A chessboard with some unit squares marked is called a $\textit{good board}$ if for any pair of rows \((s, t)\), a rook placed on a marked square in row \(s\) can reach a marked square in row \(t\) in several moves by only moving to marked squares above, below, or to the right of its current position. Consider a chessboard with 220 rows and 12 columns, where exactly 9 unit squares in each row are marked. Regardless of how the marked squares are chosen, if it is possible to delete \(k\) columns and rearrange the remaining columns to form a $\textit{good board}$ determine the maximum possible value of \(k\).

Does there exist a sequence of positive real numbers $\{a_i\}_{i=1}^{\infty}$ satisfying: \[ \sum_{i=1}^{n} a_i \geq n^2 \quad \text{and} \quad \sum_{i=1}^{n} a_i^2 \leq n^3 + 2025n \]for all positive integers $n$.

For a positive integer $n$, let $S_n$ be the set of positive integers that do not exceed $n$ and are coprime to $n$. Define $f(n)$ as the smallest positive integer that allows $S_n$ to be partitioned into $f(n)$ disjoint subsets, each forming an arithmetic progression. Prove that there exist infinitely many pairs $(a, b)$ satisfying $a, b > 2025$, $a \mid b$, and $f(a) \nmid f(b)$.

Find all positive integers $n$ such that the number \[ \frac{3 + \sqrt{4n + 9}}{2} \]is the sixth smallest positive divisor of $n$.

In triangle \( ABC \), the incircle is tangent to side \( BC \) at point \( D \), the excircle opposite vertex \( B \) is tangent to line \( AB \) at point \( X \), and the excircle opposite vertex \( C \) is tangent to line \( AC \) at point \( Y \). If \( T \) is the midpoint of segment \( [AD] \) and \( U \) is the circumcenter of triangle \( AXY \), show that \( UT \perp BC \).

In a chess tournament with 200 participants, 700 matches are arranged such that among any 100 participants, the number of matches played between them is at least \( N \). Determine the maximum possible value of \( N \).