AtCoder Regular Contest 082 E

Problem Statement

You are given N points (xi,yi) located on a two-dimensional plane. Consider a subset S of the N points that forms a convex polygon. Here, we say a set of points S forms a convex polygon when there exists a convex polygon with a positive area that has the same set of vertices as S. All the interior angles of the polygon must be strictly less than 180°.

#include<cstdio>
#include<cstring>
#include<algorithm>
const int M=207,mod=998244353;
int read(){
int ans=0,f=1,c=getchar();
while(c<'0'||c>'9'){if(c=='-') f=-1; c=getchar();}
while(c>='0'&&c<='9'){ans=ans*10+(c-'0'); c=getchar();}
return ans*f;
}
int n,f[M],sz[M];
int find(int x){while(f[x]!=x) x=f[x]=f[f[x]]; return x;}
int gcd(int x,int y){return y?gcd(y,x%y):x;}
struct pos{int x,y;}q[M];
int cnt;
struct node{
int u,v,w;
bool operator <(const node &x)const{return w<x.w;}
void calc(){
int p=find(u),q=find(v);
if(p!=q) f[q]=p,sz
+=sz[q];
}
}e[M*M];
int pw[M],ans;
void prepare(){
pw[0]=1;
for(int i=1;i<=n;i++) pw[i]=(pw[i-1]<<1)%mod;
}
int main(){
n=read();
prepare(); ans=(pw
-n-1)%mod;
for(int i=1;i<=n;i++) q[i].x=read(),q[i].y=read();
for(int i=1;i<=n;i++)
for(int j=1;j<i;j++){
int x=q[i].x-q[j].x,y=q[i].y-q[j].y,g=gcd(x,y);
x/=g; y/=g;
if(!x) y=1;
if(!y) x=1;
if(x<0) x=-x,y=-y;
e[++cnt]=(node){i,j,x*30000+y};
}
std::sort(e+1,e+1+cnt);
for(int i=1,j=1;i<=cnt;i=j){
for(int k=1;k<=n;k++) sz[f[k]=k]=1;
while(j<=cnt&&e[j].w==e[i].w) e[j++].calc();
for(int k=1;k<=n;k++) if(f[k]==k&&sz[k]>=2) ans=(ans-pw[sz[k]]+sz[k]+1)%mod;
}printf("%d\n",(ans+mod)%mod);
return 0;
}