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 <complex>
#include <vector>
#include <cstdio>
#include <algorithm>
#include <cmath>
 
using namespace std;
 
typedef long double ldb;
typedef complex<ldb> P;
 
const ldb eps = 1e-8;
// math.hのM_PIでもOK?
const ldb pi = acos(-1);
// 誤差を考慮した比較 sig(a-b) のように使う
// ret<0 なら a<b, ret>0 なら a>b, ret=0 なら a=b
int sig(ldb a) {return (a<-eps)?-1:(a>eps)?1:0;}
// 誤差を考慮した比較
bool eq(ldb a,ldb b){return abs(a-b)<eps;}
// 内積
ldb dot(P a, P b) {
  return (a.real() * b.real() + a.imag() * b.imag());
}
// 外積
ldb cross(P a, P b) {
  return (a.real() * b.imag() - a.imag() * b.real());
}
 
// 直線,線分 (半直線は未確認だけど線分で無限に長いものを作れば良さそう)
struct L {
  P a,b;
  L(){}
  L(P a,P b):a(a),b(b){}
  ldb len(){return abs(a-b);}
  P vec(){return b-a;}
};
 
// 線分abに対する点xの位置
enum CCW{FRONT = 1, RIGHT = 2, BACK = 4, LEFT = 8, ON = 16};
int ccw(L s, P p) {
  P a = s.vec();
  p -= s.a;
  ldb cr = cross(a, p);
  if (eq(cr, 0)) {
    if (sig(dot(a, p)) < 0) return BACK;
    if (sig(norm(a) - norm(p)) < 0) return FRONT;
    return ON;
  }
  return (cr > 0) ? LEFT : RIGHT;
}
 
// "線分"a と 点b の距離
double dLP2(L a,P b) {
  if ( dot(a.b-a.a, b-a.a) < eps ) return abs(b-a.a);
  if ( dot(a.a-a.b, b-a.b) < eps ) return abs(b-a.b);
  return abs(cross(a.b-a.a, b-a.a)) / abs(a.b-a.a);
}
 
int main(void)
{
  int n;
  ldb r;
  int done[111];
  ldb x[111],y[111],s[111],v[111];
  ldb vx[111],vy[111];
  for(;;) {
    scanf("%Lf%d",&r,&n); if( r+n == 0 ) break;
    for( int i = 0; i < n; i++ ) {
      scanf("%Lf%Lf%Lf%Lf",x+i,y+i,s+i,v+i);
      ldb ang = atan2(y[i],x[i]);
      vx[i] = -v[i]*cos(ang);
      vy[i] = -v[i]*sin(ang);
      done[i] = 0;
    }
    int res = 0;
    for(;;) {
      for( int i = 0; i < n; i++ ) {
        if( done[i] ) continue;
        ldb nx = x[i] + vx[i];
        ldb ny = y[i] + vy[i];
        ldb dist = dLP2(L(P(x[i],y[i]),P(nx,ny)),P(0,0));
        if( sig(dist-r) <= 0 ) {
          ++res;
          done[i] = 2;
        }
        x[i] = nx;
        y[i] = ny;
      }
      int mi = -1;
      for( int i = 0; i < n; i++ ) {
        if( done[i] ) continue;
        if( mi == -1 || sig(abs(P(x[i],y[i]))-abs(P(x[mi],y[mi]))) < 0 ) {
          mi = i;
        }
      }
      //printf("%d\n",mi);
      if( mi < 0 ) break;
      const ldb w = 1e7;
      P a = P(x[mi],y[mi]);
      a = a/abs(a);
      L l(a*r+a*eps,a*w);
      for( int i = 0; i < n; i++ ) {
        if( done[i] ) continue;
        ldb dist = dLP2(l,P(x[i],y[i]));
        //printf("%Lf %Lf %Lf\n",dist,x[i],y[i]);
        if( sig(dist-s[i]) <= 0 ){
          done[i] = 1;
        }
      }
      /*
      for( int i = 0; i < n; i++ ) {
        printf("%d:%c ",i,done[i]==0?'o':(done[i]==1?'x':'c'));
      }
      puts("");
      //*/
    }
    printf("%d\n",res);
  }
  return 0;
}