Question bank 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 $1 = d_1 \lt d_2 \lt \dots \lt d_k = n!$, then we have

$$d_2 - d_1 \leq d_3 - d_2 \leq … \leq d_k - d_{k-1}.$$

Proposed by Luke Robitaille. One nonofficial editorial point; manual essay grading.

Let $S_1$, $S_2$, …, $S_{100}$ 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 $|T|$.
What is the smallest possible number of elements which are in at least $50$ sets?

Proposed by Rishabh Das. One nonofficial editorial point; manual essay grading.

Let $(m,n)$ be positive integers with $n \ge 3$ and draw a regular $n$-gon.
We wish to triangulate this $n$-gon into $n-2$ triangles,
each colored one of $m$ colors, so that each color has an equal sum of areas.
For which $(m,n)$ is such a triangulation and coloring possible?

Proposed by Krit Boonsiriseth. One nonofficial editorial point; manual essay grading.

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. One nonofficial editorial point; manual essay grading.

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. One nonofficial editorial point; manual essay grading.

Let $n > 2$ be an integer and let $\ell \in \{1, 2,\dots, n\}$.
A collection $A_1$, …, $A_k$ 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 $x_1$, $x_2$, …, $x_k$,
and all $\ell$-large collections $A_1$, $A_2$, …, $A_k$
of subsets of $\{1,2,\dots,n\}$.

Proposed by Titu Andreescu, Gabriel Dospinescu. One nonofficial editorial point; manual essay grading.