Olympiad problems

2021 Pan-African, 3

Let $(a_i)_{i\in \mathbb{N}}$ and $(p_i)_{i\in \mathbb{N}}$ be two sequences of positive integers such that the following conditions hold: $\bullet ~~a_1\ge 2$. $\bullet~~ p_n$ is the smallest prime divisor of $a_n$ for every integer $n\ge 1$ $\bullet~~ a_{n+1}=a_n+\frac{a_n}{p_n}$ for every integer $n\ge 1$ Prove that there is a positive integer $N$ such that $a_{n+3}=3a_n$ for every integer $n>N$

2021 Pan-African, 4

Tags: PAMO , number theory , Divisibility

Find all integers $m$ and $n$ such that $\frac{m^2+n}{n^2-m}$ and $\frac{n^2+m}{m^2-n}$ are both integers.

2017 Pan-African Shortlist, I?

Tags: algebraic inequality , PAMO , inequalities

Let $x,y$, and $z$ be positive real numbers such that $xy+yz+zx=3xyz$. Prove that $$x^2y+y^2z+z^2x \geq 2(x+y+z)-3.$$ In which cases do we have equality?

2004 Pan African, 2

Tags: PAMO , 2004

Each of the digits $1$, $3$, $7$ and $9$ occurs at least once in the decimal representation of some positive integers. Prove that one can permute the digits of this integer such that the resulting integer is divisible by $7$.

2023 Pan-African, 3

Tags: algebra , PAMO

Consider a sequence of real numbers defined by: \begin{align*} x_{1} & = c \\ x_{n+1} & = cx_{n} + \sqrt{c^{2} - 1}\sqrt{x_{n}^{2} - 1} \quad \text{for all } n \geq 1. \end{align*} Show that if $c$ is a positive integer, then $x_{n}$ is an integer for all $n \geq 1$. [i](South Africa)[/i]

2019 Pan-African Shortlist, N6

Tags: binomial coefficients , PAMO , number theory

Find the $2019$th strictly positive integer $n$ such that $\binom{2n}{n}$ is not divisible by $5$.

2023 Pan-African, 1

Tags: PAMO , geometry

In a triangle $ABC$ with $AB<AC$, $D$ is a point on segment $AC$ such that $BD = CD$. A line parallel to $BD$ meets segment $BC$ at $E$ and line $AB$ at $F$. Point $G$ is the intersection of $AE$ and $BD$. Show that $\angle BCG = \angle BCF$. [i](Côte d’Ivoire)[/i]

2019 Pan-African Shortlist, G4

Tags: geometry , Circumcenter , PAMO , Equilateral Triangle , circumcircle

Let $ABC$ be a triangle, and $D$, $E$, $F$ points on the segments $BC$, $CA$, and $AB$ respectively such that $$ \frac{BD}{DC} = \frac{CE}{EA} = \frac{AF}{FB}. $$ Show that if the centres of the circumscribed circles of the triangles $DEF$ and $ABC$ coincide, then $ABC$ is an equilateral triangle.

2017 Pan-African Shortlist, A?

Tags: algebra , PAMO

We consider the real sequence $(x_n)$ defined by $x_0=0, x_1=1$ and $x_{n+2}=3x_{n+1}-2x_n$ for $n=0,1,...$ We define the sequence $(y_n)$ by $y_n=x_n^2+2^{n+2}$ for every non negative integer $n$. Prove that for every $n>0$, $y_n$ is the square of an odd integer

2021 Pan-African, 1

Tags: geometry , trapezoid , PAMO

Let $n$ be an integer greater than $3$. A square of side length $n$ is divided by lines parallel to each side into $n^2$ squares of length $1$. Find the number of convex trapezoids which have vertices among the vertices of the $n^2$ squares of side length $1$, have side lengths less than or equal $3$ and have area equal to $2$ Note: Parallelograms are trapezoids.

2017 Pan African, Problem 2

Tags: algebraic inequality , PAMO , inequalities

Let $x,y$, and $z$ be positive real numbers such that $xy+yz+zx=3xyz$. Prove that $$x^2y+y^2z+z^2x \geq 2(x+y+z)-3.$$ In which cases do we have equality?

2019 Pan-African, 2

Tags: number theory , prime numbers , Sums and Products , PAMO

Let $k$ be a positive integer. Consider $k$ not necessarily distinct prime numbers such that their product is ten times their sum. What are these primes and what is the value of $k$?

2019 Pan-African Shortlist, N2

Tags: number theory , prime numbers , Sums and Products , PAMO

Let $k$ be a positive integer. Consider $k$ not necessarily distinct prime numbers such that their product is ten times their sum. What are these primes and what is the value of $k$?

2019 Pan-African, 1

Tags: recurrence relation , Linear Recurrences , algebra , induction , PAMO

Let $(a_n)_{n=0}^{\infty}$ be a sequence of real numbers defined as follows: [list] [*] $a_0 = 3$, $a_1 = 2$, and $a_2 = 12$; and [*] $2a_{n + 3} - a_{n + 2} - 8a_{n + 1} + 4a_n = 0$ for $n \geq 0$. [/list] Show that $a_n$ is always a strictly positive integer.

2023 Pan-African, 2

Tags: PAMO , number theory

Find all positive integers $m$ and $n$ with no common divisor greater than 1 such that $m^3 + n^3$ divides $m^2 + 20mn + n^2$. [i](Professor Yongjin Song)[/i]

2019 Pan-African Shortlist, G1

Tags: geometry , PAMO , circumcircle

The tangents to the circumcircle of $\triangle ABC$ at $B$ and $C$ meet at $D$. The circumcircle of $\triangle BCD$ meets sides $AC$ and $AB$ again at $E$ and $F$ respectively. Let $O$ be the circumcentre of $\triangle ABC$. Show that $AO$ is perpendicular to $EF$.

2019 Pan-African, 5

Tags: combinatorics , broken line , PAMO

A square is divided into $N^2$ equal smaller non-overlapping squares, where $N \geq 3$. We are given a broken line which passes through the centres of all the smaller squares (such a broken line may intersect itself). [list] [*] Show that it is possible to find a broken line composed of $4$ segments for $N = 3$. [*] Find the minimum number of segments in this broken line for arbitrary $N$. [/list]

2019 Pan-African Shortlist, C3

Tags: combinatorics , broken line , PAMO

A square is divided into $N^2$ equal smaller non-overlapping squares, where $N \geq 3$. We are given a broken line which passes through the centres of all the smaller squares (such a broken line may intersect itself). [list] [*] Show that it is possible to find a broken line composed of $4$ segments for $N = 3$. [*] Find the minimum number of segments in this broken line for arbitrary $N$. [/list]

2019 Pan-African, 6

Tags: binomial coefficients , PAMO , number theory

Find the $2019$th strictly positive integer $n$ such that $\binom{2n}{n}$ is not divisible by $5$.

2017 Pan-African Shortlist, N?

Tags: number theory , PAMO

Let $n$ be a positive integer. - Find, in terms of $n$, the number of pairs $(x,y)$ of positive integers that are solutions of the equation : $$x^2-y^2=10^2.30^{2n}$$ - Prove further that this number is never a square

2017 Pan African, Problem 3

Tags: number theory , PAMO

Let $n$ be a positive integer. - Find, in terms of $n$, the number of pairs $(x,y)$ of positive integers that are solutions of the equation : $$x^2-y^2=10^2.30^{2n}$$ - Prove further that this number is never a square

2021 Pan-African, 2

Tags: geometry , circumcircle , geometric transformation , reflection , PAMO

Let $\Gamma$ be a circle, $P$ be a point outside it, and $A$ and $B$ the intersection points between $\Gamma$ and the tangents from $P$ to $\Gamma$. Let $K$ be a point on the line $AB$, distinct from $A$ and $B$ and let $T$ be the second intersection point of $\Gamma$ and the circumcircle of the triangle $PBK$.Also, let $P'$ be the reflection of $P$ in point $A$. Show that $\angle PBT=\angle P'KA$

2017 Pan African, Problem 1

Tags: algebra , PAMO

We consider the real sequence $(x_n)$ defined by $x_0=0, x_1=1$ and $x_{n+2}=3x_{n+1}-2x_n$ for $n=0,1,...$ We define the sequence $(y_n)$ by $y_n=x_n^2+2^{n+2}$ for every non negative integer $n$. Prove that for every $n>0$, $y_n$ is the square of an odd integer

2019 Pan-African Shortlist, A1

Tags: recurrence relation , Linear Recurrences , algebra , induction , PAMO

Let $(a_n)_{n=0}^{\infty}$ be a sequence of real numbers defined as follows: [list] [*] $a_0 = 3$, $a_1 = 2$, and $a_2 = 12$; and [*] $2a_{n + 3} - a_{n + 2} - 8a_{n + 1} + 4a_n = 0$ for $n \geq 0$. [/list] Show that $a_n$ is always a strictly positive integer.

2019 Pan-African, 3

Tags: geometry , Circumcenter , PAMO , Equilateral Triangle , circumcircle

Let $ABC$ be a triangle, and $D$, $E$, $F$ points on the segments $BC$, $CA$, and $AB$ respectively such that $$ \frac{BD}{DC} = \frac{CE}{EA} = \frac{AF}{FB}. $$ Show that if the centres of the circumscribed circles of the triangles $DEF$ and $ABC$ coincide, then $ABC$ is an equilateral triangle.