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-10
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+EPS<0)
	{
		return dist(l[0], c.p)<c.r+EPS;
	}
	
	double vv=dot(l[1]-l[0], l[1]-l[0]);
	
	if(vc > vv)
	{
		double d=dist(l[1], c.p);
		
		return d*d<c.r*c.r+EPS;
	}
	
	double cc=dot(c.p-l[0], c.p-l[0]);
	
	return cc-(vc/vv)*vc < c.r*c.r+EPS;
}


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

int main()
{
	int R,N;
	while(cin >> R >> N,(R||N))
	{
		priority_queue<UFO> q;
		while(N--)
		{
			double x,y,r,v;
			cin >> x >> y >> r >> v;
			double dx=1.0*v*x/sqrt(x*x+y*y), dy=1.0*v*y/sqrt(x*x+y*y);
			double d=dist(P(0,0),P(x,y));
			q.push(UFO(C(P(x-dx,y-dy),r), v, d-v, dx,dy,x,y));
		}
		
		int cnt=0,t=1;
		while(!q.empty())
		{
			t++;
			
			UFO u=q.top(); q.pop();
			if(u.d < R+EPS)
			{
				cnt++;
				continue;
			}
			
			priority_queue<UFO> tq;
			L s(P(R*u.x/sqrt(u.x*u.x+u.y*u.y),R*u.y/sqrt(u.x*u.x+u.y*u.y)),P(u.c.p.real()*3000, u.c.p.imag()*3000));
			while(!q.empty())
			{
				UFO uf=q.top(); q.pop();
				if(intersectSC(s,uf.c)) continue;
				
				tq.push(UFO(C(P(uf.x-uf.dx*t,uf.y-uf.dy*t), uf.c.r), uf.v, uf.d-uf.v, uf.dx, uf.dy, uf.x, uf.y));
			}
			
			while(!tq.empty())
			{
				q.push(tq.top()); tq.pop();
			}
		}
		
		cout << cnt << endl;
	}
}

C++: // This file is a "Hello, world!" in C++ language by GCC for wandbox.
#include <iostream>
#include <cstdlib>
#include <vector>
#include <cmath>
#include <algorithm>

int main()
{
    struct P {
        float x, y;
    };
    
    struct UFO {
        float x, y;
        float v;
        float r;
        float sita;
        float d;
    };
    
    while (true) {
        int count = 0;
        int R, N;
        std::cin >> R >> N;
        
        if (R == 0 && N == 0) {
            break;
        }
        
        std::vector<UFO> ufos(N);
        
        for (int i = 0; i < N; i++) {
            std::cin >> ufos[i].x >> ufos[i].y >> ufos[i].r >> ufos[i].v;
            ufos[i].sita = std::atan2(ufos[i].y, ufos[i].x);
            ufos[i].d = std::sqrt(ufos[i].x * ufos[i].x + ufos[i].y * ufos[i].y);
        }
        
        while (!ufos.empty()) {
            for (std::size_t i = 0; i < ufos.size(); i++) {
                ufos[i].d = std::max(ufos[i].d - ufos[i].v, 0.0f);
                ufos[i].x = std::cos(ufos[i].sita) * ufos[i].d;
                ufos[i].y = std::sin(ufos[i].sita) * ufos[i].d;
            }
            
            count += std::count_if(ufos.begin(), ufos.end(), [R](auto& elem){
                return elem.d <= R;
            });
            
            {
                auto rm_it = std::remove_if(ufos.begin(), ufos.end(), [R](auto& elem){
                    return elem.d <= R;
                });
                ufos.erase(rm_it, ufos.end());
            }
            
            std::sort(ufos.begin(), ufos.end(), [](auto& lhs, auto& rhs){
                return lhs.d < rhs.d;
            });
            
            if (ufos.empty()) {
                break;
            }
            
            float a = ufos[0].y / ufos[0].x;
            {
                auto rm_it = std::remove_if(ufos.begin(), ufos.end(), [R, a, &ufos](auto& elem){
                    float d = std::abs(-a * elem.x + elem.y) / std::sqrt(a * a + 1);
                    return elem.r >= d && (ufos[0].sita * elem.sita) >= 0;
                });
                ufos.erase(rm_it, ufos.end());
            }
        }
        printf("%d\n", count);
    }
}