Navvye/Olympiad-Problems-CSV huggingface dataset:
2021 Federal Competition For Advanced Students, P2
Let $a$ be a real number. Determine all functions $f: R \to R$ with $f (f (x) + y) = f (x^2 - y) + af (x) y$ for all $x, y \in R$.



(Walther Janous)
2021 Federal Competition For Advanced Students, P2
Let $ABCD$ be a convex cyclic quadrilateral with diagonals $AC$ and $BD$. Each of the four vertixes are reflected across the diagonal on which the do not lie.

(a) Investigate when the four points thus obtained lie on a straight line and give as simple an equivalent condition as possible to the cyclic quadrilateral $ABCD$ for it.

(b) Show that in all other cases the four points thus obtained lie on one circle.



(Theresia Eisenkölbl)
2021 Federal Competition For Advanced Students, P2
Let $p$ be an odd prime number and $M$ a set derived from $\frac{p^2 + 1}{2}$ square numbers.

Investigate whether $p$ elements can be selected from this set whose arithmetic mean is an integer.



(Walther Janous)
2019 Finnish National High School Mathematics Comp
Solve  $x(8\sqrt{1-x}+\sqrt{1+x}) \le 11\sqrt{1+x}-16\sqrt{1-x}$ when $0<x\le 1$
2019 Finnish National High School Mathematics Comp
Prove that the number $\lfloor (2+\sqrt5)^{2019} \rfloor$ is not prime.
2019 Finnish National High School Mathematics Comp
Let $ABCD$ be a cyclic quadrilateral whose side $AB$ is at the same time the diameter of the circle. The lines $AC$ and $BD$ intersect at point $E$ and the extensions of lines $AD$ and $BC$ intersect at point $F$.  Segment $EF$ intersects the circle at $G$ and the extension of segment $EF$ intersects $AB$ at $H$. Show that if $G$ is the midpoint of $FH$, then $E$ is the midpoint of $GH$.
2019 Finnish National High School Mathematics Comp
Define a sequence $ a_n = n^n + (n - 1)^{n+1}$ when $n$ is a positive integer.

Define all those positive integer $m$ , for which this sequence of numbers is eventually periodic modulo $m$, e.g. there are such positive integers $K$ and $s$ such that $a_k \equiv a_{k+s}$ ($mod \,m$),  where $k$ is an integer with $k \ge K$.
2019 Finnish National High School Mathematics Comp
A teacher is known to have $2^k$ apples for some $k \in \mathbb{N}$. He ets one of the apples and distributes the rest of the apples to his students $A$ and $B$. The students do not see how many apples the other gets, and they do not know the number $k$. However, they have pre-selected a discreet way to reveal one another something about the number of apples: each of the students scratches their head either by their right, left or both hands, depending on the number of apples they have received. To the teacher's surprise, the students will always know which one of the students got more apples, or that the teacher ate the only apple by herself. How is this possible?
2018 Finnish National High School Mathematics Comp
Eve and Martti have a whole number of euros.

Martti said to Eve: ''If you give give me three euros, so I have $n$ times the money compared to you. '' Eve in turn said to Martti: ''If you give me $n$ euros then I have triple the amount of money compared to you'' . Suppose, that both claims are valid.

What values can a positive integer $n$ get?
2018 Finnish National High School Mathematics Comp
The sides of triangle ABC are $a = | BC |, b = | CA |$ and $c = | AB |$. Points $D, E$ and $F$ are the points on the sides $BC, CA$ and $AB$ such that $AD, BE$ and $CF$ are the angle bisectors of the triangle $ABC$. Determine the lengths of the segments $AD, BE$, and $CF$ in terms of $a, b$, and $c$.
2018 Finnish National High School Mathematics Comp
The chords $AB$ and $CD$ of a circle intersect at $M$, which is the midpoint of the chord $PQ$. The points $X$ and $Y$ are the intersections of the segments $AD$ and $PQ$, respectively, and $BC$ and $PQ$, respectively. Show that $M$ is the midpoint of $XY$.
2018 Finnish National High School Mathematics Comp
Define $f : \mathbb{Z}_+ \to \mathbb{Z}_+$ such that $f(1) = 1$ and $f(n) $ is the greatest prime divisor of $n$ for $n > 1$.

Aino and Väinö play a game, where each player has a pile of stones. On each turn the player to turn with $m$ stones in his pile may remove at most $f(m)$ stones from the opponent's pile, but must remove at least one stone. (The own pile stays unchanged.) The first player to clear the opponent's pile wins the game. Prove that there exists a positive integer $n$ such that Aino loses, when both players play optimally, Aino starts, and initially both players have $n$ stones.
2018 Finnish National High School Mathematics Comp
Solve the diophantine equation $x^{2018}-y^{2018}=(xy)^{2017}$  when $x$ and $y$ are non-negative integers.
2017 Finnish National High School Mathematics Comp
By dividing the integer $m$ by the integer $n, 22$ is the quotient and $5$ the remainder.

As the division of the remainder with $n$ continues, the new quotient is $0.4$  and the new remainder is $0.2$.

Find $m$ and $n$.
2017 Finnish National High School Mathematics Comp
Determine $x^2+y^2$ and $x^4+y^4$,  when $x^3+y^3=2$ and $x+y=1$
2017 Finnish National High School Mathematics Comp
Consider positive integers $m$ and $n$ for which $m> n$ and the  number $22 220 038^m-22 220 038^n$ has are eight zeros at the end. Show that $n> 7$.
2017 Finnish National High School Mathematics Comp
Let $m$ be a positive integer.

Two players, Axel and Elina play the game HAUKKU ($m$)  proceeds as follows:

Axel starts and the players choose integers alternately. Initially, the set of integers is the set of positive divisors of a positive integer $m$ .The player in turn chooses one of the remaining numbers, and removes that number and all of its multiples from the list of selectable numbers. A player who has to choose number $1$, loses. Show that the beginner player, Axel, has a winning strategy in the HAUKKU ($m$) game for all $m \in Z_{+}$.



PS. As member Loppukilpailija noted, it should be written $m>1$,  as the statement does not hold for $m = 1$.
2017 Finnish National High School Mathematics Comp
Let $A$ and $B$ be two arbitrary points on the circumference of the circle such that $AB$ is not the diameter of the circle. The tangents to the circle drawn at points $A$ and $B$ meet at $T$. Next, choose the diameter $XY$ so that the segments $AX$ and $BY$ intersect. Let this be the intersection of $Q$. Prove that the points $A, B$, and $Q$ lie on a circle with center  $T$.
2016 Finnish National High School Mathematics Comp
Which triangles satisfy the equation $\frac{c^2-a^2}{b}+\frac{b^2-c^2}{a}=b-a$ when $a, b$ and $c$ are sides of a triangle?
2016 Finnish National High School Mathematics Comp
Suppose that $y$ is a positive integer written only with digit $1$, in base $9$ system. Prove that $y$ is a triangular number, that is, exists positive integer $n$ such that the number $y$ is the sum of the $n$ natural numbers from $1$ to $n$.
2016 Finnish National High School Mathematics Comp
From the foot of one altitude of the acute triangle, perpendiculars are drawn on the other two sides, that meet the other sides at $P$ and $Q$. Show that the length of $PQ$ does not depend on which of the three altitudes  is selected.
2016 Finnish National High School Mathematics Comp
How many pairs $(a, b)$ of positive integers $a,b$ solutions of the equation $(4a-b)(4b-a )=1770^n$  exist , if $n$ is a positive integer?
2016 Finnish National High School Mathematics Comp
The ruler of Laputa will set up a train network between cities in the state, which satisfies the following conditions:

- Uniformity: From one city to another, by train, possibly through exchanges.

- Prohibition N: There exist no four cities $A, B, C, D$ such that there are direct routes between $A$ and $B, B$ and $C$, and $C$ and $D$, but taking a shortcut is not possible, that is, there are no direct rout between $A$ and $C, B$ and $D$, or $A$ and $D$.

In addition, a direct airliner connection will be established exactly between their city pairs, with no direct train connection.

Prove that the airline network is not connected when there is more than one city.
2015 Finnish National High School Mathematics Comp
Solve the equation $\sqrt{1+\sqrt {1+x}}=\sqrt[3]{x}$ for $x \ge 0$.
2015 Finnish National High School Mathematics Comp
The lateral edges of a right square pyramid are of length $a$.

Let $ABCD$ be the base of the pyramid, $E$ its top vertex and $F$ the midpoint of $CE$.

Assuming that $BDF$ is an equilateral triangle, compute the volume of the pyramid.
2015 Finnish National High School Mathematics Comp
Determine the largest integer $k$ for which $12^k$ is a factor of $120! $
2015 Finnish National High School Mathematics Comp
Let $n$ be a positive integer. Every square in a $n \times n$-square grid is either white or black.

How many such colourings exist, if every $2 \times 2$-square consists of exactly two white and two black squares?

The squares in the grid are identified as e.g. in a chessboard, so in general colourings obtained from each other by rotation are different.
2015 Finnish National High School Mathematics Comp
Mikko takes a multiple choice test with ten questions. His only goal is to pass the test, and this requires seven points. A correct answer is worth one point, and answering wrong results in the deduction of one point. Mikko knows for sure that he knows the correct answer in the six first questions. For the rest, he estimates that he can give the correct answer to each problem with probability $p, 0 < p < 1$. How many questions Mikko should try?
2014 Finnish National High School Mathematics
Determine the value of the expression $x^2 + y^2 + z^2$,



if $x + y + z = 13$ , $xyz= 72$ and $\frac1x + \frac1y + \frac1z = \frac34$.
2014 Finnish National High School Mathematics
The center of the circumcircle of the acute triangle $ABC$ is $M$, and the circumcircle of $ABM$ meets $BC$ and $AC$ at $P$ and $Q$ ($P\ne B$). Show that the extension of the line segment $CM$ is perpendicular to $PQ$.
2014 Finnish National High School Mathematics
The points $P = (a, b)$ and $Q = (c, d)$ are in the first quadrant of the $xy$ plane, and $a, b, c$ and $d$ are integers satisfying $a < b, a < c, b < d$  and $c < d$. A route from point $P$ to point $Q$ is a broken line consisting of unit steps in the directions of the positive coordinate axes. An allowed route is a route not touching the line $x = y$. Tetermine the number of allowed routes.
2014 Finnish National High School Mathematics
The radius $r$ of a circle with center at the origin is an odd integer.

There is a point ($p^m, q^n$) on the circle, with $p,q$ prime numbers and $m,n$ positive integers.

Determine $r$.
2014 Finnish National High School Mathematics
Determine the smallest number $n \in Z_+$, which can be written as $n = \Sigma_{a\in A}a^2$, where $A$ is a finite set of positive integers and $\Sigma_{a\in A}a= 2014$.

In other words: what is the smallest positive number which can be written as a sum of squares of different positive integers summing to $2014$?
2013 Finnish National High School Mathematics Competition
The coefficients $a,b,c$ of a polynomial $f:\mathbb{R}\to\mathbb{R}, f(x)=x^3+ax^2+bx+c$ are mutually distinct integers and different from zero. Furthermore, $f(a)=a^3$ and $f(b)=b^3.$ Determine $a,b$ and $c$.
2013 Finnish National High School Mathematics Competition
In a particular European city, there are only $7$ day tickets and $30$ day tickets to the public transport. The former costs $7.03$ euro and the latter costs $30$ euro. Aina the Algebraist decides to buy at once those tickets that she can travel by the public transport the whole three year (2014-2016, 1096 days) visiting in the city. What is the cheapest solution?
2013 Finnish National High School Mathematics Competition
The points $A,B,$ and $C$ lies on the circumference of the unit circle. Furthermore, it is known that $AB$ is a diameter of the circle and $$\frac{|AC|}{|CB|}=\frac{3}{4}.$$ The bisector of $ABC$ intersects the circumference at the point $D$. Determine the length of the $AD$.
2013 Finnish National High School Mathematics Competition
A subset $E$ of the set $\{1,2,3,\ldots,50\}$ is said to be special if it does not contain any pair of the form $\{x,3x\}.$ A special set $E$ is superspecial if it contains as many elements as possible. How many element there are in a superspecial set and how many superspecial sets there are?
2013 Finnish National High School Mathematics Competition
Find all integer triples $(m,p,q)$ satisfying $$2^mp^2+1=q^5$$ where $m>0$ and both $p$ and $q$ are prime numbers.
2012 Finnish National High School Mathematics Competition
A secant line splits a circle into two segments. Inside those segments, one draws two squares such that both squares has two corners on a secant line and two on the circumference. The ratio of the square's side lengths is $5:9$. Compute the ratio of the secant line versus circle radius.
2012 Finnish National High School Mathematics Competition
Let $x\ne 1,y\ne 1$ and $x\ne y.$ Show that if $$\frac{yz-x^2}{1-x}=\frac{zx-y^2}{1-y},$$ then $$\frac{yz-x^2}{1-x}=\frac{zx-y^2}{1-y}=x+y+z.$$
2012 Finnish National High School Mathematics Competition
Prove that for all integers $k\geq 2,$ the number $k^{k-1}-1$ is divisible by $(k-1)^2.$
2012 Finnish National High School Mathematics Competition
Let $k,n\in\mathbb{N},0<k<n.$ Prove that $$\sum_{j=1}^k\binom{n}{j}=\binom{n}{1}+ \binom{n}{2}+\ldots + \binom{n}{k}\leq n^k.$$
2012 Finnish National High School Mathematics Competition
The Collatz's function is a mapping $f:\mathbb{Z}_+\to\mathbb{Z}_+$ satisfying $$

f(x)=\begin{cases}

3x+1,& \mbox{as }x\mbox{ is odd}\\
x/2, & \mbox{as }x\mbox{ is even.}\\
\end{cases}

$$ In addition, let us define the notation $f^1=f$ and inductively $f^{k+1}=f\circ f^k,$ or to say in another words, $f^k(x)=\underbrace{f(\ldots (f}_{k\text{ times}}(x)\ldots ).$



Prove that there is an $x\in\mathbb{Z}_+$ satisfying $$f^{40}(x)> 2012x.$$
2011 Finnish National High School Mathematics Competition
An equilateral triangle has been drawn inside the circle. Split the triangle to two parts with equal area by a line segment parallel to the triangle side. Draw an inscribed circle inside this smaller triangle. What is the ratio of the area of this circle compared to the area of original circle.
2011 Finnish National High School Mathematics Competition
Find all integers $x$ and $y$ satisfying the inequality $$x^4-12x^2+x^2y^2+30\leq 0.$$
2011 Finnish National High School Mathematics Competition
Points $D$ and $E$ divides the base $BC$ of an isosceles triangle $ABC$ into three equal parts and $D$ is between $B$ and $E.$ Show that $\angle BAD<\angle DAE.$
2011 Finnish National High School Mathematics Competition
Show that there is a perfect square (a number which is a square of an integer) such that sum of its digits is $2011.$
2011 Finnish National High School Mathematics Competition
Two players, the builder and the destroyer, plays the following game. Builder starts and players chooses alternatively different elements from the set $\{0,1,\ldots,10\}.$ Builder wins if some four integer of those six integer he chose forms an arithmetic sequence. Destroyer wins if he can prevent to form such an arithmetic four-tuple. Which one has a winning strategy?
2010 Finnish National High School Mathematics Competition
Let $ABC$ be right angled triangle with sides $s_1,s_2,s_3$ medians $m_1,m_2,m_3$. Prove that $m_1^2+m_2^2+m_3^2=\frac{3}{4}(s_1^2+s_2^2+s_3^2)$.
2010 Finnish National High School Mathematics Competition
Determine the least $n\in\mathbb{N}$ such that $n!=1\cdot 2\cdot 3\cdots (n-1)\cdot n$ has at least $2010$ positive factors.
2010 Finnish National High School Mathematics Competition
Let $P(x)$ be a polynomial with integer coefficients and roots $1997$ and $2010$. Suppose further that $|P(2005)|<10$. Determine what integer values $P(2005)$ can get.
2010 Finnish National High School Mathematics Competition
In a football season, even number $n$ of teams plays a simple series, i.e. each team plays once against each other team. Show that ona can group the series into $n-1$ rounds such that in every round every team plays exactly one match.
2010 Finnish National High School Mathematics Competition
Let $S$ be a non-empty subset of a plane. We say that the point $P$ can be seen from $A$ if every point from the line segment $AP$ belongs to $S$. Further, the set $S$ can be seen from $A$ if every point of $S$ can be seen from $A$. Suppose that $S$ can be seen from $A$, $B$ and $C$ where $ABC$ is a triangle. Prove that $S$ can also be seen from any other point of the triangle $ABC$.
2009 Finnish National High School Mathematics Competition
In a plane, the point $(x,y)$ has temperature $x^2+y^2-6x+4y$. Determine the coldest point of the plane and its temperature.
2009 Finnish National High School Mathematics Competition
A polynomial $P$ has integer coefficients and $P(3)=4$ and $P(4)=3$. For how many $x$ we might have $P(x)=x$?
2009 Finnish National High School Mathematics Competition
The circles $\mathcal{Y}_0$ and $\mathcal{Y}_1$ lies outside each other. Let $O_0$ be the center of $\mathcal{Y}_0$ and $O_1$ be the center of $\mathcal{Y}_1$. From $O_0$, draw the rays which are tangents to $\mathcal{Y}_1$ and similarty from $O_1$, draw the rays which are tangents to $\mathcal{Y}_0$. Let the intersection points of rays and circle $\mathcal{Y}_i$ be $A_i$ and $B_i$. Show that the line segments $A_0B_0$ and $A_1B_1$ have equal lengths.
2009 Finnish National High School Mathematics Competition
We say that the set of step lengths $D\subset \mathbb{Z}_+=\{1,2,\ldots\}$ is excellent if it has the following property: If we split the set of integers into two subsets $A$ and $\mathbb{Z}\setminus{A}$, at least other set contains element $a-d,a,a+d$ (i.e. $\{a-d,a,a+d\} \subset A$ or $\{a-d,a,a+d\}\in \mathbb{Z}\setminus A$ from some integer $a\in \mathbb{Z},d\in D$.) For example the set of one element $\{1\}$ is not excellent as the set of integer can be split into even and odd numbers, and neither of these contains three consecutive integer. Show that the set $\{1,2,3,4\}$ is excellent but it has no proper subset which is excellent.
2009 Finnish National High School Mathematics Competition
As in the picture below, the rectangle on the left hand side has been divided into four parts by line segments which are parallel to a side of the rectangle. The areas of the small rectangles are $A,B,C$ and $D$. Similarly, the small rectangles on the right hand side have areas $A^\prime,B^\prime,C^\prime$ and $D^\prime$. It is known that $A\leq A^\prime$, $B\leq B^\prime$, $C\leq C^\prime$ but $D\leq B^\prime$.

[asy]

import graph; size(12cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-4.3,xmax=12.32,ymin=-10.68,ymax=6.3; 

draw((0,3)--(0,0)); draw((3,0)--(0,0)); draw((3,0)--(3,3)); draw((0,3)--(3,3)); draw((2,0)--(2,3)); draw((0,2)--(3,2)); label("$A$",(0.86,2.72),SE*lsf); label("$B$",(2.38,2.7),SE*lsf); label("$C$",(2.3,1.1),SE*lsf); label("$D$",(0.82,1.14),SE*lsf); draw((5,2)--(11,2)); draw((5,2)--(5,0)); draw((11,0)--(5,0)); draw((11,2)--(11,0)); draw((8,0)--(8,2)); draw((5,1)--(11,1)); label("$A'$",(6.28,1.8),SE*lsf); label("$B'$",(9.44,1.82),SE*lsf); label("$C'$",(9.4,0.8),SE*lsf); label("$D'$",(6.3,0.86),SE*lsf); 

dot((0,3),linewidth(1pt)+ds); dot((0,0),linewidth(1pt)+ds); dot((3,0),linewidth(1pt)+ds); dot((3,3),linewidth(1pt)+ds); dot((2,0),linewidth(1pt)+ds); dot((2,3),linewidth(1pt)+ds); dot((0,2),linewidth(1pt)+ds); dot((3,2),linewidth(1pt)+ds); dot((5,0),linewidth(1pt)+ds); dot((5,2),linewidth(1pt)+ds); dot((11,0),linewidth(1pt)+ds); dot((11,2),linewidth(1pt)+ds); dot((8,0),linewidth(1pt)+ds); dot((8,2),linewidth(1pt)+ds); dot((5,1),linewidth(1pt)+ds); dot((11,1),linewidth(1pt)+ds); 

clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); 

[/asy]

Prove that the big rectangle on the left hand side has area smaller or equal to the area of the big rectangle on the right hand side, i.e. $A+B+C+D\leq A^\prime+B^\prime+C^\prime+D^\prime$.
2008 Finnish National High School Mathematics Competition
Foxes, wolves and bears arranged a big rabbit hunt. There were $45$ hunters catching $2008$ rabbits.

Every fox caught $59$ rabbits, every wolf $41$ rabbits and every bear $40$ rabbits.

How many foxes, wolves and bears were there in the hunting company?
2008 Finnish National High School Mathematics Competition
The incentre of the triangle $ABC$ is $I.$ The lines $AI, BI$ and $CI$ meet the circumcircle of the triangle $ABC$ also at points $D, E$ and $F,$ respectively.

Prove that $AD$ and $EF$ are perpendicular.
2008 Finnish National High School Mathematics Competition
Solve the diophantine equation $$x^{2008}- y^{2008} = 2^{2009}.$$
2008 Finnish National High School Mathematics Competition
Eight football teams play matches against each other in such a way that no two teams meet twice and no three teams play all of the three possible matches.

What is the largest possible number of matches?
2008 Finnish National High School Mathematics Competition
The closed line segment $I$ is covered by finitely many closed line segments.

Show that one can choose a subfamily $S$ of the family of line segments having the properties:

(1) the chosen line segments are disjoint,

(2) the sum of the lengths of the line segments of S is more than half of the length of $I.$



Show that the claim does not hold any more if the line segment $I$ is replaced by a circle and other occurences of the compound word ''line segment" by the word ''circular arc".
2007 Finnish National High School Mathematics Competition
Show: when a prime number is divided by $30,$ the remainder is either $1$ or a prime number. Is a similar claim true, when the divisor is $60$ or $90$?
2007 Finnish National High School Mathematics Competition
Determine the number of real roots of the equation $$x^8 - x^7 + 2x^6 - 2x^5 + 3x^4 - 3x^3 + 4x^2 - 4x +\frac{5}{2}= 0.$$
2007 Finnish National High School Mathematics Competition
There are five points in the plane, no three of which are collinear. Show that some four of these points are the vertices of a convex quadrilateral.
2007 Finnish National High School Mathematics Competition
The six offices of the city of Salavaara are to be connected to each other by a communication network which utilizes modern picotechnology. Each of the offices is to be connected to all the other ones by direct cable connections. Three operators compete to build the connections, and there is a separate competition for every connection.

When the network is finished one notices that the worst has happened: the systems of the three operators are incompatible. So the city must reject connections built by two of the operators, and these are to be chosen so that the damage is minimized. What is the minimal number of offices which still can be connected to each other, possibly through intermediate offices, in the worst possible case.
2007 Finnish National High School Mathematics Competition
Show that there exists a polynomial $P(x)$ with integer coefficients, such that the equation $P(x) = 0$ has no integer solutions, but for each positive integer $n$ there is an $x \in \Bbb{Z}$ such that $n \mid P(x).$
2006 Finnish National High School Mathematics Competition
Determine all pairs $(x, y)$ of positive integers for which the equation $$x + y + xy = 2006$$ holds.
2006 Finnish National High School Mathematics Competition
Show that the inequality $$3(1 + a^2 + a^4)\geq  (1 + a + a^2)^2$$

holds for all real numbers $a.$
2006 Finnish National High School Mathematics Competition
The numbers $p, 4p^2 + 1,$ and $6p^2 + 1$ are primes. Determine $p.$
2006 Finnish National High School Mathematics Competition
Two medians of a triangle are perpendicular. Prove that the medians of the triangle are the sides of a right-angled triangle.
2006 Finnish National High School Mathematics Competition
The game of Nelipe is played on a $16\times16$-grid as follows: The two players write in turn numbers $1, 2,..., 16$ in different squares. The numbers on each row, column, and in every one of the 16 smaller squares have to be different. The loser is the one who is not able to write a number. Which one of the players wins, if both play with an optimal strategy?
2005 Finnish National High School Mathematics Competition
In the figure below, the centres of four squares have been connected by two line

segments. Prove that these line segments are perpendicular.