MathArena

Consider a $2025 \times 2025$ grid of unit squares. Matilda wishes to place on the grid some rectangular tiles, possibly of different sizes, such that each side of every tile lies on a grid line and every unit square is covered by at most one tile.
Determine the minimum number of tiles Matilda needs to place so that each row and each column of the grid has exactly one unit square that is not covered by any tile.

Let $\omega$ be a circle with diameter $\overline{A B}$, center $O$, and cyclic quadrilateral $A B C D$ inscribed in it, with $C$ and $D$ on the same side of $\overline{A B}$. Let $A B=20, B C=13, A D=7$. Let $\overleftrightarrow{B C}$ and $\overleftrightarrow{A D}$ intersect at $E$. Let the $E$-excircle of $E C D$ have its center at $L$. Find $O L$.

Let $(X,\omega)$ be a compact Kähler surface (complex dimension $2$), and let $g_\omega$ be the associated Riemannian metric. Let $S(\omega):X\to\mathbb R$ denote the Chern scalar curvature of $\omega$ (so that $\operatorname{scal}(g_\omega)=2S(\omega)$), and assume $S(\omega)(x)>0$ for all $x\in X$. Define
\[
\sys_2(\omega):=\inf\{\operatorname{Mass}_{g_\omega}(Z):\ Z\text{ is an integral }2\text{-cycle with }[Z]\neq 0\in H_2(X;\mathbb Z)\}.
\]
Let $m:=\min_{x\in X} S(\omega)(x)$. Determine the smallest universal constant $C\in\mathbb R$ such that for every such $(X,\omega)$ one has $m\,\sys_2(\omega)\le C$.

Define the function $f$ on positive integers

$$
f(n)= \begin{cases}\frac{n}{2} & \text { if } n \text { is even } \\ n+1 & \text { if } n \text { is odd }\end{cases}
$$

Let $S(n)$ equal the smallest positive integer $k$ such that $f^{k}(n)=1$. How many positive integers satisfy $S(n)=11$ ?

Suppose $n$ integers are placed in a circle such that each of the following conditions is satisfied:
\begin{itemize}
\item at least one of the integers is $0$;
\item each pair of adjacent integers differs by exactly $1$; and
\item the sum of the integers is exactly $24$.
\end{itemize}
Compute the smallest value of $n$ for which this is possible.

Find the sum of the $10$th terms of all arithmetic sequences of integers that have first term equal to $4$ and include both $24$ and $34$ as terms.