Go: package main

import (
	"bufio"
	"os"
	"math"
	"strings"
	"strconv"
	"fmt"
)

var scanner = bufio.NewScanner(os.Stdin)

const (
	INF = 0x7FFFFFFF
	EPS = 1e-10
)

type Point struct {
	x, y float64
}
func (p *Point) plus(q *Point) *Point {
	return &Point{p.x + q.x, p.y + q.y}
}
func (p *Point) minus(q *Point) *Point {
	return &Point{p.x - q.x, p.y - q.y}
}
func (p *Point) multi(a float64) *Point {
	return &Point{p.x * a, p.y * a}
}
func (p *Point) div(a float64) *Point {
	return &Point{p.x / a, p.y / a}
}
func (p *Point) abs() float64 {
	return math.Sqrt(p.norm())
}
func (p *Point) norm() float64 {
	return p.x*p.x + p.y*p.y
}
func (p *Point) isLessThan(q *Point) bool {
	if p.x != q.x {
		return p.x < q.x && p.y < q.y
	}
	return true
}
func (p *Point) isEqual(q *Point) bool {
	return math.Abs(p.x - q.x) < EPS && math.Abs(p.y - q.y) < EPS
}

type Segment struct {
	p1, p2 *Point
}
func (s *Segment) norm() float64 {
	return (s.p2.x - s.p1.x)*(s.p2.x - s.p1.x) +
		(s.p2.y - s.p1.y)*(s.p2.y - s.p1.y)
}
func (s *Segment) abs() float64 {
	return math.Sqrt(s.norm())
}

func norm(a *Point) float64 {
	return a.x*a.x + a.y*a.y
}

func abs(a *Point) float64 {
	return math.Sqrt(norm(a))
}

func polar(a, r float64) *Point {
	return &Point{math.Cos(r)*a, math.Sin(r)*a}
}

func getDistance(a, b *Point) float64 {
	return abs(a.minus(b))
}

func dot(a, b *Point) float64 {
	return a.x*b.x + a.y*b.y
}

func cross(a, b *Point) float64 {
	return a.x*b.y - a.y*b.x
}

func getDistanceLP(s Segment, p *Point) float64 {
	return cross(s.p2.minus(s.p1), p.minus(s.p1))/ abs(s.p2.minus(s.p1))
}

func getDistanceSP(s Segment, p *Point) float64 {
	if dot(s.p2.minus(s.p1), p.minus(s.p1)) < 0 {
		return abs(p.minus(s.p1))
	}
	if dot(s.p1.minus(s.p2), p.minus(s.p2)) < 0 {
		return abs(p.minus(s.p2))
	}
	return getDistanceLP(s, p)
}

func main() {
	for scanner.Scan() {
		buf := strings.Split(scanner.Text(), " ")
		R, _ := strconv.Atoi(buf[0])
		N, _ := strconv.Atoi(buf[1])
		if R == 0 && N == 0 {
			break
		}
		p := make([]*Point, N)
		r := make([]float64, N)
		v := make([]float64, N)
		d := make([]float64, N)
		nd := make([]float64, N)
		die := make([]bool, N)
		for i := 0; i < N; i++ {
			scanner.Scan()
			buf_ := strings.Split(scanner.Text(), " ")
			_x, _ := strconv.ParseFloat(buf_[0], 64)
			_y, _ := strconv.ParseFloat(buf_[1], 64)
			_r, _ := strconv.ParseFloat(buf_[2], 64)
			_v, _ := strconv.ParseFloat(buf_[3], 64)
			tmp := &Point{_x, _y}
			p[i] = tmp
			r[i] = _r
			v[i] = _v
			nd[i] = getDistance(p[i], &Point{0, 0})
			d[i] = getDistance(p[i], &Point{0, 0})
			die[i] = false
		}
		for {
			for i := 0; i < N; i++ {
				nd[i] = math.Max(float64(R), nd[i]-v[i])
			}
			mini := float64(INF)
			id := -1
			for i := 0; i < N; i++ {
				if die[i] {
					continue
				}
				if nd[i] < EPS+float64(R) {
					continue
				}
				if nd[i] > mini {
					continue
				}
				mini = nd[i]
				id = i
			}
			if id == -1 {
				break
			}
			s := Segment{p[id], p[id]}
			s.p1 = s.p1.div(d[id])
			s.p1 = s.p1.multi(float64(R))
			s.p2 = s.p2.div(d[id])
			s.p2 = s.p2.multi(10000000.0)
			for i := 0; i < N; i++ {
				if nd[i] < EPS + float64(R) {
					continue
				}
				a := p[i]
				a = a.div(d[i])
				a = a.multi(nd[i])
				if getDistanceSP(s, a) < EPS + r[i] {
					die[i] = true
				}
			}
		}
		ans := 0
		for i := 0; i < N; i++ {
			if !die[i] {
				ans++
			}
		}
		fmt.Println(ans)
	}
}

C++: #include <iostream>
#include <complex>
#include <vector>
#include <cmath>
#include <queue>
using namespace std;

#define EPS 1e-7
typedef complex<double> P;

struct L : public vector<P>
{
  L(const P &a, const P &b) {
    push_back(a); push_back(b);
  }
};

struct C
{
	P p; double r;
	C(const P &p, double r) 
	:p(p),r(r) 
	{}
};

double cross(const P& a, const P& b) {
  return imag(conj(a)*b);
}

double dot(const P& a, const P& b) {
  return real(conj(a)*b);
}

double dist(const P& a, const P& b)
{
	return sqrt((a.real()-b.real())*(a.real()-b.real())+(a.imag()-b.imag())*(a.imag()-b.imag()));
}

bool intersectSC(const L& l, const C& c)
{	
	double vc=dot(l[1]-l[0], c.p-l[0]);
	if(vc<0)
	{
		double dif=c.r-dist(l[0], c.p);
		return dif+EPS > 0;
	}
	
	double vv=dot(l[1]-l[0], l[1]-l[0]);
	
	if(vc > vv)
	{
		double d=dist(l[1], c.p);
		double dif=c.r*c.r-d*d;
		
		return dif+EPS > 0;
	}
	
	double cc=dot(c.p-l[0], c.p-l[0]);
	double dif=c.r*c.r-(cc-(vc/vv)*vc);
	
	return dif+EPS > 0;
}

int T;

class UFO
{
public:
	double v,d,dx,dy,x,y,r;
	UFO(double v, double d, double dx, double dy, double x, double y, double r)
	:v(v),d(d),dx(dx),dy(dy),x(x),y(y),r(r)
	{}
	
	bool operator<(const UFO& u) const
	{
		return d-v*T>u.d-u.v*T;
	}
};

int main()
{
	int R,N;
	while(cin >> R >> N,(R||N))
	{
		T=1;
		priority_queue<UFO> q;
		while(N--)
		{
			double x,y,r,v;
			cin >> x >> y >> r >> v;
			double d=sqrt(x*x+y*y), dx=v*x/d, dy=v*y/d;
			q.push(UFO(v,d,dx,dy,x,y,r));
		}
		
		int cnt=0;
		while(!q.empty())
		{
			UFO u=q.top(); q.pop();
			if(u.d-u.v*T < R+EPS)
			{
				cnt++;
				continue;
			}
			
			priority_queue<UFO> tq;
			double nx=u.x-u.dx*T, ny=u.y-u.dy*T, r=sqrt(nx*nx+ny*ny);
			L s(P(R*nx/r,R*ny/r),P(nx*3000, ny*3000));
			while(!q.empty())
			{
				UFO uf=q.top(); q.pop();
				if(intersectSC(s,C(P(uf.x-uf.dx*T, uf.y-uf.dy*T), uf.r))) continue;
				
				tq.push(uf);
			}
			
			while(!tq.empty())
			{
				q.push(tq.top()); tq.pop();
			}
			
			T++;
		}
		
		cout << cnt << endl;
	}
}

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

#define EPS (1e-6)
#define EQUALS(a, b) (fabs((a) - (b)) < EPS)
#define SQR(n) ((n) * (n))
#define DIST(ax, bx, ay, by) (sqrt(SQR((ax) - (bx)) + SQR((ay) - (by))))

struct UFO
{
    double x, y, r, v;

    UFO () {}
    UFO (double x, double y, double r, double v) : x(x), y(y), r(r), v(v) {}

    void move()
    {
        double dx, dy;
        double rr = dist();

        dx = v * (x / rr), dy = v * (y / rr);
        if ( fabs(x) < fabs(dx) + EPS ) {
            x = EPS, y = EPS;
        } else {
            x -= dx, y -= dy;
        }
    }

    double dist()
    {
        return ( sqrt(SQR(x) + SQR(y)) );
    }

    bool laser(double bx, double by)
    {
        double dot = x * bx + y * by;
        double gx = bx * dot, gy = by * dot;

        return ( DIST(x, gx, y, gy) < r - EPS );
    }

};


int main()
{
    double R;
    int N;

    UFO ufo[105];

    while ( cin >> R >> N, R || N ) {

        for ( int i = 0; i < N; i++ ) {
            double x, y, r, v;
            cin >> x >> y >> r >> v;
            ufo[i] = UFO(x, y, r, v);
        }

        int cnt = 0;
        bool isfinished[105] = {};
        for ( int t = 1; t <= 1000; t++ ) {
            int minp = -1;
            double minv;

            for ( int i = 0; i < N; i++ ) {
                ufo[i].move();
                //cout << ufo[i].x << " " << ufo[i].y << endl;
                double d = ufo[i].dist();
                if ( !isfinished[i] && d > R + EPS && (minp == -1 || minv > d) ) {
                    minv = ufo[i].dist();
                    minp = i;
                }
            }
            //cout << minp << endl;
            if ( minp == -1 ) {
                break;
            } else {
                double x = ufo[minp].x / ufo[minp].dist(), y = ufo[minp].y / ufo[minp].dist();

                isfinished[minp] = true;
                for ( int i = 0; i < N; i++ ) {
                    if ( ufo[i].dist() > R + EPS && !isfinished[i] && ufo[i].laser(x, y) ) {
                        isfinished[i] = true;
                    }
                }
            }
        }
        for ( int i = 0; i < N; i++ ) {
            if ( !isfinished[i] ) {
                cnt++;
            }
        }
        cout << cnt << endl;
    }

    return ( 0 );
}