Navvye/Olympiad-Problems-CSV huggingface dataset:
1996 Swedish Mathematical Competition
In the country of Postonia, one wants to have only two values of stamps. These values should be integers greater than $1$ with the difference $2$, and should have the property that one can combine the stamps for any postage which is greater than or equal to the sum of these two values. What values can be chosen?
1996 Swedish Mathematical Competition
For every positive integer $n$, we define the function $p_n$ for $x\ge 1$ by

$$p_n(x) = \frac12 \left(\left(x+\sqrt{x^2-1}\right)^n+\left(x-\sqrt{x^2-1}\right)^n\right).$$Prove that $p_n(x) \ge 1$ and that $p_{mn}(x) = p_m(p_n(x))$.
1996 Swedish Mathematical Competition
The angles at $A,B,C,D,E$ of a pentagon $ABCDE$ inscribed in a circle form an increasing sequence. Show that the angle at $C$ is greater than $\pi/2$, and that this lower bound cannot be improved.
1996 Swedish Mathematical Competition
Let $n \ge 1$. Prove that it is possible to select some of the integers $1,2,...,2^n$ so that for each $p = 0,1,...,n - 1$ the sum of the $p$-th powers of the selected numbers is equal to the sum of the $p$-th powers of the remaining numbers.
1996 Swedish Mathematical Competition
A rectangle is tiled with rectangles of size $6\times 1$. Prove that one of its side lengths is divisible by $6$.
1997 Swedish Mathematical Competition
Let $AC$ be a diameter of a circle and $AB$ be tangent to the circle. The segment $BC$ intersects the circle again at $D$. Show that if $AC = 1$, $AB = a$, and $CD = b$, then $$\frac{1}{a^2+ \frac12 }< \frac{b}{a}< \frac{1}{a^2}$$
1997 Swedish Mathematical Competition
Let $D$ be the point on side $AC$ of a triangle $ABC$ such that $BD$ bisects $\angle B$, and $E$ be the point on side $AB$ such that $3\angle  ACE = 2\angle  BCE$. Suppose that $BD$ and $CE$ intersect at a point $P$ with $ED = DC = CP$. Determine the angles of the triangle.
1997 Swedish Mathematical Competition
Let $A$ and $B$ be integers with an odd sum. Show that every integer can be written in the form $x^2 -y^2 +Ax+By$, where $x,y$ are integers.
1997 Swedish Mathematical Competition
Players $A$ and $B$ play the following game. Each of them throws a dice, and if the outcomes are $x$ and $y$ respectively, a list of all two digit numbers $10a + b$ with $a,b\in \{1,..,6\}$ and $10a + b \le 10x + y$ is created. Then the players alternately reduce the list by replacing a pair of numbers in the list by their absolute difference, until only one number remains. If the remaining number is of the same parity as the outcome of $A$’s throw, then $A$ is proclaimed the winner. What is the probability that $A$ wins the game?
1997 Swedish Mathematical Competition
Let $s(m)$ denote the sum of (decimal) digits of a positive integer $m$. Prove that for every integer $n > 1$ not equal to $10$ there is a unique integer $f(n) \ge 2$ such that $s(k)+s(f(n)-k) = n$ for all integers $k$ with $0 < k < f(n)$.
1997 Swedish Mathematical Competition
Assume that a set $M$ of real numbers is the union of finitely many disjoint intervals with the total length greater than $1$. Prove that $M$ contains a pair of distinct numbers whose difference is an integer.
1998 Swedish Mathematical Competition
Find all positive integers $a, b, c$, such that $(8a-5b)^2 + (3b-2c)^2 + (3c-7a)^2 = 2$.
1998 Swedish Mathematical Competition
$ABC$ is a triangle. Show that $c \ge (a+b) \sin \frac{C}{2}$
1998 Swedish Mathematical Competition
A cube side $5$ is made up of unit cubes. Two small cubes are adjacent  if they have a common face. Can we start at a cube adjacent to a corner cube and move through all the cubes just once? (The path must always move from a cube to an adjacent cube).
1998 Swedish Mathematical Competition
$ABCD$ is a quadrilateral with $\angle A = 90o$, $AD = a$, $BC = b$, $AB = h$, and area $\frac{(a+b)h}{2}$. What can we say about $\angle B$?
1998 Swedish Mathematical Competition
Show that for any $n > 5$ we can find positive integers $x_1, x_2, ... , x_n$ such that $\frac{1}{x_1} + \frac{1}{x_2} +... + \frac{1}{x_n} = \frac{1997}{1998}$. Show that in any such equation there must be two of the $n$ numbers with a common divisor ($> 1$).
1998 Swedish Mathematical Competition
Show that for some $c > 0$, we have $\left|\sqrt[3]{2} - \frac{m}{n}\right | > \frac{c}{n^3}$ for all integers $m, n$ with $n \ge 1$.
1999 Swedish Mathematical Competition
Solve $|||||x^2-x-1| - 2| - 3| - 4| - 5| = x^2 + x - 30$.
1999 Swedish Mathematical Competition
Circle $C$ center $O$ touches externally circle $C'$ center $O'$. A line touches $C$ at $A$ and $C'$ at $B$. $P$ is the midpoint of $AB$. Show that $\angle OPO' = 90^o$.
1999 Swedish Mathematical Competition
Find non-negative integers $a, b, c, d$ such that $5^a + 6^b + 7^c + 11^d = 1999$.
1999 Swedish Mathematical Competition
An equilateral triangle of side $x$ has its vertices on the sides of a square side $1$. What are the possible values of $x$?
1999 Swedish Mathematical Competition
$x_i$ are non-negative reals. $x_1 + x_2 + ...+ x_n = s$. Show that $x_1x_2 + x_2x_3 + ... + x_{n-1}x_n \le \frac{s^2}{4}$.
1999 Swedish Mathematical Competition
$S$ is any sequence of at least $3$ positive integers. A move is to take any $a, b$ in the sequence such that neither divides the other and replace them by gcd $(a,b)$ and lcm $(a,b)$. Show that only finitely many moves are possible and that the final result is independent of the moves made, except possibly for order.
2000 Swedish Mathematical Competition
Each of the numbers $1, 2, ... , 10$ is colored red or blue. $5$ is red and at least one number is blue. If $m, n$ are different colors and $m+n \le  10$, then $m+n$ is blue. If $m, n$ are different colors and $mn \le 10$, then $mn$ is red. Find all the colors.
2000 Swedish Mathematical Competition
$p(x)$  is a polynomial such that $p(y^2+1) = 6y^4 - y^2 + 5$. Find $p(y^2-1)$.
2000 Swedish Mathematical Competition
Are there any integral solutions to $n^2 + (n+1)^2 + (n+2)^2 = m^2$ ?
2000 Swedish Mathematical Competition
The vertices of a triangle are three-dimensional lattice points. Show that its area is at least $\frac12$.
2000 Swedish Mathematical Competition
Let $f(n)$ be defined on the positive integers and satisfy:

$f(prime) = 1$, $f(ab) = a f(b) + f(a) b$.

Show that $f$ is unique and find all $n$ such that $n = f(n)$.
2000 Swedish Mathematical Competition
Solve $$\left\{ \begin{array}{l} y(x+y)^2 = 9 \\
y(x^3-y^3) = 7  \\
 \end{array} \right.

$$
2001 Swedish Mathematical Competition
Show that if we take any six numbers from the following array, one from each row and column, then the product is always the same:



4   6  10  14   22   26

6   9  15  21   33   39

10  15  25  35   55   65

16  24  40  56   88  104

18  27  45  63   99  117

20  30  50  70  110  130
2001 Swedish Mathematical Competition
Show that $\sqrt[3]{\sqrt{52} + 5}- \sqrt[3]{\sqrt{52}- 5}$ is rational.
2001 Swedish Mathematical Competition
Show that if $b = \frac{a+c}{2}$ in the triangle $ABC$, then $\cos (A-C) + 4 \cos B = 3$.
2001 Swedish Mathematical Competition
$ABC$ is a triangle. A circle through $A$ touches the side $BC$ at $D$ and intersects the sides $AB$ and $AC$ again at $E, F$ respectively. $EF$ bisects $\angle AFD$ and $\angle ADC = 80^o$. Find $\angle ABC$.
2001 Swedish Mathematical Competition
Find all polynomials $p(x)$ such that $p'(x)^2 = c p(x) p''(x)$ for some constant $c$.
2001 Swedish Mathematical Competition
A chessboard is covered with $32$ dominos. Each domino covers two adjacent squares. Show that the number of horizontal dominos with a white square on the left equals the number with a white square on the right.
2002 Swedish Mathematical Competition
$268$ numbers are written around a circle. The $17$th number is $3$, the $83$rd is $4$ and the $144$th is $9$. The sum of every $20$ consecutive numbers is $72$. Find the $210$th number.
2002 Swedish Mathematical Competition
$A, B, C$ can walk at $5$ km/hr. They have a car that can accomodate any two of them whch travels at $50$ km/hr. Can they reach a point $62$ km away in less than $3$ hrs?
2002 Swedish Mathematical Competition
$C$ is the circle center $(0,1)$, radius $1$. $P$ is the parabola $y = ax^2$. They meet at $(0, 0)$. For what values of $a$ do they meet at another point or points?
2002 Swedish Mathematical Competition
For which integers $n \ge 8$ is $n^{\frac{1}{n-7}}$ an integer?
2002 Swedish Mathematical Competition
The reals $a, b$ satisfy $a^3 - 3a^2 + 5a - 17 = 0$, $b^3 - 3b^2 + 5b + 11 = 0$.  Find $a+b$.
2002 Swedish Mathematical Competition
A tetrahedron has five edges of length $3$ and circumradius $2$. What is the length of the sixth edge?
2003 Swedish Mathematical Competition
If $x, y, z, w$ are nonnegative real numbers satisfying $$\left\{ \begin{array}{l}y = x -  2003 \\ z = 2y - 2003 \\ w = 3z -  2003 \\
 \end{array} \right.

$$find the smallest possible value of $x$ and the values of $y, z, w$  corresponding to it.
2003 Swedish Mathematical Competition
In a lecture hall some  chairs are placed in rows and columns, forming a rectangle. In each row there are $6$ boys sitting and in each column there are $8$ girls sitting, whereas $15$ places are not taken. What can be said about the number of rows and that of columns?
2003 Swedish Mathematical Competition
Find all real solutions $x$ of the equation $$\lfloor x^2-2 \rfloor +2 \lfloor x \rfloor  = \lfloor x \rfloor ^2. $$.
2003 Swedish Mathematical Competition
Determine all polynomials $P$ with real coeffients such that $1 + P(x) = \frac12 (P(x -1) + P(x + 1))$ for all real $x$.
2003 Swedish Mathematical Competition
Given two positive numbers $a, b$, how many non-congruent plane quadrilaterals are there such that $AB = a$, $BC = CD = DA = b$ and $\angle B = 90^o$ ?
2003 Swedish Mathematical Competition
Consider an infinite square board with an integer written in each square. Assume that for each square the integer in it is equal to the sum of its neighbor to the left and its neighbor above. Assume also that there exists a row $R_0$ in the board such that all numbers in $R_0$ are positive. Denote by $R_1$ the row below $R_0$ , by $R_2$ the row below $R_1$ etc. Show that for each $N \ge 1$ the row $R_N$  cannot  contain more than $N$ zeroes.
2004 Swedish Mathematical Competition
Two circles in the plane, both of radius $R$, intersect at a right angle. Compute the area of the intersection of the interiors of the two circles.
2004 Swedish Mathematical Competition
In one country there are coins of value $1,2,3,4$ or $5$. Nisse wants to buy a pair of shoes. While paying, he tells the seller that he has $100$ coins in the bag, but that he does not know the exact number of coins of each value. ”Fine, then you will have the exact amount”, the seller responds. What is the price of the shoes, and how did the seller conclude that Nisse would have the exact amount?
2004 Swedish Mathematical Competition
A function $f$ satisfies $f(x)+x f(1-x) = x^2$ for all real $x$. Determine $f$ .
2004 Swedish Mathematical Competition
If $0 < v <\frac{\pi}{2}$ and $\tan v = 2v$, decide whether $sinv < \frac{20}{21}$.
2004 Swedish Mathematical Competition
A square of side $n \ge  2$ is divided into $n^2$ unit squares ($n \in  N$). One draws $n-1$ lines so that the interior of each of the unit squares is cut by at least one of these lines.

(a) Give an example of such a configuration for some $n$.

(b) Show that some two of the lines must meet inside the square.
2004 Swedish Mathematical Competition
Prove that every convex $n$-gon of area $1$ contains a quadrilateral of area at least $\frac12 $.

.
2005 Swedish Mathematical Competition
Find all integer solutions $x$,$y$ of the equation $(x+y^2)(x^2+y)=(x+y)^3$.
2005 Swedish Mathematical Competition
There are 12 people in a line in a bank. When the desk closes, the people form a new line at a newly opened desk. In how many ways can they do this in such a way that none of the 12 people changes his/her position in the line by more than one?
2005 Swedish Mathematical Competition
In a triangle $ABC$ the bisectors of angles $A$ and $C$ meet the opposite sides at $D$ and $E$ respectively. Show that if the angle at $B$ is greater than $60^\circ$, then $AE +CD <AC$.
2005 Swedish Mathematical Competition
The zeroes of a fourth degree polynomial $f(x)$ form an arithmetic progression. Prove that the three zeroes of the polynomial $f'(x)$ also form an arithmetic progression.
2005 Swedish Mathematical Competition
Every cell of a $2005 \times 2005$  square board is colored white or black so that every $2 \times 2$ subsquare contains an odd number of black cells.

Show that among the corner cells there is an even number of black ones. How many ways are there to color the board in this manner?
2005 Swedish Mathematical Competition
A regular tetrahedron of edge length $1$ is orthogonally projected onto a plane. Find the largest possible area of its image.
2006 Swedish Mathematical Competition
If positive integers $a$ and $b$ have 99 and 101 different positive divisors respectively (including 1 and the number itself), can the product $ab$ have exactly 150 positive divisors?
2006 Swedish Mathematical Competition
In a triangle $ABC$, point $P$ is the incenter and $A'$, $B'$, $C'$ its orthogonal projections on $BC$, $CA$, $AB$, respectively. Show that $\angle B'A'C'$ is acute.
2006 Swedish Mathematical Competition
A cubic polynomial $f$ with a positive leading coefficient has three different positive zeros. Show that $f'(a)+ f'(b)+ f'(c) > 0$.
2006 Swedish Mathematical Competition
Saskia and her sisters have been given a large number of pearls. The pearls are white, black and red, not necessarily the same number of each color. Each white pearl is worth $5$ Ducates, each black one is worth $7$, and each red one is worth $12$. The total worth of the pearls is $2107$ Ducates. Saskia and her sisters split the pearls so that each of them gets the same number of pearls and the same total worth, but the color distribution may vary among the sisters. Interestingly enough, the total worth in Ducates that each of the sisters holds equals the total number of pearls split between the sisters. Saskia is particularly fond of the red pearls, and therefore makes sure that she has as many of those as possible. How many pearls of each color has Saskia?
2006 Swedish Mathematical Competition
In each square of an $m \times n$ rectangular board there is a nought or a cross. Let $f(m,n)$ be the number of such arrangements that contain a row or a column consisting of noughts only. Let $g(m,n)$ be the number of arrangements that contain a row consisting of noughts only, or a column consisting of crosses only. Which of the numbers $f(m,n)$ and $g(m,n)$ is larger?
2006 Swedish Mathematical Competition
Determine all positive integers $a,b,c$ satisfying $a^{(b^c)}=(b^a)^c$
2007 Swedish Mathematical Competition
Solve the following system

$$

\left\{ \begin{array}{l}

 xyzu-x^3=9 \\ 

 x+yz=\frac{3}{2}u \\ 

\end{array} \right.

$$in positive integers $x$, $y$, $z$ and $u$.
2007 Swedish Mathematical Competition
A number of flowers are distributed between $n$ persons so that the first of them, Andreas, gets one flower, the other gets two flowers, the third gets three flowers, etc., to $n$-th person who gets $n$ flowers. Andreas then walks around shaking hands with each other of the others, in any order. In order to do  so, he receives a flower from everyone which he hangs on to and which has more flowers than himself at the moment they shake hands. Which is the smallest number of flowers Andreas can have after shaking hands with everyone?
2007 Swedish Mathematical Competition
Let $\alpha$, $\beta$, $\gamma$ be the angles of a triangle. If $a$, $b$, $c$ are the side length of the triangle and $R$ is the circumradius, show that

$$

\cot \alpha + \cot \beta +\cot \gamma =\frac{R\left(a^2+b^2+c^2\right)}{abc}

$$
2007 Swedish Mathematical Competition
There are a number of arcs on the edge of a circular disk. Each pair of arcs has the least one point in common. Show that on the circle you can choose two diametrical opposites points such that each arc contains at least one of these two points.
2007 Swedish Mathematical Competition
Anna and Brian play a game where they put the domino tiles (of size $2 \times 1$) in a  boards composed of $n \times 1$ boxes. Tiles must be placed so that they cover exactly two boxes. Players take turnslaying each tile and the one laying last tile wins. They play once for each $n$, where $n = 2, 3,\dots,2007$. Show that Anna wins at least $1505$ of the games if she always starts first and they both always play optimally, ie if they do their best to win in every move.
2007 Swedish Mathematical Competition
In the plane, a triangle is given. Determine all points $P$ in the plane such that each line through $P$ that divides the triangle into two parts with the same area must pass through one of the vertices of the triangle.
2008 Swedish Mathematical Competition
A rhombus is inscribed in a convex quadrilateral. The sides of the rhombus are parallel with the diagonals of the quadrilateral, which have the lengths $d_1$ and $d_2$. Calculate the length of side of the rhombus , expressed in terms of $d_1$ and $d_2$.
2008 Swedish Mathematical Competition
Determine the smallest integer $n \ge 3$ with the property that you can choose two of the numbers $1,2,\dots, n$ in such a way that their product is equal to the sum of the other $n - 2$ languages. What are the two numbers?
2008 Swedish Mathematical Competition
The function $f(x)$ has the property that $\frac{f(x)}{x}$ is increasing for $x>0$. Show that

$$

f(x)+f(y) \leq f(x+y) \qquad , \qquad \text{for all } x,y>0

$$
2008 Swedish Mathematical Competition
A convex $n$-side polygon has angles $v_1,v_2,\dots,v_n$ (in degrees), where all $v_k$ ($k = 1,2,\dots,n$) are positive integers divisible by $36$.

(a) Determine the largest $n$ for which this is possible.

(b) Show that if $n>5$, two of the sides of the $n$-polygon must be parallel.
2008 Swedish Mathematical Competition
Anna and Orjan play the following game: they start with a positive integer $n>1$,  Anna writes it as the sum of two other positive integers, $n = n_1+n_2$. Orjan deletes one of them, $n_1$ or $n_2$. If the remaining number is larger than $1$, the process is repeated, i.e. Anna writes it as the sum of two positive integers, $ n_3+n_4$, Orjan deletes one of them etc. The game ends when the last number is $1$. Orjan is the winner if there are two equal numbers among the numbers he has deleted, otherwise Anna wins. Who is winning the game if n = 2008 and they both play optimally?
2008 Swedish Mathematical Competition
A sum decomposition of the number 100 is given by a positive integer $n$ and $n$ positive integers $x_1<x_2<\cdots <x_n$ such that  $x_1 + x_2 + \cdots + x_n = 100$. Determine the largest possible value of the product $x_1x_2\cdots x_n$, and $n$ , as $x_1, x_2,\dots, x_n$ vary among all sum decompositions of the number $100$.
2009 Swedish Mathematical Competition
Five square carpets have been bought for a square hall with a side of $6$ m , two with the side $2$ m, one with the side $2.1$ m and two with the side $2.5$ m. Is it possible to place the five carpets so that they do not overlap in any way each other? The edges of the carpets do not have to be parallel to the cradles in the hall.
2009 Swedish Mathematical Competition
Find all real solutions of the equation

$$

\left(1+x^2\right)\left(1+x^3\right)\left(1+x^5\right)=8x^5

$$
2009 Swedish Mathematical Competition
An urn contain a number of yellow and green balls. You extract two balls from the urn (without adding them back) and calculate the probability of both balls being green. Can you choose the number of yellow and green balls such that this probability to be $\frac{1}{4}$?
2009 Swedish Mathematical Competition
Determine all integers solutions of the equation $x + x^3 = 5y^2$.
2009 Swedish Mathematical Competition
A semicircular arc and a diameter $AB$ with a length of $2$ are given. Let $O$ be the midpoint of the diameter. On the radius perpendicular to the diameter, we select a point $P$ at the distance $d$ from the midpoint of the diameter $O$, $0 <d <1$. A line through $A$ and $P$ intersects the semicircle at point $C$. Through point $P$ we draw another line at right angle against $AC$ that intersects the semicircle at point $D$.  Through point $C$ we draw a line $l_1$, parallel to $PD$ and then a line $l_2$, through $D$ parallel to $PC$. The lines $l_1$ and $l_2$ intersect at point $E$. Show that the distance between $O$ and $E$ is equal to $\sqrt{2- d^2}$
2009 Swedish Mathematical Competition
On a table lie  $289$ coins that form a square array $17 \times 17$. All coins are facing with the crown up. In one move, it is possible to reverse any five coins lying in a row: vertical, horizontal or diagonal. Is it possible that after a number of such moves, all the coins to be arranged with tails up?
2010 Swedish Mathematical Competition
Exists a triangle whose three altitudes have lengths $1, 2$ and $3$ respectively?
2010 Swedish Mathematical Competition
Consider the four lines $y = mx-k^2$ for different integer $k$. Let $(x_i,y_i)$, $i = 1,2,3,4$ be four different points , such that each belongs to two different lines and on each line pass through just the two of them. Lat $x_1\leq x_2\leq x_3\leq x_4$. Show that $x_1 + x_4 =x_2+x_3$ and $y_1y_4 =y_2y_3$.
2010 Swedish Mathematical Competition
Find all natural numbers $n \ge 1$ such that there is a polynomial $p(x)$ with integer coefficients for which $p (1) = p (2) = 0$ and where $p (n)$ is a prime number .
2010 Swedish Mathematical Competition
We create a sequence by setting $a_1 = 2010$ and requiring that $a_n-a_{n-1}\leq n$ and $a_n$ is also divisible by $n$.

Show that $a_{100},a_{101},a_{102},\dots$ form an arithmetic sequence.
2010 Swedish Mathematical Competition
Consider the number of triangles where the side lengths $a,b,c$ satisfy $(a + b + c) (a + b -c) = 2b^2$.

Determine the angles in the triangle for which the angle opposite to the side with the length $a$  is as big as possible.
2010 Swedish Mathematical Competition
An infinite number of squares on an infinitely square grid paper are painted red. Show that you can draw a number of squares on the paper, with sides along the grid lines, such that:



(1) no square in the grid belongs to more than one square (an edge, on the other hand, may belong to more than one square)



(2) each red square is located in one of the squares and the number of red squares in such square is at least $1/5$ and at most  $4/5$ of the number of squares in the square.



original wordingEtt andligt antal rutor pa ett oandligt rutat papper ar malade roda. Visa att man pa papperet kan rita in ett antal kvadrater, med sidor utefter rutnatets linjer, sadana att :

(1) ingen ruta i natet tillhor mer an en kvadrat (en kant kan daremot tillhora mer an en kvadrat),

(2) varje rod ruta ligger i nagon av kvadraterna och antalet roda rutor i en sadan kvadratar minst 1/5  och hogst 4/5  av antalet rutor i kvadraten.



source



PS. I always post the original wording when I doubt about my (using Google) translation.
2011 Swedish Mathematical Competition
Determine all positive integers $k$, $\ell$, $m$ and $n$, such that $$\frac{1}{k!}+\frac{1}{\ell!}+\frac{1}{m!} =\frac{1}{n!} $$
2011 Swedish Mathematical Competition
Given a triangle $ABC$, let $P$ be a point inside the triangle such that $| BP | > | AP |, | BP | > | CP |$. Show that $\angle ABC <90^o$
2011 Swedish Mathematical Competition
Find all positive real numbers $x, y, z$, such that

$$x - \frac{1}{y^2} = y - \frac{1}{z^2}= z - \frac{1}{x^2}$$
2011 Swedish Mathematical Competition
Towns $A$, $B$ and $C$ are connected with a telecommunications cable. If you for example want to send a message from $A$ to $B$ is assigned to either a direct line between $A$ and $B$, or if necessary, a line via $C$. There are $43$ lines between $A$ and $B$, including those who go through $C$, and $29$ lines between $B$ and $C$, including those who go via $A$. How many lines, are there between $A$ and $C$ (including those who go via $B$)?