On each turn, the current player gets rid of one of the buttons available to them. At the same time, if you press the common button, the enemy will not be able to press it as well. Since each player wants to leave their opponent without buttons to press before they run out of those themselves, they will click on the common buttons as long as there is at least one available.

The number of the player who clicks on their (not the general) button first depends on $$$c$$$. It can be noticed that this player will win if and only if they have strictly more buttons than their opponent.

t = int(input())
for i in range(t):
    a, b, c = map(int, input().split())
    if c % 2 == 0:
        if a > b:
            print("First")
        else:
            print("Second")
    else:
        if b > a:
            print("Second")
        else:
            print("First")

First, let's calculate how many cookies Petya will eat if we don't remove the cookie sellers at all (we will later refer to this value as $$$res$$$). Note that since the cookie sellers reset the time elapsed since the last eaten cookie, the number of cookies eaten on all segments between adjacent cookie sellers are counted independently. Therefore, we can easily calculate $$$res$$$: let's iterate through the cookie sellers from $$$1$$$ to $$$m-1$$$. For each of them, we should add the number $$$\left( \left\lfloor\frac{s_{i+1}-s_{i}-1}{d}\right\rfloor + 1 \right)$$$ to $$$res$$$. We should also carefully take into account the cookies that Petya will eat on the segments $$$[1;s_{1}-1]$$$ and $$$[s_{m}, n]$$$ (it might help to assume that there are two more cookie sellers at positions $$$1-d$$$ and $$$n+1$$$).

In order to find the minimum possible number of cookies eaten by Petya, we will fix the cookie seller that we remove. Let it be the cookie seller $$$i$$$. Then the number of cookies eaten by Petya will be

(for the first and last cookie sellers, the formulas may differ slightly). Thus, we can find the number of cookies that Petya will eat if a certain cookie seller is removed in $$$O(1)$$$. After going through all $$$m$$$ options, we will be able to find the answer to the problem.

The final complexity of the solution is $$$\mathcal{O}(m)$$$ (because array $$$s$$$ is sorted in the input).

#include <bits/stdc++.h>

using namespace std;

int solve(int d, vector<int> x)
{
    int ans = 0;
    for (int i = 1; i < x.size(); i++)
    {
        ans += (x[i] &mdash; x[i &mdash; 1] &mdash; 1) / d;
    }
    ans += int(x.size()) &mdash; 2;
    return ans;
}

void solve()
{
    #define tests

    int n, m, d;
    cin >> n >> m >> d;
    vector<int> r(m);
    for (int i = 0; i < m; i++) cin >> r[i];
    r.push_back(1 - d);
    r.push_back(n + 1);

    sort(r.begin(), r.end());

    int ans = 2e9;
    vector<int> res;
    for (int i = 1; i <= m; i++)
    {
        int A = r[i] - r[i - 1] - 1;
        int B = r[i + 1] - r[i] - 1;
        int C = r[i + 1] - r[i - 1] - 1;
        int D = C / d - (A / d + B / d);
        if (D < ans)
        {
            ans = D;
            res.clear();
        }
        if (D == ans)
        {
            res.push_back(r[i]);
        }
    }
    cout << ans + solve(d, r) - 1 << ' ' << res.size() << endl;
}

int main()
{
    int t = 1;
    #ifdef tests
    cin >> t;
    #endif
    while (t--)
    {
        solve();
    }
}

It is impossible to get $$$d_i = \gcd(a_i, a_{(i\mod n) + 1}) > \left\lfloor \frac{n}{2}\right\rfloor$$$: otherwise, at least one of the numbers in $$$a$$$ would be divisible by $$$d_i$$$ and would be greater than $$$d_i$$$ at the same time, so it would be at least $$$2 \cdot d_i$$$, which is greater than $$$n$$$. Therefore, the maximum possible score is no more than $$$\left\lfloor\frac{n}{2}\right\rfloor$$$. Actually, we can always get a score equal to $$$\left\lfloor\frac{n}{2}\right\rfloor$$$.

How do we get such score? Let's set $$$a_1= 1$$$. After that, we put the powers of $$$2$$$ less or equal $$$n$$$ sequentially. Then we put $$$3$$$ and powers of $$$2$$$ multiplied by $$$3$$$, then $$$5$$$ and so on (for example, for $$$n=12$$$, we will get an array $$$a=[1,2,4,8,3,6,12,5,10,7,9,11]$$$). Then, for each number $$$a_i = x \leq \lfloor \frac{n}{2} \rfloor$$$, the next number will be $$$a_{(i\bmod n) + 1} = x \cdot 2 \leq n$$$. Their $$$\gcd$$$ will be exactly $$$x$$$, so there will be a pair of adjacent elements of $$$a$$$ with greatest common divisor equal to $$$x$$$ for all $$$1 \leq x \leq \lfloor\frac{n}{2}\rfloor$$$.

#include<iostream>
#include<vector>

using namespace std;

int main() {
    int t;
    cin >> t;
    while (t--) {
        int n;
        cin >> n;
        vector<int> a(n);
        int cur = 0;
        for (int i = 1; i <= n; i += 2) {
            for (int j = i; j <= n; j *= 2) {
                a[cur++] = j;
            }
        }
        for (int i = 0; i<n; ++i) {
            cout << a[i] << " ";
        }
        cout << '\n';
    }
    return 0;
}

There are many various dynamic programming solutions of this problem. We will describe one of them. Let's calculate the dynamics $$$pref_{i, \ j}$$$ = the length of the longest subsegment of zeros that can be obtained on the prefix up to $$$i$$$, which ends at index $$$i$$$ and costs exactly $$$j$$$ operations. Similarly, $$$suf_{i, \ j}$$$ is the length of the longest subsegment of zeros on the suffix starting at $$$i$$$, which starts at index $$$i$$$ and costs exactly $$$j$$$ operations. Such dynamics can be easily computed:

In the first case, we simply prolong the existing subsegment of zeros, in the second case, we change the current $$$1$$$ to $$$0$$$, spending one operation on it (so if have $$$0$$$ operations left ($$$j = 0$$$), we cannot do anything and the value of the dynamics is $$$0$$$, meaning the segment has ended). $$$suf_{i,j}$$$ can be calculated similarly. Let's update both dynamics in such a way that $$$pref_{i,j}$$$ will mean the maximum length of a subsegment of zeros that ends no later than $$$i$$$ and costs no more than $$$j$$$ operations. This can be easily done by updating $$$pref_{i,j}$$$ with the value of $$$pref_{i - 1,j}$$$, and then with $$$pref_{i,j - 1}$$$. Similarly, we update the second dynamics.

Now let's consider a subsegment [$l, \ r]$ that we want to convert into a segment of ones. We can easily calculate the number of operations $$$x$$$ that we will need (we'll just need to calculate the number of zeros in such a segment). Now, calculate the new dynamics $$$dp_{len}$$$ for the length $$$len = r - l + 1$$$ of the segment of ones, which equals the maximum length of a subsegment of zeros that we can obtain. Update this value with $$$\max(pref_{l - 1, k - x}, suf_{r + 1, k - x})$$$.

Then, to answer the question for a fixed number $$$a$$$, we can iterate over the length $$$len$$$ of the segment of ones that will be in our answer and update the answer with the value $$$a \cdot dp_{len} + len$$$, if there exists a value for $$$len$$$ in the dynamics $$$dp$$$.

The complexity is $$$O(nk + n^2)$$$. Solutions with complexity $$$O(nk \log n)$$$ and $$$O(nk)$$$ using various optimizations of the dynamics (\textit{convex hull trick, D&Q}) also exist.

#include <bits/stdc++.h>

#define int long long

using namespace std;
using ll = long long;

void solve();

template<typename ...Args>
void println(Args... args) {
    apply([](auto &&... args) { ((cout << args << ' '), ...); }, tuple(args...));
    cout << '\n';
}

int32_t main() {
    cin.tie(nullptr);
    ios_base::sync_with_stdio(false);
    int t = 1;
    cin >> t;
    for (int tc = 0; tc < t; ++tc) {
        solve();
    }
    return 0;
}

void solve() {
    int n, k;
    cin >> n >> k;
    string s;
    cin >> s;
    vector<int> max0by1(n + 1, -1e9);
    vector<vector<int>> max0pref(n + 1, vector<int>(n + 1));
    vector<vector<int>> max0suf(n + 1, vector<int>(n + 1));
    for (int l = 0; l < n; ++l) {
        int cnt1 = 0;
        for (int r = l + 1; r <= n; ++r) {
            cnt1 += s[r &mdash; 1] == '1';
            max0pref[r][cnt1] = max(max0pref[r][cnt1], r &mdash; l);
            max0suf[l][cnt1] = max(max0suf[l][cnt1], r &mdash; l);
        }
    }
    for (int r = 0; r <= n; ++r) {
        for (int cnt = 0; cnt <= n; ++cnt) {
            if (r) max0pref[r][cnt] = max(max0pref[r][cnt], max0pref[r &mdash; 1][cnt]);
            if (cnt) max0pref[r][cnt] = max(max0pref[r][cnt], max0pref[r][cnt &mdash; 1]);
        }
    }
    for (int l = n; l >= 0; --l) {
        for (int cnt = 0; cnt <= n; ++cnt) {
            if (l + 1 <= n) max0suf[l][cnt] = max(max0suf[l][cnt], max0suf[l + 1][cnt]);
            if (cnt) max0suf[l][cnt] = max(max0suf[l][cnt], max0suf[l][cnt &mdash; 1]);
        }
    }
    vector<int> ans(n + 1, -1e9);
    for (int l = 0; l < n; ++l) {
        int cnt0 = 0;
        for (int r = l; r <= n; ++r) {
            if (r > l) cnt0 += s[r &mdash; 1] == '0';
            if (cnt0 > k) break;
            max0by1[r &mdash; l] = max(max0by1[r &mdash; l], max0pref[l][k &mdash; cnt0]);
            max0by1[r &mdash; l] = max(max0by1[r &mdash; l], max0suf[r][k &mdash; cnt0]);
        }
    }
    for (int i = 0; i <= n; ++i) {
        for (int a = 1; a <= n; ++a) ans[a] = max(ans[a], i + max0by1[i] * a);
    }
    for (int i = 1; i <= n; ++i) cout << ans[i] << ' ';
    cout << '\n';
}

First, let's learn how to solve the problem without rollbacks. Let $$$b$$$ be an array of the same length as $$$a$$$, where $$$b_i=1$$$ if $$$i$$$ is the minimum position at which the number $$$a_i$$$ is in the array $$$a$$$, and $$$b_i=0$$$ otherwise. Then the number of different numbers in the array $$$a$$$ is equal to the sum of all the elements of the array $$$b$$$.

The $$$b$$$ array can be maintained using a Fenwick tree, a segment tree, or any other data structure that supports point updates and range sum queries. The author's solution uses the Fenwick tree.

Let's use the method that is often used when implementing a stack or a deque: we create a large array $$$A$$$ and at each moment we store the index of the last existing element $$$r$$$ (this is just the size of the array $$$a$$$ at this moment). The array $$$a$$$ itself will be a prefix of $$$A$$$, that is, $$$a_{i}=A_{i}$$$ for $$$i\leq len(a)$$$. Also, for each value $$$val$$$ we maintain std,2023-08-15::set of indexes on which the $$$val$$$ value is located in the $$$A$$$ array (in $$$A$$$, not in $$$a$$$).

Then for the operation of removing $$$k$$$ elements from the end of the array $$$a$$$, it is enough to reduce the value of the index $$$r$$$ by $$$k$$$. This operation works in $$$O(1)$$$.

When adding one element $$$x$$$ to the end of the array, we need to check if it has been encountered before, and (if it has not been encountered) change one element in the Fenwick tree. This can be done in $$$O(\log{q})$$$ using std,2023-08-15::set for the corresponding value. You also need to increase $$$r$$$ by 1, assign $$$A_{r} =x$$$ after that, and update the corresponding std,2023-08-15::set. This operation works for $$$O(\log{q})$$$.

In order to find the number of different numbers in $$$a$$$, we need to find the sum in the Fenwick tree on the prefix of length $$$r$$$ in the array $$$b$$$. This operation works in $$$O(\log{q})$$$.

Now, we need to learn how to roll back operations. Note that we perform the deletion operation in $$$O(1)$$$, and the addition operation in $$$O(\log{q})$$$, so we can roll back these operations with the same asymptotics. We can just store a stack of all changes, and remember everything that we changed during the operations.

The final asymptotics is $$$O(q\log{q})$$$.

#include <iostream>
#include <vector>
#include <numeric>
#include <set>

using namespace std;

const int maxn = 1e6 + 1;

int f[maxn];

int get(int i) {
    int ans = 0;
    while (i >= 0) {
        ans += f[i];
        i = (i & (i + 1)) &mdash; 1;
    }
    return ans;
}

void upd(int i, int x) {
    while (i < maxn) {
        f[i] += x;
        i = i | (i + 1);
    }
}

int a[maxn];
int rev[maxn];
set<int> ids[maxn];

int32_t main() {
    ios_base::sync_with_stdio(false);
    cin.tie(0);
    cout.tie(0);
    fill(rev, rev + maxn, -1);
    fill(a, a + maxn, -1);
    int q;
    cin >> q;
    int ptr = -1;
    vector<pair<pair<int, int>, int>> changes;
    while (q--) {
        char t;
        cin >> t;
        if (t == '?') {
            cout << get(ptr) << endl;
        } else if (t == '+') {
            int x;
            cin >> x;
            int mem = a[ptr + 1];
            if (a[ptr + 1] != -1) {
                if (ids[a[ptr + 1]].size()) {
                    upd(*ids[a[ptr + 1]].begin(), -1);
                    ids[a[ptr + 1]].erase(ptr + 1);
                }
                if (ids[a[ptr + 1]].size()) {
                    upd(*ids[a[ptr + 1]].begin(), 1);
                }
            }
            a[ptr + 1] = x;
            if (a[ptr + 1] != -1) {
                if (ids[a[ptr + 1]].size()) {
                    upd(*ids[a[ptr + 1]].begin(), -1);
                }
                ids[x].insert(ptr + 1);
                if (ids[a[ptr + 1]].size()) {
                    upd(*ids[a[ptr + 1]].begin(), 1);
                }
            }
            ++ptr;
            changes.push_back({ { 1, mem }, -1 });
        } else if (t == '-') {
            int k;
            cin >> k;
            ptr -= k;
            changes.push_back({ { -1, k }, -1 });
        } else {
            if (changes.back().first.first == 1) {
                if (a[ptr] != -1) {
                    if (ids[a[ptr]].size()) {
                        upd(*ids[a[ptr]].begin(), -1);
                        ids[a[ptr]].erase(ptr);
                    }
                    if (ids[a[ptr]].size()) {
                        upd(*ids[a[ptr]].begin(), 1);
                    }
                }
                a[ptr] = changes.back().first.second;
                --ptr;
                if (a[ptr + 1] != -1) {
                    if (ids[a[ptr + 1]].size()) {
                        upd(*ids[a[ptr + 1]].begin(), -1);
                    }
                    ids[a[ptr + 1]].insert(ptr + 1);
                    if (ids[a[ptr + 1]].size()) {
                        upd(*ids[a[ptr + 1]].begin(), 1);
                    }
                }
            } else {
                ptr += changes.back().first.second;
            }
            changes.pop_back();
        }
    }
}