USAMO 2024

Find all integers $n \geq 3$ such that the following property holds: if we list the divisors of $n!$ in increasing order as $1 = d_1 < d_2 < \dots < d_k = n!$ , then we have $$d_2 - d_1 \leq d_3 - d_2 \leq \dots \leq d_k - d_{k-1}.$$

Proposed by Luke Robitaille.

Let $S_1, S_2, \ldots, S_{100}$ be finite sets of integers whose intersection is not empty. For each non-empty $T \subseteq \{S_1, S_2, \ldots, S_{100}\},$ the size of the intersection of the sets in $T$ is a multiple of the number of sets in $T$ . What is the least possible number of elements that are in at least $50$ sets?

Proposed by Rishabh Das

Let $m$ be a positive integer. A triangulation of a polygon is $m$ -balanced if its triangles can be colored with $m$ colors in such a way that the sum of the areas of all triangles of the same color is the same for each of the $m$ colors. Find all positive integers $n$ for which there exists an $m$ -balanced triangulation of a regular $n$ -gon.

Note: A triangulation of a convex polygon $\mathcal{P}$ with $n \ge 3$ sides is any partitioning of $\mathcal{P}$ into $n-2$ triangles by $n-3$ diagonals of $\mathcal{P}$ that do not intersect in the polygon's interior.

Proposed by Krit Boonsiriseth

Let $m$ and $n$ be positive integers. A circular necklace contains $mn$ beads, each either red or blue. It turned out that no matter how the necklace was cut into $m$ blocks of $n$ consecutive beads, each block had a distinct number of red beads. Determine, with proof, all possible values of the ordered pair $(m, n)$ .

Proposed by Rishabh Das

Point $D$ is selected inside acute $\triangle ABC$ so that $\angle DAC = \angle ACB$ and $\angle BDC = 90^{\circ} + \angle BAC$ . Point $E$ is chosen on ray $BD$ so that $AE = EC$ . Let $M$ be the midpoint of $BC$ .

Show that line $AB$ is tangent to the circumcircle of triangle $BEM$ .

Proposed by Anton Trygub

Let $n > 2$ be an integer and let $\ell \in \{1, 2,\dots, n\}$ . A collection $A_1,\dots,A_k$ of (not necessarily distinct) subsets of $\{1, 2,\dots, n\}$ is called $\ell$ -large if $|A_i| \ge \ell$ for all $1 \le i \le k$ . Find, in terms of $n$ and $\ell$ , the largest real number $c$ such that the inequality $$\sum_{i=1}^k\sum_{j=1}^k x_ix_j\frac{|A_i\cap A_j|^2}{|A_i|\cdot|A_j|}\ge c\left(\sum_{i=1}^k x_i\right)^2$$ holds for all positive integer $k$ , all nonnegative real numbers $x_1,x_2,\dots,x_k$ , and all $\ell$ -large collections $A_1,A_2,\dots,A_k$ of subsets of $\{1,2,\dots,n\}$ .

Proposed by Titu Andreescu and Gabriel Dospinescu