HMIC 2024

In an empty $100 \times 100$ grid, 300 cells are colored blue, 3 in each row and each column. Compute the largest positive integer $k$ such that you can always recolor $k$ of these blue cells red so that no contiguous $2 \times 2$ square has four red cells.

Suppose that $a, b, c$, and $d$ are real numbers such that $a+b+c+d=8$. Compute the minimum possible value of

$$20\left(a^{2}+b^{2}+c^{2}+d^{2}\right)-\sum_{\text {sym }} a^{3} b,$$

where the sum is over all 12 symmetric terms.

Let $S$ be a set of nonnegative integers such that

• there exist two elements $a$ and $b$ in $S$ such that $a, b>1$ and $\operatorname{gcd}(a, b)=1$; and
• for any (not necessarily distinct) element $x$ and nonzero element $y$ in $S$, both $x y$ and the remainder when $x$ is divided by $y$ are in $S$.

Prove that $S$ contains every nonnegative integer.

Given a positive integer $n$, let $[n]=\{1,2, \ldots, n\}$.

• Let $a_{n}$ denote the number of functions $f:[n] \rightarrow[n]$ such that $f(f(i)) \geq i$ for all $i$.
• Let $b_{n}$ denote the number of ordered set partitions of [ $n$ ], i.e., the number of ways to pick an integer $k$ and an ordered $k$-tuple of pairwise disjoint nonempty sets $\left(A_{1}, \ldots, A_{k}\right)$ whose union is [ $n$ ].

Prove that $a_{n}=b_{n}$.

Let $A B C$ be an acute, scalene triangle with circumcenter $O$ and symmedian point $K$. Let $X$ be the point on the circumcircle of triangle $B O C$ such that $\angle A X O=90^{\circ}$. Assume that $X \neq K$. The hyperbola passing through $B, C, O, K$, and $X$ intersects the circumcircle of triangle $A B C$ at points $U$ and $V$, distinct from $B$ and $C$. Prove that $U V$ is the perpendicular bisector of $A X$. The symmedian point of triangle $A B C$ is the intersection of the reflections of $B$-median and $C$-median across the angle bisectors of $\angle A B C$ and $\angle A C B$, respectively.