PAMO 2024

Find all positive intgers $a,b$ and $c$ such that $\frac{a+b}{a+c}=\frac{b+c}{b+a}$ and $ab+bc+ca$ is a prime number

In triangle $ABC$ ,let $M$ be the midpoint of the side $BC$ ,and $N$ is the midpoint of the segment $AM$ ,the circle going through $N$ and tangent the line $AC$ at $A$ intersects the segment $AB$ again in $P$ .

prove that the circumcircle of triangle $BPM$ is tangent the line $AM$

Given an integer $n \geq 1$, Jo-Ané alternately writes crosses ($\mathcal{X}$) and circles ($\mathcal{O}$) in the cells of a square grid with $2n + 1$ rows and $2n + 1$ columns: she first writes a cross in a cell, then a circle in a second cell, then a cross in a third cell, and so on. When the table is completely filled, her score is calculated as the sum $\mathcal{X}+ \mathcal{O}$, where $\mathcal{X}$ is the number of rows containing more crosses than circles and $\mathcal{O}$ is the number of columns containing more circles than crosses.

Determine, in terms of $n$, the highest possible score that Jo-Ané can obtain..

Consider $m$ segments on the real line. Each segment has its two endpoints in the set of integers $\{1, 2, \ldots, 2024\}$ , and no two segments have the same length. No segment is entirely contained in another segment, but two segments may partially overlap each other.

What is the maximum value of $m$ ?

Let $\mathbb{R}$ denote the set of real numbers. Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that

$$f(x^2) - y f(y) = f(x+y)(f(x) - y)$$

for all real numbers $x$ and $y$.

Find all integers $n$ for which $n^7-41$ is the square of an integer