2024 Pan-African

Part of Pan African

Subcontests

(6)

1

Number Theory

Find all integers

n

n^7-41

is the square of an integer

1

Functional equations

\mathbb{R}

denote the set of real numbers. Find all functions

f: \mathbb{R} \to \mathbb{R}

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

for all real numbers

x

y

1

Combinatorics

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

1

Combinatorics

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

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}

\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..

1

Geometry

ABC

M

be the midpoint of the side

BC

N

is the midpoint of the segment

AM

,the circle going through

N

and tangent the line

AC

A

intersects the segment

AB

P

.
prove that the circumcircle of triangle

BPM

is tangent the line

AM

1

Number Theory

Find all positive intgers

a,b

c

\frac{a+b}{a+c}=\frac{b+c}{b+a}

ab+bc+ca

is a prime number