Navvye/Olympiad-Problems-CSV huggingface dataset:
2005 Finnish National High School Mathematics Competition
There are $12$ seats at a round table in a restaurant. A group of five women and seven men arrives at the table. How many ways are there for choosing the sitting order, provided that every woman ought to be surrounded by two men and two orders are regarded as different, if at least one person has a different neighbour on one's right side.
2005 Finnish National High School Mathematics Competition
Solve the group of equations: $$\begin{cases} (x + y)^3 = z \\ (y + z)^3 = x \\ (z + x)^3 = y \end{cases}$$
2005 Finnish National High School Mathematics Competition
The numbers $1, 3, 7$ and $9$ occur in the decimal representation of an integer.

Show that permuting the order of digits one can obtain an integer divisible by $7.$
2005 Finnish National High School Mathematics Competition
A finite sequence is said to be disorderly, if no two terms of the sequence have their average in between them. For example, $(0, 2, 1)$ is disorderly, for $1 = \frac{0+2}{2}$ is not in between $0$ and $2$, and the other averages $\frac{0+1}{2} = \frac{1}{2}$ and $\frac{2+1}{2} = 1\frac{1}{2}$ do not even occur in the sequence.

Prove that for every $n \in \Bbb{N}$ there is a disorderly sequence enumerating the numbers $0, 1,\ldots , n$ without repetitions.
2004 Finnish National High School Mathematics Competition
The equations $x^2 +2ax+b^2 = 0$ and $x^2 +2bx+c^2 = 0$ both have two different real roots.

Determine the number of real roots of the equation $x^2 + 2cx + a^2 = 0.$
2004 Finnish National High School Mathematics Competition
$a, b$ and $c$ are positive integers and $$\frac{a\sqrt{3} + b}{b\sqrt{3} + c}$$

is a rational number.

Show that $$\frac{a^2 + b^2 + c^2}{a + b + c}$$ is an integer.
2004 Finnish National High School Mathematics Competition
Two circles with radii $r$ and $R$ are externally tangent.

Determine the length of the segment cut from the common tangent of the circles by the other common tangents.
2004 Finnish National High School Mathematics Competition
The numbers $2005! + 2, 2005! + 3, ... , 2005! + 2005$ form a sequence of $2004$ consequtive integers, none of which is a prime number.

Does there exist a sequence of $2004$ consequtive integers containing exactly $12$ prime numbers?
2004 Finnish National High School Mathematics Competition
Finland is going to change the monetary system again and replace the Euro by the Finnish Mark.

The Mark is divided into $100$ pennies.

There shall be coins of three denominations only, and the number of coins a person has to carry in order to be able

to pay for any purchase less than one mark should be minimal.

Determine the coin denominations.
2003 Finnish National High School Mathematics Competition
The incentre of the triangle $ABC$ is $I.$ The rays $AI, BI$ and $CI$ intersect the circumcircle of the triangle $ABC$ at the points $D, E$ and $F,$ respectively. Prove that $AD$ and $EF$ are perpendicular.
2003 Finnish National High School Mathematics Competition
Find consecutive integers bounding the expression $$\frac{1}{x_1 + 1}+\frac{1}{x_2 + 1}+\frac{1}{x_3 + 1}+... +\frac{1}{x_{2001} + 1}+\frac{1}{x_{2002} + 1}$$

where $x_1 = 1/3$ and $x_{n+1} = x_n^2 + x_n.$
2003 Finnish National High School Mathematics Competition
There are six empty purses on the table. How many ways are there to put 12 two-euro coins in purses in such a way that at most one purse remains empty?
2003 Finnish National High School Mathematics Competition
Find pairs of positive integers $(n, k)$ satisfying $$(n + 1)^k - 1 = n!$$
2003 Finnish National High School Mathematics Competition
Players Aino and Eino take turns choosing numbers from the set $\{0,..., n\}$ with $n\in \Bbb{N}$ being fixed in advance.

The game ends when the numbers picked by one of the players include an arithmetic progression of length $4.$

The one who obtains the progression wins.

Prove that for some $n,$ the starter of the game wins. Find the smallest such $n.$
2002 Finnish National High School Mathematics Competition
A function $f$ satisfies $f(\cos x) = \cos (17x)$ for every real $x$. Show that $f(\sin x) =\sin (17x)$ for every $x \in \mathbb{R}.$
2002 Finnish National High School Mathematics Competition
Show that if $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a + b + c},$ then also $$\frac{1}{a^n} +\frac{1}{b^n} +\frac{1}{c^n} =\frac{1}{a^n + b^n + c^n},$$

provided $n$ is an odd positive integer.
2002 Finnish National High School Mathematics Competition
$n$ pairs are formed from $n$ girls and $n$ boys at random.

What is the probability of having at least one pair of girls? For which $n$ the probability is over $0,9?$
2002 Finnish National High School Mathematics Competition
Convex figure $\mathcal{K}$ has the following property:

if one looks at $\mathcal{K}$ from any point of the certain circle $\mathcal{Y}$, then $\mathcal{K}$ is seen in the right angle.

Show that the figure is symmetric with respect to the centre of $\mathcal{Y.}$
2002 Finnish National High School Mathematics Competition
There is a regular $17$-gon $\mathcal{P}$ and its circumcircle $\mathcal{Y}$ on the plane.

The vertices of $\mathcal{P}$ are coloured in such a way that $A,B \in \mathcal{P}$ are of different colour, if the shorter arc connecting $A$ and $B$ on $\mathcal{Y}$ has $2^k+1$ vertices, for some $k \in \mathbb{N},$ including $A$ and $B.$

What is the least number of colours which suffices?
2001 Finnish National High School Mathematics Competition
In the right triangle $ABC,$ $CF$ is the altitude based on the hypotenuse $AB.$

The circle centered at $B$ and passing through $F$ and the circle with centre $A$ and the same radius intersect at a point of $CB.$

Determine the ratio $FB : BC.$
2001 Finnish National High School Mathematics Competition
Equations of non-intersecting curves are $y = ax^2 + bx + c$ and $y = dx^2 + ex + f$ where $ad < 0.$

Prove that there is a line of the plane which does not meet either of the curves.
2001 Finnish National High School Mathematics Competition
Numbers $a, b$ and $c$ are positive integers and $\frac{1}{a}+\frac{1}{b}+\frac{ 1}{c}< 1.$ Show that $$\frac{1}{a}+\frac{1}{b}+\frac{ 1}{c}\leq \frac{41}{42}.$$
2001 Finnish National High School Mathematics Competition
A sequence of seven digits is randomly chosen in a weekly lottery. Every digit can be any of the digits $0, 1, 2, 3, 4, 5, 6, 7, 8, 9.$

What is the probability of having at most five different digits in the sequence?
2001 Finnish National High School Mathematics Competition
Determine $n \in \Bbb{N}$ such that $n^2 + 2$ divides $2 + 2001n.$
2000 Finnish National High School Mathematics Competition
Two circles are externally tangent at the point $A$. A common tangent of the circles meets one circle at the point $B$ and another at the point $C$ ($B \ne C)$. Line segments $BD$ and $CE$ are diameters of the circles. Prove that the points $D, A$ and $C$ are collinear.
2000 Finnish National High School Mathematics Competition
Prove that the integral part of the decimal representation of the number $(3+\sqrt{5})^n$ is odd, for every positive integer $n.$
2000 Finnish National High School Mathematics Competition
Determine the positive integers $n$ such that the inequality $$n! > \sqrt{n^n}$$ holds.
2000 Finnish National High School Mathematics Competition
There are seven points on the plane, no three of which are collinear. Every pair of points is connected with a line segment, each of which is either blue or red. Prove that there are at least four monochromatic triangles in the figure.
2000 Finnish National High School Mathematics Competition
Irja and Valtteri are tossing coins. They take turns, Irja starting. Each of them has a pebble which reside on opposite vertices of a square at the start. If a player gets heads, she or he moves her or his pebble on opposite vertex. Otherwise the player in turn moves her or his pebble to an adjacent vertex so that Irja proceeds in positive and Valtteri in negative direction. The winner is the one who can move his pebble to the vertex where opponent's pebble lies. What is the probability that Irja wins the game?
1999 Finnish National High School Mathematics Competition
Show that the equation $x^3 + 2y^2 + 4z = n$ has an integral solution $(x, y, z)$ for all integers $n.$
1999 Finnish National High School Mathematics Competition
Suppose that the positive numbers $a_1, a_2,.. , a_n$ form an arithmetic progression; hence $a_{k+1}- a_k = d,$ for $k = 1, 2,... , n - 1.$

Prove that $$\frac{1}{a_1a_2}+\frac{1}{a_2a_3}+...+\frac{1}{a_{n-1}a_n}=\frac{n-1}{a_1a_n}.$$
1999 Finnish National High School Mathematics Competition
Determine how many primes are there in the sequence $$101, 10101, 1010101 ....$$
1999 Finnish National High School Mathematics Competition
Three unit circles have a common point $O.$ The other points of (pairwise) intersection are $A, B$ and $C$. Show that the points $A, B$ and $C$ are located on some unit circle.
1999 Finnish National High School Mathematics Competition
An ordinary domino tile can be identified as a pair $(k,m)$ where numbers $k$ and $m$ can get values $0, 1, 2, 3, 4, 5$ and $6.$

Pairs $(k,m)$ and $(m, k)$ determine the same tile. In particular, the pair $(k, k)$ determines one tile.

We say that two domino tiles match, if they have a common component.

Generalized n-domino tiles $m$ and $k$ can get values $0, 1,... , n.$

What is the probability that two randomly chosen $n$-domino tiles match?
1998 Finnish National High School Mathematics Competition
Show that points $A, B, C$ and $D$ can be placed on the plane in such a way that the quadrilateral $ABCD$ has an area which is twice the area of the quadrilateral $ADBC.$
1998 Finnish National High School Mathematics Competition
There are $11$ members in the competetion committee. The problem set is kept in a safe having several locks.

The committee members have been provided with keys in such a way that every six members can open the safe, but no five members can do that.

What is the smallest possible number of locks, and how many keys are needed in that case?
1998 Finnish National High School Mathematics Competition
Consider the geometric sequence $1/2, \ 1 / 4, \ 1 / 8,...$

Can one choose a subsequence, finite or infinite, for which the ratio of consecutive terms is not $1$ and whose sum is $1/5?$
1998 Finnish National High School Mathematics Competition
There are $110$ points in a unit square. Show that some four of these points reside in a circle whose radius is $1/8.$
1998 Finnish National High School Mathematics Competition
$15\times 36$-checkerboard is covered with square tiles. There are two kinds of tiles, with side $7$ or $5.$

Tiles are supposed to cover whole squares of the board and be non-overlapping.

What is the maximum number of squares to be covered?
1997 Finnish National High School Mathematics Competition
Determine the real numbers $a$ such that the equation $a 3^x + 3^{-x} = 3$ has exactly one solution $x.$
1997 Finnish National High School Mathematics Competition
Circles with radii $R$ and $r$ ($R > r$) are externally tangent. Another common tangent of the circles in drawn.

This tangent and the circles bound a region inside which a circle as large as possible is drawn.

What is the radius of this circle?
1997 Finnish National High School Mathematics Competition
$12$ knights are sitting at a round table. Every knight is an enemy with two of the adjacent knights but with none of the others.

$5$ knights are to be chosen to save the princess, with no enemies in the group. How many ways are there for the choice?
1997 Finnish National High School Mathematics Competition
Count the sum of the four-digit positive integers containing only odd digits in their decimal representation.
1997 Finnish National High School Mathematics Competition
For an integer $n\geq 3$, place $n$ points on the plane in such a way that all the distances between the points are at most one and exactly $n$ of the pairs of points have the distance one.
2015 Flanders Math Olympiad
The sides and vertices of a pentagon are labelled with the numbers $1$ through $10$ so that the sum of the numbers on every side is the same. What is the smallest possible value of this sum?
2015 Flanders Math Olympiad
Consider two points $Y$ and $X$ in a plane and a variable point $P$ which is not on $XY$. Let the parallel line to $YP$ through $X$ intersect the internal angle bisector of $\angle XYP$ in $A$, and let the parallel line to $XP$ through $Y$ intersect the internal angle bisector of $\angle YXP$ in $B$. Let $AB$ intersect $XP$ and $YP$ in $S$ and $T$ respectively. Show that the product $|XS|*|YT|$ does not depend on the position of $P$.
2015 Flanders Math Olympiad
A group of people is divided over two busses in such a way that there are as many seats in total as people. The chance that two friends are seated on the same bus is $\frac{1}{2}$.

a) Show that the number of people in the group is a square.

b) Show that the number of seats on each bus is a triangular number.
2015 Flanders Math Olympiad
Show that for $n \geq 5$, the integers $1, 2, \ldots n$ can be split into two groups so that the sum of the integers in one group equals the product of the integers in the other group.
2007 Flanders Math Olympiad
1. The numbers $1,2, \ldots$ are placed in a triangle as following:



$$

\begin{matrix}

1 &  &  &  \\ 

2 & 3 &  &  \\ 

4 & 5 & 6 &  \\ 

7 & 8 & 9 & 10 \\ 

 \ldots 

\end{matrix}

$$



What is the sum of the numbers on the $n$-th row?
2007 Flanders Math Olympiad
Given is a half circle with midpoint $O$ and diameter $AB$. Let $Z$ be a random point inside the half circle, and let $X$ be the intersection of $OZ$ and the half circle, and $Y$ the intersection of $AZ$ and the half circle.



If $P$ is the intersection of $BY$ with the tangent line in $X$ to the half circle, show that $PZ \perp BX$.
2007 Flanders Math Olympiad
Let $ABCD$ be a square with side $10$. Let $M$ and $N$ be the midpoints of $[AB]$ and $[BC]$ respectively. Three circles are drawn: one with midpoint $D$ and radius $|AD|$, one with midpoint $M$ and radius $|AM|$, and one with midpoint $N$ and radius $|BN|$. The three circles intersect in the points $R, S$ and $T$ inside the square. Determine the area of $\triangle RST$.
2007 Flanders Math Olympiad
If $f,g: \mathbb{R} \to \mathbb{R}$ are functions that satisfy $f(x+g(y)) = 2x+y $ $\forall x,y \in \mathbb{R}$, then determine $g(x+f(y))$.
2006 Flanders Math Olympiad
(a) Solve for $\theta\in\mathbb{R}$: $\cos(4\theta) = \cos(3\theta)$



(b) $\cos\left(\frac{2\pi}{7}\right)$, $\cos\left(\frac{4\pi}{7}\right)$ and $\cos\left(\frac{6\pi}{7}\right)$ are the roots of an equation of the form $ax^3+bx^2+cx+d = 0$ where $a, b, c, d$ are integers. Determine $a, b, c$ and $d$.
2006 Flanders Math Olympiad
Let $\triangle ABC$ be an equilateral triangle and let $P$ be a point on $\left[AB\right]$.

$Q$ is the point on $BC$ such that $PQ$ is perpendicular to $AB$. $R$ is the point on $AC$ such that $QR$ is perpendicular to $BC$. And $S$ is the point on $AB$ such that $RS$ is perpendicular to $AC$.

$Q'$ is the point on $BC$ such that $PQ'$ is perpendicular to $BC$. $R'$ is the point on $AC$ such that $Q'R'$ is perpendicular to $AC$. And $S'$ is the point on $AB$ such that $R'S'$ is perpendicular to $AB$.

Determine $\frac{|PB|}{|AB|}$ if $S=S'$.
2006 Flanders Math Olympiad
Elfs and trolls are seated at a round table, 60 creatures in total. Trolls always lie, and all elfs always speak the truth, except when they make a little mistake.

Everybody claims to sit between an elf and a troll, but exactly two elfs made a mistake! How many trolls are there at this table?
2006 Flanders Math Olympiad
Find all functions $f: \mathbb{R}\backslash\{0,1\} \rightarrow \mathbb{R}$ such that



$$ f(x)+f\left(\frac{1}{1-x}\right) = 1+\frac{1}{x(1-x)}. $$
2005 Flanders Math Olympiad
For all positive integers $n$, find the remainder of $\dfrac{(7n)!}{7^n \cdot n!}$ upon division by 7.
2005 Flanders Math Olympiad
We can obviously put 100 unit balls in a $10\times10\times1$ box.

How can one put $105$ unit balls in? How can we put $106$ unit balls in?
2005 Flanders Math Olympiad
Prove that $2005^2$ can be written in at least $4$ ways as the sum of 2 perfect (non-zero) squares.
2005 Flanders Math Olympiad
If $n$ is an integer, then find all values of $n$ for which $\sqrt{n}+\sqrt{n+2005}$ is an integer as well.
2004 Flanders Math Olympiad
Consider a triangle with side lengths 501m, 668m, 835m. How many lines can be drawn with the property that such a line halves both area and perimeter?
2004 Flanders Math Olympiad
Two bags contain some numbers, and the total number of numbers is prime.



When we tranfer the number 170 from 1 bag to bag 2, the average in both bags increases by one.



If the total sum of all numbers is 2004, find the number of numbers.
2004 Flanders Math Olympiad
A car has a 4-digit integer price, which is written digitally. (so in digital numbers, like on your watch probably)



While the salesmen isn't watching, the buyer turns the price upside down and gets the car for 1626 less. How much did the car initially cost?
2004 Flanders Math Olympiad
Each cell of a beehive is constructed from a right regular 6-angled prism, open at the bottom and closed on the top by a regular 3-sided pyramidical mantle. The edges of this pyramid are connected to three of the rising edges of the prism and its apex $T$ is on the perpendicular line through the center $O$ of the base of the prism (see figure). Let $s$ denote the side of the base, $h$ the height of the cell and $\theta$ the angle between the line $TO$ and $TV$.

(a) Prove that the surface of the cell consists of 6 congruent trapezoids and 3 congruent rhombi.

(b) the total surface area of the cell is given by the formula $6sh - \dfrac{9s^2}{2\tan\theta} + \dfrac{s^2 3\sqrt{3}}{2\sin\theta}$
2003 Flanders Math Olympiad
Playing soccer with 3 goes as follows: 2 field players try to make a goal past the goalkeeper, the one who makes the goal stands goalman for next game, etc.



Arne, Bart and Cauchy played this game. Later, they tell their math teacher that A stood 12 times on the field, B 21 times on the field, C 8 times in the goal. Their teacher knows who made the 6th goal.



Who made it?
2003 Flanders Math Olympiad
Two circles $C_1$ and $C_2$ intersect at $S$.

The tangent in $S$ to $C_1$ intersects $C_2$ in $A$ different from $S$.

The tangent in $S$ to $C_2$ intersects $C_1$ in $B$ different from $S$.

Another circle $C_3$ goes through $A, B, S$.

The tangent in $S$ to $C_3$ intersects $C_1$ in $P$ different from $S$ and $C_2$ in $Q$ different from $S$.

Prove that the distance $PS$ is equal to the distance $QS$.
2003 Flanders Math Olympiad
A number consists of 3 different digits. The sum of the 5 other numbers formed with those digits is 2003. Find the number.
2003 Flanders Math Olympiad
Consider all points with integer coordinates in the carthesian plane. If one draws a circle with M(0,0) and a well-chose radius r, the circles goes through some of those points. (like circle with $r=2\sqrt2$ goes through 4 points)



Prove that $\forall n\in \mathbb{N}, \exists r$ so that the circle with midpoint 0,0 and radius $r$ goes through at least $n$ points.
2002 Flanders Math Olympiad
Is it possible to number the $8$ vertices of a cube from $1$ to $8$ in such a way that the value of the sum on every edge is different?
2002 Flanders Math Olympiad
Determine all functions $f: \mathbb{R}\rightarrow\mathbb{R}$ so that $\forall x: x\cdot f(\frac x2) - f(\frac2x) = 1$
2002 Flanders Math Olympiad
show that $\frac1{15} < \frac12\cdot\frac34\cdots\frac{99}{100} < \frac1{10}$
2002 Flanders Math Olympiad
A lamp is situated at point $A$ and shines inside the cube.

A (massive) square is hung on the midpoints of the $4$ vertical faces.

What's the area of its shadow?
2001 Flanders Math Olympiad
may be challenge for beginner section, but anyone is able to solve it if you really try.



show that for every natural $n > 1$ we have: $(n-1)^2|\ n^{n-1}-1$
2001 Flanders Math Olympiad
Consider a triangle and 2 lines that each go through a corner and intersects the opposing segment, such that the areas are as on the attachment.



Find the "?"
2001 Flanders Math Olympiad
In a circle we enscribe a regular $2001$-gon and inside it a regular $667$-gon with shared vertices.



Prove that the surface in the $2001$-gon but not in the $667$-gon is of the form $k.sin^3\left(\frac{\pi}{2001}\right).cos^3\left(\frac{\pi}{2001}\right)$ with $k$ a positive integer. Find $k$.
2001 Flanders Math Olympiad
A student concentrates on solving quadratic equations in $\mathbb{R}$. He starts with a first quadratic equation $x^2 + ax + b = 0$ where $a$ and $b$ are both different from 0. If this first equation has solutions $p$ and $q$ with $p  \leq  q$, he forms a second quadratic equation $x^2 + px + q = 0$. If this second equation has solutions, he forms a third quadratic equation in an identical way. He continues this process as long as possible. Prove that he will not obtain more than five equations.
2000 Flanders Math Olympiad
An integer consists of 7 different digits, and is a multiple of each of its digits.



What digits are in this nubmer?
2000 Flanders Math Olympiad
Given two triangles and such that the lengths of the sides of the first triangle are the lengths of the medians of the second triangle. Determine the ratio of the areas of these triangles.
2000 Flanders Math Olympiad
Let $p_n$ be the $n$-th prime. ($p_1=2$)

Define the sequence $(f_j)$ as follows:

- $f_1=1, f_2=2$

- $\forall j\ge 2$: if $f_j = kp_n$ for $k<p_n$ then $f_{j+1}=(k+1)p_n$

- $\forall j\ge 2$: if $f_j = p_n^2$ then $f_{j+1}=p_{n+1}$



(a) Show that all $f_i$ are different

(b) from which index onwards are all $f_i$ at least 3 digits?

(c) which integers do not appear in the sequence?

(d) how many numbers with less than 3 digits appear in the sequence?
2000 Flanders Math Olympiad
Solve for $x \in [0,2\pi[$: $$\sin x < \cos x < \tan x < \cot x$$
1999 Flanders Math Olympiad
Determine all 6-digit numbers $(abcdef)$ so that $(abcdef) = (def)^2$ where $\left( x_1x_2...x_n \right)$ is no multiplication but an n-digit number.
1999 Flanders Math Olympiad
Let $[mn]$ be a diameter of the circle $C$ and $[AB]$ a chord with given length on this circle. $[AB]$ neither coincides nor is perpendicular to $[MN]$.

Let $C,D$ be the orthogonal projections of $A$ and $B$ on $[MN]$ and $P$ the midpoint of $[AB]$.

Prove that $\angle CPD$ does not depend on the chord $[AB]$.
1999 Flanders Math Olympiad
Determine all $f: \mathbb{R}\rightarrow\mathbb{R}$ for which

$$ 2\cdot f(x)-g(x)=f(y)-y \textrm{ and } f(x)\cdot g(x) \geq x+1. $$
1999 Flanders Math Olympiad
Let $a,b,m,n$ integers greater than 1. If $a^n-1$ and $b^m+1$ are both primes, give as much info as possible on $a,b,m,n$.
1998 Flanders Math Olympiad
Prove there exist positive integers a,b,c for which $a+b+c=1998$, the gcd is maximized, and $0<a<b\leq c<2a$.

Find those numbers.

Are they unique?
1998 Flanders Math Olympiad
Given a cube with edges of length 1, $e$ the midpoint of $[bc]$, and $m$ midpoint of the face $cdc_1d_1$, as on the figure.

Find the area of intersection of the cube with the plane through the points $a,m,e$.
1998 Flanders Math Olympiad
a magical $3\times3$ square is a $3\times3$ matrix containing all number from 1 to 9, and of which the sum of every row, every column, every diagonal, are all equal.



Determine all magical $3\times3$ square
1998 Flanders Math Olympiad
A billiard table. (see picture)

A white ball is on $p_1$ and a red ball is on $p_2$.

The white ball is shot towards the red ball as shown on the pic, hitting 3 sides first.

Find the minimal distance the ball must travel.
1997 Flanders Math Olympiad
Write the number 1997 as the sum of positive integers for which the product is maximal, and prove there's no better solution.
1997 Flanders Math Olympiad
In the cartesian plane, consider the curves $x^2+y^2=r^2$ and $(xy)^2=1$. Call $F_r$ the convex polygon with vertices the points of intersection of these 2 curves. (if they exist)



(a) Find the area of the polygon as a function of $r$.

(b) For which values of $r$ do we have a regular polygon?