Greedy from small to large.

We can delete the numbers from small to large. Thus, previously removed numbers will not affect future choices (if $$$x<y$$$, then $$$x$$$ cannot be a multiple of $$$y$$$). So an integer $$$x$$$ ($$$l\le x\le r$$$) can be removed if and only if $$$k\cdot x\le r$$$, that is, $$$x\le \left\lfloor\frac{r}{k}\right\rfloor$$$. The answer is $$$\max\left(\left\lfloor\frac{r}{k}\right\rfloor-l+1,0\right)$$$.

Time complexity: $$$\mathcal{O}(1)$$$ per test case.

#include <bits/stdc++.h>
#define all(s) s.begin(), s.end()
using namespace std;
using ll = long long;
using ull = unsigned long long;
 
const int _N = 1e5 + 5;
 
int T;
 
void solve() {
	int l, r, k; cin >> l >> r >> k;
	cout << max(r / k - l + 1, 0) << endl;
	return;
}
 
int main() {
	ios::sync_with_stdio(false), cin.tie(0), cout.tie(0);
	cin >> T;
	while (T--) {
		solve();
	}
}

for _ in range(int(input())):
    l, r, k = map(int, input().split())
    print(max(r // k - l + 1, 0))

• Amazing problem:
• Good problem:
• Average problem:
• Bad problem:
• Didn't solve:

($$$\texttt{01}$$$ or $$$\texttt{10}$$$ exists) $$$\Longleftrightarrow$$$ (both $$$\texttt{0}$$$ and $$$\texttt{1}$$$ exist).

Each time we do an operation, if $$$s$$$ consists only of $$$\texttt{0}$$$ or $$$\texttt{1}$$$, we surely cannot find any valid indices. Otherwise, we can always perform the operation successfully. In the $$$i$$$-th operation, if $$$t_i=\texttt{0}$$$, we actually decrease the number of $$$\texttt{1}$$$-s by $$$1$$$, and vice versa. Thus, we only need to maintain the number of $$$\texttt{0}$$$-s and $$$\texttt{1}$$$-s in $$$s$$$. If any of them falls to $$$0$$$ before the last operation, the answer is NO, otherwise, the answer is YES.

Time complexity: $$$\mathcal{O}(n)$$$ per test case.

#include <bits/stdc++.h>
#define all(s) s.begin(), s.end()

using namespace std;
using ll = long long;

const int _N = 1e5 + 5;

void solve() {
	int n; cin >> n;
	string s, t; cin >> s >> t;
	int cnt0 = count(all(s), '0'), cnt1 = n - cnt0;
	for (int i = 0; i < n - 1; i++) {
		if (cnt0 == 0 || cnt1 == 0) {
			cout << "NO" << '\n';
			return;
		}
		if (t[i] == '1') cnt0--;
		else cnt1--;
	}
	cout << "YES" << '\n';
}

int main() {
	int T; cin >> T;
	while (T--) {
		solve();
	}
}

for _ in range(int(input())):
    n = int(input())
    s = input()
    one = s.count("1")
    zero = s.count("0")
    ans = "YES"
    for ti in input():
        if one == 0 or zero == 0:
            ans = "NO"
            break
        one -= 1
        zero -= 1
        if ti == "1":
            one += 1
        else:
            zero += 1
    print(ans)

• Amazing problem:
• Good problem:
• Average problem:
• Bad problem:
• Didn't solve:

#include <bits/stdc++.h>
#define all(s) s.begin(), s.end()
using namespace std;
using ll = long long;
using ull = unsigned long long;

const int _N = 1e5 + 5;

int T;

void solve() {
	int n; cin >> n;
	vector<int> a(n + 1);
	for (int i = 1; i <= n; i++) cin >> a[i];
	vector<int> pre(n + 1);
	int curf = 0;
	for (int i = 1; i <= n; i++) {
		if (curf < a[i]) curf++;
		else if (curf > a[i]) curf--;
		pre[i] = max(pre[i - 1], curf);
	}
	auto check = [&](int k) {
		int curg = k;
		for (int i = n; i >= 1; i--) {
			if (pre[i - 1] >= curg) return true;
			if (a[i] < curg) curg++;
			else curg--;
		}
		return false;
	};
	int L = 0, R = n + 1;
	while (L < R) {
		int mid = (L + R + 1) >> 1;
		if (check(mid)) L = mid;
		else R = mid - 1;
	}
	cout << L << '\n';
	return;
}

int main() {
	ios::sync_with_stdio(false), cin.tie(0), cout.tie(0);
	cin >> T;
	while (T--) {
		solve();
	}
}

for _ in range(int(input())):
    n = int(input())
    
    def f(a, x):
        return a + (a < x) - (a > x)
    
    dp = [0, -n, -n]
    for x in map(int, input().split()):
        dp[2] = max(f(dp[1], x), f(dp[2], x))
        dp[1] = max(dp[1], dp[0])
        dp[0] = f(dp[0], x)

    print(max(dp[1], dp[2]))

• Amazing problem:
• Good problem:
• Average problem:
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• Didn't solve:

Try to make the graph into a forest first.

($$$deg_i\le 1$$$ for every $$$i$$$) $$$\Longrightarrow$$$ (The graph is a forest).

Let $$$d_i$$$ be the degree of vertex $$$i$$$.

First, we keep doing the following until it is impossible:

• Choose a vertex $$$u$$$ with $$$d_u\ge 2$$$, then find any two vertices $$$v,w$$$ adjacent to $$$u$$$. Perform the operation on $$$(u,v,w)$$$.

Since each operation decreases the number of edges by at least $$$1$$$, at most $$$m$$$ operations will be performed. After these operations, $$$d_i\le 1$$$ holds for every $$$i$$$. Thus, the resulting graph consists only of components with size $$$\le 2$$$.

If there are no edges, the graph is already cool, and we don't need to do any more operations.

Otherwise, let's pick an arbitrary edge $$$(u,v)$$$ as the base of the final tree, and then merge everything else to it.

For a component with size $$$=1$$$ (i.e. it is a single vertex $$$w$$$), perform the operation on $$$(u, v, w)$$$, and set $$$(u, v) \gets (u, w)$$$.

For a component with size $$$=2$$$ (i.e. it is an edge connecting $$$a$$$ and $$$b$$$), perform the operation on $$$(u, a, b)$$$.

It is clear that the graph is transformed into a tree now.

In the author's solution, we used some data structures to maintain the edges, thus, the time complexity is $$$\mathcal{O}(n+m\log m)$$$ per test case.

#include <bits/stdc++.h>
using namespace std;
#ifdef DEBUG
#include "debug.hpp"
#else
#define debug(...) (void)0
#endif

using i64 = int64_t;
constexpr bool test = false;

int main() {
  cin.tie(nullptr)->sync_with_stdio(false);
  int t;
  cin >> t;
  for (int ti = 0; ti < t; ti += 1) {
    int n, m;
    cin >> n >> m;
    vector<set<int>> adj(n + 1);
    for (int i = 0, u, v; i < m; i += 1) {
      cin >> u >> v;
      adj[u].insert(v);
      adj[v].insert(u);
    }
    vector<tuple<int, int, int>> ans;
    for (int i = 1; i <= n; i += 1) {
      while (adj[i].size() >= 2) {
        int u = *adj[i].begin();
        adj[i].erase(adj[i].begin());
        int v = *adj[i].begin();
        adj[i].erase(adj[i].begin());
        adj[u].erase(i);
        adj[v].erase(i);
        ans.emplace_back(i, u, v);
        if (adj[u].contains(v)) {
          adj[u].erase(v);
          adj[v].erase(u);
        } else {
          adj[u].insert(v);
          adj[v].insert(u);
        }
      }
    }
    vector<int> s;
    vector<pair<int, int>> p;
    for (int i = 1; i <= n; i += 1) {
      if (adj[i].size() == 0) {
        s.push_back(i);
      } else if (*adj[i].begin() > i) {
        p.emplace_back(i, *adj[i].begin());
      }
    }
    if (not p.empty()) {
      auto [x, y] = p.back();
      p.pop_back();
      for (int u : s) {
        ans.emplace_back(x, y, u);
        tie(x, y) = pair(x, u);
      }
      for (auto [u, v] : p) {
        ans.emplace_back(y, u, v);
      }
    }
    println("{}", ans.size());
    for (auto [x, y, z] : ans) println("{} {} {}", x, y, z);
  }
}

• Amazing problem:
• Good problem:
• Average problem:
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$$$2$$$ is powerful.

Consider primes.

How did you prove that $$$2$$$ can generate every integer except odd primes? Can you generalize it?

In this problem, we do not take the integer $$$1$$$ into consideration.

Claim 1. $$$2$$$ can generate every integer except odd primes.

Proof. For a certain non-prime $$$x$$$, let $$$\operatorname{mind}(x)$$$ be the minimum divisor of $$$x$$$. Then $$$x-\operatorname{mind}(x)$$$ must be an even number, which is $$$\ge 2$$$. So $$$x-\operatorname{mind}(x)$$$ can be generated by $$$2$$$, and $$$x$$$ can be generated by $$$x-\operatorname{mind}(x)$$$. Thus, $$$2$$$ is a generator of $$$x$$$.

Claim 2. Primes can only generated by themselves.

According to the above two claims, we can first check if there exist primes in the array $$$a$$$. If not, then $$$2$$$ is a common generator. Otherwise, let the prime be $$$p$$$, the only possible generator should be $$$p$$$ itself. So we only need to check whether $$$p$$$ is a generator of the rest integers.

For an even integer $$$x$$$, it is easy to see that, $$$p$$$ is a generator of $$$x$$$ if and only if $$$x\ge 2\cdot p$$$.

Claim 3. For a prime $$$p$$$ and an odd integer $$$x$$$, $$$p$$$ is a generator of $$$x$$$ if and only if $$$x - \operatorname{mind}(x)\ge 2\cdot p$$$.

Proof. First, $$$x-\operatorname{mind}(x)$$$ is the largest integer other than $$$x$$$ itself that can generate $$$x$$$. Moreover, only even numbers $$$\ge 2\cdot p$$$ can be generated by $$$p$$$ ($$$x-\operatorname{mind}(x)$$$ is even). That ends the proof.

Thus, we have found a good way to check if a certain number can be generated from $$$p$$$. We can use the linear sieve to pre-calculate all the $$$\operatorname{mind}(i)$$$-s.

Time complexity: $$$\mathcal{O}(\sum n+V)$$$, where $$$V=\max a_i$$$.

Some other solutions with worse time complexity can also pass, such as $$$\mathcal{O}(V\log V)$$$ and $$$\mathcal{O}(t\sqrt{V})$$$.

#include <bits/stdc++.h>
#define all(s) s.begin(), s.end()
using namespace std;
using ll = long long;
using ull = unsigned long long;

const int _N = 4e5 + 5;

int vis[_N], pr[_N], cnt = 0;

void init(int n) {
	vis[1] = 1;
	for (int i = 2; i <= n; i++) {
		if (!vis[i]) {
			pr[++cnt] = i;
		}
		for (int j = 1; j <= cnt && i * pr[j] <= n; j++) {
			vis[i * pr[j]] = pr[j];
			if (i % pr[j] == 0) continue;
		}
	}
}

int T;

void solve() {
	int n; cin >> n;
	vector<int> a(n + 1);
	for (int i = 1; i <= n; i++) cin >> a[i];
	int p = 0;
	for (int i = 1; i <= n; i++) {
		if (!vis[a[i]]) p = a[i];
	}
	if (!p) {
		cout << 2 << '\n';
		return;
	}
	for (int i = 1; i <= n; i++) {
		if (a[i] == p) continue;
		if (vis[a[i]] == 0) {
			cout << -1 << '\n';
			return;
		}
		if (a[i] & 1) {
			if (a[i] - vis[a[i]] < 2 * p) {
				cout << -1 << '\n';
				return;
			}
		} else {
			if (a[i] < 2 * p) {
				cout << -1 << '\n';
				return;
			}
		}
	}
	cout << p << '\n';
	return;
}

int main() {
	ios::sync_with_stdio(false), cin.tie(0), cout.tie(0);
	init(400000);
	cin >> T;
	while (T--) {
		solve();
	}
}

• Amazing problem:
• Good problem:
• Average problem:
• Bad problem:
• Didn't solve:

If there are both consecutive $$$\texttt{R}$$$-s and $$$\texttt{B}$$$-s, does the condition hold for all $$$(i,j)$$$? Why?

Suppose that there are only consecutive $$$\texttt{R}$$$-s, check the parity of the number of $$$\texttt{R}$$$-s in each consecutive segment of $$$\texttt{R}$$$-s.

If for each consecutive segment of $$$\texttt{R}$$$-s, the parity of the number of $$$\texttt{R}$$$-s is odd, does the condition hold for all $$$(i,j)$$$? Why?

If for at least two of the consecutive segments of $$$\texttt{R}$$$-s, the parity of the number of $$$\texttt{R}$$$-s is even, does the condition hold for all $$$(i,j)$$$? Why?

Is this the necessary and sufficient condition? Why?

Don't forget some trivial cases like $$$\texttt{RRR...RB}$$$ and $$$\texttt{RRR...R}$$$.

#include <bits/stdc++.h>
using namespace std;
void solve(){
	int n; cin>>n;
	string s; cin>>s;
	int visr=0,visb=0,ok=0; 
	for(int i=0;i<n;i++){
		if(s[i]==s[(i+1)%n]){
			if(s[i]=='R') visr=1;
			else visb=1;
			ok++;
		}
	}
	if(visr&visb){
		cout<<"NO\n";
		return ;
	}
	if(ok==n){
		cout<<"YES\n";
		return ;
	}
	if(visb) for(int i=0;i<n;i++) s[i]='R'+'B'-s[i];
	int st=0;
	for(int i=0;i<n;i++) if(s[i]=='B') st=(i+1)%n;
	vector<int> vc;
	int ntot=0,cnt=0;
	for(int i=0,j=st;i<n;i++,j=(j+1)%n){
		if(s[j]=='B') vc.push_back(ntot),cnt+=(ntot&1)^1,ntot=0;
		else ntot++;
	}
	if(vc.size()==1||cnt==1){
		cout<<"YES\n";
		return ;
	}
	cout<<"NO\n";
	return ;
}
signed main(){
	int t; cin>>t;
	while(t--) solve();
	return 0;
}

• Amazing problem:
• Good problem:
• Average problem:
• Bad problem:
• Didn't solve:

Coming soon.

Coming soon.

• Amazing problem:
• Good problem:
• Average problem:
• Bad problem:
• Didn't solve:

It's hard to calculate the expected number. Try to change it to the probability.

Consider a $$$\mathcal{O}(3^n)$$$ dp first.

Use inclusion and exclusion.

Write out the transformation. Try to optimize it.

Let $$$dp_S$$$ be the probability such that exactly the points in $$$S$$$ has the message.

The answer is $$$\sum dp_S\cdot tr_S$$$, where $$$tr_S$$$ is the expected number of days before at least one vertex out of $$$S$$$ to have the message (counting from the day that exactly the points in $$$S$$$ has the message).

For transformation, enumerate $$$S,T$$$ such that $$$S\cap T=\varnothing$$$. Transfer $$$dp_S$$$ to $$$dp_{S\cup T}$$$. It's easy to precalculate the coefficient, and the time complexity differs from $$$\mathcal{O}(3^n)$$$, $$$\mathcal{O}(n\cdot 3^n)$$$ to $$$\mathcal{O}(n^2\cdot 3^n)$$$, according to the implementation.

The transformation is hard to optimize. Enumerate $$$T$$$ and calculate the probability such that vertices in $$$S$$$ is only connected with vertices in $$$T$$$, which means that the real status $$$R$$$ satisfies $$$S\subseteq R\subseteq (S\cup T)$$$. Use inclusion and exclusion to calculate the real probability.

List out the coefficient:

(Note that $$$w_e$$$ denotes the probability of the appearance of the edge $$$e$$$)

We can express it as $$$\mathrm{Const}\cdot f_{S\cup T}\cdot g_S\cdot h_T$$$. Use a subset convolution to optimize it. The total time complexity is $$$\mathcal{O}(2^n\cdot n^2)$$$.

It's easy to see that all $$$dp_S\neq0$$$ satisfies $$$1\in S$$$, so the time complexity can be $$$\mathcal{O}(2^{n-1}\cdot n^2)$$$ if well implemented.

#include <bits/stdc++.h>
#define int long long 
using namespace std;
const int N=(1<<21),mod=998244353;
const int Lim=8e18;
inline void add(signed &i,int j){
	i+=j;
	if(i>=mod) i-=mod;
}
int qp(int a,int b){
	int ans=1;
	while(b){
		if(b&1) (ans*=a)%=mod;
		(a*=a)%=mod;
		b>>=1;
	}
	return ans;
}
int dp[N],f[N],g[N],h[N];
int s1[N],s2[N];
signed pre[22][N/2],t[22][N/2],pdp[N][22];
int p[N],q[N],totp,totq;
signed main(){
	int n,m; cin>>n>>m;
	int totprod=1;
	for(int i=0;i<(1<<n);i++) s1[i]=s2[i]=1;
	for(int i=1;i<=m;i++){
		int u,v,p,q; cin>>u>>v>>p>>q;
		int w=p*qp(q,mod-2)%mod;
		(s1[(1<<(u-1))+(1<<(v-1))]*=(mod+1-w))%=mod;
		(s2[(1<<(u-1))+(1<<(v-1))]*=qp(mod+1-w,mod-2))%=mod;
		(totprod*=(mod+1-w))%=mod;
	}
	for(int j=1;j<=n;j++) for(int i=0;i<(1<<n);i++) if((i>>(j-1))&1) (s1[i]*=s1[i^(1<<(j-1))])%=mod,(s2[i]*=s2[i^(1<<(j-1))])%=mod;
	for(int i=0;i<(1<<n);i++) f[i]=s2[i],g[i]=totprod*s2[((1<<n)-1)^i]%mod*qp(mod+1-totprod*s2[i]%mod*s2[((1<<n)-1)^i]%mod,mod-2)%mod,h[i]=s1[i];
	for(int i=1;i<(1<<n);i++) pre[__builtin_popcount(i)][i>>1]=h[i];
	dp[1]=1;
	for(int j=1;j<=n;j++){
		if(!((1>>(j-1))&1)) add(pdp[1|(1<<(j-1))][j],mod-(dp[1]*g[1]%mod*f[1]%mod));
	}
	t[0][0]=dp[1]*g[1]%mod;
	for(int k=1;k<=n;k++) for(int j=1;j<n;j++) for(int i=0;i<(1<<(n-1));i++) if((i>>(j-1))&1) add(pre[k][i],pre[k][i^(1<<(j-1))]);
	for(int j=1;j<n;j++) for(int i=0;i<(1<<(n-1));i++) if((i>>(j-1))&1) add(t[0][i],t[0][i^(1<<(j-1))]);
	for(int k=1;k<n;k++){
		totp=totq=0;
		for(int i=0;i<(1<<(n-1));i++) if(__builtin_popcount(i)<=k) p[++totp]=i; else q[++totq]=i;
		for(int l=1,i=p[l];l<=totp;l++,i=p[l]) for(int j=0;j<k;j++) add(t[k][i],1ll*t[j][i]*pre[k-j][i]%mod);
		for(int i=0;i<(1<<(n-1));i++) t[k][i]%=mod;
		for(int j=1;j<n;j++) for(int l=1,i=p[l];l<=totp;l++,i=p[l]) if((i>>(j-1))&1) add(t[k][i],mod-t[k][i^(1<<(j-1))]);
		for(int i=0;i<(1<<(n-1));i++){
			if(__builtin_popcount(i)==k){
				add(pdp[(i<<1)|1][0],t[k][i]*f[(i<<1)|1]%mod);
				int pre=0;
				for(int j=1;j<=n;j++){
					(pre+=pdp[(i<<1)|1][j-1])%=mod;
					if(!((((i<<1)|1)>>(j-1))&1)) add(pdp[(i<<1)|1|(1<<(j-1))][j],mod-pre);
				}
				(pre+=pdp[(i<<1)|1][n])%=mod;
				dp[(i<<1)|1]=pre;
				for(int j=1;j<=n;j++){
					if(!((((i<<1)|1)>>(j-1))&1)) add(pdp[(i<<1)|1|(1<<(j-1))][j],mod-(dp[(i<<1)|1]*g[(i<<1)|1]%mod*f[(i<<1)|1]%mod));
				}
				t[k][i]=pre*g[(i<<1)|1]%mod;
			}
			else t[k][i]=0;
		}
		for(int j=1;j<n;j++) for(int l=1,i=q[l];l<=totq;l++,i=q[l]) if((i>>(j-1))&1) add(t[k][i],t[k][i^(1<<(j-1))]);
	}
	int ans=0;
	for(int i=1;i<(1<<n)-1;i+=2) (ans+=dp[i]*qp(mod+1-totprod*s2[i]%mod*s2[((1<<n)-1)^i]%mod,mod-2)%mod)%=mod;
	cout<<ans;
	return 0;
}

• Amazing problem:
• Good problem:
• Average problem:
• Bad problem:
• Didn't solve:

The intended solution has nothing to do with dynamic programming.

This is the key observation to the problem. Suppose we have an array $$$b$$$ and a function $$$f(x)=\sum (b_i-x)^2$$$, then the minimum value of $$$f(x)$$$ is the variance of $$$b$$$.

Using the observation mentioned in Hint 2, we can reduce the problem to minimizing $$$\sum(a_i-x)^2$$$ for a given $$$x$$$. There are only $$$\mathcal{O}(nm)$$$ possible $$$x$$$-s.

The rest of the problem is way easy. Try flows, or just find a greedy algorithm!

If you are trying flows: the quadratic function is always convex.

Coming soon.

#include <bits/stdc++.h>
#define all(s) s.begin(), s.end()
using namespace std;
using ll = long long;
using ull = unsigned long long;

const int _N = 1e5 + 5;

int T;

void solve() {
	ll n, m, k; cin >> n >> m >> k;
	vector<ll> a(n + 1);
	for (int i = 1; i <= n; i++) cin >> a[i];
	vector<ll> kans(m + 1, LLONG_MAX);
	vector<__int128> f(n + 1), g(n + 1), v(n + 1);
	vector<int> vis(n + 1), L(n + 1), L2(n + 1);
	ll sum = 0;
	for (int i = 1; i <= n; i++) sum += a[i];
	__int128 pans = 0;
	for (int i = 1; i <= n; i++) pans += n * a[i] * a[i];
	auto work = [&](ll s) {
		__int128 ans = pans;
		ans += s * s - 2ll * sum * s;
		f.assign(n + 1, LLONG_MAX);
		g.assign(n + 1, LLONG_MAX);
		for (int i = 1; i <= n; i++) {
			v[i] = n * (2 * a[i] * k + k * k) - 2ll * s * k;
			vis[i] = 0;
		}
		for (int i = 1; i <= m; i++) {
			for (int j = 1; j <= n; j++) {
				L[j] = L2[j] = j;
				if (f[j - 1] < 0) f[j] = f[j - 1] + v[j], L[j] = L[j - 1];
				else f[j] = v[j];
				if (!vis[j]) {
					g[j] = LLONG_MAX;
					continue;
				}
				if (g[j - 1] < 0) g[j] = g[j - 1] + 2ll * n * k * k - v[j], L2[j] = L2[j - 1];
				else g[j] = 2ll * n * k * k - v[j];
			}
			__int128 min_sum = LLONG_MAX;
			int l = 1, r = n, type = 0;
			for (int j = 1; j <= n; j++) {
				if (f[j] < min_sum) {
					min_sum = f[j], r = j, l = L[j];
				}
			}
			for (int j = 1; j <= n; j++) {
				if (g[j] < min_sum) {
					min_sum = g[j], r = j, l = L2[j];
					type = 1;
				}
			}
			ans += min_sum;
			if (type == 0) {
				for (int j = l; j <= r; j++) vis[j]++, v[j] += 2 * n * k * k;
			} else {
				for (int j = l; j <= r; j++) vis[j]--, v[j] -= 2 * n * k * k;
			}
			kans[i] = min((__int128)kans[i], ans);
		}
	};
	
	for (ll x = sum; x <= sum + n * m * k; x += k) {
		work(x);
	}
	for (int i = 1; i <= m; i++) cout << kans[i] << " \n"[i == m];
	return;
}

int main() {
	ios::sync_with_stdio(false), cin.tie(0), cout.tie(0);
	cin >> T;
	while (T--) {
		solve();
	}
}

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