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