OPM 2025

Let $ABCD$ be a quadrilateral such that $\angle A$ and $\angle D$ are acute and $\overline{AB} = \overline{BC} = \overline{CD}$ . Suppose that $\angle BDA = 30^\circ$ , prove that $\angle DAC= 30^\circ$ .

A computer science teacher has asked his students to write a program that, given a list of $n$ numbers $a_1, a_2, ..., a_n$ , calculates the list $b_1, b_2, ..., b_n$ where $b_k$ is the number of times the number $a_k$ appears in the list. So, for example, for the list $1,2,3,1$ , the program returns the list $2,1,1,2$ .

Next, the teacher asked Alexandre to run the program for a list of $2025$ numbers. Then he asked him to apply the program to the resulting list, and so on, until a number greater than or equal to $k$ appears in the list. Find the largest value of $k$ for which, whatever the initial list of $2025$ positive integers $a_1, a_2, ..., a_{2025}$ , it is possible for Alexander to do what the teacher asked him to do.

Let $ABCD$ be a square with $2cm$ side length and with center $T$ . A rhombus $ARTE$ is drawn where point $E$ lies on line $DC$ . What is the area of $ARTE$ ?

An integer number $n \geq 2$ is called feirense if it is possible to write on a sheet of paper some integers such that every positive divisor of $n$ less than $n$ is the difference between two numbers on the sheet, and no other positive number is. Find all the feirense numbers.