Navvye/Olympiad-Problems-CSV huggingface dataset:
2020 Dutch IMO TST
For a positive integer $n$, we consider an $n \times n$ board and tiles with dimensions $1 \times 1, 1  \times 2, ..., 1  \times n$. In how many ways exactly can $\frac12 n (n + 1)$ cells of the board are colored red, so that the red squares can all be covered by placing the $n$ tiles all horizontally, but also by placing all $n$ tiles vertically?



Two colorings that are not identical, but by rotation or reflection from the board into each other count as different.
2020 Dutch IMO TST
Let $a, b \ge 2$ be positive integers with $gcd (a, b) = 1$. Let $r$ be the smallest positive value assumed by an expression of the form $\frac{a}{b}-  \frac{c}{d}$, with $c$ and $d$ positive integers satisfying $c \le  a$ and $d \le b$. Prove that $\frac{1}{r}$ is an integer.



original wordingLaat a, b ≥ 2 positieve gehele getallen met ggd(a, b) = 1 zijn. Zij r de kleinste positieve waarde die aangenomen wordt bij een uitdrukking van de vorm a/b −c/d, met c en d positieve gehele getallen die voldoen aan c ≤ a en d ≤ b. Bewijs dat 1/rgeheel is .
2020 Dutch IMO TST
Given are real numbers $a_1, a_2,..., a_{2020}$, not necessarily different.

For every $n \ge 2020$,  define $a_{n + 1}$ as the smallest real zero of the polynomial $$P_n (x) = x^{2n} + a_1x^{2n - 2} + a_2x^{2n - 4} +... + a_{n -1}x^2 + a_n$$, if it exists. Assume that $a_{n + 1}$ exists for all $n \ge  2020$.

Prove that $a_{n + 1} \le a_n$ for all $n \ge 2021$.
2020 Dutch IMO TST
Ward and Gabrielle play a game on a large piece of paper. In the beginning, there are the integers $1-999$  written on the paper, all once . Ward and Gabrielle take turns, where Ward begins. A player whose turn is, takes two numbers $a$ and $b$ from the paper for which $gcd (a, b) = 1$, erase these numbers and writes the number $a + b$ on paper. The first person who can no longer make a move loses. Determine who from Ward and Gabrielle  can win this game with certainty.





original wording, in case my translation has flaws, it might be incorrectWard en Gabrielle spelen een spel op een groot vel papier. In het begin staan er 999 enen op het papier geschreven. Ward en Gabrielle zijn om en om aan de beurt, waarbij Ward begint. Een speler die aan de beurt is, mag twee getallen a en b van het papier uitkiezen waarvoor geldt ggd(a, b) = 1, deze getallen weggummen en het getal a + b erbij schrijven. De eerste die geen zet meer kan doen, verliest. Bepaal wie van Ward en Gabrielle dit spel met zekerheid kan winnen
2020 Dutch IMO TST
Find all pairs $(a, b)$ of positive integers for which $a + b = \phi (a) +  \phi (b) + gcd (a, b)$.



Here $ \phi (n)$ is the number of numbers $k$ from $\{1, 2,. . . , n\}$ with $gcd (n, k) = 1$.
2020 Dutch IMO TST
Let $ABC$ be an acute-angled triangle and let $P$ be the intersection of the tangents at $B$ and $C$ of the circumscribed circle of $\vartriangle ABC$. The line through $A$ perpendicular on $AB$ and cuts the line perpendicular on $AC$ through $C$  at $X$. The line through $A$ perpendicular on $AC$ cuts the line perpendicular on $AB$ through $B$ at $Y$. Show that $AP \perp XY$.
2020 Dutch IMO TST
For a positive number $n$,  we write $d (n)$ for the number of positive divisors of $n$.

Determine all positive integers $k$ for which exist positive integers $a$ and $b$  with the property $k = d (a) = d (b) = d (2a + 3b)$.
2020 Dutch IMO TST
Given is a triangle $ABC$ with its circumscribed circle and $| AC | <| AB |$. On the short arc $AC$, there is a variable point $D\ne A$. Let $E$ be the reflection of $A$ wrt the inner bisector of $\angle BDC$. Prove that the line $DE$ passes through a fixed point, regardless of point $D$.
2020 Dutch IMO TST
Find all functions $f: Z \to Z$ that satisfy $$f(-f (x) - f (y))= 1 -x - y$$for all $x, y \in Z$
2020 Dutch IMO TST
Given are two positive integers $k$ and $n$ with $k \le n \le 2k - 1$. Julian has a large stack of rectangular $k \times 1$ tiles. Merlin calls a positive integer $m$ and receives $m$ tiles from Julian to place on an $n \times n$ board. Julian first writes on every tile  whether it should be a horizontal or a vertical tile. Tiles may be used the board should not overlap or protrude. What is the largest number $m$ that Merlin can call if he wants to make sure that he has all tiles according to the rule of Julian can put on the plate?
2019 Dutch IMO TST
Let $P(x)$ be a quadratic polynomial with two distinct real roots.

For all real numbers $a$ and $b$ satisfying $|a|,|b| \ge 2017$, we have $P(a^2+b^2) \ge P(2ab)$.

Show that at least one of the roots of $P$ is negative.
2019 Dutch IMO TST
Write $S_n$ for the set $\{1, 2,..., n\}$. Determine all positive integers $n$ for which there exist functions $f : S_n \to S_n$ and $g : S_n \to S_n$ such that for every $x$ exactly one of the equalities $f(g(x)) = x$ and $g(f(x)) = x$ holds.
2019 Dutch IMO TST
Let $n$ be a positive integer. Determine the maximum value of $gcd(a, b) + gcd(b, c) + gcd(c, a)$ for positive integers $a, b, c$ such that $a + b + c = 5n$.
2019 Dutch IMO TST
Let $\Delta ABC$ be a scalene triangle. Points $D,E$ lie on side $\overline{AC}$ in the order, $A,E,D,C$. Let the parallel through $E$ to $BC$ intersect $\odot (ABD)$ at $F$, such that, $E$ and $F$ lie on the same side of $AB$. Let the parallel through $E$ to $AB$ intersect $\odot (BDC)$ at $G$, such that, $E$ and $G$ lie on the same side of $BC$. Prove, Points $D,F,E,G$ are concyclic
2019 Dutch IMO TST
In each of the different grades of a high school there are an odd number of pupils. Each pupil has a best friend (who possibly is in a different grade). Everyone is the best friend of their best friend. In the upcoming school trip, every pupil goes to either Rome or Paris. Show that the pupils can be distributed over the two destinations in such a way that

(i) every student goes to the same destination as their best friend;

(ii) for each grade the absolute difference between the number of pupils that are going to Rome and that of those who are going to Paris is equal to $1$.
2019 Dutch IMO TST
Determine all $4$-tuples $(a,b, c, d)$ of positive real numbers satisfying $a + b +c + d = 1$ and

$\max (\frac{a^2}{b},\frac{b^2}{a}) \cdot \max (\frac{c^2}{d},\frac{d^2}{c}) = (\min (a + b, c + d))^4$
2019 Dutch IMO TST
Let $ABC$ be an acute angles triangle with $O$ the center of the circumscribed circle. Point $Q$ lies on the circumscribed circle of $\vartriangle BOC$ so that $OQ$ is a diameter. Point $M$ lies on $CQ$ and point $N$ lies internally on line segment $BC$ so that $ANCM$ is a parallelogram. Prove that the circumscribed circle of $\vartriangle BOC$ and the lines $AQ$ and $NM$ pass through the same point.
2019 Dutch IMO TST
Find all functions $f : Z \to Z$ satisfying

$\bullet$ $ f(p) > 0$ for all prime numbers $p$,

$\bullet$ $p| (f(x) + f(p))^{f(p)}- x$ for all $x \in Z$ and all prime numbers $p$.
2019 Dutch IMO TST
Let $ABCD$ be a cyclic quadrilateral (In the same order) inscribed into the circle $\odot (O)$. Let $\overline{AC}$ $\cap$ $\overline{BD}$ $=$ $E$. A randome line $\ell$ through $E$ intersects $\overline{AB}$ at $P$ and $BC$ at $Q$.  A circle $\omega$ touches $\ell$ at $E$ and passes through $D$. Given, $\omega$ $\cap$ $\odot (O)$ $=$ $R$. Prove,  Points $B,Q,R,P$ are concyclic.
2019 Dutch IMO TST
Let $n$ be a positive integer. Prove that $n^2 + n + 1$ cannot be written as the product of two positive integers of which the difference is smaller than $2\sqrt{n}$.
2019 Dutch IMO TST
Find all functions $f : Z \to Z$ satisfying the following two conditions:

(i) for all integers $x$ we have $f(f(x)) = x$,

(ii) for all integers $x$ and y such that $x + y$ is odd, we have $f(x) + f(y) \ge x + y$.
2019 Dutch IMO TST
There are $300$ participants to a mathematics competition. After the competition some of the contestants play some games of chess. Each two contestants play at most one game against each other. There are no three contestants, such that each of them plays against each other. Determine the maximum value of $n$ for which it is possible to satisfy the following conditions at the same time: each contestant plays at most $n$ games of chess, and for each $m$ with $1 \le m \le n$, there is a contestant playing exactly $m$ games of chess.
2018 Dutch IMO TST
Suppose a grid with $2m$ rows and $2n$ columns is given, where $m$ and $n$ are positive integers. You may place one pawn on any square of this grid, except the bottom left one or the top right one. After placing the pawn, a snail wants to undertake a journey on the grid. Starting from the bottom left square, it wants to visit every square exactly once, except the one with the pawn on it, which the snail wants to avoid. Moreover, it wants to finish in the top right square. It can only move horizontally or vertically on the grid.

On which squares can you put the pawn for the snail to be able to finish its journey?
2018 Dutch IMO TST
Suppose a triangle $\vartriangle ABC$ with $\angle C = 90^o$ is given. Let $D$ be the midpoint of $AC$, and let $E$ be the foot of the altitude through $C$ on $BD$. Show that the tangent in $C$ of the circumcircle of $\vartriangle AEC$ is perpendicular to $AB$.
2018 Dutch IMO TST
Let $n \ge  0$ be an integer. A sequence $a_0,a_1,a_2,...$ of integers is defined as follows:

we have $a_0 = n$ and for $k \ge 1, a_k$ is the smallest integer greater than $a_{k-1}$ for which $a_k +a_{k-1}$ is the square of an integer.

Prove that there are exactly $\lfloor \sqrt{2n}\rfloor$  positive integers that cannot be written in the form $a_k - a_{\ell}$ with $k > \ell\ge  0$.
2018 Dutch IMO TST
Let $A$ be a set of functions $f : R\to R$.

For all $f_1, f_2 \in A$ there exists a $f_3 \in A$ such that $f_1(f_2(y) - x)+ 2x = f_3(x + y)$ for all $x, y \in R$.

Prove that for all $f \in A$, we have $f(x - f(x))= 0$ for all $x \in R$.
2018 Dutch IMO TST
(a) If $c(a^3+b^3) = a(b^3+c^3) = b(c^3+a^3)$ with $a, b, c$ positive real numbers,

does $a = b = c$ necessarily hold?

(b) If $a(a^3+b^3) = b(b^3+c^3) = c(c^3+a^3)$ with $a, b, c$  positive real numbers,

does $a = b = c$ necessarily hold?
2018 Dutch IMO TST
Find all positive integers $n$, for which there exists a positive integer $k$ such that for every positive divisor $d$ of $n$, the number $d - k$ is also a (not necessarily positive) divisor of $n$.
2018 Dutch IMO TST
Let $ABC$ be an acute triangle, and let $D$ be the foot of the altitude through $A$. On $AD$, there are distinct points $E$ and $F$ such that $|AE| = |BE|$ and $|AF| =|CF|$. A point$ T \ne D$ satises $\angle BTE = \angle CTF = 90^o$. Show that $|TA|^2 =|TB| \cdot  |TC|$.
2018 Dutch IMO TST
In the classroom of at least four students the following holds: no matter which four of them take seats around a round table, there is always someone who either knows both of his neighbours, or does not know either of his neighbours. Prove that it is possible to divide the students into two groups such that in one of them, all students know one another, and in the other, none of the students know each other.



(Note: if student A knows student B, then student B knows student A as well.)
2018 Dutch IMO TST
A set of lines in the plan is called nice i f every line in the set intersects an odd number of other lines in the set.

Determine the smallest integer $k \ge  0$ having the following property:

for each $2018$ distinct lines $\ell_1, \ell_2, ..., \ell_{2018}$ in the plane, there exist lines $\ell_{2018+1},\ell_{2018+2}, . . . , \ell_{2018+k}$ such that the lines $\ell_1, \ell_2, ..., \ell_{2018+k}$ are distinct and form a nice  set.
2018 Dutch IMO TST
Find all functions $f : R \to  R$ such that $f(x^2)-f(y^2) \le   (f(x)+y) (x-f(y))$ for all $x, y \in  R$.
2018 Dutch IMO TST
Determine all pairs $(a,b)$ of positive integers such that $(a+b)^3-2a^3-2b^3$ is a power of two.
2018 Dutch IMO TST
In a non-isosceles triangle $ABC$ the centre of the incircle is denoted by $I$. The other intersection point of the angle bisector of $\angle BAC$ and the circumcircle of $\vartriangle ABC$ is $D$. The line through $I$ perpendicular to $AD$ intersects $BC$ in $F$. The midpoint of the circle arc $BC$ on which $A$ lies, is denoted by $M$. The other intersection point of the line $MI$ and the circle through $B, I$ and $C$, is denoted by $N$. Prove that $FN$ is tangent to the circle through $B, I$ and $C$.
2017 Dutch IMO TST
Let $n$ be a positive integer. Suppose that we have disks of radii $1, 2, . . . , n.$ Of each size there are two disks: a transparent one and an opaque one. In every disk there is a small hole in the centre, with which we can stack the

disks using a vertical stick. We want to make stacks of disks that satisfy the following conditions:

$i)$ Of each size exactly one disk lies in the stack.

$ii)$ If we look at the stack from directly above, we can see the edges of all of the $n$ disks in the stack. (So if there is an opaque disk in the stack,no smaller disks may lie beneath it.)

Determine the number of distinct stacks of disks satisfying these conditions.

(Two stacks are distinct if they do not use the same set of disks, or, if they do use the same set of disks and the orders in which the disks occur are different.)
2017 Dutch IMO TST
Let $n \geq 4$ be an integer. Consider a regular $2n-$gon for which to every vertex, an integer is assigned, which we call the value of said vertex. If four distinct vertices of this $2n-$gon form a rectangle, we say that the sum of the values of these vertices is a rectangular sum.

Determine for which (not necessarily positive) integers $m$ the integers $m + 1, m + 2, . . . , m + 2n$ can be assigned to the vertices (in some order) in such a way that every rectangular sum is a prime number. (Prime numbers are positive by definition.)
2017 Dutch IMO TST
let $x,y$ be non-zero reals such that : $x^3+y^3+3x^2y^2=x^3y^3$

find all values of $\frac{1}{x}+\frac{1}{y}$
2017 Dutch IMO TST
Let $ABC$ be a triangle, let $M$ be the midpoint of $AB$, and let $N$ be the midpoint of $CM$. Let $X$ be a point satisfying both $\angle XMC = \angle MBC$ and $\angle XCM =  \angle MCB$ such that $X$ and $B$ lie on opposite sides of $CM$. Let $\omega$ be the circumcircle of triangle $AMX$.

$(a)$ Show that $CM$ is tangent to $\omega$.

$(b)$ Show that the lines $NX$ and $AC$ intersect on $\omega$
2017 Dutch IMO TST
Let $a, b,c$ be distinct positive integers, and suppose that $p = ab+bc+ca$ is a prime number.

$(a)$ Show that $a^2,b^,c^2$  give distinct remainders after division by $p$.

(b) Show that $a^3,b^3,c^3$  give distinct remainders after division by $p$.
2017 Dutch IMO TST
The incircle of a non-isosceles triangle $ABC$ has centre $I$ and is tangent to  $BC$ and $CA$ in $D$ and $E$, respectively. Let $H$ be the orthocentre of $ABI$, let $K$ be the intersection of $AI$ and $BH$ and let $L$ be the intersection of $BI$ and $AH$. Show that the circumcircles of $DKH$ and $ELH$ intersect on the incircle of $ABC$.
2017 Dutch IMO TST
Let $k > 2$ be an integer. A positive integer $l$  is said to be $k-pable$ if the numbers $1, 3, 5, . . . , 2k - 1$ can be partitioned into two subsets $A$ and $B$ in such a way that the sum of the elements of $A$ is exactly $l$  times as large as the sum of the elements of $B$.

Show that the smallest $k-pable$ integer is coprime to $k$.
2017 Dutch IMO TST
Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that

$$(y + 1)f(x) + f(xf(y) + f(x + y))= y$$for all $x, y \in \mathbb{R}$.
2017 Dutch IMO TST
A circle $\omega$ with diameter $AK$ is given. The point $M$ lies in the interior of the circle, but not on $AK$. The line $AM$ intersects $\omega$ in $A$ and $Q$. The tangent to $\omega$ at $Q$ intersects the line through $M$ perpendicular to $AK$, at $P$. The point $L$ lies on $\omega$, and is such that $PL$ is tangent to $\omega$ and $L\neq Q$.

Show that $K, L$, and $M$ are collinear.
2017 Dutch IMO TST
let $a_1,a_2,...a_n$ a sequence of real numbers such that $a_1+....+a_n=0$.

define $b_i=a_1+a_2+....a_i$ for all $1 \leq i \leq n$ .suppose $b_i(a_{j+1}-a_{i+1}) \geq 0$ for all $1 \leq i \leq j \leq n-1$.

Show that $$\max_{1 \leq l \leq n} |a_l| \geq \max_{1 \leq m \leq n} |b_m|$$
2017 Dutch IMO TST
Compute the product of all positive integers $n$ for which $3(n!+1)$ is divisible by $2n - 5$.
2017 Dutch IMO TST
Let $n \geq 2$ be an integer. Find the smallest positive integer $m$ for which the following holds: given $n$ points in the plane, no three on a line, there are $m$ lines such that no line passes through any of the given points, and

for all points $X  \neq Y$ there is a line with respect to which $X$ and $Y$ lie on opposite sides
2016 Dutch IMO TST
Let $\triangle ABC$ be a acute triangle. Let $H$ the foot of the C-altitude in $AB$ such that $AH=3BH$, let $M$ and $N$ the midpoints of $AB$ and $AC$ and let $P$ be a point such that $NP=NC$ and $CP=CB$ and $B$, $P$ are located on different sides of the line $AC$. Prove that $\measuredangle APM=\measuredangle PBA$.
2016 Dutch IMO TST
In a $2^n \times 2^n$ square with $n$ positive integer is covered with at least two non-overlapping rectangle pieces with integer dimensions and a power of two as surface. Prove that two rectangles of the covering have the same dimensions  (Two rectangles have the same dimensions as they have the same width and the same height, wherein they, not allowed to be rotated.)
2016 Dutch IMO TST
Find all positive integers $k$ for which the equation: $$ \text{lcm}(m,n)-\text{gcd}(m,n)=k(m-n)$$has no solution in integers positive $(m,n)$ with $m\neq n$.
2016 Dutch IMO TST
Find all funtions $f:\mathbb R\to\mathbb R$ such that: $$f(xy-1)+f(x)f(y)=2xy-1$$for all $x,y\in \mathbb{R}$.
2016 Dutch IMO TST
Prove that for all positive reals $a, b,c$ we have: $a +\sqrt{ab}+ \sqrt[3]{abc}\le \frac43 (a + b + c)$
2016 Dutch IMO TST
Determine all pairs $(a, b)$ of integers having the following property:

there is an integer $d \ge  2$ such that $a^n + b^n + 1$ is divisible by $d$ for all positive integers $n$.
2016 Dutch IMO TST
Let $\vartriangle ABC$ be an isosceles triangle with $|AB| = |AC|$. Let $D, E$ and $F$ be points on line segments $BC, CA$ and $AB$, respectively, such that $|BF| = |BE|$ and such that $ED$ is the internal angle bisector of $\angle BEC$. Prove that  $|BD|= |EF|$ if and only if $|AF| = |EC|$.



2016 Dutch IMO TST2 P3
2016 Dutch IMO TST
Determine the number of sets $A = \{a_1,a_2,...,a_{1000}\}$ of positive integers satisfying $a_1 < a_2 <...< a_{1000} \le 2014$, for which we have that the set

$S = \{a_i + a_j | 1 \le i, j \le 1000$ with $i + j \in A\}$ is a subset of $A$.
2016 Dutch IMO TST
Let $n$ be a positive integer. In a village, $n$ boys and $n$ girls are living.

For the yearly ball, $n$ dancing couples need to be formed, each of which consists of one boy and one girl. Every girl submits a list, which consists of the name of the boy with whom she wants to dance the most, together with zero or more names of other boys with whom she wants to dance. It turns out that $n$ dancing couples can be formed in such a way that every girl is paired with a boy who is on her list.

Show that it is possible to form $n$ dancing couples in such a way that every girl is paired with a boy who is on her list, and at least one girl is paired with the boy with whom she wants to dance the most.
2016 Dutch IMO TST
For distinct real numbers $a_1,a_2,...,a_n$, we calculate the $\frac{n(n-1)}{2}$ sums $a_i +a_j$ with $1 \le i < j \le  n$, and sort them in ascending order. Find all integers $n \ge  3$ for which there exist $a_1,a_2,...,a_n$, for which this sequence of $\frac{n(n-1)}{2}$ sums form an arithmetic progression (i.e. the dierence between consecutive terms is constant).
2016 Dutch IMO TST
Let $k$ be a positive integer, and let $s(n)$ denote the sum of the digits of $n$.

Show that among the positive integers with $k$ digits, there are as many numbers $n$ satisfying $s(n) < s(2n)$ as there are numbers $n$ satisfying $s(n) > s(2n)$.
2016 Dutch IMO TST
Let $\Gamma_1$ be a circle with centre $A$ and $\Gamma_2$ be a circle with centre $B$, with $A$ lying on $\Gamma_2$. On $\Gamma_2$ there is a (variable) point $P$ not lying on $AB$. A line through $P$ is a tangent of $\Gamma_1$ at $S$, and it intersects $\Gamma_2$  again in $Q$, with $P$ and $Q$ lying on the same side of $AB$. A different line through $Q$ is tangent to $\Gamma_1$ at $T$. Moreover, let $M$ be the foot of the perpendicular to $AB$ through $P$. Let $N$ be the intersection of $AQ$ and $MT$.

Show that $N$ lies on a line independent of the position of $P$ on $\Gamma_2$.



2016 Dutch IMO TST3 P4
2015 Dutch IMO TST
In a quadrilateral $ABCD$ we have $\angle A = \angle C = 90^o$. Let $E$ be a point in the interior of $ABCD$. Let $M$ be the midpoint of $BE$. Prove that $\angle ADB = \angle EDC$ if and only if  $|MA| = |MC|$.
2015 Dutch IMO TST
Determine all polynomials P(x) with real coefficients such that

[(x + 1)P(x − 1) − (x − 1)P(x)] is a constant polynomial.
2015 Dutch IMO TST
Let $n$ be a positive integer.

Consider sequences $a_0, a_1, ..., a_k$ and $b_0, b_1,,..,b_k$ such that $a_0 = b_0 = 1$ and $a_k = b_k = n$ and such that for all $i$ such that $1 \le  i \le k $, we have that $(a_i, b_i)$ is either equal to $(1 + a_{i-1}, b_{i-1})$ or $(a_{i-1}; 1 + b_{i-1})$.

Consider for $1 \le  i \le k$ the number $c_i = \begin{cases} a_i \,\,\, if \,\,\, a_i = a_{i-1} \\
b_i \,\,\, if \,\,\, b_i = b_{i-1}\end{cases}$

Show that $c_1 + c_2 + ... + c_k = n^2 - 1$.
2015 Dutch IMO TST
Let $\Gamma_1$ and $\Gamma_2$ be circles - with respective centres $O_1$ and $O_2$ - that intersect each other in $A$ and $B$. The line $O_1A$ intersects $\Gamma_2$ in $A$ and $C$ and the line $O_2A$ intersects $\Gamma_1$ in $A$ and $D$. The line through $B$ parallel to $AD$ intersects $\Gamma_1$ in $B$ and $E$. Suppose that $O_1A$ is parallel to $DE$. Show that $CD$ is perpendicular to $O_2C$.
2015 Dutch IMO TST
For a positive integer $n$, we dene $D_n$ as the largest integer that is a divisor  of $a^n + (a + 1)^n + (a + 2)^n$ for all positive integers $a$.

1. Show that for all positive integers $n$, the number $D_n$ is of the form $3^k$ with $k \ge 0$ an integer.

2. Show that for all integers $k \ge 0$ there exists a positive integer n such that $D_n = 3^k$.
2015 Dutch IMO TST
Let $a$ and $b$ be two positive integers satifying $gcd(a, b) = 1$. Consider a pawn standing on the grid point $(x, y)$.

A step of type A consists of moving the pawn to one of the following grid points: $(x+a, y+a),(x+a,y-a), (x-a, y + a)$ or $(x - a, y - a)$.

A step of type B consists of moving the pawn to $(x + b,y + b),(x + b,y - b), (x - b,y + b)$ or $(x - b,y - b)$.

Now put a pawn on $(0, 0)$. You can make a (nite) number of steps, alternatingly of type A and type B, starting with a step of type A. You can make an even or odd number of steps, i.e., the last step could be of either type A or type B.

Determine the set of all grid points $(x,y)$ that you can reach with such a series of steps.
2015 Dutch IMO TST
Determine all positive integers $n$ for which there exist positive integers $a_1,a_2, ..., a_n$



with $a_1 + 2a_2 + 3a_3 +... + na_n = 6n$ and $\frac{1}{a_1}+\frac{2}{a_2}+\frac{3}{a_3}+ ... +\frac{n}{a_n}= 2 + \frac1n$
2015 Dutch IMO TST
An equilateral triangle $ABC$ is given. On the line through $B$ parallel to $AC$ there is a point $D$, such that $D$ and $C$ are on the same side of the line $AB$. The perpendicular bisector of $CD$ intersects the line $AB$ in $E$. Prove that triangle $CDE$ is equilateral.
2015 Dutch IMO TST
Each of the numbers $1$ up to and including $2014$ has to be coloured; half of them have to be coloured red the other half blue. Then you consider the number $k$ of positive integers that are expressible as the sum of a red and a blue number. Determine the maximum value of $k$ that can be obtained.
2015 Dutch IMO TST
Let $N$ be the set of positive integers.

Find all the functions $f: N\to N$ with $f (1) = 2$ and such that $max \{f(m)+f(n), m+n\}$ divides $min\{2m+2n,f (m+ n)+1\}$ for all $m, n$ positive integers
2014 Dutch IMO TST
Determine all pairs $(a,b)$ of positive integers satisfying

$$a^2+b\mid a^2b+a\quad\text{and}\quad b^2-a\mid ab^2+b.$$
2014 Dutch IMO TST
Let $\triangle ABC$ be a triangle. Let $M$ be the midpoint of $BC$ and let $D$ be a point on the interior of side $AB$. The intersection of $AM$ and $CD$ is called $E$. Suppose that $|AD|=|DE|$. Prove that $|AB|=|CE|$.