Navvye/Olympiad-Problems-CSV huggingface dataset:
2008 Nordic
Find all reals $A,B,C$ such that there exists a real function $f$ satisfying $f(x+f(y))= Ax+By+C$ for all reals $x,y$.
2008 Nordic
Assume that $n\ge 3$ people with different names sit around a round table. We call any unordered pair of them, say $M,N$, dominating if

1) they do not sit in adjacent seats

2) on one or both arcs connecting $M,N$ along the table, all people have names coming alphabetically after $M,N$.



Determine the minimal number of dominating pairs.
2008 Nordic
Let $ABC$ be a triangle and $D,E$ be points on $BC,CA$ such that $AD,BE$ are angle bisectors of $\triangle ABC$. Let $F,G$ be points on the circumcircle of $\triangle ABC$ such that $AF||DE$ and $FG||BC$. Prove that $\frac{AG}{BG}= \frac{AB+AC}{AB+BC}$.
2008 Nordic
The difference between the cubes of two consecutive positive integers is equal to $n^2$ for a positive integer $n$. Show that $n$ is the sum of two squares.
2007 Nordic
Find a solution to the equation $x^2-2x-2007y^2=0$ in positive integers.
2007 Nordic
Three given rectangles cover the sides of a triangle completely and each rectangle has a side parallel to a given line. Show that the rectangles also cover the interior of the triangle.
2007 Nordic
The number $10^{2007}$ is written on the blackboard. Anne and Berit play a two player game in which the player in turn performs one of the following operations:

1) replace a number $x$ on the blackboard with two integers $a,b>1$ such that $ab=x$.

2) strike off one or both of two equal numbers on the blackboard.



The person who cannot perform any operation loses.

Who has the winning strategy if Anne starts?
2007 Nordic
A line through $A$ intersects a circle at points $B,C$ with $B$ between $A,C$. The two tangents from $A$ intersect the circle at $S,T$. $ST$ and $AC$ intersect at $P$. Show that $\frac{AP}{PC}=2\frac{AB}{BC}$.
2006 Nordic
Points $B,C$ vary on two fixed rays emanating from point $A$ such that $AB+AC$ is constant. Show that there is a point $D$, other than $A$, such that the circumcircle of triangle $ABC$ passes through $D$ for all possible choices of $B, C$.
2006 Nordic
Real numbers $x,y,z$ are not all equal and satisfy $x+\frac{1}{y} = y + \frac{1}{z} = z + \frac{1}{x}=k$. Find all possible values of $k$.
2006 Nordic
A sequence $(a_n)$ of positive integers is defined by $a_0=m$ and $a_{n+1}= a_n^5 +487$ for all $n\ge 0$.

Find all positive integers $m$ such that the sequence contains the maximum possible number of perfect squares.
2006 Nordic
Each square of a $100\times 100$ board is painted with one of $100$ different colours, so that each colour is used exactly $100$ times. Show that there exists a row or column of the chessboard in which at least $10$ colours are used.
2005 Nordic
Find all positive integers $k$ such that the product of the digits of $k$, in decimal notation, equals $$\frac{25}{8}k-211$$
2005 Nordic
Let $a,b,c$ be positive real numbers. Prove that $$\frac{2a^2}{b+c} + \frac{2b^2}{c+a} + \frac{2c^2}{a+b} \geq a+b+c$$(this is, of course, a joke!)



EDITED with exponent 2 over c
2005 Nordic
There are $2005$ young people sitting around a large circular table. Of these, at most $668$ are boys. We say that a girl $G$ has a strong position, if, counting from $G$ in either direction, the number of girls is always strictly larger than the number of boys ($G$ is herself included in the count). Prove that there is always a girl in a strong position.
2005 Nordic
The circle $\zeta_{1}$ is inside the circle $\zeta_{2}$, and the circles touch each other at $A$. A line through $A$ intersects $\zeta_{1}$ also at $B$, and $\zeta_{2}$ also at $C$. The tangent to $\zeta_{1}$ at $B$ intersects $\zeta_{2}$ at $D$ and $E$. The tangents of $\zeta_{1}$ passing thorugh $C$ touch $\zeta_{2}$ at $F$ and $G$. Prove that $D$, $E$, $F$ and $G$ are concyclic.
2004 Nordic
Twenty-seven balls labelled from $1$ to $27$ are distributed in three bowls: red, blue, and yellow. What are the possible values of the number of balls in the red bowl if the average labels in the red, blue and yellow bowl are $15$, $3$, and $18$, respectively?
2004 Nordic
The Fibonacci sequence is defined by $f_1 = 0, f_2 = 1$, and $f_{n+2} = f_{n+1}+f_n$ for $n\ge 1$. Prove that there is a strictly increasing arithmetic progression whose no term is in the Fibonacci sequence.
2004 Nordic
Given a finite sequence $x_{1,1}, x_{2,1}, \dots , x_{n,1}$ of integers $(n\ge 2)$, not all equal, define the sequences $x_{1,k}, \dots , x_{n,k}$ by

$$ x_{i,k+1}=\frac{1}{2}(x_{i,k}+x_{i+1,k})\quad\text{where }x_{n+1,k}=x_{1,k}.$$

Show that if $n$ is odd, then not all $x_{j,k}$ are integers. Is this also true for even $n$?
2004 Nordic
Let $a, b, c$ be the sides and $R$ be the circumradius of a triangle. Prove that

$$\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}\ge\frac{1}{R^2}.$$
2003 Nordic
The squares of a rectangular chessboard with 10 rows and 14 columns are colored alternatingly black and white in the usual manner. Some stones

are placed the board (possibly more than one on the same square) so that there are an odd number of stones in each row and each column.

Show that the total number of stones on black squares is even.
2003 Nordic
Find all triples of integers ${(x, y, z)}$ satisfying ${x^3 + y^3 + z^3 - 3xyz = 2003}$
2003 Nordic
The point ${D}$ inside the equilateral triangle ${\triangle ABC}$ satisfies ${\angle ADC = 150^o}$. Prove that a triangle with side lengths ${|AD|, |BD|, |CD|}$ is necessarily a right-angled triangle.
2003 Nordic
Let ${R^* = R-\{0\}}$ be the set of non-zero real numbers. Find all functions ${f : R^* \rightarrow R^*}$ satisfying ${f(x) + f(y) = f(xy f(x + y))}$, for ${x, y \in R^*}$  and ${ x + y\ne  0 }$.
2002 Nordic
The trapezium ${ABCD}$, where ${AB}$ and ${CD}$ are parallel and ${AD < CD}$, is inscribed in the circle ${c}$. Let ${DP}$ be a chord of the circle, parallel to ${AC}$. Assume that the tangent to ${c}$ at ${D}$ meets the line ${AB}$ at ${E}$ and that ${PB}$ and ${DC}$ meet at ${Q}$. Show that ${EQ = AC}$.
2002 Nordic
In two bowls there are in total ${N}$ balls, numbered from ${1}$ to ${N}$. One ball is moved from one of the bowls into the other. The average of the numbers in the bowls is increased in both of the bowls by the same amount, ${x}$. Determine the largest possible value of ${x}$.
2002 Nordic
Let ${a_1, a_2, . . . , a_n,}$ and ${b_1, b_2, . . . , b_n}$ be real numbers with ${a_1, a_2, . . . , a_n}$ distinct. Show that if  the product ${(a_i + b_1)(a_i + b_2) \cdot \cdot \cdot (a_i + b_n)}$ takes the same value for every ${ i = 1, 2, . . . , n, }$ , then the product  ${(a_1 + b_j)(a_2 + b_j)  \cdot \cdot \cdot (a_n + b_j)}$ also takes the same value for every ${j = 1, 2, . . . , n, }$ .
2002 Nordic
Eva, Per and Anna play with their pocket calculators. They choose different integers and check, whether or not they are divisible by ${11}$. They only look at nine-digit numbers consisting of all the digits ${1, 2, . . . , 9}$. Anna claims that the probability of such a number to be a multiple of ${11}$ is exactly ${1/11}$. Eva has a different opinion: she thinks the probability is less than ${1/11}$. Per thinks the probability is more than ${1/11}$. Who is correct?
2001 Nordic
Let ${A}$ be a finite collection of squares in the coordinate plane such that the vertices of all squares that belong to ${A}$ are ${(m, n), (m + 1, n), (m, n + 1)}$, and ${(m + 1, n + 1)}$ for some integers ${m}$ and ${n}$. Show that there exists a subcollection ${B}$ of ${A}$ such that ${B}$ contains at least ${25 \% }$ of the squares in ${A}$, but no two of the squares in ${B}$ have a common vertex.
2001 Nordic
Let ${f}$ be a bounded real function defined for all real numbers and satisfying for all real numbers ${x}$  the condition ${ f \Big(x+\frac{1}{3}\Big) + f \Big(x+\frac{1}{2}\Big)=f(x)+ f \Big(x+\frac{5}{6}\Big)}$ . Show that ${f}$ is periodic.
2001 Nordic
Determine the number of real roots of the equation

${x^8 -x^7 + 2x^6- 2x^5 + 3x^4 - 3x^3 + 4x^2 - 4x + \frac{5}{2}= 0}$
2001 Nordic
Let ${ABCDEF}$ be a convex hexagon, in which each of the diagonals ${AD, BE}$ , and ${CF}$ divides the hexagon into two quadrilaterals of equal area. Show that ${AD, BE}$ , and ${CF}$ are concurrent.
2000 Nordic
In how many ways can the number $2000$ be written as a sum of three positive, not necessarily different integers? (Sums like $1 + 2 + 3$ and $3 + 1 + 2$ etc. are the same.)
2000 Nordic
The persons $P_1, P_2, . . . , P_{n-1}, P_n$ sit around a table, in this order, and each one of them has a number of coins. In the start, $P_1$ has one coin more than $P_2, P_2$ has one coin more than $P_3$, etc., up to $P_{n-1}$ who has one coin more than $P_n$. Now $P_1$ gives one coin to $P_2$, who in turn gives two coins to $P_3 $ etc., up to $ Pn$ who gives n coins to $ P_1$. Now the process continues in the same way: $P_1$ gives $n+ 1$ coins to $P_2$, $P_2$ gives $n+2$ coins to $P_3$; in this way the transactions go on until someone has not enough coins, i.e. a person no more can give away one coin more than he just received. At the moment when the process comes to an end in this manner, it turns out that there are two neighbours at the table such that one of them has exactly five times as many coins as the other. Determine the number of persons and the number of coins circulating around the table.
2000 Nordic
In the triangle $ABC$, the bisector of angle $\angle B$ meets $AC$ at $D$ and the bisector of angle $\angle C$ meets $AB$ at $E$. The bisectors meet each other at $O$. Furthermore, $OD = OE$. Prove that either $ABC$ is isosceles or $\angle BAC = 60^\circ$.
2000 Nordic
The real-valued function $f$ is defined for $0 \le x \le 1, f(0) = 0, f(1) = 1$, and $\frac{1}{2} \le \frac{ f(z) - f(y)}{f(y) - f(x)} \le 2$ for all $0 \le x < y < z \le 1$ with $z - y = y -x$. Prove that $\frac{1}{7} \le  f (\frac{1}{3} ) \le \frac{4}{7}$.
1999 Nordic
The function $f$ is defined for non-negative integers and satisfies the condition $f(n) = f(f(n + 11))$, if $n \le 1999$ and

$f(n) =  n - 5$, if $n > 1999$. Find all solutions of the equation $f(n) = 1999$.
1999 Nordic
Consider $7$-gons inscribed in a circle such that all sides of the $7$-gon are of different length. Determine the maximal number of $120^\circ$ angles in this kind of a $7$-gon.
1999 Nordic
The infinite integer plane $Z\times  Z = Z^2$ consists of all number pairs $(x, y)$, where $x$ and $y$ are integers. Let $a$ and $b$ be non-negative integers. We call any move from a point $(x, y)$ to any of the points $(x\pm a, y \pm b)$ or $(x \pm b, y  \pm a) $ a $(a, b)$-knight move. Determine all numbers $a$ and $b$, for which it is possible to reach all points of the integer plane from an arbitrary starting point using only $(a, b)$-knight moves.
1999 Nordic
Let $a_1, a_2, . . . , a_n$ be positive real numbers and $n \ge 1$. Show that

$n (\frac{1}{a_1}+...+\frac{1}{a_n}) \ge (\frac{1}{1+a_1}+...+\frac{1}{1+a_n})(n+\frac{1}{a_1}+...+\frac{1}{a_n})$

When does equality hold?
1998 Nordic
Determine all functions $ f$ defined in the set of rational numbers and taking their values in the same set such that the equation $ f(x + y) + f(x - y) = 2f(x) + 2f(y)$ holds for all rational numbers $x$ and $y$.
1998 Nordic
Let $C_1$ and $C_2$ be two circles intersecting at $A $ and $B$. Let $S$ and $T $ be the centres of $C_1 $ and $C_2$, respectively. Let $P$ be a point on the segment $AB$ such that $ |AP|\ne  |BP|$ and $P\ne A, P \ne  B$. We draw a line perpendicular to $SP$ through $P$ and denote by $C$ and $D$ the points at which this line intersects $C_1$. We likewise draw a line perpendicular to $TP$ through $P$ and denote by $E$ and F the points at which this line intersects $C_2$. Show that $C, D, E,$ and $F$ are the vertices of a rectangle.
1998 Nordic
(a) For which positive numbers $n$ does there exist a sequence $x_1, x_2, ..., x_n$, which contains each of the numbers $1, 2, ..., n$ exactly once and for which $x_1 + x_2 +... + x_k$ is divisible by $k$ for each $k = 1, 2,...., n$?

(b) Does there exist an infinite sequence $x_1, x_2, x_3, ..., $ which contains every positive integer exactly once and such that $x_1 + x_2 +... + x_k$ is divisible by $k$ for every positive integer $k$?
1998 Nordic
Let $n$ be a positive integer. Count the number of numbers $k \in \{0, 1, 2, . . . , n\}$ such that $\binom{n}{k}$ is odd. Show that this number is a power of two, i.e. of the form $2^p$ for some nonnegative integer $p$.
1997 Nordic
Let $A$ be a set of seven positive numbers. Determine the maximal number of triples $(x, y, z)$ of elements of $A$

satisfying $x < y$ and $x + y = z$.
1997 Nordic
Let $ABCD$ be a convex quadrilateral. We assume that there exists a point $P$ inside the quadrilateral such that

the areas of the triangles $ABP, BCP, CDP$, and $DAP$ are equal. Show that at least one of the diagonals of the

quadrilateral bisects the other diagonal.
1997 Nordic
Let $A, B, C$, and $D$ be four different points in the plane. Three of the line segments $AB, AC, AD, BC, BD$,

and $CD$ have length $a$. The other three have length $b$, where $b > a$. Determine all possible values of the quotient $\frac{b}{a}$.

.
1997 Nordic
Let f be a function defined in the set $\{0, 1, 2,...\}$ of  non-negative integers, satisfying $f(2x) = 2f(x), f(4x+1) =

4f(x) + 3$, and $f(4x-1) = 2f(2x - 1) -1$.

Show that $f $ is an injection, i.e. if $f(x) = f(y)$, then $x = y$.
1996 Nordic
Show that there exists an integer divisible by $1996$ such that the sum of the its decimal digits is $1996$.
1996 Nordic
Determine all real numbers $x$, such that $x^n+x^{-n}$ is an integer for all integers $n$.
1996 Nordic
The circle whose diameter is the altitude dropped from the vertex $A$ of the triangle $ABC$ intersects the sides

$AB$ and $AC$ at $D$ and $E$, respectively $(A\ne D, A \ne E)$. Show that the circumcentre of $ABC$ lies on the altitude dropped from the vertex $A$ of the triangle $ADE$, or on its extension.
1996 Nordic
The real-valued function $f$ is defined for positive integers, and the positive integer $a$ satisfies

$f(a) = f(1995), f(a+1) = f(1996), f(a+2) = f(1997), f(n + a) = \frac{f(n) - 1}{f(n) + 1}$ for all positive integers $n$.

(i) Show that $f(n+ 4a) = f(n)$ for all positive integers $n$.

(ii) Determine the smallest possible $a$.
1995 Nordic
Let $AB$ be a diameter of a circle with centre $O$. We choose a point $C$ on the circumference of the circle such that $OC$ and $AB$ are perpendicular to each other. Let $P$ be an arbitrary point on the (smaller) arc $BC$ and let the lines $CP$ and $AB$ meet at $Q$. We choose $R$ on $AP$ so that $RQ$ and $AB$ are perpendicular to each other. Show that $BQ =QR$.
1995 Nordic
Messages are coded using sequences consisting of zeroes and ones only. Only sequences with at most two consecutive ones or zeroes are allowed. (For instance the sequence $011001$ is allowed, but $011101$ is not.) Determine the number of sequences consisting of exactly $12$ numbers.
1995 Nordic
Let $n \ge 2$ and let $x_1, x_2, ..., x_n$ be real numbers satisfying $x_1 +x_2 +...+x_n \ge 0$ and $x_1^2+x_2^2+...+x_n^2=1$. Let $M = max \{x_1, x_2,... , x_n\}$. Show that $M \ge  \frac{1}{\sqrt{n(n-1)}}$      (1) .When does equality hold in (1)?
1995 Nordic
Show that there exist infinitely many mutually non- congruent triangles $T$, satisfying

(i) The side lengths of $T $ are consecutive integers.

(ii) The area of $T$ is an integer.
1994 Nordic
Let $O$ be an interior point in the equilateral triangle $ABC$, of side length $a$. The lines $AO, BO$, and $CO$ intersect the sides of the triangle in the points $A_1, B_1$, and $C_1$. Show that  $OA_1 + OB_1 + OC_1 < a$.
1994 Nordic
We call a finite plane set $S$ consisting of points with integer coefficients a two-neighbour set, if for each point $(p, q)$ of $S$ exactly two of the points $(p +1, q), (p, q +1), (p-1, q), (p, q-1)$ belong to $S$. For which integers $n$ there exists a two-neighbour set which contains exactly $n$ points?
1994 Nordic
A piece of paper is the square $ABCD$. We fold it by placing the vertex $D$ on the point $D' $  of the side $BC$. We assume that $AD$ moves on the segment $A' D'$ and that $A'  D' $  intersects $AB$ at $E$. Prove that the perimeter of the triangle $EBD' $  is one half of the perimeter of the square.
1994 Nordic
Determine all positive integers $n < 200$, such that $n^2 + (n+ 1)^2$ is the square of an integer.
1993 Nordic
Let $F$ be an increasing real function defined for all $x, 0 \le x \le 1$, satisfying the conditions

(i) $F (\frac{x}{3}) = \frac{F(x)}{2}$.

(ii) $F(1- x) = 1 - F(x)$.

Determine $F(\frac{173}{1993})$ and   $F(\frac{1}{13})$ .
1993 Nordic
A hexagon is inscribed in a circle of radius $r$. Two of the sides of the hexagon have length $1$, two have length

$2$ and two have length $3$. Show that $r$ satisfies the equation $2r^3 - 7r - 3 = 0$.
1993 Nordic
Find all solutions of the system of equations

$\begin{cases} s(x) + s(y) = x \\ x + y + s(z) = z \\ s(x) + s(y) + s(z) = y - 4 \end{cases}$

where $x, y$, and $z$ are positive integers, and $s(x), s(y)$, and $s(z)$ are the numbers of digits in the decimal representations of $x, y$, and $z$, respectively.
1993 Nordic
Denote by $T(n)$ the sum of the digits of the decimal representation of a positive integer $n$.

a) Find an integer $N$, for which $T(k \cdot  N)$ is even for all $k, 1 \le  k \le 1992, $ but $T(1993 \cdot N)$ is odd.

b) Show that no positive integer $N$ exists such that $T(k \cdot N)$ is even for all positive integers $k$.
1992 Nordic
Determine all real numbers $x > 1, y > 1$, and $z > 1$,satisfying the equation

$x+y+z+\frac{3}{x-1}+\frac{3}{y-1}+\frac{3}{z-1}=2(\sqrt{x+2}+\sqrt{y+2}+\sqrt{z+2})$
1992 Nordic
Prove that among all triangles with inradius $1$, the equilateral one has the smallest perimeter .
1992 Nordic
Peter has many squares of equal side. Some of the squares are black, some are white. Peter wants to assemble

a big square, with side equal to $n$ sides of the small squares, so that the big square has no rectangle formed by the small squares such that all the squares in the vertices of the rectangle are of equal colour. How big a square is Peter able to assemble?
1991 Nordic
Determine the last two digits of the number $2^5 + 2^{5^{2}} + 2^{5^{3}} +... + 2^{5^{1991}}$ , written in decimal notation.
1991 Nordic
In the trapezium $ABCD$ the sides $AB$ and $CD$ are parallel, and $E$ is a fixed point on the side $AB$. Determine the point $F$ on the side $CD$ so that the area of the intersection of the triangles $ABF$ and $CDE$ is as large as possible.
1991 Nordic
Show that  $ \frac{1}{2^2}  +\frac{1}{3^2}  +...+\frac{1}{n^2} <\frac{2}{3}$ for all $n \ge 2 $.
1991 Nordic
Let $f(x)$ be a polynomial with integer coefficients. We assume that there exists a positive integer $k$ and $k$ consecutive integers $n, n+1, ... , n+k -1$ so that none of the numbers $f(n), f(n+ 1),... , f(n + k - 1)$ is divisible by $k$.

Show that the zeroes of $f(x)$ are not integers.
1990 Nordic
Let $m, n,$ and $p$ be odd positive integers. Prove that the number $\sum\limits_{k=1}^{{{(n-1)}^{p}}}{{{k}^{m}}}$  is divisible by $n$
1990 Nordic
Let $a_1, a_2, . . . , a_n$ be real numbers. Prove

$\sqrt[3]{a_1^3+ a_2^3+ . . . + a_n^3} \le  \sqrt{a_1^2+ a_2^2+ . . . + a_n^2} $ (1)

When does equality hold in (1)?
1990 Nordic
Let $ABC$ be a triangle and let $P$ be an interior point of $ABC$. We assume that a line $l$, which passes through $P$, but not through $A$, intersects $AB$ and $AC$ (or their extensions over $B$ or $C$) at $Q$ and $R$, respectively. Find $l$ such that the perimeter of the triangle $AQR$ is as small as possible.
1990 Nordic
It is possible to perform three operations $f, g$, and $h$ for positive integers: $f(n) = 10n, g(n) = 10n + 4$, and $h(2n) = n$; in other words, one may write $0$ or $4$ in the end of the number and one may divide an even number by $2$. Prove: every positive integer can be constructed starting from $4$ and performing a finite number of the operations $f,

g,$ and $h$ in some order.
1989 Nordic
Find a polynomial $P$ of lowest possible degree such that

(a) $P$ has integer coefficients,

(b) all roots of $P$ are integers,

(c) $P(0) = -1$,

(d) $P(3) = 128$.
1989 Nordic
Three sides of a tetrahedron are right-angled triangles having the right angle at their common vertex. The areas

of these sides are $A, B$, and $C$. Find the total surface area of the tetrahedron.
1989 Nordic
Let $S$ be the set of all points $t$ in the closed interval $[-1, 1]$ such that for the sequence $x_0, x_1, x_2, ...$ defined by the equations $x_0 = t, x_{n+1} = 2x_n^2-1$, there exists a positive integer $N$ such that $x_n = 1$ for all $n \ge N$. Show that the set $S$ has infinitely many elements.
1989 Nordic
For which positive integers $n$ is the following statement true:

if $a_1, a_2, ... , a_n$ are positive integers, $a_k \le n$ for all $k$ and $\sum\limits_{k=1}^{{n}}{a_k}=2n$

then it is always possible to choose $a_{i1} , a_{i2} , ..., a_{ij}$ in such a way that

the indices $i_1, i_2,... , i_j$ are different numbers, and  $\sum\limits_{k=1}^{{{j}}}{a_{ik}}=n$?
1988 Nordic
The positive integer $ n$ has the following property:

if the three last digits of $n$ are removed, the number $\sqrt[3]{n}$ remains.

Find $n$.
1988 Nordic
Let $a, b,$ and $c$ be non-zero real numbers and let $a \ge  b  \ge  c$. Prove the inequality

$\frac{a^3 - c^3}{3} \ge  abc (\frac{a- b}{c}+ \frac{b- c}{a})$ . When does equality hold?
1988 Nordic
Two concentric spheres have radii $r$ and $R,r < R$. We try to select points $A, B$ and $C$ on the surface of the

larger sphere such that all sides of the triangle $ABC$ would be tangent to the surface of the smaller sphere. Show that

the points can be selected if and only if $R \le  2r$.
1988 Nordic
Let $m_n$ be the smallest value of the function ${{f}_{n}}\left( x \right)=\sum\limits_{k=0}^{2n}{{{x}^{k}}}$

Show that $m_n  \to \frac{1}{2}$, as $n \to \infty.$
1987 Nordic
Nine journalists from different countries attend a press conference. None of these speaks more than three

languages, and each pair of the journalists share a common language. Show that there are at least five journalists sharing a common language.
1987 Nordic
Let $ABCD$ be a parallelogram in the plane. We draw two circles of radius $R$, one through the points $A$ and

$B$, the other through $B$ and $C$. Let $E$ be the other point of intersection of the circles. We assume that $E$ is not a vertex of the parallelogram. Show that the circle passing through $A, D$, and $E$ also has radius $R$.
1987 Nordic
Let $f$ be a strictly increasing function defined in the set of natural numbers satisfying the conditions $f(2) = a >

2$ and $f(mn) = f(m)f(n)$ for all natural numbers $m$ and $n$. Determine the smallest possible value of $a$.
1987 Nordic
Let $a, b$, and $c$ be positive real numbers. Prove: $\frac{a}{b}+ \frac{b}{c}+ \frac{c}{a}\le  \frac{a^2}{b^2} + \frac{b^2}{c^2} + \frac{c^2}{a^2}$ .
2021 Pan-African
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.
2021 Pan-African
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 $\angle PBT=\angle P'KA$
2021 Pan-African
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
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.
2021 Pan-African
Find all functions $f$ $:$ $\mathbb{R} \rightarrow \mathbb{R}$  such that $\forall x,y \in \mathbb{R}$ :

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