Navvye/Olympiad-Problems-CSV huggingface dataset:
1936 Moscow Mathematical Olympiad
How many ways are there to represent $10^6$ as the product of three factors?

Factorizations which only differ in the order of the factors are considered to be distinct.
1936 Moscow Mathematical Olympiad
Given three planes and a ball in space. In space, find the number of different ways of placing another ball so that it would be tangent the three given planes and the given ball.
1935 Moscow Mathematical Olympiad
Find the ratio of two numbers if the ratio of their arithmetic mean to their geometric mean is $25 : 24$
1935 Moscow Mathematical Olympiad
Given the lengths of two sides of a triangle and that of the bisector of the angle between these sides, construct the triangle.
1935 Moscow Mathematical Olympiad
The base of a pyramid is an isosceles triangle with the vertex angle $\alpha$. The pyramid’s lateral edges are at angle $\phi$ to the base. Find the dihedral angle $\theta$ at the edge connecting the pyramid’s vertex to that of angle $\alpha$.
1935 Moscow Mathematical Olympiad
A train passes an observer in $t_1$ sec. At the same speed the train crosses a bridge $\ell$ m long. It takes the train $t_2$ sec to cross the bridge from the moment the locomotive drives onto the bridge until the last car leaves it. Find the length and speed of the train.
1935 Moscow Mathematical Olympiad
Given three parallel straight lines. Construct a square three of whose vertices belong to these lines.
1935 Moscow Mathematical Olympiad
The base of a right pyramid is a quadrilateral whose sides are each of length $a$. The planar angles at the vertex of the pyramid are equal to the angles between the lateral edges and the base. Find the volume of the pyramid.
1935 Moscow Mathematical Olympiad
Find four consecutive terms $a, b, c, d$ of an arithmetic progression and four consecutive terms $a_1, b_1, c_1, d_1$ of a geometric progression such that $a + a_1 = 27, b + b_1 = 27, c + c_1 = 39$, and $d + d_1 = 87$.
1935 Moscow Mathematical Olympiad
Prove that if the lengths of the sides of a triangle form an arithmetic progression, then the radius of the inscribed circle is one third of one of the heights of the triangle.
1935 Moscow Mathematical Olympiad
The height of a truncated cone is equal to the radius of its base. The perimeter of a regular hexagon circumscribing its top is equal to the perimeter of an equilateral triangle inscribed in its base. Find the angle $\phi$ between the cone’s generating line and its base.
1935 Moscow Mathematical Olympiad
Solve the system  $\begin{cases} x^2 + y^2 - 2z^2 = 2a^2 \\
x + y + 2z = 4(a^2 + 1) \\
z^2 - xy = a^2

\end{cases}$
1935 Moscow Mathematical Olympiad
In $\vartriangle ABC$, two straight lines drawn from an arbitrary point $D$ on $AB$ are parallel to $AC$ and BC and intersect $BC$ and $AC$ at $F$ and $G$, respectively. Prove that the sum of the circumferences of the circles circumscribed around $\vartriangle ADG$ and $\vartriangle BDF$ is equal to the circumference of the circle circumscribed around $\vartriangle ABC$.
1935 Moscow Mathematical Olympiad
The unfolding of the lateral surface of a cone is a sector of angle $120^o$. The angles at the base of a pyramid constitute an arithmetic progression with a difference of $15^o$. The pyramid is inscribed in the cone. Consider a lateral face of the pyramid with the smallest area. Find the angle $\alpha$ between the plane of this face and the base.
1935 Moscow Mathematical Olympiad
The median, bisector, and height, all originate at the same vertex of a triangle. Given the intersection points of the median, bisector, and height with the circumscribed circle, construct the triangle.
1935 Moscow Mathematical Olympiad
Find the locus of points on the surface of a cube that serve as the vertex of the smallest angle that subtends the diagonal.
1935 Moscow Mathematical Olympiad
Triangles $\vartriangle ABC$ and $\vartriangle A_1B_1C_1$ lie on different planes. Line $AB$ intersects line $A_1B_1$, line $BC$ intersects line $B_1C_1$ and line $CA$ intersects line $C_1A_1$. Prove that either the three lines $AA_1, BB_1, CC_1$ meet at one point or that they are all parallel.
1935 Moscow Mathematical Olympiad
How many real solutions does the following system have? $\begin{cases} x+y=2  \\
 xy - z^2 = 1 \end{cases}$
1935 Moscow Mathematical Olympiad
Solve the system $\begin{cases} x^3 - y^3 = 26 \\
x^2y - xy^2 = 6  

\end{cases}$



other versionsolved below

Solve the system $\begin{cases} x^3 - y^3 = 2b \\
x^2y - xy^2 = b  

\end{cases}$
1935 Moscow Mathematical Olympiad
Evaluate the sum: $1^3 + 3^3 + 5^3 +... + (2n - 1)^3$.
1935 Moscow Mathematical Olympiad
a) How many distinct ways are there are there of painting the faces of a cube six different colors?

(Colorations are considered distinct if they do not coincide when the cube is rotated.)



b)* How many distinct ways are there are there of painting the faces of a dodecahedron $12$ different colors?

(Colorations are considered distinct if they do not coincide when the cube is rotated.)
1935 Moscow Mathematical Olympiad
How many ways are there of representing a positive integer $n$ as the sum of three positive integers?

Representations which differ only in the order of the summands are considered to be distinct.
1935 Moscow Mathematical Olympiad
Denote by $M(a, b, c, . . . , k)$ the least common multiple and by $D(a, b, c, . . . , k)$ the greatest common divisor of $a, b, c, . . . , k$. Prove that:

a) $M(a, b)D(a, b) = ab$,

b) $\frac{M(a, b, c)D(a, b)D(b, c)D(a, c)}{D(a, b, c)}= abc$.
2018 Nepal National Olympiad
Problem Section #1

a) A set contains four numbers. The six pairwise sums of distinct elements of the set, in no particular order, are $189, 320, 287, 264, x$, and y. Find the greatest possible value of: $x + y$.



NOTE: There is a high chance that this problems was copied.
2018 Nepal National Olympiad
Problem Section #1

b) Let $a, b$ be positive integers such that $b^n +n$ is a multiple of $a^n + n$ for all positive integers $n$. Prove that $a = b.$
2018 Nepal National Olympiad
Problem Section #1

c) Find all pairs $(m, n)$ of non-negative integers for which $m^2+2.3^n=m(2^{n+1}-1).$
2018 Nepal National Olympiad
Problem Section #2

a) If $$ax+by=7$$$$ax^2+by^2=49$$$$ax^3+by^3=133$$$$ax^4+by^4=406$$,

find the value of    $2014(x+y-xy)-100(a+b).$
2018 Nepal National Olympiad
Problem Section #2

b) Find the maximal value of $(x^3+1)(y^3+1)$, where $x,y \in \mathbb{R}$, $x+y=1$.
2018 Nepal National Olympiad
Problem Section #2



c). Denote by $\mathbb{Q^+}$ the set of all positive rational numbers. Determine all functions $f:\mathbb{Q^+}\to\mathbb{Q^+}$ which satisfy the following equation for all

$x,y \in \mathbb{Q^+} : f(f(x)^2.y)=x^3.f(xy)$.
2018 Nepal National Olympiad
Problem Section #3





a) Circles $O_1$ and $O_2$ interest at two points $B$ and $C$, and $BC$ is the diameter of circle $O_1$. Construct a

tangent line of circle $O_1$ at $C$ and intersecting circle $O_2$ at another point $A$. Join $AB$ to intersect circle

$O_1$ at point $E$, then join $CE$ and extend it to intersect circle $O_2$ at point $F$. Assume $H$ is an arbitrary

point on line segment $AF$. Join $HE$ and extend it to intersect circle $O_1$ at point $G$, and then join $BG$

and extend it to intersect the extend of $AC$ at point $D$. Prove: $\frac{AH}{HF}=\frac{AC}{CD}$.
2018 Nepal National Olympiad
 Problem Section #3



NOTE: Neglect that HF and CD.
2018 Nepal National Olympiad
Problem Section #3



c) Let $ABCDE$ be a convex pentagon such that $BC \parallel AE, AB = BC + AE$, and $\angle{ABC} =\angle{CDE}$. Let $M$ be the midpoint of $CE$, and let $O$ be the circumcenter of triangle $BCD$. Given that $\angle{DMO}=90^{o}$, prove that  $2\angle{BDA} =\angle{CDE}$.
2018 Nepal National Olympiad
Problem Section #4



a) There is a $6 * 6$ grid, each square filled with a grasshopper. After the bell rings, each grasshopper

jumps to an adjacent square (A square that shares a side). What is the maximum number of empty

squares possible?
2018 Nepal National Olympiad
Problem Section #4



b) Let $A$ be a unit square. What is the largest area of a triangle whose vertices lie on the perimeter of

$A$? Justify your answer.
2018 Nepal National Olympiad
Problem Section #4



c) A special deck of cards contains $49$ cards, each labeled with a number from $1$ to $7$ and colored with

one of seven colors. Each number-color combination appears on exactly one card. John will select

a set of eight cards from the deck at random. Given that he gets at least one card of each color and

at least one card with each number, the probability that John can discard one of his cards and still

have at least one card of each color and at least one card with each number is $\frac{p}{q}$, where $p$ and $q$ are

relatively prime positive integers. Find $p+q$.
1993 Abels Math Contest (Norwegian MO)
Let $ABCD$ be a convex quadrilateral and $A',B'C',D'$ be the midpoints of $AB,BC,CD,DA$, respectively. Let $a,b,c,d$ denote the areas of quadrilaterals into which lines $A'C'$ and $B'D'$ divide the quadrilateral $ABCD$ (where a corresponds to vertex $A$ etc.). Prove that $a+c = b+d$.
1993 Abels Math Contest (Norwegian MO)
Given a triangle with sides of lengths $a,b,c$, prove that $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}< 2$.
1993 Abels Math Contest (Norwegian MO)
If $a,b,c,d$ are real numbers with $b < c < d$, prove that $(a+b+c+d)^2 > 8(ac+bd)$.
1993 Abels Math Contest (Norwegian MO)
The Fermat-numbers are defined by $F_n = 2^{2^n}+1$ for $n\in N$.

(a) Prove that $F_n = F_{n-1}F_{n-2}....F_1F_0 +2$ for $n > 0$.

(b) Prove that any two different Fermat numbers are coprime
1993 Abels Math Contest (Norwegian MO)
Each of the $8$ vertices of a given cube is given a value $1$ or $-1$.

Each of the $6$ faces is given the value of product of its four vertices.

Let $A$ be the sum of all the $14$ values. Which are the possible values of $A$?
1994 Abels Math Contest (Norwegian MO)
In a half-ball of radius $3$ is inscribed a cylinder with base lying on the base plane of the half-ball, and another such cylinder with equal volume. If the base-radius of the first cylinder is $\sqrt3$, what is the base-radius of the other one?
1994 Abels Math Contest (Norwegian MO)
Let $C$ be a point on the prolongation of the diameter $AB$ of a circle. A line through $C$ is tangent to the circle at point $N$. The bisector of $\angle ACN$ meets the lines $AN$ and $BN$ at $P$ and $Q$ respectively. Prove that $PN = QN$.
1994 Abels Math Contest (Norwegian MO)
Find all primes $p,q,r$ and natural numbers $n$ such that $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=\frac{1}{n}$.
1994 Abels Math Contest (Norwegian MO)
Find all integers $x,y,z$ such that $x^3 +5y^3 = 9z^3$.
1994 Abels Math Contest (Norwegian MO)
Let $x_1,x_2,...,x_{1994}$ be positive real numbers. Prove that

$$\left(\frac{x_1}{x_2}\right)^{\frac{x_1}{x_2}}\left(\frac{x_2}{x_3}\right)^{\frac{x_2}{x_3}}...\left(\frac{x_{1993}}{x_{1994}}\right)^{\frac{x_{1993}}{x_{1994}}} \ge \left(\frac{x_1}{x_2}\right)^{\frac{x_2}{x_1}}\left(\frac{x_2}{x_3}\right)^{\frac{x_3}{x_2}}...\left(\frac{x_{1993}}{x_{1994}}\right)^{\frac{x_{1994}}{x_{1993}}}$$
1994 Abels Math Contest (Norwegian MO)
Prove that there is no function $f : Z \to Z$ such that $f(f(x)) = x+1$ for all $x$.
1994 Abels Math Contest (Norwegian MO)
In a group of $20$ people, each person sends a letter to $10$ of the others.

Prove that there are two persons who send a letter to each other.
1994 Abels Math Contest (Norwegian MO)
Finitely many cities are connected by one-way roads. For any two cities it is possible to come from one of them to the other (with possible transfers), but not necessarily both ways. Prove that there is a city which can be reached from any other city, and that there is a city from which any other city can be reached.
1995 Abels Math Contest (Norwegian MO)
Let a function $f$ satisfy f$(1) = 1$ and $f(1)+ f(2)+...+ f(n) = n^2f(n)$ for all $n \in N$. Determine $f(1995)$.
1995 Abels Math Contest (Norwegian MO)
Prove that if  $(x+\sqrt{x^2 +1})(y+\sqrt{y^2 +1})= 1$ for real numbers $x,y$, then $x+y = 0$.
1995 Abels Math Contest (Norwegian MO)
Two circles $k_1,k_2$ touch each other at $P$, and touch a line $\ell$ at $A$ and $B$ respectively.

Line $AP$ meets $k_2$ at $C$. Prove that $BC$ is perpendicular to $\ell$.
1995 Abels Math Contest (Norwegian MO)
Two circles of the same radii intersect in two distinct points $A$ and $B$. A line $P$, not touching any of the circles, intersects the circles again at $A$ and $B$. Prove that $Q$ lies on the perpendicular bisector of $AB$.
1995 Abels Math Contest (Norwegian MO)
Show that there exists a sequence $x_1,x_2,...$ of natural numbers in which every natural number occurs exactly once, such that the sums $\sum_{i=1}^n \frac{1}{x_i}$, $n = 1,2,3,...$, include all natural numbers.
1995 Abels Math Contest (Norwegian MO)
Let $x_i,y_i$ be positive real numbers, $i = 1,2,...,n$. Prove that

$$\left( \sum_{i=1}^n (x_i +y_i)^2\right)\left( \sum_{i=1}^n\frac{1}{x_iy_i}\right)\ge 4n^2$$
1996 Abels Math Contest (Norwegian MO)
Let $S$ be a circle with center $C$ and radius $r$, and let $P \ne C$ be an arbitrary point.

A line $\ell$ through $P$ intersects the circle in $X$ and $Y$. Let $Z$ be the midpoint of $XY$.

Prove that the points $Z$, as $\ell$ varies, describe a circle. Find the center and radius of this circle.
1996 Abels Math Contest (Norwegian MO)
Prove that $[\sqrt{n}+\sqrt{n+1}]=[\sqrt{4n+1}]$ for all $n \in  N$.
1996 Abels Math Contest (Norwegian MO)
Per and Kari each have $n$ pieces of paper. They both write down the numbers from $1$ to $2n$ in an arbitrary order, one number on each side. Afterwards, they place the pieces of paper on a table showing one side. Prove that they can always place them so that all the numbers from $1$ to $2n$ are visible at once.
1996 Abels Math Contest (Norwegian MO)
Let $f : N \to  N$ be a function such that $f(f(1995)) = 95, f(xy) = f(x)f(y)$ and $f(x) \le x$ for all $x,y$.

Find all possible values of $f(1995)$.
1998 Abels Math Contest (Norwegian MO)
Let $a_0,a_1,a_2,...$ be an infinite sequence of positive integers such that $a_0 = 1$ and $a_i^2 > a_{i-1}a_{i+1}$ for all $i > 0$.

(a) Prove that $a_i < a_1^i$ for all $i > 1$.

(b) Prove that $a_i > i$ for all $i$.
1998 Abels Math Contest (Norwegian MO)
Let be given an $n \times n$ chessboard, $n \in N$. We wish to tile it using particular tetraminos which can be rotated. For which $n$ is this possible if we use

(a) $T$-tetraminos

(b) both kinds of $L$-tetraminos?
1998 Abels Math Contest (Norwegian MO)
Let $n$ be a positive integer.

(a) Prove that $1^5 +3^5 +5^5 +...+(2n-1)^5$ is divisible by $n$.

(b) Prove that $1^3 +3^3 +5^3 +...+(2n-1)^3$ is divisible by $n^2$.
1998 Abels Math Contest (Norwegian MO)
Let $A,B,P$ be points on a line $\ell$, with $P$ outside the segment $AB$. Lines $a$ and $b$ pass through $A$ and $B$ and are perpendicular to $\ell$. A line $m$ through $P$, which is neither parallel nor perpendicular to $\ell$, intersects $a$ and $b$ at $Q$ and $R$, respectively. The perpendicular from $B$ to $AR$ meets $a$ and $AR$ at $S$ and $U$, and the perpendicular from $A$ to $BQ$ meets $b$ and $BQ$ at $T$ and $V$, respectively.

(a) Prove that $P,S,T$ are collinear.

(b) Prove that $P,U,V$ are collinear.
1999 Abels Math Contest (Norwegian MO)
Find a function $f$ such that $f(t^2 +t +1) = t$ for all real $t \ge 0$
1999 Abels Math Contest (Norwegian MO)
If $a,b,c,d,e$ are real numbers, prove the inequality $a^2 +b^2 +c^2 +d^2+e^2 \ge  a(b+c+d+e)$.
1999 Abels Math Contest (Norwegian MO)
Find all integers $m$ and $n$ such that $2m^2 +n^2 = 2mn+3n$
1999 Abels Math Contest (Norwegian MO)
If $a,b,c$ are positive integers such that $b | a^3, c | b^3$ and $a | c^3$ , prove that $abc | (a+b+c)^{13}$
1999 Abels Math Contest (Norwegian MO)
An isosceles triangle $ABC$ with $AB = AC$ and $\angle A = 30^o$ is inscribed in a circle with center $O$. Point $D$ lies on the shorter arc $AC$ so that $\angle DOC = 30^o$, and point $G$ lies on the shorter arc $AB$ so that $DG = AC$ and $AG < BG$. The line $BG$ intersects $AC$ and $AB$ at $E$ and $F$, respectively.

(a) Prove that triangle $AFG$ is equilateral.

(b) Find the ratio between the areas of triangles $AFE$ and $ABC$.
1999 Abels Math Contest (Norwegian MO)
For every nonempty subset $R$ of $S = \{1,2,...,10\}$, we define the alternating sum $A(R)$ as follows:

If $r_1,r_2,...,r_k$ are the elements of $R$ in the increasing order, then $A(R) = r_k -r_{k-1} +r_{k-2}- ... +(-1)^{k-1}r_1$.

(a) Is it possible to partition $S$ into two sets having the same alternating sum?

(b) Determine the sum $\sum_{R} A(R)$, where $R$ runs over all nonempty subsets of $S$.
2000 Abels Math Contest (Norwegian MO)
Show that any odd number can be written as the difference between two perfect squares.
2000 Abels Math Contest (Norwegian MO)
Determine if there is an infinite sequence $a_1,a_2,a_3,...,a_n$ of positive integers such that for all $n\ge 1$ the sum $a_1^2+a_2^2+a_3^2+...^2+a_n^2$ is a perfect square
2000 Abels Math Contest (Norwegian MO)
Let $x, y$ and $z$ be real numbers such that $x + y + z = 0$. Show that $x^3 + y^3 + z^3 = 3xyz$.
2000 Abels Math Contest (Norwegian MO)
Let $a,b,c$ and $d$ be non-negative real numbers such that $a+b+c+d = 4$.

Show that $\sqrt{a+b+c}+\sqrt{b+c+d}+\sqrt{c+d+a}+\sqrt{d+a+b}\ge 6$.
2000 Abels Math Contest (Norwegian MO)
a) Each point, on the perimeter of a square, is colored either red, or blue. Show that, there is a right-angled triangle where all the corners are on the square of the square and so that all the corners are on points of the same color.

b) Show that, it is possible to color each point on the perimeter of one square, red, white, or blue so that, there is not a right-angled triangle where all the three corners are at points of same color.
2000 Abels Math Contest (Norwegian MO)
For some values of c, the equation $x^c + y^c = z^c$ can be illustrated geometrically.

For example, the case $c = 2$ can be illustrated by a right-angled triangle. By this we mean that, x, y, z is a solution of the equation $x^2 + y^2 = z^2$ if and only if there exists a right-angled triangle with catheters $x$ and $y$ and hypotenuse $z$.

In this problem we will look at the cases $c = -\frac{1}{2}$ and $c = - 1$.

a) Let $x, y$ and $z$ be the radii of three circles intersecting each other and a line, as shown, in the figure. Show that,

$x^{-\frac{1}{2}}+ y^{-\frac{1}{2}} = z^{-\frac{1}{2}}$



b) Draw a geometric figure that illustrates the case in a similar way, $c = - 1$. The figure must be able to be constructed with a compass and a ruler. Describe such a construction and prove that, in the figure, lines $x, y$ and $z$ satisfy $x^{-1}+ y^{-1} = z^{-1}$. (All positive solutions of this equation should be possible values for $x, y$, and $z$ on such a figure, but you don't have to prove that.)
2001 Abels Math Contest (Norwegian MO)
Suppose that $a, b, c$ are real numbers such that $a + b + c> 0$, and so the equation $ax^2 + bx + c = 0$ has no real solutions. Show that $c> 0$.
2001 Abels Math Contest (Norwegian MO)
Suppose that $x$ and $y$ are positive real numbers such that $x^3, y^3$ and $x + y$ are all rational numbers. Show that the numbers $xy, x^2+y^2, x$ and $y$ are also rational
2001 Abels Math Contest (Norwegian MO)
Let $A$ be a set, and let $P (A)$ be the powerset of all non-empty subsets of $A$. (For example, $A = \{1,2,3\}$, then $P (A) = \{\{1\},\{2\} ,\{3\},\{1,2\}, \{1,3\},\{2,3\}, \{1,2,3\}\}$.)

A subset $F$ of P $(A)$ is called strong   if the following is true:

If $B_1$ and  $B_2$ are elements of $F$, then $B_1 \cup B_2$ is also an element of $F$.

Suppose that $F$ and $G$ are strong subsets of $P (A)$.

a) Is the union $F \cup  G$ necessarily strong?

b) Is the intersection  $F \cap G$ necessarily strong?
2001 Abels Math Contest (Norwegian MO)
What is the largest possible area of a quadrilateral with sidelengths $1, 4, 7$ and $8$ ?
2001 Abels Math Contest (Norwegian MO)
The diagonals $AC$ and $BD$ in the convex quadrilateral $ABCD$ intersect in $S$. Let $F_1$ and $F_2$ be the areas of  $\vartriangle  ABS$ and $\vartriangle CSD$. and let $F$ be the area of the quadrilateral $ABCD$. Show that $\sqrt{ F_1 }+\sqrt{ F_2}\le \sqrt{ F}$
2001 Abels Math Contest (Norwegian MO)
At a two-day team competition in chess, three schools with $15$ pupils each attend. Each student plays one game against each player on the other two teams, ie a total of $30$ chess games per student.

a) Is it possible for each student to play exactly $15$ games after the first day?

b) Show that it is possible for each student to play exactly $16$ games after the first day.

c) Assume that each student has played exactly $16$ games after the first day. Show that there are three students, one from each school, who have played their three parties
2002 Abels Math Contest (Norwegian MO)
Find all integers $k$ such that both $k + 1$ and $16k + 1$ are perfect squares.
2002 Abels Math Contest (Norwegian MO)
Find all integers $c$ such that the equation $(2a+b) (2b+a) =5^c$ has integer solutions.
2002 Abels Math Contest (Norwegian MO)
a) Let $x$ be a positive real number. Show that $x + 1 / x\ge 2$.

b) Let $n\ge  2$ be a positive integer and let $x _1,y_1,x_2,y_2,...,x_n,y_n$ be positive real numbers such that $x _1+x _2+...+x _n \ge x _1y_1+x _2y_2+...+x _ny_n$.



Show that $x _1+x _2+...+x _n \le  \frac{x _1}{y_1}+\frac{x _2}{y_2}+...+\frac{x _n}{y_n}$
2002 Abels Math Contest (Norwegian MO)
If $a$ and $b$ are real numbers such that $a^3-3ab^2 = 8$ and $b^3-3a^2b = 11$, then what is $a^2+b^2$?
2002 Abels Math Contest (Norwegian MO)
A circle with center in $O$ is given. Two parallel tangents tangent to the circle at points $M$ and $N$. Another tangent intersects the first two tangents at points $K$ and $L$. Show that the circle having the line segment $KL$ as diameter passes through $O$.
2002 Abels Math Contest (Norwegian MO)
Six line segments of lengths $17, 18, 19, 20, 21$ and $23$ form the side edges of a triangular pyramid (also called a tetrahedron). Can there exist a sphere tangent to all six lines?
2002 Abels Math Contest (Norwegian MO)
An integer is given $N> 1$. Arne and Britt play the following game:

(1) Arne says a positive integer $A$.

(2) Britt says an integer $B> 1$ that is either a divisor of $A$ or a multiple of $A$. ($A$ itself is a possibility.)

(3) Arne says a new number $A$ that is either $B - 1, B$ or $B + 1$.

The game continues by repeating steps 2 and 3. Britt wins if she is okay with being told the number $N$ before the $50$th has been said. Otherwise, Arne wins.

a) Show that Arne has a winning strategy if $N = 10$.

b) Show that Britt has a winning strategy if $N = 24$.

c) For which $N$ does Britt have a winning strategy?
2003 Abels Math Contest (Norwegian MO)
Let $x$ and $y$ are real numbers such that $x + y = 2$ and $x^3 + y^3 = 3$. What is $x^2+y^2$?
2003 Abels Math Contest (Norwegian MO)
Let $x_1,x_2,...,x_n$ be real numbers in an interval $[m,M]$  such that  $\sum_{i=1}^n x_i = 0$. Show that  $\sum_{i=1}^n x_i ^2 \le -nmM$
2003 Abels Math Contest (Norwegian MO)
Find all pairs $(x, y)$ of integers  numbers such that $y^3+5=x(y^2+2)$
2003 Abels Math Contest (Norwegian MO)
Let $a_1,a_2,...,a_n$ be $n$  different positive integers where $n\ge  1$. Show that  $$\sum_{i=1}^n a_i^3 \ge \left(\sum_{i=1}^n a_i\right)^2$$
2003 Abels Math Contest (Norwegian MO)
Let $ABC$ be a triangle with $AC> BC$, and let $S$ be the circumscribed circle of the triangle. $AB$ divides $S$ into two arcs. Let $D$  be the midpoint of the arc containing $C$.

(a) Show that $\angle ACB +2 \cdot \angle ACD = 180^o$.

(b) Let $E$ be the foot of the altitude from $D$ on $AC$. Show that $BC +CE = AE$.
2003 Abels Math Contest (Norwegian MO)
$25$ boys and $25$ girls sit around a table. Show that there is a person who has a girl sitting on either side of them.
2003 Abels Math Contest (Norwegian MO)
Let $m> 3$ be an integer. At a camp there are more than $m$ participants. The camp manager discovers that every time he picks out the camp participants, they say they have exactly one mutual friend among the participants. Which is the largest possible number of participants at the camp?

(If $A$ is a friend of $B, B$ is also a friend of $A$. A person is not considered a friend of himself.)