# Codeforces Round #636 (Div. 3)

## A. Candies

time limit per test1 second memory limit per test256 megabytes

Recently Vova found n candy wrappers. He remembers that he bought x candies during the first day, 2x candies during the second day, 4x candies during the third day, $$\dots, 2^{k-1} x$$ candies during the k-th day. But there is an issue: Vova remembers neither x nor k but he is sure that x and k are positive integers and k > 1.

Vova will be satisfied if you tell him any positive integer x so there is an integer k>1 that $$x + 2x + 4x + \dots + 2^{k-1} x = n$$. It is guaranteed that at least one solution exists. Note that k > 1.

You have to answer t independent test cases.Input

The first line of the input contains one integer t $$(1 \le t \le 10^4)$$ — the number of test cases. Then t test cases follow.

The only line of the test case contains one integer n $$(3 \le n \le 10^9)$$ — the number of candy wrappers Vova found. It is guaranteed that there is some positive integer x and integer k>1 that $$x + 2x + 4x + \dots + 2^{k-1} x = n$$.Output

Print one integer — any positive integer value of x so there is an integer k>1 that $$x + 2x + 4x + \dots + 2^{k-1} x = n$$.ExampleinputCopy

7
3
6
7
21
28
999999999
999999984


outputCopy

1
2
1
7
4
333333333
333333328


Note

In the first test case of the example, one of the possible answers is x=1, k=2. Then $$1 \cdot 1 + 2 \cdot 1$$ equals n=3.

In the second test case of the example, one of the possible answers is x=2, k=2. Then $$1 \cdot 2 + 2 \cdot 2$$ equals n=6.

In the third test case of the example, one of the possible answers is x=1, k=3. Then 1 \cdot 1 + 2 \cdot 1 + 4 \cdot 1 equals n=7.

In the fourth test case of the example, one of the possible answers is x=7, k=2. Then 1 \cdot 7 + 2 \cdot 7 equals n=21.

In the fifth test case of the example, one of the possible answers is x=4, k=3. Then 1 \cdot 4 + 2 \cdot 4 + 4 \cdot 4 equals n=28.

#### 签到

#include<bits/stdc++.h>
using namespace std;
#define rep(i,a,b) for(auto i=(a);i<=(b);++i)
#define dep(i,a,b) for(auto i=(a);i>=(b);--i)
#define pb push_back
typedef long long ll;
const int maxn=(int)1e6+100;
const int mod=(int)1e9+7;
void solve(){
ll n;cin>>n;
for(ll i=2;;i++){
int now=(1ll<<i)-1;
if(n%now==0) return (void)printf("%lld\n",n/now);
}
}
int main(){
cin.tie(0)->sync_with_stdio(0);
int T;cin>>T;
while(T--) solve();
}


## B. Balanced Array

time limit per test1 second memory limit per test256 megabytes

You are given a positive integer n, it is guaranteed that n is even (i.e. divisible by 2).

You want to construct the array a of length n such that:

• The first \frac{n}{2} elements of a are even (divisible by 2);
• the second \frac{n}{2} elements of a are odd (not divisible by 2);
• all elements of a are distinct and positive;
• the sum of the first half equals to the sum of the second half $$(\sum\limits_{i=1}^{\frac{n}{2}} a_i = \sum\limits_{i=\frac{n}{2} + 1}^{n} a_i)$$.

If there are multiple answers, you can print any. It is not guaranteed that the answer exists.

You have to answer t independent test cases.Input

The first line of the input contains one integer t $$(1 \le t \le 10^4)$$ — the number of test cases. Then t test cases follow.

The only line of the test case contains one integer n$$(2 \le n \le 2 \cdot 10^5)) — the length of the array. It is guaranteed that that n is even (i.e. divisible by 2). It is guaranteed that the sum of n over all test cases does not exceed (2 \cdot 10^5 (\sum n \le 2 \cdot 10^5)).Output For each test case, print the answer — "NO" (without quotes), if there is no suitable answer for the given test case or "YES" in the first line and any suitable array (a_1, a_2, \dots, a_n (1 \le a_i \le 10^9)$$ satisfying conditions from the problem statement on the second line.ExampleinputCopy

5
2
4
6
8
10


outputCopy

NO
YES
2 4 1 5
NO
YES
2 4 6 8 1 3 5 11
NO

#### 签到

#include<bits/stdc++.h>
using namespace std;
#define rep(i,a,b) for(auto i=(a);i<=(b);++i)
#define dep(i,a,b) for(auto i=(a);i>=(b);--i)
#define pb push_back
typedef long long ll;
const int maxn=(int)1e6+100;
const int mod=(int)1e9+7;
void solve(){
int n;cin>>n;
if((n/2)%2) return (void)puts("NO");
puts("YES");
rep(i,1,n/2) printf("%d ",i*2);
rep(i,1,n/2-1) printf("%d ",i*2-1);
printf("%lld\n",1ll*(2+n)*n/4-(1ll*(n-2)*(n/2-1)/2));
}
int main(){
cin.tie(0)->sync_with_stdio(0);
int T;cin>>T;
while(T--) solve();
}


## C. Alternating Subsequence

time limit per test1 second memory limit per test256 megabytes

Recall that the sequence b is a a subsequence of the sequence a if b can be derived from a by removing zero or more elements without changing the order of the remaining elements. For example, if a=[1, 2, 1, 3, 1, 2, 1], then possible subsequences are: [1, 1, 1, 1],  and [1, 2, 1, 3, 1, 2, 1], but not [3, 2, 3] and [1, 1, 1, 1, 2].

You are given a sequence a consisting of n positive and negative elements (there is no zeros in the sequence).

Your task is to choose maximum by size (length) alternating subsequence of the given sequence (i.e. the sign of each next element is the opposite from the sign of the current element, like positive-negative-positive and so on or negative-positive-negative and so on). Among all such subsequences, you have to choose one which has the maximum sum of elements.

In other words, if the maximum length of alternating subsequence is k then your task is to find the maximum sum of elements of some alternating subsequence of length k.

You have to answer t independent test cases.Input

The first line of the input contains one integer $$t (1 \le t \le 10^4)$$ — the number of test cases. Then t test cases follow.

The first line of the test case contains one integer $$n (1 \le n \le 2 \cdot 10^5)$$— the number of elements in a. The second line of the test case contains n integers$$a_1, a_2, \dots, a_n (-10^9 \le a_i \le 10^9, a_i \ne 0)$$, where a_i is the i-th element of a.

It is guaranteed that the sum of n over all test cases does not exceed $$2 \cdot 10^5 (\sum n \le 2 \cdot 10^5)$$.Output

For each test case, print the answer — the maximum sum of the maximum by size (length) alternating subsequence of a.ExampleinputCopy

4
5
1 2 3 -1 -2
4
-1 -2 -1 -3
10
-2 8 3 8 -4 -15 5 -2 -3 1
6
1 -1000000000 1 -1000000000 1 -1000000000


outputCopy

2
-1
6
-2999999997


Note

In the first test case of the example, one of the possible answers is( [1, 2, \underline{3}, \underline{-1}, -2]\).

In the second test case of the example, one of the possible answers is $$[-1, -2, \underline{-1}, -3]$$.

In the third test case of the example, one of the possible answers is $$[\underline{-2}, 8, 3, \underline{8}, \underline{-4}, -15, \underline{5}, \underline{-2}, -3, \underline{1}]$$.

In the fourth test case of the example, one of the possible answers is$$[\underline{1}, \underline{-1000000000}, \underline{1}, \underline{-1000000000}, \underline{1}, \underline{-1000000000}]$$.

#### 每段连续的符号中取最大

#include<bits/stdc++.h>
using namespace std;
#define rep(i,a,b) for(auto i=(a);i<=(b);++i)
#define dep(i,a,b) for(auto i=(a);i>=(b);--i)
#define pb push_back
typedef long long ll;
const int maxn=(int)1e6+100;
const int mod=(int)1e9+7;
ll a[maxn];
void solve(){
int n;cin>>n;
rep(i,1,n) cin>>a[i];
int op=a>0;
ll ans=0;
vector<int> v;v.pb(1);
int pos=1;
while(1){
if(op){
while(pos<=n&&a[pos]>0) pos++;
v.pb(pos);op^=1;
}else{
while(pos<=n&&a[pos]<0) pos++;
v.pb(pos);op^=1;
}
if(pos>n) break;
}
int sz=(int)v.size();
rep(i,0,sz-2){
ll tmp=-2e9;
rep(j,v[i],v[i+1]-1) tmp=max(tmp,a[j]);
ans+=tmp;
} cout<<ans<<endl;

}
int main(){
cin.tie(0)->sync_with_stdio(0);
int T;cin>>T;
while(T--) solve();
}


## D. Constant Palindrome Sum

time limit per test1 second memory limit per test256 megabytes

You are given an array a consisting of n integers (it is guaranteed that n is even, i.e. divisible by 2). All a_i does not exceed some integer k.

Your task is to replace the minimum number of elements (replacement is the following operation: choose some index i from 1 to n and replace a_i with some integer in range [1; k]) to satisfy the following conditions:

• after all replacements, all a_i are positive integers not greater than k;
• for all i from 1 to \frac{n}{2} the following equation is true: a_i + a_{n - i + 1} = x, where x should be the same for all $$\frac{n}{2}$$ pairs of elements.

You have to answer t independent test cases.Input

The first line of the input contains one integer $$t (1 \le t \le 10^4)$$ — the number of test cases. Then t test cases follow.

The first line of the test case contains two integers n and k $$(2 \le n \le 2 \cdot 10^5, 1 \le k \le 2 \cdot 10^5)$$ — the length of a and the maximum possible value of some a_i correspondingly. It is guratanteed that n is even (i.e. divisible by 2). The second line of the test case contains n integers $$a_1, a_2, \dots, a_n (1 \le a_i \le k)$$, where a_i is the i-th element of a.

It is guaranteed that the sum of n (as well as the sum of k) over all test cases does not exceed$$2 \cdot 10^5 (\sum n \le 2 \cdot 10^5, \sum k \le 2 \cdot 10^5)$$.Output

For each test case, print the answer — the minimum number of elements you have to replace in a to satisfy the conditions from the problem statement.ExampleinputCopy

4
4 2
1 2 1 2
4 3
1 2 2 1
8 7
6 1 1 7 6 3 4 6
6 6
5 2 6 1 3 4


outputCopy

0
1
4
2

#### 差分做一下；c[i]表示的是取x为i时的最小次数，很显然x的取值范围是[2,2*k]，那么我们对于每对数算一下贡献即可；

#include<bits/stdc++.h>
using namespace std;
#define rep(i,a,b) for(auto i=(a);i<=(b);++i)
#define dep(i,a,b) for(auto i=(a);i>=(b);--i)
#define pb push_back
typedef long long ll;
const int maxn=(int)5e5+100;
const int mod=(int)1e9+7;
int a[maxn];
void solve(){
int n,k;cin>>n>>k;
rep(i,1,n) cin>>a[i];
int c[maxn]={0,};
rep(i,1,n/2){
int x=a[i],y=a[n-i+1];
c[x+y+1]++;c[max(x,y)+k+1]--;
c[max(x,y)+k+1]+=2;c[2*k+1]-=2;
c[min(x,y)+1]++;c[x+y]--;
c+=2;c[min(x,y)+1]-=2;
}int ans=mod;
rep(i,2,2*k) c[i]+=c[i-1],ans=min(ans,c[i]);
cout<<ans<<endl;
}
int main(){
cin.tie(0)->sync_with_stdio(0);
int T;cin>>T;
while(T--) solve();
}


## E. Weights Distributing

time limit per test2 seconds memory limit per test256 megabytes

You are given an undirected unweighted graph consisting of n vertices and m edges (which represents the map of Bertown) and the array of prices p of length m. It is guaranteed that there is a path between each pair of vertices (districts).

Mike has planned a trip from the vertex (district) a to the vertex (district) b and then from the vertex (district) b to the vertex (district) c. He can visit the same district twice or more. But there is one issue: authorities of the city want to set a price for using the road so if someone goes along the road then he should pay the price corresponding to this road (he pays each time he goes along the road). The list of prices that will be used p is ready and they just want to distribute it between all roads in the town in such a way that each price from the array corresponds to exactly one road.

You are a good friend of Mike (and suddenly a mayor of Bertown) and want to help him to make his trip as cheap as possible. So, your task is to distribute prices between roads in such a way that if Mike chooses the optimal path then the price of the trip is the minimum possible. Note that you cannot rearrange prices after the start of the trip.

You have to answer t independent test cases.Input

The first line of the input contains one integer t$$(1 \le t \le 10^4)$$— the number of test cases. Then t test cases follow.

The first line of the test case contains five integers n, m, a, b and $$c (2 \le n \le 2 \cdot 10^5, n-1 \le m \le min(\frac{n(n-1)}{2}, 2 \cdot 10^5), 1 \le a, b, c \le n)$$ — the number of vertices, the number of edges and districts in Mike's trip.

The second line of the test case contains m integers $$p_1, p_2, \dots, p_m (1 \le p_i \le 10^9)$$, where p_i is the i-th price from the array.

The following m lines of the test case denote edges: edge i is represented by a pair of integers $$v_i, u_i (1 \le v_i, u_i \le n, u_i \ne v_i)$$, which are the indices of vertices connected by the edge. There are no loops or multiple edges in the given graph, i. e. for each pair (v_i, u_i) there are no other pairs (v_i, u_i) or (u_i, v_i) in the array of edges, and for each pair (v_i, u_i) the condition\$$v_i \ne u_i$$ is satisfied. It is guaranteed that the given graph is connected.

It is guaranteed that the sum of m does not exceed (2 \cdot 10^5 (\sum m \le 2 \cdot 10^5)\).Output

For each test case, print the answer — the minimum possible price of Mike's trip if you distribute prices between edges optimally.ExampleinputCopy

2
4 3 2 3 4
1 2 3
1 2
1 3
1 4
7 9 1 5 7
2 10 4 8 5 6 7 3 3
1 2
1 3
1 4
3 2
3 5
4 2
5 6
1 7
6 7


outputCopy

7
12


Note

One of the possible solution to the first test case of the example:

One of the possible solution to the second test case of the example:

#### 乱搞题，先求出a和c到每个点的最短路，然后再从b点开始跑dijk，最终的路线必定是a->u,u->b,b->u,u->c；那么我们只要每次遇到一个u就更新一次答案即可；

#include<bits/stdc++.h>
using namespace std;
#define rep(i,a,b) for(auto i=(a);i<=(b);++i)
#define dep(i,a,b) for(auto i=(a);i>=(b);--i)
#define pb push_back
typedef long long ll;
const int maxn=(int)1e6+100;
const int mod=(int)1e9+7;
int n,m,a,b,c;
ll aa[maxn],pre[maxn],dep[maxn],dis[maxn],vis[maxn],ans;
struct Edge{int v,w,nex;}g[maxn<<1];
void bfs(int s,int id){
rep(i,1,n) dep[id][i]=mod,vis[i]=0;
queue<int> q; q.push(s);dep[id][s]=0;vis[s]=1;
while(!q.empty()){
int u=q.front();q.pop();
int v=g[i].v;
if(dep[id][v]>dep[id][u]+1&&!vis[v]){
dep[id][v]=dep[id][u]+1;
q.push(v);vis[v]=1;
}
}
}
}
void dijk(int s){
rep(i,1,n) dis[i]=mod,vis[i]=0;
queue<int> q;q.push(s);dis[s]=0;vis[s]=1;
while(!q.empty()){
int u=q.front();q.pop();
if(dis[u]+dep[u]+dep[u]<=m)
ans=min(ans,pre[dis[u]]+pre[dis[u]+dep[u]+dep[u]]);
int v=g[i].v;
if(dis[v]>dis[u]+1&&!vis[v]){
dis[v]=dis[u]+1;
q.push(v);vis[v]=1;
}
}
}
}
void solve(){
scanf("%d%d%d%d%d",&n,&m,&a,&b,&c);init();ans=(ll)2e18;
rep(i,1,m) scanf("%lld",&aa[i]); sort(aa+1,aa+1+m);
rep(i,1,m) pre[i]=pre[i-1]+aa[i];
rep(i,1,n) dep[i]=dep[i]=0;
rep(i,1,m){
int u,v;scanf("%d%d",&u,&v);
} bfs(a,0); bfs(c,1); dijk(b);
printf("%lld\n",ans);
}
int main(){
int T;cin>>T;
while(T--) solve();
}


## F. Restore the Permutation by Sorted Segments

time limit per test2 seconds memory limit per test256 megabytes

We guessed a permutation p consisting of n integers. The permutation of length n is the array of length n where each element from 1 to n appears exactly once. This permutation is a secret for you.

For each position r from 2 to n we chose some other index l (l < r) and gave you the segment $$p_l, p_{l + 1}, \dots, p_r) in sorted order (i.e. we rearranged the elements of this segment in a way that the elements of this segment are sorted). Thus, you are given exactly n-1 segments of the initial permutation but elements inside each segment are sorted. The segments are given to you in random order. For example, if the secret permutation is p=[3, 1, 4, 6, 2, 5] then the possible given set of segments can be: • [2, 5, 6] • [4, 6] • [1, 3, 4] • [1, 3] • [1, 2, 4, 6] Your task is to find any suitable permutation (i.e. any permutation corresponding to the given input data). It is guaranteed that the input data corresponds to some permutation (i.e. such permutation exists). You have to answer t independent test cases.Input The first line of the input contains one integer \(t (1 \le t \le 100) )— the number of test cases. Then t test cases follow. The first line of the test case contains one integer \(n (2 \le n \le 200)$$— the length of the permutation.

The next n-1 lines describe given segments.

The i-th line contains the description of the i-th segment. The line starts with the integer $$k_i (2 \le k_i \le n)$$ — the length of the i-th segment. Then k_i integers follow. All integers in a line are distinct, sorted in ascending order, between 1 and n, inclusive.

It is guaranteed that the required p exists for each test case.

It is also guaranteed that the sum of n over all test cases does not exceed $$200 (\sum n \le 200)$$.Output

For each test case, print the answer: n integers (p_1, p_2, \dots, p_n (1 \le p_i \le n, all p_i should be distinct)\) — any suitable permutation (i.e. any permutation corresponding to the test case input).ExampleinputCopy

5
6
3 2 5 6
2 4 6
3 1 3 4
2 1 3
4 1 2 4 6
5
2 2 3
2 1 2
2 1 4
2 4 5
7
3 1 2 6
4 1 3 5 6
2 1 2
3 4 5 7
6 1 2 3 4 5 6
3 1 3 6
2
2 1 2
5
2 2 5
3 2 3 5
4 2 3 4 5
5 1 2 3 4 5


outputCopy

3 1 4 6 2 5
3 2 1 4 5
2 1 6 3 5 4 7
1 2
2 5 3 4 1 

#### 看到那个each了吗（反正我一开始没看到），那也就是说我如果能找到一个恰当的顺序重新排列题给的这些序列，那么每一次新增的那个数就是我们要找的；现在假设我们知道了答案的第i位填的是x，那么只需要枚举所有n-1个序列，如果有某个序列满足：1、有且仅有一个数之前没填过；2、其余的所有数在之前填的过程中的下标范围都在[i-sz,i-1]（sz为序列大小）那么这个序列就是我们要找的下一个序列了，同时那个没填过的数也就是第i位的数；举个例子，我们看第一个样例，假设现在已经知道了前三位的答案为「3，1，4」，现在要获得第四位，我们就枚举所有的5个序列，发现第二个序列「4，6」满足条件（4在之前填的位置为3，大于（i-sz=2）），于是第四位就是6；既然如此，我们就可以枚举第一个数，然后一路check下去；

#include<bits/stdc++.h>
using namespace std;
#define rep(i,a,b) for(auto i=(a);i<=(b);++i)
#define dep(i,a,b) for(auto i=(a);i>=(b);--i)
#define pb push_back
typedef long long ll;
const int maxn=(int)1e6+100;
const int mod=(int)1e9+7;
int n,ans,pos;
vector<int> v;
bool check(){
rep(i,2,n){//第i位
rep(j,2,n){
int sz=v[j],num=0,x=0;
rep(k,1,sz){int u=v[j][k];
if(!pos[u]) x=u;
else if(pos[u]>i-sz) num++;
}if(num==sz-1&&x){ans[i]=x;pos[x]=i;break;}
}if(!ans[i]) return 0;
}return 1;
}
void solve(){
scanf("%d",&n);
rep(i,1,n) v[i].clear();
rep(i,2,n){
int k;scanf("%d",&k);v[i].pb(k);
rep(j,1,k){int x;scanf("%d",&x);v[i].pb(x);}
}
rep(i,1,n){
memset(ans,0,sizeof(ans));
memset(pos,0,sizeof(pos));
ans=i;pos[i]=1;
if(check()){
rep(j,1,n) printf(j==n?"%d\n":"%d ",ans[j]);return;
}
}
}
int main(){
int T;cin>>T;
while(T--) solve();
}


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