Problem: Example

Input

8 900 0
40 100
70 -80
350 30
680 -20
230 230
300 400
530 130
75 -275


Output

1210.99416

Solution (in CPP):

Solution:
#include <bits/stdc++.h>
using namespace std;

//#define dump(n) cout<<"# "<<#n<<'='<<(n)<<endl
#define dump(n) cout<<"\e[0;32mL"<<setw(3)<<__LINE__<<":\e[0m "<<#n<<'='<<(n)<<endl
#define repi(i,a,b) for(int i=int(a);i<int(b);i++)
#define peri(i,a,b) for(int i=int(b);i-->int(a);)
#define rep(i,n) repi(i,0,n)
#define per(i,n) peri(i,0,n)
#define all(c) begin(c),end(c)
#define mp make_pair
#define mt make_tuple

typedef unsigned int uint;
typedef long long ll;
typedef unsigned long long ull;
typedef pair<int,int> pii;
typedef vector<int> vi;
typedef vector<vi> vvi;
typedef vector<ll> vl;
typedef vector<vl> vvl;
typedef vector<double> vd;
typedef vector<vd> vvd;
typedef vector<string> vs;

const int INF=1e9;
const int MOD=1e9+7;
const double EPS=1e-9;
const double PI=acos(-1);

template<typename T1,typename T2>
ostream& operator<<(ostream& os,const pair<T1,T2>& p){
	return os<<'('<<p.first<<','<<p.second<<')';
}
template<typename T>
ostream& operator<<(ostream& os,const vector<T>& a){
	os<<'[';
	rep(i,a.size()) os<<(i?" ":"")<<a[i];
	return os<<']';
}

struct Point{
	double x,y;
	Point(){}
	Point(double x,double y):x(x),y(y){}
	Point& operator+=(Point p){x+=p.x,y+=p.y; return *this;}
	Point& operator-=(Point p){x-=p.x,y-=p.y; return *this;}
	Point& operator*=(double c){x*=c,y*=c; return *this;}
	Point& operator/=(double c){x/=c,y/=c; return *this;}
};
Point operator+(Point a,Point b){return a+=b;}
Point operator-(Point a,Point b){return a-=b;}
Point operator*(Point a,double c){return a*=c;}
Point operator*(double c,Point a){return a*=c;}
Point operator/(Point a,double c){return a/=c;}
bool operator==(Point a,Point b){return abs(a.x-b.x)<EPS && abs(a.y-b.y)<EPS;}
bool operator!=(Point a,Point b){return !(a==b);}

struct Line{
	Point pos,dir;
	Line(){}
	Line(Point p,Point d):pos(p),dir(d){}
	Line(double x,double y,double u,double v):pos(x,y),dir(u,v){}
};

struct Segment{
	Point pos,dir;
	Segment(){}
	Segment(Point p,Point d):pos(p),dir(d){}
	Segment(double x,double y,double u,double v):pos(x,y),dir(u,v){}
	explicit Segment(Line l):pos(l.pos),dir(l.dir){}
	explicit operator Line()const{return Line(pos,dir);}
};

struct Circle{
	Point center;
	double radius;
	Circle(){}
	Circle(Point c,double r):center(c),radius(r){}
	Circle(double x,double y,double r):center(x,y),radius(r){}
};
bool operator==(Circle a,Circle b){
	return a.center==b.center && abs(a.radius-b.radius)<EPS;
}
bool operator!=(Circle a,Circle b){
	return !(a==b);
}

int Signum(double x){
	return x<-EPS?-1:x>EPS?1:0;
}

double Abs(Point p){
	return sqrt(p.x*p.x+p.y*p.y);
}
double Abs2(Point p){
	return p.x*p.x+p.y*p.y;
}
double Arg(Point p){
	return atan2(p.y,p.x);
}
double Dot(Point a,Point b){
	return a.x*b.x+a.y*b.y;
}
double Cross(Point a,Point b){
	return a.x*b.y-a.y*b.x;
}
Point Rot(Point p,double t){
	return Point(cos(t)*p.x-sin(t)*p.y,sin(t)*p.x+cos(t)*p.y);
}

bool Orthogonal(Line a,Line b){
	return abs(Dot(a.dir,b.dir))<EPS;
}
bool Parallel(Line a,Line b){
	return abs(Cross(a.dir,b.dir))<EPS;
}
Point Proj(Line l,Point p){
	Point a=p-l.pos,b=l.dir;
	return l.pos+Dot(a,b)/Abs2(b)*b;
}

int CCW(Point a,Point b,Point c){
	b-=a,c-=a;
	if(int sign=Signum(Cross(b,c)))
		return sign; // 1:ccw,-1:cw
	if(Dot(b,c)<-EPS)
		return -2;   // c-a-b
	if(Abs2(b)<Abs2(c)-EPS)
		return 2;    // a-b-c
	return 0;        // a-c-b (inclusive)
}

vector<Line> Intangent(Circle c1,Circle c2)
{
	double r1=c1.radius,r2=c2.radius;
	double d=Abs(c1.center-c2.center);
	vector<Line> res;
	if(d>r1+r2+EPS){
		double t=acos((r1+r2)/d);
		rep(i,2){
			Point p1=c1.center+Rot(r1/d*(c2.center-c1.center),(i?1:-1)*t);
			Point p2=c2.center+Rot(r2/d*(c1.center-c2.center),(i?1:-1)*t);
			res.push_back(Line(p1,p2-p1));
		}
	}
	else if(d>r1+r2-EPS){
		Point p=c1.center+r1/d*(c2.center-c1.center);
		res.push_back(Line(p,Rot(p-c1.center,PI/2)));
	}
	return res;
}
vector<Line> Extangent(Circle c1,Circle c2)
{
	double r1=c1.radius,r2=c2.radius;
	double d=Abs(c1.center-c2.center);
	vector<Line> res;
	if(d>abs(r2-r1)+EPS){
		double t=acos((r1-r2)/d);
		rep(i,2){
			Point p1=c1.center+Rot(r1/d*(c2.center-c1.center),(i?1:-1)*t);
			Point p2=c2.center+Rot(r2/d*(c2.center-c1.center),(i?1:-1)*t);
			res.push_back(Line(p1,p2-p1));
		}
	}
	else if(d>abs(r2-r1)-EPS){
		Point p=c1.center+r1/d*(c2.center-c1.center);
		res.push_back(Line(p,Rot(p-c1.center,PI/2)));
	}
	return res;
}

double DistLP(Line l,Point p){
	return Abs(Proj(l,p)-p);
}
double DistSP(Segment s,Point p){
	int ccw=CCW(s.pos,s.pos+s.dir,Proj(Line(s),p));
	if(ccw==-2) return Abs(s.pos-p);
	if(ccw== 2) return Abs(s.pos+s.dir-p);
	return DistLP(Line(s),p);
}

pair<Point,Point> InterPointCC(Circle c1,Circle c2){
	Point p1=c1.center,p2=c2.center;
	double r1=c1.radius,r2=c2.radius;
	double d=Abs(p1-p2);
	double a=(d*d+r1*r1-r2*r2)/(2*d);
	double s=sqrt(r1*r1-a*a);
	return mp(p1+a/d*(p2-p1)+s*Rot((p2-p1)/d,PI/2),
			  p1+a/d*(p2-p1)-s*Rot((p2-p1)/d,PI/2));
}

// xが弧ab(反時計周り)の区間にあるかどうか．a,b,xはradian．
bool InArc(double a,double b,double x)
{
	if(a<b) return a<x && x<b;
	else    return a<x || x<b;
}
// 半径rの円についての弧ab(反時計回り)の長さ
double ArcLength(double r,double a,double b)
{
	if(a<b) return r*(b-a);
	else    return r*(b-a+2*PI);
}

struct Edge{
	int src,dst;
	double weight;
	Edge(){}
	Edge(int s,int d,double w):src(s),dst(d),weight(w){}
};
bool operator<(Edge a,Edge b){return a.weight<b.weight;}
bool operator>(Edge a,Edge b){return a.weight>b.weight;}
typedef vector<vector<Edge>> Graph;

void Dijkstra(const Graph& g,int v,vd& dist)
{
	priority_queue<Edge,vector<Edge>,greater<Edge>> pq;
	pq.emplace(-1,v,0);
	while(pq.size()){
		Edge cur=pq.top(); pq.pop();
		if(dist[cur.dst]!=INF) continue;
		dist[cur.dst]=cur.weight;
		for(Edge e:g[cur.dst])
			pq.emplace(e.src,e.dst,cur.weight+e.weight);
	}
}

int main()
{
	const double R=100;
	
	for(int n;cin>>n && n;){
		Point start(0,0),goal; cin>>goal.x>>goal.y;
		vector<Point> poles(n); // poles
		for(auto& p:poles) cin>>p.x>>p.y;
		
		vector<Circle> cs(n);
		rep(i,n) cs[i]=Circle(poles[i],R);
		
		// 始点・終点から円へ引いた接線の接点
		vector<pair<Point,int>> pairs; // (頂点，どの円の周上の点であるか)
		rep(i,n){
			vector<Line> t=Extangent(cs[i],Circle(start,0));
			for(Line l:t)
				pairs.emplace_back(Proj(l,cs[i].center),i);
		}
		rep(i,n){
			vector<Line> t=Extangent(cs[i],Circle(goal,0));
			for(Line l:t)
				pairs.emplace_back(Proj(l,cs[i].center),i);
		}
		
		// 円の共通接線の接点
		rep(i,n) repi(j,i+1,n){
			vector<Line> t=Intangent(cs[i],cs[j]);
			for(Line l:t){
				pairs.emplace_back(Proj(l,cs[i].center),i);
				pairs.emplace_back(Proj(l,cs[j].center),j);
			}
			t=Extangent(cs[i],cs[j]);
			for(Line l:t){
				pairs.emplace_back(Proj(l,cs[i].center),i);
				pairs.emplace_back(Proj(l,cs[j].center),j);
			}
		}
		
		// 無駄な頂点を消しつつpairをばらす
		vector<Point> ps;
		vi is;
		for(auto tmp:pairs){
			Point p; int i; tie(p,i)=tmp;
			bool in=false;
			for(Circle c:cs)
				in|=Abs(p-c.center)<R-EPS;
			if(!in){
				ps.push_back(p);
				is.push_back(i);
			}
		}
		
		ps.push_back(start); is.push_back(-1);
		ps.push_back(goal);  is.push_back(-2);
		
		Graph g(ps.size());
		rep(i,ps.size()) repi(j,i+1,ps.size()){
			if(is[i]==is[j]){ // 円周上をたどる
				Circle c1=cs[is[i]];
				double a1=Arg(ps[i]-c1.center),a2=Arg(ps[j]-c1.center);
				rep(_,2){
					bool intersect=false;
					rep(k,n){
						Circle c2=cs[k];
						if(Abs(c2.center-c1.center)>2*R+EPS)
							continue;
						Point p1,p2; tie(p1,p2)=InterPointCC(c1,c2);
						intersect|=InArc(a1,a2,Arg(p1-c1.center));
						intersect|=InArc(a1,a2,Arg(p2-c1.center));
					}
					if(!intersect){
						g[i].emplace_back(i,j,ArcLength(R,a1,a2));
						g[j].emplace_back(j,i,ArcLength(R,a1,a2));
					}
					swap(a1,a2);
				}
			}
			else{
				Segment s(ps[i],ps[j]-ps[i]);
				bool intersect=false;
				rep(k,n) intersect|=DistSP(s,cs[k].center)<R-EPS;
				if(!intersect){
					g[i].emplace_back(i,j,Abs(s.dir));
					g[j].emplace_back(j,i,Abs(s.dir));
				}
			}
		}
		
		vd dist(ps.size(),INF);
		Dijkstra(g,dist.size()-2,dist);
		if(dist.back()==INF)
			puts("0.0");
		else
			printf("%.10f\n",dist.back());
	}
}