Navvye/Olympiad-Problems-CSV huggingface dataset:
2021 Pan-African
Let $ABCD$ be a trapezoid which is not a parallelogram, such that $AD$ is parallel to $BC$.

Let $O=BD\cap AC$ and $S$ be the second intersection of the circumcircles of triangles $AOB$ and $DOC$.

Prove that the circumcircles of triangles $ASD$ and $BSC$ are tangent.
2019 Pan-African
Let $(a_n)_{n=0}^{\infty}$ be a sequence of real numbers defined as follows:



$a_0 = 3$, $a_1 = 2$, and $a_2 = 12$; and

$2a_{n + 3} - a_{n + 2} - 8a_{n + 1} + 4a_n = 0$ for $n \geq 0$.



Show that $a_n$ is always a strictly positive integer.
2019 Pan-African
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
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.
2019 Pan-African
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
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).



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 $N$.
2019 Pan-African
Find the $2019$th strictly positive integer $n$ such that $\binom{2n}{n}$ is not divisible by $5$.
2018 Pan African
Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that $(f(x + y))^2 = f(x^2) + f(y^2)$ for all $x, y \in \mathbb{Z}$.
2018 Pan African
A chess tournament is held with the participation of boys and girls. The girls are twice as many as boys. Each player plays against each other player exactly once. By the end of the tournament, there were no draws and the ratio of girl winnings to boy winnings was $\frac{7}{9}$.



How many players took part at the tournament?
2018 Pan African
For any positive integer $x$, we set

$$

  g(x) = \text{ largest odd divisor of } x,

$$$$

  f(x) = \begin{cases}

    \frac{x}{2} + \frac{x}{g(x)} & \text{ if } x \text{ is even;} \\
    2^{\frac{x+1}{2}} & \text{ if } x \text{ is odd.}

  \end{cases}

$$

Consider the sequence $(x_n)_{n \in \mathbb{N}}$ defined by $x_1 = 1$, $x_{n + 1} = f(x_n)$. Show that the integer $2018$ appears in this sequence, determine the least integer $n$ such that $x_n = 2018$, and determine whether $n$ is unique or not.
2018 Pan African
Given a triangle $ABC$, let $D$ be the intersection of the line through $A$ perpendicular to $AB$, and the line through $B$ perpendicular to $BC$. Let $P$ be a point inside the triangle. Show that $DAPB$ is cyclic if and only if $\angle BAP = \angle CBP$.
2018 Pan African
Let $a$, $b$, $c$ and $d$ be non-zero pairwise different real numbers such that

$$

  \frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} = 4 \text{ and } ac = bd.

$$

Show that

$$

  \frac{a}{c} + \frac{b}{d} + \frac{c}{a} + \frac{d}{b} \leq -12

$$and that $-12$ is the maximum.
2018 Pan African
A circle is divided into $n$ sectors ($n \geq 3$). Each sector can be filled in with either $1$ or $0$. Choose any sector $\mathcal{C}$ occupied by $0$, change it into a $1$ and simultaneously change the symbols $x, y$ in the two sectors adjacent to $\mathcal{C}$ to their complements $1-x$, $1-y$. We repeat this process as long as there exists a zero in some sector. In the initial configuration there is a $0$ in one sector and $1$s elsewhere. For which values of $n$ can we end this process?
2017 Pan African
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
2017 Pan African
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?
2017 Pan African
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
Find all the real numbers $x$ such that $\frac{1}{[x]}+\frac{1}{[2x]}=\{x\}+\frac{1}{3}$ where $[x]$ denotes the integer part of $x$ and $\{x\}=x-[x]$.

For example, $[2.5]=2, \{2.5\} = 0.5$ and $[-1.7]= -2, \{-1.7\} = 0.3$
2017 Pan African
The numbers from $1$ to $2017$ are written on a board. Deka and Farid play the following game :

each of them, on his turn, erases one of the numbers. Anyone who erases a multiple of $2, 3$ or $5$ loses and the game is over. Is there a winning strategy for Deka ?
2017 Pan African
Let $ABC$ be a triangle with $H$ its orthocenter. The circle with diameter $[AC]$ cuts the circumcircle of triangle $ABH$ at $K$. Prove that the point of intersection of the lines $CK$ and $BH$ is the midpoint of the segment $[BH]$
2016 PAMO
Two circles $\mathcal{C}_1$ and $\mathcal{C}_2$ intersect each other at two distinct points $M$ and $N$. A common tangent lines touches $\mathcal{C}_1$ at $P$ and $\mathcal{C}_2$ at $Q$, the line being closer to $N$ than to $M$. The line $PN$ meets the circle $\mathcal{C}_2$ again at the point $R$.

Prove that the line $MQ$ is a bisector of the angle $\angle{PMR}$.
2016 PAMO
We have a pile of $2016$ cards and a hat. We take out one card, put it in the hat and then divide the remaining cards into two arbitrary non empty piles. In the next step, we choose one of the two piles, we move one card from this pile to the hat and then divide this pile into two arbitrary non empty piles.

This procedure is repeated several times : in the $k$-th step $(k>1)$ we move one card from one of the piles existing after the step $(k-1)$ to the hat and then divide this pile into two non empty piles.

Is it possible that after some number of steps we get all piles containing three cards each?
2016 PAMO
For any positive integer $n$, we define the integer $P(n)$ by :



$P(n)=n(n+1)(2n+1)(3n+1)...(16n+1)$.



Find the greatest common divisor of the integers $P(1)$, $P(2)$, $P(3),...,P(2016)$.
2016 PAMO
Let $x,y,z$ be positive real numbers such that $xyz=1$. Prove that



$\frac{1}{(x+1)^2+y^2+1}$ $+$ $\frac{1}{(y+1)^2+z^2+1}$ $+$ $\frac{1}{(z+1)^2+x^2+1}$ $\leq$ ${\frac{1}{2}}$.
2016 PAMO
Let $ABCD$ be a trapezium such that the sides $AB$ and $CD$ are parallel and the side $AB$ is longer than the side $CD$. Let $M$ and $N$ be on the segments $AB$ and $BC$ respectively, such that each of the segments $CM$ and $AN$ divides the trapezium in two parts of equal area.

Prove that the segment $MN$ intersects the segment $BD$ at its midpoint.
2016 PAMO
Consider an $n\times{n}$ grid formed by $n^2$ unit squares. We define the centre of a unit square as the intersection of its diagonals.

Find the smallest integer $m$ such that, choosing any $m$ unit squares in the grid, we always get four unit squares among them whose centres are vertices of a parallelogram.
2015 PAMO
Prove that

$$\sqrt{x-1}+\sqrt{2x+9}+\sqrt{19-3x}<9$$

for all real $x$ for which the left-hand side is well defined.
2015 PAMO
A convex hexagon $ABCDEF$ is such that

$$AB=BC \quad CD=DE \quad  EF=FA$$

and

$$\angle ABC=2\angle AEC \quad \angle CDE=2\angle CAE \quad \angle EFA=2\angle ACE$$

Show that $AD$, $CF$ and $EB$ are concurrent.
2015 PAMO
Let $a_1,a_2,...,a_{11}$ be integers. Prove that there are numbers $b_1,b_2,...,b_{11}$, each $b_i$ equal $-1,0$ or $1$, but not all being $0$, such that the number

$$N=a_1b_1+a_2b_2+...+a_{11}b_{11}$$

is divisible by $2015$.
2015 PAMO
For a positive integer $n$ denote $d(n)$ its greatest odd divisor. Find the value of the sum

$$d(1008)+d(1009)+...+d(2015)$$
2015 PAMO
There are seven cards in a hat, and on the card $k$ there is a number $2^{k-1}$, $k=1,2,...,7$. Solarin picks the cards up at random from the hat, one card at a time, until the sum of the numbers on cards in his hand exceeds $124$. What is the most probable sum he can get?
2015 PAMO
Let $ABCD$ be a quadrilateral (with non-perpendicular diagonals).



The perpendicular from $A$ to $BC$ meets $CD$ at $K$.

The perpendicular from $A$ to $CD$ meets $BC$ at $L$.

The perpendicular from $C$ to $AB$ meets $AD$ at $M$.

The perpendicular from $C$ to $AD$ meets $AB$ at $N$.



1. Prove that $KL$ is parallel to $MN$.

2. Prove that $KLMN$ is a parallelogram if $ABCD$ is cyclic.
2013 Pan African
A positive integer $n$ is such that $n(n+2013)$ is a perfect square.

a) Show that $n$ cannot be prime.

b) Find a value of $n$ such that $n(n+2013)$ is a perfect square.
2013 Pan African
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(x)f(y)+f(x+y)=xy$ for all real numbers $x$ and $y$.
2013 Pan African
Let $ABCDEF$ be a convex hexagon with $\angle A= \angle D$ and $\angle B=\angle E$ . Let $K$ and $L$

be the midpoints of the sides $AB$ and $DE$ respectively. Prove that the sum of the areas of triangles $FAK$, $KCB$ and $CFL$ is equal to half of the area of the hexagon if and only if

$$\frac{BC}{CD}=\frac{EF}{FA}.$$
2013 Pan African
Let $ABCD$ be a convex quadrilateral with $AB$ parallel to $CD$. Let $P$ and $Q$ be the midpoints of $AC$ and $BD$, respectively. Prove that if $\angle ABP=\angle CBD$, then $\angle BCQ=\angle ACD$.
2013 Pan African
The cells of an $n\times n$ board with $n\ge 5$ are coloured black or white so that no three adjacent squares in a row, column or diagonal are the same colour. Show that for any $3\times 3$ square within the board, two of its corner squares are coloured black and two are coloured white.
2013 Pan African
Let $x$, $y$, and $z$ be real numbers such that $x<y<z<6$. Solve the system of inequalities:

$$\left\{\begin{array}{cc}

\dfrac{1}{y-x}+\dfrac{1}{z-y}\le 2 \\
\dfrac{1}{6-z}+2\le x \\
\end{array}\right.$$
2012 Pan African
$AB$ is a chord (not a diameter) of a circle with centre $O$. Let $T$ be a point on segment $OB$. The line through $T$ perpendicular to $OB$ meets $AB$ at $C$ and the circle at $D$ and $E$. Denote by $S$ the orthogonal projection of $T$ onto $AB$ .

Prove that $AS \cdot BC = TE \cdot TD$.
2012 Pan African
Find all positive integers $m$ and $n$ such that $n^m - m$ divides $m^2 + 2m$.
2012 Pan African
Find all real solutions $x$ to the equation $\lfloor x^2 - 2x \rfloor + 2\lfloor x \rfloor = \lfloor x \rfloor^2$.
2012 Pan African
The numbers $\frac{1}{1}, \frac{1}{2}, \cdots , \frac{1}{2012}$ are written on the blackboard. Aïcha chooses any two numbers from the blackboard, say $x$ and $y$, erases them and she writes instead the number $x + y + xy$. She continues to do this until only one number is left on the board. What are the possible values of the final number?
2012 Pan African
Find all functions $ f:\mathbb{R}\rightarrow\mathbb{R} $ such that $f(x^2 - y^2) = (x+y)(f(x) - f(y))$ for all real numbers $x$ and $y$.
2012 Pan African
(i) Find the angles of $\triangle ABC$ if the length of the altitude through $B$ is equal to the length of the median through $C$ and the length of the altitude through $C$ is equal to the length of the median through $B$.



(ii) Find all possible values of $\angle ABC$ of $\triangle ABC$ if the length of the altitude through $A$ is equal to the length of the median through $C$ and the length of the altitude through $C$ is equal to the length of the median through $B$.
2010 Pan African
a) Show that it is possible to pair off the numbers $1,2,3,\ldots ,10$ so that the sums of each of the five pairs are five different prime numbers.

b) Is it possible to pair off the numbers $1,2,3,\ldots ,20$ so that the sums of each of the ten pairs are ten different prime numbers?
2010 Pan African
How many ways are there to line up $19$ girls (all of different heights) in a row so that no girl has a shorter girl both in front of and behind her?
2010 Pan African
In an acute-angled triangle $ABC$, $CF$ is an altitude, with $F$ on $AB$, and $BM$ is a median, with $M$ on $CA$. Given that $BM=CF$ and $\angle MBC=\angle FCA$, prove that triangle $ABC$ is equilateral.
2010 Pan African
Seven distinct points are marked on a circle of circumference $c$. Three of the points form an equilateral triangle and the other four form a square. Prove that at least one of the seven arcs into which the seven points divide the circle has length less than or equal $\frac{c}{24}$.
2010 Pan African
A sequence $a_0,a_1,a_2,\ldots ,a_n,\ldots$ of positive integers is constructed as follows:



if the last digit of $a_n$ is less than or equal to $5$ then this digit is deleted and $a_{n+1}$ is the number consisting of the remaining digits. (If $a_{n+1}$ contains no digits the process stops.)

otherwise $a_{n+1}=9a_n$.



Can one choose $a_0$ so that an infinite sequence is obtained?
2010 Pan African
Does there exist a function $f:\mathbb{Z}\to\mathbb{Z}$ such that $f(x+f(y))=f(x)-y$ for all integers $x$ and $y$?
2009 Pan African
Determine whether or not there exist numbers $x_1,x_2,\ldots ,x_{2009}$ from the set $\{-1,1\}$, such that:

$$x_1x_2+x_2x_3+x_3x_4+\ldots+x_{2008}x_{2009}+x_{2009}x_1=999$$
2009 Pan African
Point $P$ lies inside a triangle $ABC$. Let $D,E$ and $F$ be reflections of the point $P$ in the lines $BC,CA$ and $AB$, respectively. Prove that if the triangle $DEF$ is equilateral, then the lines $AD,BE$ and $CF$ intersect in a common point.
2009 Pan African
Let $x$ be a real number with the following property: for each positive integer $q$, there exists an integer $p$, such that

$$\left|x-\frac{p}{q} \right|<\frac{1}{3q}. $$

Prove that $x$ is an integer.
2009 Pan African
Consider $n$ children in a playground, where $n\ge 2$. Every child has a coloured hat, and every pair of children is joined by a coloured ribbon. For every child, the colour of each ribbon held is different, and also different from the colour of that child’s hat. What is the minimum number of colours that needs to be used?
2009 Pan African
Find all functions $f:\mathbb{N}_0\to\mathbb{N}_0$ for which $f(0)=0$ and

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

for all $x,y\in\mathbb{N}_0$ with $x>y$.
2009 Pan African
Points $C,E,D$ and $F$ lie on a circle with centre $O$. Two chords $CD$ and $EF$ intersect at a point $N$. The tangents at $C$ and $D$ intersect at $A$, and the tangents at $E$ and $F$ intersect at $B$. Prove that $ON\perp AB$.
2008 Pan African
Determine all functions $f:\mathbb{R}\to\mathbb{R}$ satisfying $f(x+y)\le f(x)+f(y)\le x+y$ for all $x,y\in\mathbb{R}$.
2008 Pan African
Let $C_1$ be a circle with centre $O$, and let $AB$ be a chord of the circle that is not a diameter. $M$ is the midpoint of $AB$. Consider a point $T$ on the circle $C_2$ with diameter $OM$. The tangent to $C_2$ at the point $T$ intersects $C_1$ at two points. Let $P$ be one of these points. Show that $PA^2+PB^2=4PT^2$.
2008 Pan African
Let $a,b,c$ be three positive integers such that $a<b<c$. Consider the the sets $A,B,C$ and $X$, defined as follows: $A=\{ 1,2,\ldots ,a \}$, $B=\{a+1,a+2,\ldots,b\}$, $C=\{b+1,b+2,\ldots ,c\}$ and $X=A\cup B\cup C$.

Determine, in terms of $a,b$ and $c$, the number of ways of placing the elements of $X$ in three boxes such that there are $x,y$ and $z$ elements in the first, second and third box respectively, knowing that:

i) $x\le y\le z$;

ii) elements of $B$ cannot be put in the first box;

iii) elements of $C$ cannot be put in the third box.
2008 Pan African
Let $x$ and $y$ be two positive reals. Prove that $xy\le\frac{x^{n+2}+y^{n+2}}{x^n+y^n}$ for all non-negative integers $n$.
2008 Pan African
A set of positive integers $X$ is called connected if $|X|\ge 2$ and there exist two distinct elements $m$ and $n$ of $X$ such that $m$ is a divisor of $n$.

Determine the number of connected subsets of the set $\{1,2,\ldots,10\}$.
2008 Pan African
Prove that for all positive integers $n$, there exists a positive integer $m$ which is a multiple of $n$ and the sum of the digits of $m$ is equal to $n$.
2007 Pan African
Find all natural numbers $N$ consisting of exactly $1112$ digits (in decimal notation) such that:

(a) The sum of the digits of $N$ is divisible by $2000$;

(b) The sum of the digits of $N+1$ is divisible by $2000$;

(c) $1$ is a digit of $N$.
2007 Pan African
Let $A$, $B$ and $C$ be three fixed points, not on the same line. Consider all triangles $AB'C'$ where $B'$ moves on a given straight line (not containing $A$), and $C'$ is determined such that $\angle B'=\angle B$ and $\angle C'=\angle C$. Find the locus of $C'$.
2007 Pan African
In a country, towns are connected by roads. Each town is directly connected to exactly three other towns. Show that there exists a town from which you can make a round-trip, without using the same road more than once, and for which the number of roads used is not divisible by $3$. (Not all towns need to be visited.)
2007 Pan African
Solve the following system of equations for real $x,y$ and $z$:

\begin{eqnarray*}

x &=& \sqrt{2y+3}\\
y &=& \sqrt{2z+3}\\
z &=& \sqrt{2x+3}.

\end{eqnarray*}
2007 Pan African
For which positive integers $n$ is  $231^n-222^n-8 ^n -1$ divisible by $2007$?
2007 Pan African
An equilateral triangle of side length 2 is divided into four pieces by two perpendicular lines that intersect in the centroid of the triangle. What is the maximum possible area of a piece?
2006 Pan African
Let $AB$ and $CD$ be two perpendicular diameters of a circle with centre $O$. Consider a point $M$ on the diameter $AB$, different from $A$ and $B$. The line $CM$ cuts the circle again at $N$. The tangent at $N$ to the circle and the perpendicular at $M$ to $AM$ intersect at $P$. Show that $OP = CM$.
2006 Pan African
Let $a, b, c$ be three non-zero integers. It is known that the sums $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$ and $\frac{b}{a}+\frac{c}{b}+\frac{a}{c}$ are integers. Find these sums.
2006 Pan African
For a real number $x$ let $\lfloor x\rfloor$ be the greatest integer less than or equal to $x$ and let $\{x\} = x - \lfloor x\rfloor$. If $a, b, c$ are distinct real numbers, prove that

$$\frac{a^3}{(a-b)(a-c)}+\frac{b^3}{(b-a)(b-c)}+\frac{c^3}{(c-a)(c-b)}$$

is an integer if and only if $\{a\} + \{b\} + \{c\}$ is an integer.
2006 Pan African
For every positive integer $k$ let $a(k)$ be the largest integer such that $2^{a(k)}$ divides $k$. For every positive integer $n$ determine $a(1)+a(2)+\cdots+a(2^n)$.
2006 Pan African
In how many ways can the integers from $1$ to $2006$ be divided into three non-empty disjoint sets so that none of these sets contains a pair of consecutive integers?
2006 Pan African
Let $ABC$ be a right angled triangle at $A$. Denote $D$ the foot of the altitude through $A$ and $O_1, O_2$ the incentres of triangles $ADB$ and $ADC$. The circle with centre $A$ and radius $AD$ cuts $AB$ in $K$ and $AC$ in $L$. Show that $O_1, O_2, K$ and $L$ are on a line.
2005 Pan African
For any positive real numbers $a,b$ and $c$, prove:

$$ \dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c} \geq \dfrac{2}{a+b} + \dfrac{2}{b+c} + \dfrac{2}{c+a} \geq \dfrac{9}{a+b+c} $$
2005 Pan African
Let $S$ be a set of integers with the property that any integer root of any non-zero polynomial with coefficients in $S$ also belongs to $S$.  If $0$ and $1000$ are elements of $S$, prove that $-2$ is also an element of $S$.
2005 Pan African
Let $ABC$ be a triangle and let $P$ be a point on one of the sides of $ABC$.  Construct a line passing through $P$ that divides triangle $ABC$ into two parts of equal area.
2005 Pan African
Let $[ {x} ]$ be the greatest integer less than or equal to $x$, and let $\{x\}=x-[x]$.

Solve the equation: $[x] \cdot \{x\} = 2005x$
2005 Pan African
Noah has to fit 8 species of animals into 4 cages of the Arc.  He planes to put two species of animal in each cage.  It turns out that, for each species of animal, there are at most 3 other species with which it cannot share a cage.  Prove that there is a way to assign the animals to the cages so that each species shares a cage with a compatible species.
2005 Pan African
Let $f: \mathbb{Z} \rightarrow \mathbb{Z}$ be a function such that:  For all $a$ and $b$ in $\mathbb{Z} - \{0\}$, $f(ab) \geq f(a) + f(b)$.  Show that for all $a \in \mathbb{Z} - \{0\}$ we have $f(a^n) = nf(a)$ for all $n \in \mathbb{N}$ if and only if $f(a^2) = 2f(a)$
2004 Pan African
Do there exist positive integers $m$ and $n$ such that:

$$ 3n^2+3n+7=m^3 $$
2004 Pan African
Is:

$$ 4\sqrt{4-2\sqrt{3}}+\sqrt{97-56\sqrt{3}} $$

an integer?
2004 Pan African
One writes 268 numbers around a circle, such that the sum of 20 consectutive numbers is always equal to 75. The number 3, 4 and 9 are written in positions 17, 83 and 144 respectively. Find the number in position 210.
2004 Pan African
Three real numbers satisfy the following statements:

(1) the square of their sum equals to the sum their squares.

(2) the product of the first two numbers is equal to the square of the third number.



Find these numbers.
2004 Pan African
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 $7$.
2004 Pan African
Let $ABCD$ be a cyclic quadrilateral such that $AB$ is a diameter of it's circumcircle. Suppose that $AB$ and $CD$ intersect at $I$, $AD$ and $BC$ at $J$, $AC$ and $BD$ at $K$, and let $N$ be a point on $AB$. Show that $IK$ is perpendicular to $JN$ if and only if $N$ is the midpoint of $AB$.
2003 Pan African
Let $N_0=\{0, 1, 2 \cdots \}$. Find all functions: $N_0 \to N_0$ such that:



(1) $f(n) < f(n+1)$, all $n \in N_0$;



(2) $f(2)=2$;



(3) $f(mn)=f(m)f(n)$, all $m, n \in N_0$.
2003 Pan African
The circumference of a circle is arbitrarily divided into four arcs. The midpoints of the arcs are connected by segments. Show that two of these segments are perpendicular.
2003 Pan African
Does there exists a base in which the numbers of the form:

$$ 10101, 101010101, 1010101010101,\cdots $$

are all prime numbers?
2003 Pan African
Let $\mathbb{N}_0=\{0,1,2 \cdots \}$. Does there exist a function $f: \mathbb{N}__0 \to \mathbb{N}_0$ such that:

$$ f^{2003}(n)=5n, \forall n \in \mathbb{N}_0  $$

where we define: $f^1(n)=f(n)$ and $f^{k+1}(n)=f(f^k(n))$, $\forall k \in \mathbb{N}_0$?
2003 Pan African
Find all positive integers $n$ such that $21$ divides $2^{2^n}+2^n+1$.
2003 Pan African
Find all functions $f: R\to R$ such that:

$$ f(x^2)-f(y^2)=(x+y)(f(x)-f(y)), x,y \in R $$
2002 Pan African
Find all functions $f: N_0 \to N_0$, (where $N_0$ is the set of all non-negative integers) such that $f(f(n))=f(n)+1$ for all $n \in N_0$ and the minimum of the set $\{ f(0), f(1), f(2) \cdots \}$ is $1$.
2002 Pan African
$\triangle{AOB}$ is a right triangle with $\angle{AOB}=90^{o}$. $C$ and $D$ are moving on $AO$ and $BO$ respectively such that $AC=BD$. Show that there is a fixed point $P$ through which the perpendicular bisector of $CD$ always passes.
2002 Pan African
Prove for every integer $n>0$, there exists an integer $k>0$ such that $2^nk$ can be written in decimal notation using only digits 1 and 2.
2002 Pan African
Seven students in a class compare their marks in 12 subjects studied and observe that no two of the students have identical marks in all 12 subjects. Prove that we can choose 6 subjects such that any two of the students have different marks in at least one of these subjects.
2002 Pan African
Let $\triangle{ABC}$ be an acute angled triangle. The circle with diameter AB intersects the sides AC and BC at points E and F respectively. The tangents drawn to the circle through E and F intersect at P.

Show that P lies on the altitude through the vertex C.
2002 Pan African
If $a_1 \geq a_2 \geq \cdots \geq a_n \geq 0$ and $a_1+a_2+\cdots+a_n=1$, then prove:

$$a_1^2+3a_2^2+5a_3^2+ \cdots +(2n-1)a_n^2 \leq 1$$
2001 Pan African
Find all positive integers $n$ such that:

$$ \dfrac{n^3+3}{n^2+7} $$

is a positive integer.