Navvye/Olympiad-Problems-CSV huggingface dataset:
2016 Dutch Mathematical Olympiad
(a) On a long pavement, a sequence of $999$ integers is written in chalk. The numbers need not be in increasing order and need not be distinct. Merlijn encircles $500$ of the numbers with red chalk. From left to right, the numbers circled in red are precisely the numbers $1, 2, 3, ...,499, 500$. Next, Jeroen encircles $500$ of the numbers with blue chalk. From left to right, the numbers circled in blue are precisely the numbers $500, 499, 498, ...,2,1$.



Prove that the middle number in the sequence of $999$ numbers is circled both in red and in blue.



(b) Merlijn and Jeroen cross the street and find another sequence of $999$ integers on the pavement. Again Merlijn circles $500$ of the numbers with red chalk. Again the numbers circled in red are precisely the numbers $1, 2, 3, ...,499, 500$ from left to right. Now Jeroen circles $500$ of the numbers, not necessarily the same as Merlijn, with green chalk. The numbers circled in green are also precisely the numbers $1, 2, 3, ...,499, 500$ from left to right.



Prove: there is a number that is circled both in red and in green that is not the middle number of the sequence of $999$ numbers.
2016 Dutch Mathematical Olympiad
For an integer $n \ge 1$ we consider sequences of $2n$ numbers, each equal to $0, -1$ or $1$. The sum product value of such a sequence is calculated by first multiplying each pair of numbers from the sequence, and then adding all the results together.

For example, if we take $n = 2$ and the sequence $0,1, 1, -1$, then we find the products $0\cdot 1, 0\cdot 1, 0\cdot -1, 1\cdot 1, 1\cdot -1, 1\cdot -1$. Adding these six results gives the sum product value of this sequence: $0+0+0+1+(-1)+(-1) = -1$. The sum product value of this sequence is therefore smaller than the sum product value of the sequence $0, 0, 0, 0$, which equals $0$.

Determine for each integer $n \ge 1$ the smallest sum product value that such a sequence of $2n$ numbers could have.



Attention: you are required to prove that a smaller sum product value is impossible.
2016 Dutch Mathematical Olympiad
Find all possible triples $(a, b, c)$ of positive integers with the following properties:

• $gcd(a, b) = gcd(a, c) = gcd(b, c) = 1$,

• $a$ is a divisor of $a + b + c$,

• $b$ is a divisor of $a + b + c$,

• $c$ is a divisor of $a + b + c$.



(Here $gcd(x,y)$ is the greatest common divisor of $x$ and $y$.)
2016 Dutch Mathematical Olympiad
In a quadrilateral $ABCD$ the intersection of the diagonals is called $P$. Point $X$ is the orthocentre of triangle $PAB$. (The orthocentre of a triangle is the point where the three altitudes of the triangle intersect.) Point $Y$ is the orthocentre of triangle $PCD$. Suppose that $X$ lies inside triangle $PAB$ and $Y$ lies inside triangle $PCD$. Moreover, suppose that $P$ is the midpoint of line segment $XY$ . Prove that $ABCD$ is a parallelogram.

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2016 Dutch Mathematical Olympiad
In the acute triangle $ABC$, the midpoint of side $BC$ is called $M$. Point $X$ lies on the angle bisector of $\angle AMB$ such that $\angle BXM = 90^o$. Point $Y$ lies on the angle bisector of $\angle AMC$ such that $\angle CYM = 90^o$. Line segments $AM$ and $XY$ intersect in point $Z$. Prove that $Z$ is the midpoint of $XY$ .

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2016 Dutch Mathematical Olympiad
Bas has coloured each of the positive integers. He had several colours at his disposal. His colouring satises the following requirements:

• each odd integer is coloured blue,

• each integer $n$ has the same colour as $4n$,

• each integer $n$ has the same colour as at least one of the integers $n+2$ and $n + 4$.

Prove that Bas has coloured all integers blue.
2015 Dutch Mathematical Olympiad
We make groups of numbers. Each group consists of five distinct numbers. A number may occur in multiple groups. For any two groups, there are exactly four numbers that occur in both groups.

(a) Determine whether it is possible to make $2015$ groups.

(b) If all groups together must contain exactly six  distinct numbers, what is the greatest number of groups that you can make?

(c) If all groups together must contain exactly seven  distinct numbers, what is the greatest number of groups that you can make?
2015 Dutch Mathematical Olympiad
On a $1000\times 1000$-board we put dominoes, in such a way that each domino covers exactly two squares on the board. Moreover, two dominoes are not allowed to be adjacent, but are allowed to touch in a vertex.

Determine the maximum number of dominoes that we can put on the board in this way.



Attention: you have to really prove that a greater number of dominoes is impossible.
2015 Dutch Mathematical Olympiad
In quadrilateral $ABCD$ sides $BC$ and $AD$ are parallel. In each of the four vertices we draw an angular bisector. The angular bisectors of angles $A$ and $B$ intersect in point $P$, those of angles $B$ and $C$ intersect in point $Q$, those of angles $C$ and $D$ intersect in point $R$, and those of angles $D$ and $A$ intersect in point S. Suppose that $PS$ is parallel to $QR$. Prove that $|AB| =|CD|$.

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Attention: the figure is not drawn to scale.
2015 Dutch Mathematical Olympiad
Points $A, B$, and $C$ are on a line in this order. Points $D$ and $E$ lie on the same side of this line, in such a way that triangles $ABD$ and $BCE$ are equilateral. The segments $AE$ and $CD$ intersect in point $S$. Prove that $\angle ASD = 60^o$.

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2015 Dutch Mathematical Olympiad
Find all pairs of prime numbers $(p, q)$ for which $7pq^2 + p = q^3 + 43p^3 + 1$
2015 Dutch Mathematical Olympiad
Given are (not necessarily positive) real numbers $a, b$, and $c$ for which $|a - b| \ge |c| , |b - c| \ge |a|$ and $|c - a| \ge |b|$ . Prove that one of the numbers $a, b$, and $c$ is the sum of the other two.
2014 Dutch Mathematical Olympiad
Determine all triples $(a,b,c)$, where $a, b$, and $c$ are positive integers that satisfy



$a \le b \le c$ and $abc = 2(a + b + c)$.
2014 Dutch Mathematical Olympiad
Let $ABCD$ be a parallelogram with an acute angle at $A$. Let $G$ be a point on the line $AB$, distinct from $B$, such that $|CG| = |CB|$. Let $H$ be a point on the line $BC$, distinct from $B$, such that $|AB| =|AH|$. Prove that triangle $DGH$ is isosceles.

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2014 Dutch Mathematical Olympiad
On the sides of triangle $ABC$, isosceles right-angled triangles $AUB, CVB$, and $AWC$ are placed. These three triangles have their right angles at vertices $U, V$ , and $W$, respectively. Triangle $AUB$ lies completely inside triangle $ABC$ and triangles $CVB$ and $AWC$ lie completely outside $ABC$. See the figure. Prove that quadrilateral $UVCW$ is a parallelogram.

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2014 Dutch Mathematical Olympiad
At a volleyball tournament, each team plays exactly once against each other team. Each game has a winning team, which gets $1$ point. The losing team gets $0$ points. Draws do not occur. In the nal ranking, only one team turns out to have the least number of points (so there is no shared last place). Moreover, each team, except for the team having the least number of points, lost exactly one game against a team that got less points in the final ranking.

a) Prove that the number of teams cannot be equal to $6$.

b) Show, by providing an example, that the number of teams could be equal to $7$.
2014 Dutch Mathematical Olympiad
A quadruple $(p, a, b, c)$ of positive integers is called a Leiden quadruple if

- $p$ is an odd prime number,

- $a, b$, and $c$ are distinct and

- $ab + 1, bc + 1$ and $ca + 1$ are divisible by $p$.

a) Prove that for every Leiden quadruple $(p, a, b, c)$ we have $p + 2  \le \frac{a+b+c}{3}$ .

b) Determine all numbers $p$ for which a Leiden quadruple  $(p, a, b, c)$ exists with $p + 2 =  \frac{a+b+c}{3} $
2014 Dutch Mathematical Olympiad
We consider the ways to divide a $1$ by $1$ square into rectangles (of which the sides are parallel to those of the square). All rectangles must have the same circumference, but not necessarily the same shape.

a) Is it possible to divide the square into 20 rectangles, each having a circumference of $2:5$?

b) Is it possible to divide the square into 30 rectangles, each having a circumference of $2$?
2013 Dutch Mathematical Olympiad
In a table consisting of $n$ by $n$ small squares some squares are coloured black and the other squares are coloured white. For each pair of columns and each pair of rows the four squares on the intersections of these rows and columns must not all be of the same colour.

What is the largest possible value of $n$?
2013 Dutch Mathematical Olympiad
Find all triples $(x, y, z)$ of real numbers satisfying:

$x + y - z = -1$ ,  $x^2 - y^2 + z^2 = 1$ and $- x^3 + y^3 + z^3 = -1$
2013 Dutch Mathematical Olympiad
The sides $BC$ and $AD$ of a quadrilateral $ABCD$ are parallel and the diagonals intersect in $O$. For this quadrilateral $|CD| =|AO|$ and $|BC| = |OD|$ hold. Furthermore $CA$ is the angular bisector of angle $BCD$. Determine the size of angle $ABC$.



Attention: the figure is not drawn to scale.
2013 Dutch Mathematical Olympiad
For a positive integer n the number $P(n)$ is the product of the positive divisors of $n$. For example, $P(20) = 8000$, as the positive divisors of $20$ are $1, 2, 4, 5, 10$ and $20$, whose product is $1 \cdot 2 \cdot 4 \cdot 5 \cdot 10 \cdot 20 = 8000$.

(a) Find all positive integers $n$ satisfying $P(n) = 15n$.

(b) Show that there exists no positive integer $n$ such that $P(n) = 15n^2$.
2013 Dutch Mathematical Olympiad
The number $S$ is the result of the following sum: $1 + 10 + 19 + 28 + 37 +...+ 10^{2013}$

If one writes down the number $S$, how often does the digit `$5$' occur in the result?
2012 Dutch Mathematical Olympiad
Let $a, b, c$, and $d$ be four distinct integers.

Prove that $(a-b)(a-c)(a-d)(b-c)(b-d)(c-d)$  is divisible by $12$.
2012 Dutch Mathematical Olympiad
We number the columns of an $n\times n$-board from $1$ to $n$. In each cell, we place a number. This is done in such a way that each row precisely contains the numbers $1$ to $n$ (in some order), and also each column contains the numbers $1$ to $n$ (in some order). Next, each cell that contains a number greater than the cell's column number, is coloured grey. In the figure below you can see an example for the case $n = 3$.



(a) Suppose that $n = 5$. Can the numbers be placed in such a way that each row contains the same number of grey cells?

(b) Suppose that $n = 10$. Can the numbers be placed in such a way that each row contains the same number of grey cells?
2012 Dutch Mathematical Olympiad
Determine all pairs $(p,m)$ consisting of a prime number $p$ and a positive integer $m$,

for which $p^3 + m(p + 2) = m^2 + p + 1$ holds.
2012 Dutch Mathematical Olympiad
We are given an acute triangle $ABC$ and points $D$ on $BC$ and $E$ on $AC$ such that $AD$ is perpendicular to $BC$ and $BE$ is perpendicular to $AC$. The intersection of $AD$ and BE is called $H$. A line through $H$ intersects line segment $BC$ in $P$, and intersects line segment $AC$ in $Q$. Furthermore, $K$ is a point on $BE$ such that $PK$ is perpendicular to $BE$, and $L$ is a point on $AD$ such that $QL$ is perpendicular to $AD$. Prove that $DK$ and $EL$ are parallel.

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2012 Dutch Mathematical Olympiad
The numbers $1$ to $12$ are arranged in a sequence.  The number of ways this can be done equals $12\times11\times 10\times ...\times 1$. We impose the condition that in the sequence there should be exactly one number that is smaller than the number directly preceding it.

How many of the $12\times11\times 10\times ...\times 1$ sequences satisfy this condition?
2011 Dutch Mathematical Olympiad
Determine all triples of positive integers $(a, b, n)$ that satisfy the following equation: $a! + b! = 2^n$
2011 Dutch Mathematical Olympiad
Let $ABC$ be a triangle.

Points $P$ and $Q$ lie on side $BC$ and satisfy $|BP| =|PQ| = |QC| = \frac13 |BC|$.

Points $R$ and $S$ lie on side $CA$ and satisfy $|CR| =|RS| = |SA| = 1 3 |CA|$.

Finally, points $T$ and $U$ lie on side $AB$ and satisfy $|AT| = |TU| = |UB| =\frac13 |AB|$.

Points $P, Q,R, S, T$ and $U$ turn out to lie on a common circle.

Prove that $ABC$ is an equilateral triangle.
2011 Dutch Mathematical Olympiad
In a tournament among six teams, every team plays against each other team exactly once. When a team wins, it receives $3$ points and the losing team receives $0$ points. If the game is a draw, the two teams receive $1$ point each.

Can the final scores of the six teams be six consecutive numbers $a,a +1,...,a + 5$?

If so, determine all values of $a$ for which this is possible.
2011 Dutch Mathematical Olympiad
Determine all pairs of positive real numbers $(a, b)$ with $a > b$ that satisfy the following equations:

$a\sqrt{a}+ b\sqrt{b} = 134$ and $a\sqrt{b}+ b\sqrt{a} = 126$.
2011 Dutch Mathematical Olympiad
The number devil has coloured the integer numbers: every integer is coloured either black or white.

The number $1$ is coloured white. For every two white numbers $a$ and $b$ ($a$ and $b$ are allowed to be equal) the numbers $a-b$ and $a + $b have different colours.

Prove that $2011$ is coloured white.
2010 Dutch Mathematical Olympiad
Consider a triangle $ABC$ such that $\angle A = 90^o, \angle C =60^o$ and $|AC|= 6$. Three circles with centers $A, B$ and $C$ are pairwise tangent in points on the three sides of the triangle.

Determine the area of the region enclosed by the three circles (the grey area in the figure).
2010 Dutch Mathematical Olympiad
A number is called polite if it can be written as   $ m + (m+1)+...+ n$, for certain positive integers  $ m <n$ . For example: $18$ is polite, since $18 =5 + 6 + 7$. A number is called a power of two if it can be written as $2^{\ell}$ for

some integer $\ell \ge 0$.

(a) Show that no number is both polite and a power of two.

(b) Show that every positive integer is polite or a power of two.
2010 Dutch Mathematical Olympiad
Consider a triangle $XYZ$ and a point $O$ in its interior. Three lines through $O$ are drawn, parallel to the respective sides of the triangle. The intersections with the sides of the triangle determine six line segments from $O$ to the sides of the triangle. The lengths of these segments are integer numbers $a, b, c, d, e$ and $f$ (see figure).

Prove that the product $a \cdot b \cdot c\cdot d \cdot e \cdot f$ is a perfect square.

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2010 Dutch Mathematical Olympiad
(a) Determine all pairs $(x, y)$ of (real) numbers with $0 < x < 1$ and $0 <y < 1$ for which $x + 3y$ and $3x + y$ are both integer. An example is $(x,y) =( \frac{8}{3}, \frac{7}{8}) $, because $  x+3y =\frac38 +\frac{21}{8} =\frac{24}{8} =  3$ and $ 3x+y = \frac98 + \frac78 =\frac{16}{8} = 2$.



(b) Determine the integer $m > 2$ for which there are exactly  $119$ pairs $(x,y)$ with $0 < x < 1$ and $0 < y < 1$ such that $x + my$ and  $mx + y$ are integers.



Remark: if $u \ne v,$ the pairs $(u, v)$ and $(v, u)$ are different.
2010 Dutch Mathematical Olympiad
Amber and Brian are playing a game using $2010$ coins. Throughout the game, the coins are divided into a number of piles of at least 1 coin each. A move consists of choosing one or more piles and dividing each of them into two smaller piles. (So piles consisting of only $1$ coin cannot be chosen.)

Initially, there is only one pile containing all $2010$ coins. Amber and Brian alternatingly take turns to make a move, starting with Amber. The winner is the one achieving the situation where all piles have only one coin.

Show that Amber can win the game, no matter which moves Brian makes.
2009 Dutch Mathematical Olympiad
In this problem, we consider integers consisting of $5$ digits, of which the rst and last one are nonzero. We say that such an integer is a palindromic product if it satises the following two conditions:

- the integer is a palindrome, (i.e. it doesn't matter if you read it from left to right, or the other way around);

- the integer is a product of two positive integers, of which the first, when read from left to right, is equal to the second, when read from right to left, like $4831$ and $1384$.

For example, $20502$ is a palindromic product, since $102 \cdot 201 = 20502$, and $20502$ itself is a palindrome.

Determine all palindromic products of $5$ digits.
2009 Dutch Mathematical Olympiad
Consider the sequence of integers $0, 1, 2, 4, 6, 9, 12,...$ obtained by starting with zero, adding $1$, then adding $1$ again, then adding $2$, and adding $2$ again, then adding $3$, and adding $3$ again, and so on. If we call the subsequent terms of this sequence $a_0, a_1, a_2, ...$, then we have $a_0 = 0$, and  $a_{2n-1} = a_{2n-2} + n$ , $a_{2n} = a_{2n-1} + n$  for all integers $n \ge 1$.

Find all integers $k \ge 0$ for which $a_k$ is the square of an integer.
2009 Dutch Mathematical Olympiad
A tennis tournament has at least three participants. Every participant plays exactly one match against every other participant. Moreover, every participant wins at least one of the matches he plays. (Draws do not occur in tennis matches.)

Show that there are three participants $A, B $ and $C$ for which the following holds: $A$ wins against $B, B$ wins against $C$, and $C$ wins against $A$.
2009 Dutch Mathematical Olympiad
Let $ABC$ be an arbitrary triangle. On the perpendicular bisector of $AB$, there is a point $P$ inside of triangle $ABC$. On the sides $BC$ and $CA$, triangles $BQC$ and $CRA$ are placed externally. These triangles satisfy $\vartriangle BPA \sim  \vartriangle BQC  \sim  \vartriangle CRA$. (So $Q$ and $A$ lie on opposite sides of $BC$, and $R$ and $B$ lie on opposite sides of $AC$.)

Show that the points $P, Q, C$ and $R$ form a parallelogram.
2009 Dutch Mathematical Olympiad
We number a hundred blank cards on both sides with the numbers $1$ to $100$. The cards are then stacked in order, with the card with the number $1$ on top.

The order of the cards is changed step by step as follows: at the $1$st step the top card is turned around, and is put back on top of the stack (nothing changes, of course), at the $2$nd step the topmost $2$ cards are turned around, and put back on top of the stack, up to the $100$th step, in which the entire stack of $100$ cards is turned around. At the $101$st step, again only the top card is turned around, at the $102$nd step, the top most $2$ cards are turned around, and so on.

Show that after a finite number of steps, the cards return to their original positions.
2008 Dutch Mathematical Olympiad
Suppose we have a square $ABCD$ and a point $S$ in the interior of this square.

Under homothety with centre $S$ and ratio of magnification $k > 1$, this square becomes another square $A'B'C'D'$.

Prove that the sum of the areas of the two quadrilaterals $A'ABB'$ and $C'CDD'$ are equal to the sum of the areas of the two quadrilaterals $B'BCC'$ and $D'DAA'$.

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2008 Dutch Mathematical Olympiad
Find all positive integers $(m, n)$ such that $3 \cdot 2^n + 1 = m^2$.
2008 Dutch Mathematical Olympiad
Suppose that we have a set $S$ of $756$ arbitrary integers between $1$ and $2008$ ($1$ and $2008$ included).

Prove that there are two distinct integers $a$ and $b$ in $S$ such that their sum $a + b$ is divisible by $8$.
2008 Dutch Mathematical Olympiad
Three circles $C_1,C_2,C_3$, with radii $1, 2, 3$ respectively, are externally tangent.

In the area enclosed by these circles, there is a circle $C_4$ which is externally tangent to all three circles.

Find the radius of $C_4$.

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2008 Dutch Mathematical Olympiad
We’re playing a game with a sequence of $2008$ non-negative integers.

A move consists of picking a integer $b$ from that sequence, of which the neighbours $a$ and $c$ are positive. We then replace $a, b$ and $c$ by $a - 1, b + 7$ and $c - 1$ respectively. It is not allowed to pick the first or the last integer in the sequence, since they only have one neighbour.  If there is no integer left such that both of its neighbours are positive, then there is no move left, and the game ends.

Prove that the game always ends, regardless of the sequence of integers we begin with, and regardless of the moves we make.
2007 Dutch Mathematical Olympiad
Consider the equilateral triangle ABC with $|BC| = |CA| = |AB| = 1$.

On the extension of side $BC$, we define points $A_1$ (on the same side as B) and $A_2$ (on the same side as C) such that $|A_1B| = |BC| = |CA_2| = 1$. Similarly, we define $B_1$ and $B_2$ on the extension of side $CA$ such that $|B_1C| = |CA| =|AB_2| = 1$, and $C_1$ and $C_2$ on the extension of side $AB$ such that $|C_1A| = |AB| = |BC_2| = 1$. Now the circumcentre of 4ABC is also the centre of the circle that passes through the points $A_1,B_2,C_1,A_2,B_1$ and $C_2$.

Calculate the radius of the circle through $A_1,B_2,C_1,A_2,B_1$ and $C_2$.

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2007 Dutch Mathematical Olympiad
Is it possible to partition the set $A = \{1, 2, 3, ... , 32, 33\}$ into eleven subsets that contain three integers each, such that for every one of these eleven subsets, one of the integers is equal to the sum of the other two? If so, give such a partition, if not, prove that such a partition cannot exist.
2007 Dutch Mathematical Olympiad
Does there exist an integer having the form $444...4443$ (all fours, and ending with a three) that is divisible by $13$?

If so, give an integer having that form that is divisible by $13$, if not, prove that such an integer cannot exist.
2007 Dutch Mathematical Olympiad
Determine the number of integers $a$ satisfying $1 \le a \le 100$ such that $a^a$ is a perfect square.

(And prove that your answer is correct.)
2007 Dutch Mathematical Olympiad
A triangle $ABC$ and a point $P$ inside this triangle are given.

Define $D, E$ and $F$ as the midpoints of $AP, BP$ and $CP$, respectively. Furthermore, let $R$ be the intersection of $AE$ and $BD, S$ the intersection of $BF$ and $CE$, and $T$ the intersection of $CD$ and $AF$.

Prove that the area of hexagon $DRESFT$ is independent of the position of $P$ inside the triangle.

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2006 Dutch Mathematical Olympiad
A palindrome is a word that doesn't matter if you read it from left to right or from right to left. Examples: OMO, lepel and parterretrap.

How many palindromes can you make with the five letters $a, b, c, d$ and $e$ under the conditions:

- each letter may appear no more than twice in each palindrome,

- the length of each palindrome is at least $3$ letters.

(Any possible combination of letters is considered a word.)
2006 Dutch Mathematical Olympiad
Given is a acute angled triangle $ABC$. The lengths of the altitudes from $A, B$ and $C$ are successively $h_A, h_B$ and $h_C$. Inside the triangle is a point $P$. The distance from $P$ to $BC$ is $1/3 h_A$ and the distance from $P$ to $AC$ is $1/4 h_B$. Express the distance from $P$ to $AB$ in terms of $h_C$.
2006 Dutch Mathematical Olympiad
$1+2+3+4+5+6=6+7+8$.

What is the smallest number $k$ greater than $6$ for which:

$1 + 2 +...+ k = k + (k+1) +...+ n$, with $n$ an integer greater than $k$ ?