Navvye/Olympiad-Problems-CSV huggingface dataset:
2004 Abels Math Contest (Norwegian MO)
If $m$ is a positive integer, prove that $2^m$ cannot be written as a sum of two or more consecutive natural numbers.
2004 Abels Math Contest (Norwegian MO)
Let $a_1,a_2,a_3,...$ be a strictly increasing sequence of positive integers. A number $a_n$ in the sequence is said to be lucky  if it is the sum of several (not necessarily distinct) smaller terms of the sequence, and unlucky otherwise. (For example, in the sequence $4,6,14,15,25,...$ numbers $4,6,15$ are unlucky, while $14 = 4+4+6$ and $25 = 4+6+15$ are lucky.) Prove that there are only finitely many unlucky numbers in the sequence.
2004 Abels Math Contest (Norwegian MO)
(a) Prove that $(x+y+z)^2 \le 3(x^2 +y^2 +z^2)$ for any real numbers $x,y,z$.

(b) If positive numbers $a,b,c$ satisfy $a+b+c \ge abc$, prove that $a^2 +b^2 +c^2 \ge \sqrt3 abc$
2004 Abels Math Contest (Norwegian MO)
In a quadrilateral $ABCD$ with $\angle A = 60^o, \angle B = 90^o, \angle C = 120^o$, the point $M$ of intersection of the diagonals satisfies $BM = 1$ and $MD = 2$.

(a) Prove that the vertices of $ABCD$ lie on a circle and find the radius of that circle.

(b) Find the area of quadrilateral $ABCD$.
2004 Abels Math Contest (Norwegian MO)
Among the $n$ inhabitants of an island, where $n$ is even, every two are either friends or enemies. Some day, the chief of the island orders that each inhabitant (including himself) makes and wears a necklace consisting of marbles, in such a way that two necklaces have a marble of the same type if and only if their owners are friends.

(a) Show that the chief’s order can be achieved by using $n^2/4$ different types of stones.

(b) Prove that this is not necessarily true with less than $n^2/4$ types.
2005 Abels Math Contest (Norwegian MO)
A positive integer $m$ is called triangular if $m = 1 + 2 + ... + n$, for an integer $n$. Show that a positive integer $m$ is  triangular  if and only if $8m + 1$ is the square of an integer.
2005 Abels Math Contest (Norwegian MO)
In a pyramid, the base is a right-angled triangle with integer sides. The height of the pyramid is also integer. Show that the volume of the pyramid is even.
2005 Abels Math Contest (Norwegian MO)
In an aquarium there are nine small fish. The aquarium is cube shaped with a side length of two meters and is completely filled with water. Show that it is always possible to find two small fish with a distance of less than $\sqrt3$ meters.
2005 Abels Math Contest (Norwegian MO)
Let $A$ be the number of all points with integer coordinates in a three-dimensional coordinate system. We assume that nine arbitrary points in $A$ will be colored blue. Show that we can always find two blue dots so that the line segment between them contains at least one point from $A$.
2005 Abels Math Contest (Norwegian MO)
In the isosceles triangle $\vartriangle ABC$ is $AB = AC$. Let $D$ be the midpoint of the segment $BC$. The points $P$ and $Q$ are respectively on the lines $AD$ and $AB$ (with $Q \ne B$) so that $PQ = PC$. Show that $\angle PQC =\frac12  \angle A $
2005 Abels Math Contest (Norwegian MO)
In the parallelogram $ABCD$, all sides are equal, and $\angle A = 60^o$. Let $F$ be a point on line $AD, H$ a point on line $DC$, and $G$ a point on diagonal $AC$ such that $DFGH$ is a parallelogram. Show that then $\vartriangle BHF$ is equilateral.
2005 Abels Math Contest (Norwegian MO)
Show that for all positive real numbers $a, b$ and $c$, the inequality $(a+b)(a+c)\ge 2\sqrt{abc(a+b+c)}$ is true.
2005 Abels Math Contest (Norwegian MO)
Let $a, b$ and $c$ be real numbers such that $ab + bc + ca> a + b + c> 0$. Show then  that  $a+b+c>3$
2006 Abels Math Contest (Norwegian MO)
Each square in an $n \times  n$ table is painted black or white. The routes where two rows meet two columns, called a quartet if the remaining squares are the same color.

(a) What is the largest possible number of black squares in a $4 \times  4$ table without quartets?

(b) Is it possible to paint a $5 \times 5$ table so that it has no quartets?
2006 Abels Math Contest (Norwegian MO)
a) Let $a$ and $b$ be two non-negative real numbers. Show that $a+b \ge \sqrt{\frac{a^2+b^2}{2}}+ \sqrt{ab}$

b) Let $a$ and $b$ be two real numbers in $[0, 3]$. Show that $\sqrt{\frac{a^2+b^2}{2}}+ \sqrt{ab} \ge \frac{(a+b)^2}{2}$
2006 Abels Math Contest (Norwegian MO)
(a) Let $a$ and $b$ be rational numbers such that line $y = ax + b$ intersects the circle $x^2 + y^2 = 5$ at two different points. Show that if one of the intersections has two rational coordinates, so does the other intersection.

(b) Show that there are infinitely many triples ($k, n, m$) that are such that $k^2 + n^2 = 5m^2$, where $k, n$ and $m$ are integers, and not all three have any in common prime factor.
2006 Abels Math Contest (Norwegian MO)
Let $\gamma$ be the circumscribed circle about a right-angled triangle $ABC$ with right angle $C$. Let $\delta$ be the circle tangent to the sides $AC$ and $BC$ and tangent to the circle $\gamma$ internally.

(a) Find the radius $i$ of $\delta$ in terms of $a$ when $AC$ and $BC$ both have length $a$.

(b) Show that the radius $i$ is twice the radius of the inscribed circle of $ABC$.
2007 Abels Math Contest (Norwegian MO) Final
We consider the sum of the digits of a positive integer.

For example, the sum of the digits of $2007$ is equal to $9$, since $2 + 0 + 0 + 7 = 9$.

(a) How many integers $n$, where $0 < n < 100 000$, have an even sum of digits?

(b) How many integers $n$, where $0 < n < 100 000$, have a sum of digits that is less than or equal to $22$?
2007 Abels Math Contest (Norwegian MO) Final
The vertices of a convex pentagon $ABCDE$ lie on a circle $\gamma_1$.

The diagonals $AC , CE, EB, BD$, and $DA$ are tangents to another circle $\gamma_2$ with the same centre as $\gamma_1$.

(a) Show that all angles of the pentagon $ABCDE$ have the same size and that all edges of the pentagon have the same length.

(b) What is the ratio of the radii of the circles $\gamma_1$ and $\gamma_2$? (The answer should be given in terms of integers, the four basic arithmetic operations and extraction of roots only.)
2007 Abels Math Contest (Norwegian MO) Final
(a) Let $x$ and $y$ be two positive integers such that $\sqrt{x} +\sqrt{y}$ is an integer.

Show that $\sqrt{x}$  and $\sqrt{y}$ are both integers.

(b) Find all positive integers $x$ and $y$ such that $\sqrt{x} +\sqrt{y}=\sqrt{2007}$.
2007 Abels Math Contest (Norwegian MO) Final
Let $a, b$ and $c$ be integers such that $a + b + c = 0$.

(a) Show that $a^4 + b^4 + c^4$ is divisible by $a^2 + b^2 + c^2$.

(b) Show that $a^{100} + b^{100} + c^{100}$ is divisible by $a^2 + b^2 + c^2$.

.
2008 Abels Math Contest (Norwegian MO) Final
Let $s(n) = \frac16 n^3 - \frac12 n^2 + \frac13 n$.

(a) Show that $s(n)$ is an integer whenever $n$ is an integer.

(b) How many integers $n$ with $0 < n \le 2008$ are such that $s(n)$ is divisible by $4$?
2008 Abels Math Contest (Norwegian MO) Final
We wish to lay down boards on a floor with width $B$ in the direction across the boards. We have $n$ boards of width $b$, and $B/b$ is an integer, and $nb \le B$. There are enough boards to cover the floor, but the boards may have different lengths. Show that we can cut the boards in such a way that every board length on the floor has at most one join where two boards meet end to end.
2008 Abels Math Contest (Norwegian MO) Final
A and B play a game on a square board consisting of $n \times n$ white tiles, where $n \ge 2$. A moves first, and the players alternate taking turns. A move consists of picking a square consisting of $2\times 2$ or $3\times 3$ white tiles and colouring all these tiles black. The first player who cannot find any such squares has lost. Show that A can always win the game if A plays the game right.
2008 Abels Math Contest (Norwegian MO) Final
a) Let $x$ and $y$ be positive numbers such that $x + y = 2$.

Show that $\frac{1}{x}+\frac{1}{y}  \le \frac{1}{x^2}+\frac{1}{y^2}$



b) Let $x,y$ and $z$ be positive numbers such that $x + y +z= 2$.

Show that $\frac{1}{x}+\frac{1}{y} +\frac{1}{z} +\frac{9}{4}  \le \frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}$

.
2008 Abels Math Contest (Norwegian MO) Final
Three distinct points $A, B$, and $C$ lie on a circle with centre at $O$.

The triangles $AOB, BOC$ , and $COA$ have equal area.

What are the possible magnitudes of the angles of the triangle $ABC$ ?
2008 Abels Math Contest (Norwegian MO) Final
A point $D$ lies on the side $BC$ , and a point $E$ on the side $AC$ , of the triangle $ABC$ , and $BD$ and $AE$ have the same length. The line through the centres of the circumscribed circles of the triangles $ADC$ and $BEC$ crosses $AC$ in $K$ and $BC$ in $L$. Show that $KC$ and $LC$ have the same length.
2009 Abels Math Contest (Norwegian MO) Final
Show that there exist infinitely many integers that cannot be written as the difference between two perfect squares.
2009 Abels Math Contest (Norwegian MO) Final
Show that the sum of three consecutive perfect cubes can always be written as the difference between two perfect squares.
2009 Abels Math Contest (Norwegian MO) Final
There are two letters in a language.

Every word consists of seven letters, and two different words always have different letters on at least three places.

a. Show that such a language cannot have more than $16$ words.

b. Can there be $16$ words in the language?
2009 Abels Math Contest (Norwegian MO) Final
In the triangle $ABC$ the edge $BC$ has length $a$, the edge $AC$ length $b$, and the edge $AB$ length $c$. Extend all the edges at both ends – by the length $a$ from the vertex $A, b$ from $B$, and $c$ from $C$. Show that the six endpoints of the extended edges all lie on a common circle.
2009 Abels Math Contest (Norwegian MO) Final
Show for any positive integer $n$ that there exists a circle in the plane such that there are exactly $n$ grid points within the circle. (A grid point is a point having integer coordinates.)
2009 Abels Math Contest (Norwegian MO) Final
Show that $\left(\frac{2010}{2009}\right)^{2009}> 2$.
2009 Abels Math Contest (Norwegian MO) Final
Let $x = 1 - 2^{-2009}$. Show that $x + x^2 + x^4 + x^8 +... + x^{2^m}< 2010$ for all positive integers $m$.
2010 Abels Math Contest (Norwegian MO) Final
The point $P$ lies on the edge $AB$ of a quadrilateral $ABCD$.

The angles $BAD, ABC$ and $CPD$ are right, and $AB = BC + AD$.

Show that $BC = BP$ or $AD = BP$.
2010 Abels Math Contest (Norwegian MO) Final
The edges of the square in the figure have length $1$.

Find the area of the marked region in terms of $a$, where $0 \le  a  \le 1$.
2010 Abels Math Contest (Norwegian MO) Final
Show that $\frac{x^2}{1 - x}+\frac{(1 - x)^2}{x} \ge  1$ for all real numbers $x$, where $0 < x < 1$
2010 Abels Math Contest (Norwegian MO) Final
Show that $abc \le  (ab + bc + ca)(a^2 + b^2 + c^2)^2$ for all positive real numbers $a, b$ and $c$ such that $a + b + c = 1$.
2010 Abels Math Contest (Norwegian MO) Final
$ a)$

There are $ 25$ participants in a mathematics contest having four problems. Each problem is considered solved or not solved (that is, partial solutions are not possible). Show that either there are four contestants having solved the same problems (or not having solved any of them), or two contestants, one of which has solved exactly the problems that the other did not solve.



$ b)$

There are $ k$ sport clubs for the students of a secondary school. The school has $ 100$ students, and for any selection of three of them, there exists a club having at least one of them, but not all, as a member. What is the least possible value of $ k$?
2010 Abels Math Contest (Norwegian MO) Final
Find all positive integers $k$ and $\ell$ such that $k^2 -\ell^2 = 1005$.
2010 Abels Math Contest (Norwegian MO) Final
Let $n > 2$ be an integer. Show that it is possible to choose $n$ points in the plane, not all of them lying on the same line, such that the distance between any pair of points is an integer (that is, $\sqrt{(x_1 -x_2)^2 +(y_1 -y_2)^2}$ is an integer for all pairs $(x_1, y_1)$ and $(x_2, y_2)$ of points).
2011 Abels Math Contest (Norwegian MO)
Let $n$ be the number that is produced by concatenating the numbers $1, 2,... , 4022$,

that is, $n = 1234567891011...40214022$.

a. Show that $n$ is divisible by $3$.

b. Let $a_1 = n^{2011}$, and let $a_i$ be the sum of the digits of $a_{i-1}$ for $i > 1$. Find $a_4$
2011 Abels Math Contest (Norwegian MO)
In the quadrilateral $ABCD$ the side $AB$ has length $7, BC$ length $14, CD$ length $26$, and $DA$ length $23$. Show that the diagonals are perpendicular.



You may assume that the quadrilateral is convex (all internal angles are less than $180^o$).
2011 Abels Math Contest (Norwegian MO)
The diagonals $AD, BE$, and $CF$ of a convex hexagon $ABCDEF$ intersect in a common point.

Show that $a(ABE) a(CDA) a(EFC) = a(BCE) a(DEA) a(FAC)$,

where $a(KLM)$ is the area of the triangle $KLM$.
2011 Abels Math Contest (Norwegian MO)
The positive numbers $a_1, a_2,...$ satisfy $a_1 = 1$ and $(m+n)a_{m+n }\le  a_m +a_n$ for all positive integers $m$ and $n$. Show that $\frac{1}{a_{200}} > 4  \cdot 10^7$ .

.
2011 Abels Math Contest (Norwegian MO)
Find all functions $f$ from the real numbers to the real numbers such that $f(xy)  \le \frac12 \left(f(x) + f(y) \right)$ for all real numbers $x$ and $y$.
2011 Abels Math Contest (Norwegian MO)
In a town there are $n$ avenues running from south to north. They are numbered $1$ through $n$ (from west to east). There are $n$ streets running from west to east – they are also numbered $1$ through $n$ (from south to north).

If you drive through the junction of the $k$th avenue and the $\ell$th street, you have to pay $k\ell$ kroner. How much do you at least have to pay for driving from the junction of the $1$st avenue and the $1$st street to the junction of the nth avenue and the $n$th street? (You also pay for the starting and finishing junctions.)
2011 Abels Math Contest (Norwegian MO)
In a group of $199$ persons, each person is a friend of exactly $100$ other persons in the group. All friendships are mutual, and we do not count a person as a friend of himself/herself. For which integers $k > 1$ is the existence of $k$ persons, all being friends of each other, guaranteed?
2012 Abels Math Contest (Norwegian MO) Final
Berit has $11$ twenty kroner coins, $14$ ten kroner coins, and $12$ five kroner coins. An exchange machine can exchange three ten kroner coins into one twenty kroner coin and two five kroner coins, and the reverse. It can also exchange two twenty kroner coins into three ten kroner coins and two five kroner coins, and the reverse.

(i) Can Berit get the same number of twenty kroner and ten kroner coins, but no five kroner coins?

(ii) Can she get the same number each of twenty kroner, ten kroner, and five kroner coins?
2012 Abels Math Contest (Norwegian MO) Final
Every integer is painted white or black, so that if $m$ is white then $m + 20$ is also white, and if $k$ is black then $k + 35$ is also black. For which $n$ can exactly $n$ of the numbers $1, 2, ..., 50$ be white?
2012 Abels Math Contest (Norwegian MO) Final
(a)Two circles $S_1$ and $S_2$ are placed so that they do not overlap each other, neither completely nor partially. They have centres in $O_1$ and $O_2$, respectively. Further, $L_1$ and $M_1$ are different points on $S_1$ so that $O_2L_1$ and $O_2M_1$ are tangent to $S_1$, and similarly $L_2$ and $M_2$ are different points on $S_2$ so that $O_1L_2$ and $O_1M_2$ are tangent to $S_2$. Show that there exists a unique circle which is tangent to the four line segments $O_2L_1, O_2M_1, O_1L_2$, and $O_1M_2$.



(b) Four circles $S_1, S_2, S_3$ and $S_4$ are placed so that none of them overlap each other, neither completely nor partially. They have centres in $O_1, O_2, O_3$, and $O_4$, respectively. For each pair $(S_i, S_j )$ of circles, with $1 \le i < j \le  4$, we find a circle $S_{ij}$ as in part a. The circle $S_{ij}$ has radius $R_{ij}$ . Show that $\frac{1}{R_{12}} + \frac{1}{R_{23}}+\frac{1}{R_{34}}+\frac{1}{R_{14}}= 2 \left(\frac{1}{R_{13}} +\frac{1}{R_{24}}\right)$
2012 Abels Math Contest (Norwegian MO) Final
Find the last three digits in the product $1 \cdot 3\cdot  5\cdot 7 \cdot  . . . \cdot  2009 \cdot  2011$.
2012 Abels Math Contest (Norwegian MO) Final
Which positive integers $m$ are such that $k^m - 1$ is divisible by $2^m$ for all odd numbers $k \ge 3$?
2012 Abels Math Contest (Norwegian MO) Final
Two positive numbers $x$ and $y$ are given. Show that $\left(1 +\frac{x}{y} \right)^3 + \left(1 +\frac{y}{x}\right)^3 \ge 16$.
2012 Abels Math Contest (Norwegian MO) Final
Positive numbers $b_1, b_2,..., b_n$ are given so that $b_1 + b_2 + ...+ b_n \le 10$.

Further, $a_1 = b_1$ and $a_m = sa_{m-1} + b_m$ for $m > 1$, where $0 \le  s < 1$.

Show that $a^2_1 + a^2_2 + ... + a^2_n  \le \frac{100}{1 - s^2} $
2013 Abels Math Contest (Norwegian MO) Final
Find all real numbers $a$ such that the inequality $3x^2 + y^2 \ge  -ax(x + y)$ holds for all real numbers $x$ and $y$.
2013 Abels Math Contest (Norwegian MO) Final
The sequence $a_1, a_2, a_3,...$ is defined so that $a_1 = 1$ and $a_{n+1} =\frac{a_1 + a_2 + ...+ a_n}{n}+1$ for $n \ge 1$. Show that for every positive real number $b$ we can find $a_k$ so that $a_k < bk$.
2013 Abels Math Contest (Norwegian MO) Final
In a triangle $T$, all the angles are less than $90^o$, and the longest side has length $s$. Show that for every point $p$ in $T$ we can pick a corner $h$ in $T$ such that the distance from $p$ to $h$ is less than or equal to $s/\sqrt3$.
2013 Abels Math Contest (Norwegian MO) Final
A prime number $p \ge 5$ is given. Write $\frac13+\frac24+... +\frac{p -3}{p - 1}=\frac{a}{b}$ for natural numbers $a$ and $b$. Show that $p$ divides $a$.
2013 Abels Math Contest (Norwegian MO) Final
An ordered quadruple $(P_1, P_2, P_3, P_4)$ of corners in a regular $2013$-gon is called crossing  if the four corners are all different, and the line segment from $P_1$ to $P_2$ intersects the line segment from $P_3$ to $P_4$. How many crossing  quadruples are there in the $2013$-gon?
2013 Abels Math Contest (Norwegian MO) Final
A total of $a \cdot b \cdot c$ cubical boxes are joined together in a $a \times b \times c$ rectangular stack, where $a, b, c  \ge  2$. A bee is found inside one of the boxes. It can fly from one box to another through a hole in the wall, but not through edges or corners. Also, it cannot fly outside the stack. For which triples $(a, b, c)$ is it possible for the bee to fly through all of the boxes exactly once, and end up in the same box where it started?
2014 Abels Math Contest (Norwegian MO) Final
Assume that $x, y \ge 0$. Show that $x^2 + y^2 + 1 \le \sqrt{(x^3 + y + 1)(y^3 + x + 1)}$.
2014 Abels Math Contest (Norwegian MO) Final
Find all functions $f : R-\{0\} \to R$ which satisfy $(1 + y)f(x) - (1 + x)f(y) = yf(x/y) - xf(y/x)$ for all real $x, y \ne 0$, and which take the values $f(1) = 32$ and $f(-1) = -4$.
2014 Abels Math Contest (Norwegian MO) Final
The points $P$ and $Q$ lie on the sides $BC$ and $CD$ of the parallelogram ABCD so that $BP = QD$.

Show that the intersection point between the lines $BQ$ and $DP$ lies on the line bisecting $\angle BAD$.
2014 Abels Math Contest (Norwegian MO) Final
A grasshopper is jumping about in a grid. From the point with coordinates $(a, b)$ it can jump to either $(a + 1, b),(a + 2, b),(a + 1, b + 1),(a, b + 2)$ or $(a, b + 1)$. In how many ways can it reach the line $x + y = 2014?$  Where the grasshopper starts in $(0, 0)$.
2014 Abels Math Contest (Norwegian MO) Final
Nine points are placed on a circle. Show that it is possible to colour the $36$ chords connecting them using four colours so that for any set of four points, each of the four colours is used for at least one of the six chords connecting the given points
2014 Abels Math Contest (Norwegian MO) Final
Find all triples $(a, b, c)$ of positive integers for which $\frac{32a + 3b + 48c}{4abc}$ is also an integer.
2015 Abels Math Contest (Norwegian MO) Final
Find all triples $(x, y, z) \in R^3$ satisfying the equations $\begin{cases} x^2 + 4y^2 = 4zx \\
y^2 + 4z^2 = 4xy \\
 z^2 + 4x^2 = 4yz \end{cases}$
2015 Abels Math Contest (Norwegian MO) Final
Find all functions $f : R \to R$ such that $x^2f(yf(x))= y^2f(x)f(f(x))$ for all real numbers $x$ and $y$.
2015 Abels Math Contest (Norwegian MO) Final
King Arthur is placing $a + b + c$ knights around a table.

$a$ knights are dressed in red, $b$ knights are dressed in brown, and $c$ knights are dressed in orange.

Arthur wishes to arrange the knights so that no knight is seated next to someone dressed in the same colour as himself. Show that this is possible if, and only if, there exists a triangle whose sides have lengths $a +\frac12, b +\frac12$, and $c +\frac12$
2015 Abels Math Contest (Norwegian MO) Final
Nils is playing a game with a bag originally containing $n$ red and one black marble.

He begins with a fortune equal to $1$. In each move he picks a real number $x$ with $0 \le x \le  y$, where his present fortune is $y$. Then he draws a marble from the bag. If the marble is red, his fortune increases by $x$, but if it is black, it decreases by $x$. The game is over after $n$ moves when there is only a single marble left.

In each move Nils chooses $x$ so that he ensures a final fortune greater or equal to $Y$ .

What is the largest possible value of $Y$?
2015 Abels Math Contest (Norwegian MO) Final
The five sides of a regular pentagon are extended to lines $\ell_1, \ell_2, \ell_3, \ell_4$, and $\ell_5$.

Denote by $d_i$ the distance from a point $P$ to $\ell_i$.

For which point(s) in the interior of the pentagon is the product $d_1d_2d_3d_4d_5$ maximal?
2015 Abels Math Contest (Norwegian MO) Final
a. Determine all nonnegative integers $x$ and $y$ so that $3^x + 7^y$ is a perfect square and $y$ is even.

b. Determine all nonnegative integers $x$ and $y$ so that $3^x + 7^y$ is a perfect square and $y$ is odd
2016 Abels Math Contest (Norwegian MO) Final
A "walking sequence" is a sequence of integers with $a_{i+1} = a_i \pm 1$ for every $i$ .Show that there exists a sequence $b_1, b_2, . . . , b_{2016}$ such that for every walking sequence $a_1, a_2, . . . , a_{2016}$ where $1 \leq a_i \leq1010$, there is for some $j$ for which $a_j = b_j$ .
2016 Abels Math Contest (Norwegian MO) Final
Find all positive integers $a, b, c, d$ with $a \le  b$ and $c \le  d$ such that $\begin{cases} a + b = cd \\
c + d = ab \end{cases}$ .
2016 Abels Math Contest (Norwegian MO) Final
Find all non-negative integers $x, y$ and $z$ such that $x^3 + 2y^3 + 4z^3 = 9!$
2016 Abels Math Contest (Norwegian MO) Final
Three circles $S_A, S_B$, and $S_C$ in the plane with centers in $A, B$, and $C$, respectively, are mutually tangential on the outside. The point of tangency between $S_A$ and $S_B$ we call $C'$, the one $S_A$ between $S_C$ we call $B'$, and the one between $S_B$ and $S_C$ we call $A'$. The common tangent between $S_A$ and $S_C$ (passing through B') we call $\ell_B$, and the common tangent between $S_B$ and $S_C$ (passing through $A'$) we call  $\ell_A$. The point of intersection of  $\ell_A$ and  $\ell_B$ is called $X$. The point $Y$ is located so that $\angle XBY$ and $\angle YAX$ are both right angles. Show that the points $X, Y$, and $C'$ lie on a line if and only if $AC = BC$.
2016 Abels Math Contest (Norwegian MO) Final
Let $ABC$ be an acute triangle with $AB < AC$. The points $A_1$ and $A_2$ are located on the line $BC$ so that $AA_1$ and $AA_2$ are the inner and outer angle bisectors at $A$ for the triangle $ABC$. Let $A_3$ be the mirror image $A_2$ with respect to $C$, and let $Q$ be a point on AA1 such that $\angle A_1QA_3 = 90^o$. Show that $QC // AB$.
2016 Abels Math Contest (Norwegian MO) Final
Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that

$$ f(x) f(y) = |x - y| \cdot f \left( \frac{xy + 1}{x - y} \right) $$Holds for all $x \not= y \in \mathbb{R}$
2017 Abels Math Contest (Norwegian MO) Final
Find all functions $f : R \to R$ which satisfy $f(x)f(y) = f(xy) + xy$ for all $x, y \in R$.
2017 Abels Math Contest (Norwegian MO) Final
Find all functions $f : R \to R$ which satisfy $f(x)f(y) = f(x + y) + xy$ for all $x, y \in R$.
2017 Abels Math Contest (Norwegian MO) Final
Let the sequence an be defined by $a_0 = 2, a_1 = 15$, and $a_{n+2 }= 15a_{n+1} + 16a_n$ for $n \ge 0$.

Show that there are infinitely many integers $k$ such that $269 | a_k$.