E - Rebuild UVALive - 7187 (二次函数极值问题)

#include <bits/stdc++.h>
#define met(a, b) memset(a, b, sizeof(a))
#define ll long long
#define ull unsigned long long
#define rep(i,a,b) for(int i=a;i<=b;i++)
using namespace std;
typedef pair<int,int>P;
const int maxn=100010;
const double eps=1e-10;
const double pi=acos(-1);
P p[maxn];
double d[maxn],f[maxn];
double dist(P a,P b)
{
  return sqrt((a.first-b.first)*(a.first-b.first)+(a.second-b.second)*(a.second-b.second));
}
int main()
{
  int T;
  cin>>T;
  while(T--){
    int n;
    cin>>n;
    for(int i=1;i<=n;i++)cin>>p[i].first>>p[i].second;
    for(int i=1;i<=n;i++){
      if(i==n)d[i]=dist(p[i],p[1]);
      else d[i]=dist(p[i],p[i+1]);
    }
    double maxx=0x3f3f3f3f,minn=0;//极值上下限
    f[1]=0;
    for(int i = 2; i <=n ; i++)
    {
      f[i] = d[i-1] - f[i-1];
      if(i%2==0 && f[i] < maxx)//若为偶数点,则该圆的半径只能减小这么多(即第一个圆的半径只能增大这么多),更新最大值下限
      {
        maxx = f[i];
      }
      if(i%2==1 && (-f[i]) > minn)//若为奇数点,且此时f[i]小与0,则必须第一个圆的半径更新为该值,更新最小值上限
      {
        minn = -f[i];
      }
    }
    if(minn >= maxx + eps )//无解
    {
      printf("IMPOSSIBLE\n");
      continue;
    }
    if(n%2==1){//奇数个点,有解则必有唯一解,否则无解
      double x=0;//第一个圆的半径x=(d1-d2+d3-d4...)/2,唯一解.
      for(int i=1;i<=n;i++){
        if(i%2==1)x+=d[i];
        else x-=d[i];
      }
      x/=2;
      if(x<=minn-eps||x>=maxx+eps){
        cout<<"IMPOSSIBLE"<<endl;
        continue;
      }
      double area=0;
      for(int i=1;i<=n;i++){
        if(i%2==1)area+=(f[i]+x)*(f[i]+x);
        else area+=(f[i]-x)*(f[i]-x);
      }
      area*=pi;
      printf("%.2f\n",area);
      for(int i=1;i<=n;i++){
        if(i%2==1)printf("%.2f\n",f[i]+x);
        else printf("%.2f\n",f[i]-x);
      }
    }
    else{//偶数情况构造二次函数,y=a*x*x+b*x+c
      double now=0;
      for(int i=1;i<=n;i++){
        if(i%2==1)now+=d[i];
        else now-=d[i];
      }
      if(fabs(now)>eps||minn-maxx>eps){
        cout<<"IMPOSSIBLE"<<endl;
        continue;
      }
      double a=n;
      double b=0,c=0;
      for(int i=1;i<=n;i++){
        if(i%2==1){
          b+=2*f[i];
        }
        else{
          b-=2*f[i];
        }
        c+=f[i]*f[i];
      }
      double x=-b/(2*a);
      if(x<minn+eps)x=minn;
      if(x>maxx-eps)x=maxx;
      double area=a*x*x+b*x+c;
      area*=pi;
      printf("%.2f\n",area);
      for(int i=1;i<=n;i++){
        if(i%2==1)printf("%.2f\n",f[i]+x);
        else printf("%.2f\n",f[i]-x);
      }
    }
  }
  return 0;
}