## Intuition The problem revolves around a game played on an array where two players take turns removing elements under specific constraints. The key insight is to determine if there exists a first move for Arseniy that guarantees Egor a winning strategy, regardless of Dabir's responses. To ensure Egor wins, Arseniy must make a move such that Dabir cannot continue the game after Egor's optimal play. This means that after Arseniy's move, Egor should have a strategy to leave Dabir without any valid moves. ## Approach 1. **Sort the Array**: Start by sorting the array. This helps in easily checking the conditions for valid moves. 2. **Check for Valid First Move**: For each possible first move (element `a[i]`), check if Egor can make a move such that Dabir has no valid moves left: - After Arseniy removes `a[i]`, Egor should be able to remove an element `a[j]` such that `0 <= a[j] - a[i] <= k`. - After Egor's move, ensure that there is no element `a[m]` such that `0 <= a[m] - a[j] <= k` for Dabir to play. 3. **Determine Winning Strategy**: If there exists any `a[i]` such that Egor can make a move and leave Dabir without options, output "YES". Otherwise, output "NO". ## Complexity - **Time Complexity**: Sorting the array takes `O(n log n)`, and checking each possible first move takes `O(n)`. Thus, the overall complexity per test case is `O(n log n)`. - **Space Complexity**: The space complexity is `O(n)` due to sorting. ## Solution (C++) ```cpp #include using namespace std; int main() { ios::sync_with_stdio(false); cin.tie(nullptr); int t; cin >> t; while (t--) { int n, k; cin >> n >> k; vector a(n); for (int i = 0; i < n; ++i) { cin >> a[i]; } sort(a.begin(), a.end()); bool canWin = false; for (int i = 0; i < n - 1; ++i) { // Check if Egor can make a move after Arseniy's first move if (a[i + 1] - a[i] <= k) { canWin = true; break; } } if (canWin) { cout << "YES\n"; } else { cout << "NO\n"; } } return 0; } ``` This solution efficiently determines whether Arseniy can make a first move that guarantees Egor a winning strategy by leveraging the sorted order of the array to quickly check valid moves.

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