.x)+sqr(one.center.y)-sqr(two.center.y)+sqr(two.radius)-sqr(one.radius));
    return cross(l,one,p1,p2);
}
// !!!a[0]==a.back(),vcs anticlockwise
template<class T>bool inside_convex_polygon(const vector<vc<T>>&a,vc<T>p){  
    int n=a.size()-1,l=1,r=a.size()-2;
    vc cur(a[0],p);
    if(vc(a[0],a[1])%cur<0||cur%vc(a[0],a[n-1])<0)return 0;
    while(r-l>1){
        int m=(l+r)/2;
        if(cur%vc(a[0],a[m])>=0)r=m;
        else l=m;
    }
    return vc(a[r],p)%vc(a[r],a[r+1])<=0&&vc(a[r-1],p)%vc(a[r-1],a[r])<=0;
}
template<class T>T area(const vector<vc<T>>&p){
    T res=0;
    for(int i=1;i+1<p.size();++i)res+=vc(p[0],p[i])%vc(p[0],p[i+1]);
    return abs(res);
}

template<class T>bool btw(T o,T d1,T d2){return(d1<o&&o<d2)||(d2<o&&o<d1);}
template<class T>bool pbtw(vc<T>o,vc<T>p1,vc<T>p2){return btw(o.x,p1.x,p2.x)&&btw(o.y,p1.y,p2.y);}
#define vc vc<int>
vc ch_start;
bool ch_comp(const vc&p1,const vc&p2){
    vc op1(ch_start,p1),op2(ch_start,p2);
    if(op1%op2==0)return sqr(op1.x)+sqr(op1.y)<sqr(op2.x)+sqr(op2.y);
    return op1%op2>0;       
}
vector<vc> convex_hull(const vector<vc>&p){
    if(p.empty())return{};
    ch_start=*min_element(p.begin(),p.end());
    vector<vc>a;
    for(const vc&i:p)if(i.x!=ch_start.x||i.y!=ch_start.y)a.pb(i);
    sort(a.begin(),a.end(),ch_comp);
    if(a.empty())return{ch_start};
    vector<vc> ans={ch_start,a[0]};
    for(uint i=1;i<a.size();++i){
        while(ans.size()>=2&&vc(ans[ans.size()-2],ans.back())%vc(ans.back(),a[i])<=0)ans.pop_back();
        ans.pb(a[i]);
    }
    return ans;
}
double ans[111];
int32_t main(){
    int n;
    cin>>n;
    vector<vc> a(n);
    forn(i,n){
        cin>>a[i];
        a[i].i=i;
    }
    auto h=convex_hull(a);
    int m=sz(h);
    h.pb(h[0]);
    h.pb(h[1]);
    for(int i=1;i<=m;++i){
        vc v1(h[i],h[i-1]);
        vc v2(h[i],h[i+1]);
        double ang=atan2(v2%v1,v1*v2);
        ans[h[i].i]=(PI-ang)/(PI+PI);
    }
    forn(i,n)cout<<ans[i]<<'\n';
    return 0;
}