2024 USAMO Problems

\begin{enumerate}

   \item[(1)] Find all integers  such that the following property holds: if we list the divisors of  in increasing order as , then we have

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

   \item[(2)] Let  be finite sets of integers whose intersection is not empty. For each non-empty , the size of the intersection of the sets in  is a multiple of the number of sets in . What is the least possible number of elements that are in at least 50 sets?\\\\\\\\\\\\

   \item[(3)] Let  be a positive integer. A triangulation of a polygon is -balanced if its triangles can be colored with  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  colors. Find all positive integers  for which there exists an -balanced triangulation of a regular -gon.

Note: A triangulation of a convex polygon $\mathcal{P}$ with $n \geq 3$ 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.

   \newpage
   \item[(4)] Let  and  be positive integers. A circular necklace contains  beads, each either red or blue. It turned out that no matter how the necklace was cut into  blocks of  consecutive beads, each block had a distinct number of red beads. Determine, with proof, all possible values of the ordered pair .\\\\\\\\\\\\\\

\item[(5)] Point $D$ is selected inside acute triangle $A B C$ so that $\angle D A C=$ $\angle A C B$ and $\angle B D C=90^{\circ}+\angle B A C$ . Point $E$ is chosen on ray $B D$ so that $A E=E C$ . Let $M$ be the midpoint of $B C$ . Show that line $A B$ is tangent to the circumcircle of triangle $B E M$ .\\\\\\\\\\\\\\

\item[(6)] Let $n>2$ be an integer and let $\ell \in\{1,2, \ldots, n\}$ . A collection $A_1, \ldots, A_k$ of (not necessarily distinct) subsets of $\{1,2, \ldots, n\}$ is called $\ell$ -large if $\left|A_i\right| \geq \ell$ for all $1 \leq i \leq k$ . Find, in terms of $n$ and $\ell$ , the largest real number $c$ such that the inequality $\sum_{i=1}^k \sum_{j=1}^k x_i x_j \frac{\left|A_i \cap A_j\right|^2}{\left|A_i\right| \cdot\left|A_j\right|} \geq c\left(\sum_{i=1}^k x_i\right)^2$ holds for all positive integers $k$ , all nonnegative real numbers $x_1, \ldots, x_k$ , and all $\ell$ -large collections $A_1, \ldots, A_k$ of subsets of $\{1,2, \ldots, n\}$ . Note: For a finite set $S,|S|$ denotes the number of elements in $S$ . \end{enumerate}

Page

Toolbox

Search

2024 USAMO Problems