20201119模拟

似乎好久没有写过模拟赛总结了

T1

problem

给出\(n,k,m\)，且满足\(\sum\limits_{i = 1}^k a_i = n\),求所有可能的\(\sum\limits_{i = 1}^k {a_i}^m\)

solution

考场写了个三维dp，\(n^4\)的dp，t了，得分20

考虑一个数对答案的贡献：他的数值\(\times\)他出现的次数。那么如何求他出现的次数呢？？

把问题转换成把\(i\)个球放在\(j\)个盒子里的方案数，每个盒子里球的数量就是我的\(a_i\),确定一个\(x\)，剩下的位置求方案数，就是他的方案数。

对于\(1\rightarrow n\)的所有数，他都可以出现\(0\rightarrow k\)次:

1. 0次说明他对答案没贡献

2. 对于出现次数多次\(k\)的情况，因为在\(k-1\)次的时候计算过一遍了，所以每次只需要加上方案数\(\times {x}^m\)

一些细节:

1. \(i\)个球放在\(j\)个盒子里的方案数：\(f_{i,j} = f_{i,j-1}+f_{i-j,j}\)，如果我有至少一个空盒子，把空盒子设为我新加的盒子，就可以从\(f_{i,j-1}\)转移，如果没有空盒子，在\(j\)个盒子里每个盒子先放上1个球，剩下的球再放在\(j\)个盒子里，就可以有空盒子了，可以从\(f_{i-j,j}\)转移

2. 因为\(a_i\)都为正整数，所以不允许有空盒子，和1的处理方式一样，现在每个盒子里都放上一个球，再求方案数

code

#include <iostream>
#include <cstdio>
#include <algorithm>
#define int long long
using namespace std;
int read(){
	int x = 1,a = 0;char ch = getchar();
	while (ch < '0'||ch > '9'){if (ch == '-')x = -1;ch = getchar();}
	while (ch >= '0'&&ch <= '9'){a = a*10+ch-'0';ch = getchar();}
	return x*a;
}
const int maxn = 4e3+10,mod = 998244353;
int ans;
int n,k,m;
int qpow(int a,int kk){
	int res = 1;a = a%mod;
	while (kk){
		if (kk&1) res = res*a%mod;
		a = a*a%mod;
		kk>>=1;
	}
	return res;
}
int f[maxn][maxn],fac[maxn];
signed main(){
	n = read(),k = read(),m = read();
	for (int  i = 0;i <= k;i++) f[0][i] = 1;
	for (int i = 1;i <= n;i++){
		for (int j = 1;j <= k;j++){
			(f[i][j] = f[i][j-1])%=mod;
			if (i-j >= 0) (f[i][j] += f[i-j][j])%=mod;
		}
	}
	for (int i = 1;i <= n;i++) fac[i] = qpow(i,m);
	for (int i = 1;i <= n;i++){
		for (int j = 1;j <= k;j++){
			int res = 0;
			if (n-j*i-k+j >= 0) (res += f[n-j*i-k+j][k-j])%=mod;
			(ans += res*fac[i])%=mod;
		}
	}
	printf("%lld\n",ans);
	return 0;
}

T3

solution

枚举点集，再枚举边，看是否与被删的点有关系，统计答案，暴力30

#include <iostream>
#include <algorithm>
#include <cstdio>
#include <vector>
#define int long long
using namespace std;
int read(){
	int x = 1,a = 0;char ch = getchar();
	while (ch < '0'||ch > '9'){if (ch == '-') x = -1;ch = getchar();}
	while (ch >= '0'&&ch <= '9'){a = a*10+ch-'0';ch = getchar();}
	return x*a;
}
const int maxn = 3005,mod = 1e9+7;
int n,m,k;
struct node{
	int to,from;
}ed[maxn*2];
int qpow(int a,int kk){
	int res = 1;a = a%mod;
	while (kk){
		if (kk&1) res = res*a%mod;
		a = a*a%mod;
		kk>>=1;
	}
	return res%mod;
}
int vis[maxn][maxn],ans,ins[maxn];
signed main(){
	freopen("cubes.in","r",stdin);
	freopen("cubes.out","w",stdout);
	int ans = 0;
	n = read(),m = read(),k = read();
	for (int i = 1;i <= m;i++){
		ed[i].from = read(),ed[i].to = read();
	}
	for (int i = 0;i < (1<<(n+1));i++){
		int res = m;
		for (int j = 1;j <= m;j++){
			if (((1<<ed[j].from)&i)||((1<<ed[j].to)&i)) res--; 
		}
		ans += qpow(res,k);
	}
	printf("%lld\n",ans/2);
	return 0;
}

T4

solution

猜的结论，当\(m=1\)的情况，答案为\({\left(\frac{n\times (n-1)}{2}\right)}^k\)，得分10

#include <iostream>
#include <algorithm>
#include <cstdio>
#include <vector>
#define int long long
using namespace std;
int read(){
	int x = 1,a = 0;char ch = getchar();
	while (ch < '0'||ch > '9'){if (ch == '-') x = -1;ch = getchar();}
	while (ch >= '0'&&ch <= '9'){a = a*10+ch-'0';ch = getchar();}
	return x*a;
}
const int mod = 998244353;
int n,m,k;
int qpow(int a,int kk){
	int res = 1;a = a%mod;
	while (kk){
		if (kk&1) res = res*a%mod;
		a = a*a%mod;
		kk>>=1;
	}
	return res%mod;
}
signed main(){
	freopen("box.in","r",stdin);
	freopen("box.out","w",stdout);
	n = read(),m = read(),k = read();
	if (m==1) printf("%lld\n",qpow(n*(n-1)/2,k)%mod);
}