...

#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
 
const ll M = 1e9+7;
const ll N = 2e6+7;
ll fact[N], invFact[N];
 
ll bigMod(ll base, ll power) {
    if (!power) return 1;
    ll result = bigMod(base, power/2);
    result = (result * result) % M;
    if (power & 1) result = (result * base) % M;
    return result;
}
 
signed main()
{
    ios_base::sync_with_stdio(0); cin.tie(0);
    ll a, b, p, x; cin >> a >> b >> p >> x;
 
    fact[0] = 1;
    for (ll i = 1; i < N; i++) fact[i] = (fact[i-1] * i) % M;
    
    invFact[N-1] = bigMod(fact[N-1], M-2);
    for (ll i = N-2; i >= 0; i--) invFact[i] = (invFact[i+1] * (i+1)) % M;
 
    ll ans = 0;
    for (ll i = x; i <= min(p, a); i++) {
        ll firstTeam = (fact[a] * ((invFact[i]*invFact[a-i]) % M)) % M; // aCi
        if (b-p+i < 0) continue;
        ll secondTeam = (fact[b] * ((invFact[p-i]*invFact[b-p+i]) % M)) % M; // bC(p-i)
        ll temp = (firstTeam*secondTeam) % M;
        ans = (ans+temp) % M;
    }
 
    cout << ans;
}

This is a Bisection problem. For time $$$t$$$ second, $$$i^{th}$$$ student can be anywhere within segment $$$[x_{i}-v_{i}t , x_{i}-v_{i}t]$$$. In the binary search we need to check within $$$t$$$ seconds whether there is a common point among all the segments.

For all segments $$$[l_{1}, r_{1}] ,..., [l_{n}, r_{n}]$$$, we can calculate min $$$r_{i}$$$ and max $$$l_{i}$$$. If $$$r_{i} < l_{i}$$$, there is no intersection. Otherwise $$$[max l_{i}, min r_{i}]$$$ is the intersection.

#include <bits/stdc++.h>
using namespace std;

// Function to check if all students can meet at a common point within time T
bool canMeet(double T, const vector<int>& positions, const vector<int>& speeds) {
    double min_meet = -1e9;
    double max_meet = 1e9;
    
    int n = positions.size();
    for (int i = 0; i < n; ++i) {
        double left_bound = positions[i] - speeds[i] * T;
        double right_bound = positions[i] + speeds[i] * T;
        min_meet = max(min_meet, left_bound);
        max_meet = min(max_meet, right_bound);
        
        if (min_meet > max_meet) {
            return false;
        }
    }
    return true;
}

int main() {
    ios_base::sync_with_stdio(0); cin.tie(0);
    int n;
    cin >> n;
    
    vector<int> positions(n);
    vector<int> speeds(n);
    
    for (int i = 0; i < n; ++i) {
        cin >> positions[i];
    }
    for (int i = 0; i < n; ++i) {
        cin >> speeds[i];
    }
    
    double left = 0;
    double right = 1e9;
    double precision = 1e-7;
    
    while (right - left > precision) {
        double mid = left + (right - left) / 2;
        if (canMeet(mid, positions, speeds)) {
            right = mid;
        } else {
            left = mid;
        }
    }
    
    cout << fixed << setprecision(10) << left;
}

...

#include<bits/stdc++.h>
using namespace std;           
 
int main(){
    ios_base::sync_with_stdio(false);cin.tie(NULL);
    int n;
    cin>>n;
    string s;
    cin>>s;
    int mx = 2;
    for(int i=0; i<s.size(); i++){
    	if(s[i]>='0' and s[i]<='9')mx = max(mx,s[i]-'0'+1);
    	else mx = max(mx, s[i]-'A'+11);
    }
    cout<<mx<<endl;
}

...

#include<bits/stdc++.h>
using namespace std;
int main(){
    int n;
    cin>>n;
    vector<int>chocolate(n);
    int total=0;
    for(int i=0;i<n;i++){
        cin>>chocolate[i];
    }
    for(int i=0;i<n;i++)total+=chocolate[i];
    vector<int>allpossiblesum(total+1,0);
    allpossiblesum[0]=1;
    for(int i=0;i<n;i++){
        for(int j=total;j>=chocolate[i];j--){
            allpossiblesum[j]|=allpossiblesum[j-chocolate[i]];
        }
    }
    int ans=total;
    for(int i=0;i<=total;i++){
        if(allpossiblesum[i]>0){
            int alice=i;
            int bob=total-alice;
            ans=min(ans,abs(alice-bob));
        }
    }
    cout<<ans<<endl;
}

...

#include<bits/stdc++.h>
using namespace std;
typedef long long ll;

int main()
{
    ios_base::sync_with_stdio(false);cin.tie(0);cout.tie(0);
    int t; cin >> t;
    while (t--) {
        int n, k;
        cin >> n >> k;
        vector<ll> ar(n);
        for (int i = 0; i < n; i++) cin >> ar[i];

        ll xor_sum = 0;
        for (int i = 1; i < n; i += 2) xor_sum ^= ar[i];

        if (xor_sum == 0) cout << "NO" << endl;
        else cout << "YES" << endl;
    }
}

...

#include<bits/stdc++.h>
using namespace std;
 
int main(){
    ios_base::sync_with_stdio(false);cin.tie(NULL);
    int n;
    cin>>n;
    string s, p;
    cin >> s >> p;
    s="#"+s;
    p="#"+p;
    
    int pre_s[n+2][26];
    int pre_p[n+2][26];
    mem(pre_s, 0);
    mem(pre_p, 0);
    
    for(int i=1; i<=n; i++){
    	pre_s[i][s[i]-'a']=1;
    	pre_p[i][p[i]-'a']=1;
    	
    	for(int j=0; j<26; j++){
    		pre_s[i][j] += pre_s[i-1][j];
    		pre_p[i][j] += pre_p[i-1][j];
    	}
    }
    
    int q;
    cin>>q;
    while(q--){
    	int l, r;
    	cin>>l>>r;
    	int ans = 0;
    	for(int i=0; i<26; i++){
    		ans += abs((pre_s[r][i] - pre_s[l-1][i]) - (pre_p[r][i] - pre_p[l-1][i]));
    	}
    	cout<<ans<<endl;
    }
}

...

#include<bits/stdc++.h>
using namespace std;         
#define mxn 2000000
ll chk[mxn+5];

void sieve(){
    for(int i=2; i<mxn; i++){
	if(chk[i]==0){
	    for(int j=i+i; j<mxn; j+=i){
		chk[j] = 1;
	    }
	}
    }
}

int main(){
    ios_base::sync_with_stdio(false);cin.tie(NULL);
    sieve();
    ll n;
    cin>>n;
    ll ans = 1000000000000000ll;
    for(ll i=2; i<mxn; i++){
    	if(chk[i]) continue;
    	ll x = i*i;
    	ans = min(ans,abs(x-n));
    }

    cout<<ans<<endl;
}

...

#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
 
signed main()
{
    ios_base::sync_with_stdio(0); cin.tie(0);
    ll n; cin >> n;
    if (n == 0 or n == 2) cout << "Same";
    else if (n == 1) cout << "Double";
    else cout << "Square";
}

...

#include<bits/stdc++.h>
using namespace std;
#define AS 250000
int i,j,k,l,m,n,p,q,r,a,b,c,u,v,x,y,z,ts=1;
int ar[AS],discover[AS],vis[AS],low[AS],br[AS];
vector<int> graph[AS];
set<int>st[AS];
vector<pair<int,int>>bridge;
void dfs(int nod,int par=-1) {
    discover[nod]=low[nod]=z++;
    vis[nod]=1;
    for(auto i:graph[nod]) {
        if(i==par)continue;
        if(vis[i])low[nod]=min(low[nod],discover[i]);
        else {
            dfs(i,nod);
            low[nod]=min(low[nod],low[i]);
 
            if(low[i]>discover[nod]) {
                bridge.push_back({nod,i});
                st[nod].erase(i);
                st[i].erase(nod);
            }
        }
    }
}
 
int component_count;
void biparted(int nod,int col)
{
    component_count++;
    vis[nod]=2;
    ar[nod]=col;
    for(auto i:st[nod]) {
        if(vis[i]==1)biparted(i,!col);
        else if(ar[nod]==ar[i])c=1;
    }
}
int main() {
    cin>>n>>m;
    for(int i=1;i<=m;i++) {
        cin>>u>>v;
        graph[u].push_back(v);
        graph[v].push_back(u);
        st[u].insert(v);
        st[v].insert(u);
    }
    c=0;
    for(int i=1;i<=n;i++) {
        if(vis[i]==0) {
            dfs(i,-1);
        }
    }
    c=0;
    int ans=0;
    for(int i=1;i<=n;i++) {
        if(vis[i]==1) {
            component_count=c=0;
            biparted(i,0);
            if(c)ans+=component_count;
        }
    }
    cout<<ans<<endl;
}