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CSP-S模拟赛八总结

news 2025/10/28 9:24:46

T1

题面：

真不知道小 D 还在上幼儿园研究数字干什么 ~~（有成为数学家的天赋）~~……

这题很简单，首先 $x$ 是一个完全平方数，那我们直接枚举 $x\sqrt{x}$ 是多少，进而算出 $x$ ，然后看看 $x$ 是否满足最后一个条件就行了。

因为 $x∈[0,n]x\isin[0,n]$ ，所以 $x∈[0,n]\sqrt{x}\isin[0,\sqrt{n}]$ ，最高时间复杂度： $O(n∣x∣)O(\sqrt{n}\mid x\mid)$ ，其中 $∣x∣\mid x\mid$ 表示 $x$ 的位数，但是常数很小，所以直接过。

代码：

#include<bits/stdc++.h>
#define int long long
#define code using
#define by namespace
#define plh std
code by plh;
int n,ans;
signed main()
{
//	freopen("num.in","r",stdin);
//	freopen("num.out","w",stdout);cin>>n;for(int i=0;i*i<=n;i+=2){int x=i*i;bool fl=false;while(x){if(x%10&1){fl=true;break;}x/=10;}if(!fl){ans++;}}cout<<ans;return 0;
}

题外话：关于我的打表同学：

#include<bits/stdc++.h>
using namespace std;
#define int long long
int n,l=1,r=2188,res=0;
int f[]={-1,0,4,64,400,484,4624,6084,6400,8464,26244,28224,40000,40804,48400,68644,88804,228484,242064,248004,446224,462400,608400,640000,806404,824464,846400,868624,2022084,2226064,2244004,2624400,2822400,2862864,4000000,4008004,4080400,4088484,4804864,4840000,4848804,6240004,6260004,6646084,6864400,8020224,8282884,8868484,8880400,20268004,20448484,20866624,22448644,22848400,24088464,24206400,24226084,24800400,26646244,28068804,42224004,44462224,44622400,46240000,48888064,60248644,60808804,60840000,62062884,64000000,68260644,80604484,80640400,82228624,82446400,82664464,82846404,84640000,86862400,202208400,208282624,222248464,222606400,222666084,224400400,240002064,244046884,248440644,262440000,282240000,284866884,286286400,400000000,400080004,400800400,400880484,408040000,408848400,424442404,440664064,444282084,448084224,466646404,480486400,484000000,484088004,484880400,600642064,606242884,620408464,624000400,624600064,626000400,628404624,644042884,662444644,664608400,680062084,686020864,686440000,802022400,806446404,828288400,840884004,864242404,866242624,882208804,886848400,888040000,2006860804,2024280064,2024820004,2026800400,2044848400,2048648644,2064066624,2068248484,2086662400,2208248064,2220482884,2224820224,2228028804,2240886244,2244864400,2248846084,2284648804,2284840000,2400804004,2408846400,2420640000,2422608400,2480040000,2620620864,2644222084,2664624400,2684068864,2806668484,2806880400,2862464004,2864604484,4002080644,4086022084,4222400400,4226040064,4246086244,4246868224,4444622224,4446222400,4462240000,4624000000,4640606884,4820802624,4888806400,6002840484,6024864400,6080880400,6084000000,6206288400,6228682084,6240684004,6242264064,6266622244,6282464644,6400000000,6460622884,6668682244,6802620484,6826064400,6888668004,8022426624,8046448804,8060448400,8064040000,8220686224,8222862400,8244640000,8262446404,8264264464,8266446400,8268628624,8284640400,8420264644,8464000000,8468048484,8486462884,8648628004,8686240000,8804444224,8866482244,8880800644,20220840000,20284026084,20448428004,20804600644,20828262400,20860602624,20880828004,20886608484,22200404004,22224846400,22228424464,22260640000,22266608400,22440040000,22862044804,24000206400,24000826084,24046084624,24404688400,24646488064,24806880004,24844064400,24888848644,26204486884,26244000000,26244648004,26468686864,26602262404,26646644644,28002006244,28048880484,28224000000,28226688064,28486688400,28628640000,40000000000,40000800004,40008000400,40008800484,40064826244,40080040000,40080840804,40088048400,40482244804,40804000000,40804808004,40884840000,42028280064,42280406884,42444240400,42464020624,42604262464,44066406400,44420620644,44428208400,44602060864,44606284804,44646844804,44800802244,44808422400,44862628864,46062602884,46228440064,46400606464,46484222404,46626628624,46664640400,48048640000,48220646464,48400000000,48400880004,48408800400,48440488464,48488040000,60024020004,60064206400,60624288400,62000004004,62040846400,62046824464,62400040000,62460006400,62466004624,62600040000,62840462400,64060622404,64404288400,64804866624,66044888064,66202260804,66222846244,66244464400,66266086084,66460840000,66686864644,68006208400,68226484804,68602086400,68644000000,68800240804,80202240000,80280422244,80424422464,80486824804,80644640400,80804884644,80844686224,80882222404,82828840000,84006066244,84046648464,84088400400,86002600644,86260864804,86424240400,86624262400,88204248064,88220880400,88222068484,88602666244,88626480804,88662208644,88684840000,88802808004,88804000000,88864802404,200084446864,200684288484,200686080400,202264268644,202428006400,202480200484,202482000400,202680040000,204220840464,204484840000,204486648804,204826846084,204864864400,206208626404,206406662400,206824848400,208666240000,220206824644,220824806400,220826686084,222048288400,222408446404,222482022400,222668846884,222802880400,224020462864,224048862244,224088624400,224486440000,224602062084,224884608400,226260046224,226488424464,228464880400,228482088004,228484000000,228600646884,228864646404,240080400400,240264668224,240884640000,242064000000,242260840000,242262808804,244864646244,246008064064,248002008004,248004000000,260020446084,260404048804,260602082064,260608208004,262062086400,264204224064,264422208400,264640282624,266088842244,266462440000,268406886400,268668062224,280262242404,280442266624,280666848400,280688040000,280802488464,282468864484,282600686404,282664482244,284086868004,286222860004,286246400400,286460448400,288624668644,288822406084,288882600484,400028420484,400084080484,400208064400,400686468004,402042228624,402288284644,402460822404,402600402064,402846628804,406408800004,408062884804,408602208400,408804226884,422240040000,422604006400,422840668644,424064044804,424608624400,424686822400,426006846864,440242866064,442440486244,442448468224,444446222224,444462222400,444622240000,446224000000,448808884624,462400000000,464060688400,466082020804,466486268004,468400622404,468666206464,480246228004,480260088064,482080262400,482660888644,484404864064,484446624484,48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signed main()
{
//	freopen("num.in","r",stdin);
//	freopen("num.out","w",stdout);ios::sync_with_stdio(0);cin.tie(0);cout.tie(0);cin>>n;while(l<=r){int mid=(l+r)/2;if(f[mid]<=n){l=mid+1;res=max(res,mid);}else r=mid-1;}cout<<res<<"\n";return 0;
}

T2

题面：

水题一道，但是要注意要开二倍空间！！！

其实我们最终的答案表达式如下：

$\sum a_i+\sum b_i$

现在我们把后面那一块反过来：

$\sum a_i+s-\sum b_i$

也就是用总和减去不用的。

然后合并一下：

$s+\sum(a_i-b_i)$

现在我们已知 $s$ 是 $∑b\sum b$ ，是一个定值，那只需要后面那里最大就行了，因此直接贪心。

题外话：考试的时候忘了是 $2 n$ 个数，空间开小了，直接 RE 了……

代码：

#include<bits/stdc++.h>
#define int long long
#define code using
#define by namespace
#define plh std
code by plh;
struct kc{int x,y;
}a[1000006];
int n;
signed main()
{
//	freopen("course.in","r",stdin);
//	freopen("course.out","w",stdout);cin>>n;n*=2;for(int i=1;i<=n;i++){cin>>a[i].x>>a[i].y;}sort(a+1,a+n+1,[&](kc x,kc y){return x.x-x.y>y.x-y.y;});int ans=0;for(int i=1;i<=n;i++){if(i<=n/2){ans+=a[i].x;}else{ans+=a[i].y;}}cout<<ans;return 0;
}

T3

题面：

首先做这题我们需要知道这样一条性质：

$B^x\equiv1\pmod{B-1}$

这个其实也很好证明：

证明：已知 $B≡1(modB−1)B\equiv1\pmod{B-1}$ ，对于任意的整数 $x$ ，有：

$B^x\equiv B\cdot B^{x-1}\equiv B^{x-1}\pmod{B-1}$

因此可知 $Bx≡1(modB−1)B^x\equiv1\pmod{B-1}$ 。

然后这道题就好做了。

我们设构建的这个数是 $a1a2…an‾\overline{a_1a_2\dots a_n}$ ，通过数学式子可以表示为这样：

$a_n+a_{n-1}\cdot B+\dots+a_1\cdot B^{n-1}$

现在把这个数对 $B - 1$ 取余数，按照我们上面的性质，最终答案应该是这个：

$\sum_{i=1}^na_i$

因此我们只需要尽可能多的选数，并让这些数的和是 $B - 1$ 的倍数就行了。

那我们可以先把所有数全选上，然后删掉 $(B−1)sum\bmod (B-1)$ 来保持和是 $B - 1$ 的倍数（当然余数是 $0$ 就不用删了），然后记录下每个数用的个数。

对于每次查询，我们可以先预处理前缀和一下，然后二分找答案，时间复杂度 $O(qlog⁡n)O(q\log n)$ 。当然，也可以用离线做法，这时就需要排序再用双指针解决，时间复杂度 $O (q)$ 左右。

代码（离线）：

#include<bits/stdc++.h>
#define int long long
#define code using
#define by namespace
#define plh std
code by plh;
struct card{int x,id;
}b[1000006];
int n,q,a[1000006],ans[1000006];
inline int read()
{int z=0,f=1;char c=getchar();while(c<'0'||c>'9'){if(c=='-'){f=-1;}c=getchar();}while(c>='0'&&c<='9'){z=z*10+c-'0';c=getchar();}return z*f;
}
inline void write(int x)
{if(x<0){putchar('-');x=-x;}static int top=0,stk[10006];while(x){stk[++top]=x%10;x/=10;}if(!top){stk[++top]=0;}while(top){putchar(char(stk[top--]+'0'));}
}
signed main()
{
//	freopen("card.in","r",stdin);
//	freopen("card.out","w",stdout);n=read(),q=read();for(int i=0;i<n;i++){a[i]=read();}int sum=0;for(int i=0;i<n;i++){sum+=a[i]*i;}sum%=(n-1);if(sum!=0){a[sum]--;}for(int i=1;i<n;i++){a[i]+=a[i-1];}for(int i=1;i<=q;i++){b[i].x=read();b[i].x++;b[i].id=i;}sort(b+1,b+q+1,[&](card x,card y){return x.x<y.x;});for(int i=1,j=0;i<=q;i++){while(j<n&&a[j]<b[i].x){j++;}if(j>=n){break;}ans[b[i].id]=j+1;}for(int i=1;i<=q;i++){write(ans[i]?ans[i]-1:-1);putchar('\n');}return 0;
}

题外话：离线真的很快，直接拿下时间榜第一。

T4

题面：

正解这里我就不过多赘述了，大家自己看题解：

这里我讲一下歪解（目前还没证出来，暂且算是歪解吧）。

其实在做这道题的时候，我先开始打了一个 10pts 的暴力：

#include<bits/stdc++.h>
#define int long long
#define code using
#define by namespace
#define plh std
code by plh;
const int mod=1e7+7;
int n,ans,a[5006],b[5006];
signed main()
{
//	freopen("eat.in","r",stdin);
//	freopen("eat.out","w",stdout);cin>>n;for(int i=1;i<=n;i++){a[i]=i;}do{for(int i=1;i<=n;i++){b[i]=1;}bool fl=true;for(int i=1;i<=n;i++){for(int j=1;j<i;j++){if(a[j]<a[i]){b[i]=max(b[i],b[j]+1);}}if(b[i]>2){fl=false;break;}}if(fl){ans++;}}while(next_permutation(a+1,a+n+1));cout<<ans<<" ";return 0;
}

然后我就试了一下前 5 位数都是多少：

1 2 3 4 5
1 2 5 14 42

如果你的数感比较好，其实你很容易看出来这是 Catalan 数的前 5 项，为了验证，我又算了 6 到 10 之间的数，这里不一一列举了，相信结果大家肯定都知道了：答案就是第 $n$ 项 Catalan 数！

于是我们直接使用 Catalan 数的递推式，交上去：直接 AC！

代码：

#include<bits/stdc++.h>
using namespace std;
const int mod=1e7+7;
long long n,f[5006];
signed main()
{cin>>n;f[0]=1;for(int i=1;i<=n;i++){for(int j=0;j<i;j++){f[i]+=f[j]*f[i-1-j];f[i]%=mod;}}cout<<f[n];return 0;
}

注意：这里的模数是 $10^7+7$ ，它不是一个质数，所以不好用 $O (n)$ 的组合数做……

题外话：考试的时候我发现了，但是我不会 Catalan 数的递推式！！！（这辈子最后悔的事就是没有好好学 Catalan 数）。

T5

题面：

看到异或，其实首先最容易想到的就是转化成二进制数位分离。

看到这个 $min⁡\min$ ，感觉非常烦，于是我们可以尝试把这个 $min⁡\min$ 拆开。

那么我们直接从最高位开始考虑 $a$ 和 $b$ 。

这里就不列举了，反正每个数只有 0 和 1 两种情况，两个数异或起来相同为 0、不同为 1，那么在下面四种情况下会有贡献：

其余还有两种情况没法通过最高位比较：

对于这种情况，我们直接递归到下一位。

然后这道题就能非常完美的 AC 了。

时间复杂度： $O(n\log^2\max(a))$ 。

代码：

#include<bits/stdc++.h>
#define int long long
#define code using
#define by namespace
#define plh std
code by plh;
int n,ans,mxbit,a[200006],b[200006];
int bit(int x,int y)
{return (x>>y)&1;
}
void dfs(int pos,vector<pair<int,int>>v)
{if(v.size()<2){return;}if(pos<0){for(int i=mxbit;i>=0;i--){int cnt0=0,cnt1=0;for(auto j:v){int b=bit(j.first,i);cnt0+=(b==0);cnt1+=(b==1);}ans+=(cnt0*cnt1)*(1<<i);}return;}vector<pair<int,int>>vv[6];for(auto i:v){vv[bit(i.first,pos)*2+bit(i.second,pos)].push_back(i);}vector<tuple<int,int,int>>vvv={{0,1,0},{0,2,1},{3,2,0},{3,1,1}};for(auto i:vvv){int t1,t2,t3;tie(t1,t2,t3)=i;if(vv[t1].empty()||vv[t2].empty()){continue;}for(int j=mxbit;j>=0;j--){int cnt0=0,cnt1=0;for(auto k:vv[t1]){int b=bit((t3==0?k.first:k.second),j);cnt0+=(b==0);cnt1+=(b==1);}int cntt0=0,cntt1=0;for(auto k:vv[t2]){int b=bit((t3==0?k.first:k.second),j);cntt0+=(b==0);cntt1+=(b==1);}ans+=(cnt0*cntt1+cnt1*cntt0)*(1<<j);}}vector<pair<int,int>>m1,m2;for(auto i:vv[0]){m1.push_back(i);}for(auto i:vv[3]){m1.push_back(i);}for(auto i:vv[1]){m2.push_back(i);}for(auto i:vv[2]){m2.push_back(i);}dfs(pos-1,m1);dfs(pos-1,m2);
}
signed main()
{vector<pair<int,int>>v;cin>>n;int mx=0;for(int i=1;i<=n;i++){cin>>a[i];mx=max(mx,a[i]);}for(int i=1;i<=n;i++){cin>>b[i];v.push_back(make_pair(a[i],b[i]));mx=max(mx,b[i]);}mxbit=log2(mx);dfs(mxbit,v);cout<<ans;return 0;
}