Navvye/Olympiad-Problems-CSV huggingface dataset:
1992 Czech And Slovak Olympiad IIIA
Let $S$ be the total area of a tetrahedron whose edges have lengths $a,b,c,d, e, f$ . Prove that $S \le \frac{\sqrt3}{6} (a^2 +b^2 +...+ f^2)$
1992 Czech And Slovak Olympiad IIIA
Let $S(n)$ denote the sum of digits of $n \in N$. Find all $n$ such that $S(n) = S(2n) = S(3n) =... = S(n^2)$
1992 Czech And Slovak Olympiad IIIA
Solve the equation $\cos 12x = 5\sin 3x+9\ tan ^2x+\ cot ^2x$
1992 Czech And Slovak Olympiad IIIA
The function $f : (0,1) \to  R$ is defined by

$f(x) = x$ if $x$ is irrational,

$f(x) = \frac{p+1}{q}$ if $x =\frac{p}{q}$ , where $(p,q) = 1$.

Find the maximum value of $f$ on the interval $(7/8,8/9)$.
1992 Czech And Slovak Olympiad IIIA
Let $ABC$ be an acute triangle. The altitude from $B$ meets the circle with diameter $AC$ at points $P,Q$, and the altitude from $C$ meets the circle with diameter $AB$ at $M,N$. Prove that the points $M,N,P,Q$ lie on a circle.
1991 Czech And Slovak Olympiad IIIA
Prove that for any real numbers $p,q,r,\phi$,:

$$\cos^2\phi+q \sin \phi \cos \phi +r\sin^2 \phi \ge \frac12 (p+r-\sqrt{(p-r)^2+q^2})$$
1991 Czech And Slovak Olympiad IIIA
A museum has the shape of a (not necessarily convex) 3$n$-gon.

Prove that $n$ custodians can be positioned so as to control all of the museum’s space.
1991 Czech And Slovak Olympiad IIIA
For any permutation $p$ of the set $\{1,2,...,n\}$, let us denote $d(p) = |p(1)-1|+|p(2)-2|+...+|p(n)-n|$.

Let $i(p)$ be the number of inversions of $p$, i.e. the number of pairs $1 \le i < j \le  n$ with $p(i) > p(j)$.

Prove that $i(p) \le  d(p)$.
1991 Czech And Slovak Olympiad IIIA
Prove that in all triangles $ABC$ with $\angle A = 2\angle B$ the distance from $C$ to $A$ and to the perpendicular bisector of $AB$ are in the same ratio.
1991 Czech And Slovak Olympiad IIIA
In a group of mathematicians everybody has at least one friend (friendship is a symmetric relation). Show that there is a mathematician all of whose friends have average number of friends not smaller than the average number of friends in the whole group.
1991 Czech And Slovak Olympiad IIIA
The set $N$ is partitioned into three (disjoint) subsets $A_1,A_2,A_3$.

Prove that at least one of them has the following property: There exists a positive number $m$ such that for any $k$ one can find numbers $a_1 < a_2 < ... < a_k$ in that subset satisfying $a_{j+1} -a_j \le m$ for $j = 1,...,k -1$.
1987 Czech and Slovak Olympiad III A
Given a trapezoid, divide it by a line into two quadrilaterals in such a way that both of them are cyclic with the same circumradius. Discuss conditions of solvability.
1987 Czech and Slovak Olympiad III A
Given a prime $p>3$ and an odd integer $n>0$, show that the equation $$xyz=p^n(x+y+z)$$has at least $3(n+1)$ different solutions up to symmetry. (That is, if $(x',y',z')$ is a solution and $(x'',y'',z'')$ is a permutation of the previous, they are considered to be the same solution.)
1987 Czech and Slovak Olympiad III A
Let $f:(0,\infty)\to(0,\infty)$ be a function satisfying $f\bigl(xf(y)\bigr)+f\bigl(yf(x)\bigr)=2xy$ for all $x,y>0$. Show that $f(x) = x$ for all positive $x$.
1987 Czech and Slovak Olympiad III A
Given an integer $n\ge3$ consider positive integers $x_1,\ldots,x_n$ such that $x_1<x_2<\cdots<x_n<2x_1$. If $p$ is a prime and $r$ is a positive integer such that $p^r$ divides the product $x_1\cdots x_n$, prove that $$\frac{x_1\cdots x_n}{p^r}>n!.$$
1987 Czech and Slovak Olympiad III A
Consider a table with three rows and eleven columns. There are zeroes prefilled in the cell of the first row and the first column and in the cell of the second row and the last column. Determine the least real number $\alpha$ such that the table can be filled with non-negative numbers and the following conditions hold simultaneously:

(1) the sum of numbers in every column is one,

(2) the sum of every two neighboring numbers in the first row is at most one,

(3) the sum of every two neighboring numbers in the second row is at most one,

(4) the sum of every two neighboring numbers in the third row is at most $\alpha$.
1987 Czech and Slovak Olympiad III A
Let $AA',BB',CC'$ be parallel lines not lying in the same plane. Denote $U$ the intersection of the planes $A'BC,AB'C,ABC'$ and $V$ the intersection of the planes $AB'C',A'BC',A'B'C$. Show that the line $UV$ is parallel with $AA'$.
1986 Czech And Slovak Olympiad IIIA
Given $n \in N$, let $A$ be a family of subsets of $\{1,2,...,n\}$. If for every two sets $B,C \in A$ the set $(B \cap C) -(B \cap C)$ has an even number of elements, find the largest possible number of elements of $A$ .
1986 Czech And Slovak Olympiad IIIA
Let $P(x)$ be a polynomial with integer coefficients of degree $n \ge 3$.

If $x_1,...,x_m$ ($m \ge 3$) are different integers such that $P(x_1) = P(x_2) = ... = P(x_m) = 1$, prove that $P$ cannot have integer roots.
1986 Czech And Slovak Olympiad IIIA
Prove that the entire space can be partitioned into “crosses” made of seven unit cubes as shown in the picture.
1986 Czech And Slovak Olympiad IIIA
Let $C_1,C_2$, and $C_3$ be points inside a bounded convex planar set $M$. Rays $l_1,l_2,l_3$ emanating from $C_1,C_2,C_3$ respectively partition the complement of the set $M \cup l_1 \cup  l_2 \cup l_3$ into three regions $D_1,D_2,D_3$. Prove that if the convex sets $A$ and $B$ satisfy $A\cap l_j  =\emptyset = B\cap l_j$ and $A\cap D_j \ne  \emptyset \ne B\cap D_j$ for $j = 1,2,3$, then $A\cap B \ne \emptyset$
1986 Czech And Slovak Olympiad IIIA
A sequence of natural numbers $a_1,a_2,...$ satisfies $a_1 = 1, a_{n+2} = 2a_{n+1} - a_n +2$ for $n \in N$.

Prove that for every natural $n$ there exists a natural $m$ such that $a_na_{n+1} = a_m$.
1986 Czech And Slovak Olympiad IIIA
Assume that $M \subset N$ has the property that every two numbers $m,n$ of $M$ satisfy $|m-n| \ge mn/25$.

Prove that the set $M$ contains no more than $9$ elements.
1983 Czech and Slovak Olympiad III A
Let $n$ be a positive integer and $k\in[0,n]$ be a fixed real constant. Find the maximum value of $$\left|\sum_{i=1}^n\sin(2x_i)\right|$$where $x_1,\ldots,x_n$ are real numbers satisfying  $$\sum_{i=1}^n\sin^2(x_i)=k.$$
1983 Czech and Slovak Olympiad III A
Given a triangle $ABC$, prove that for every inner point $P$ of the side $AB$ the inequality $$PC\cdot AB<PA\cdot BC+PB\cdot AC$$holds.
1983 Czech and Slovak Olympiad III A
An $8\times 8$ chessboard is made of unit squares. We put a rectangular piece of paper with sides of length 1 and 2. We say that the paper and a single square overlap if they share an inner point. Determine the maximum number of black squares that can overlap the paper.
1983 Czech and Slovak Olympiad III A
Consider an arithmetic progression $a_0,\ldots,a_n$ with $n\ge2$. Prove that $$\sum_{k=0}^n(-1)^k\binom{n}{k}a_k=0.$$
1983 Czech and Slovak Olympiad III A
Find all pair $(x,y)$ of positive integers satisfying $$\left|\frac{x}{y}-\sqrt2\right|<\frac{1}{y^3}.$$
1983 Czech and Slovak Olympiad III A
Consider a circle $k$ with center $S$ and radius $r$. Denote $\mathsf M$ the set of all triangles with incircle $k$ such that the largest inner angle is twice bigger than the smallest one. For a triangle $\mathcal T\in\mathsf M$ denote its vertices $A,B,C$ in way that $SA\ge SB\ge SC$. Find the locus of points $\{B\mid\mathcal T\in\mathsf M\}$.
1957 Czech and Slovak Olympiad III A
Find all real numbers $p$ such that the equation $$\sqrt{x^2-5p^2}=px-1$$has a root $x=3$. Then, solve the equation for the determined values of $p$.
1957 Czech and Slovak Olympiad III A
Consider a (right) square pyramid $ABCDV$ with the apex $V$ and the base (square) $ABCD$. Denote $d=AB/2$ and $\varphi$ the dihedral angle between planes $VAD$ and $ABC$.

(1) Consider a line $XY$ connecting the skew lines $VA$ and $BC$, where $X$ lies on line $VA$ and $Y$ lies on line $BC$. Describe a construction of line $XY$ such that the segment $XY$ is of the smallest possible length. Compute the length of segment $XY$ in terms of $d,\varphi$.

(2) Compute the distance $v$ between points $V$ and $X$ in terms of $d,\varphi.$
1957 Czech and Slovak Olympiad III A
Find all real numbers $\alpha$ such that both values $\cot(\alpha)$ and $\cot(2\alpha)$ are integers.
1957 Czech and Slovak Olympiad III A
Consider a non-zero convex angle $\angle POQ$ and its inner point $M$. Moreover, let $m>0$ be given. Construct a trapezoid $ABCD$ satisfying the following conditions:

(1) vertices $A, D$ lie on ray $OP$ and vertices lie on ray $OQ$,

(2) diagonals $AC$ and $BD$ intersect in $M$,

(3) $AB=m$.

Prove that your construction is correct and discuss conditions of solvability.
1952 Czech and Slovak Olympiad III A
Let $a,b,c$ be positive rational numbers such that $\sqrt a+\sqrt b=c$. Show that $\sqrt a$ and $\sqrt b$ are also rational.
1952 Czech and Slovak Olympiad III A
Consider a triangular table of positive integers

$$

\begin{matrix}

 & & & a_{11} & a_{12} & a_{13} & & &  \\
 & & a_{21} & a_{22} & a_{23} & a_{24} & a_{25} & & \\
 & a_{31} & a_{32} & a_{33} & a_{34} & a_{35} & a_{36} & a_{37} & \\
\iddots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots

\end{matrix}

$$The first row consists of odd numbers only. For $i>1,j\ge1$ we have $$a_{ij}=a_{i-1,j-2}+a_{i-1,j-1}+a_{i-1,j}.$$If we get out of range with the second index, we consider such $a$ to be zero (eg. $a_{22}=0+a_{11}+a_{12}$ and $a_{37}=a_{25}+0+0$). Show that for every $i>1$ there is $j\in\{1,\ldots,2i+1\}$ such that $a_{ij}$ is even.
1952 Czech and Slovak Olympiad III A
Let $ABCD$ be a convex quadrilateral with $AB=CD$. Let $R,S$ be midpoints of sides $AD,BC$ respectively. Consider rays $AU, DV$ parallel with ray $RS$ and all of them point in the same direction. Show that $\angle BAU=\angle CDV$.
1952 Czech and Slovak Olympiad III A
Let $p,q$ be positive integers. Consider a rectangle $ABCD$ with lengths of sides $p$ and $q$ that consists of $pq$ unital squares. How many of these squares intersect with diagonal $AC$ if

(1) $\gcd(p,q)=1,$

(2) $\gcd(p,q)>1?$
2021 Dutch Mathematical Olympiad
Niek has $16$ square cards that are yellow on one side and red on the other. He puts down the cards to form a $4 \times 4$-square. Some of the cards show their yellow side and some show their red side. For a colour pattern he calculates the monochromaticity  as follows. For every pair of adjacent cards that share a side he counts $+1$ or $-1$ according to the following rule: $+1$ if the adjacent cards show the same colour, and $-1$ if the adjacent cards show different colours. Adding this all together gives the monochromaticity (which might be negative). For example, if he lays down the cards as below, there are $15$ pairs of adjacent cards showing the same colour, and $9$ such pairs showing different colours.



The monochromaticity of this pattern is thus $15 \cdot (+1) + 9 \cdot (-1) = 6$. Niek investigates all possible colour patterns and makes a list of all possible numbers that appear at least once as a value of the monochromaticity. That is, Niek makes a list with all numbers such that there exists a colour pattern that has this number as its monochromaticity.



(a) What are the three largest numbers on his list?

(Explain your answer. If your answer is, for example, $ 12$, $9$ and $6$, then you have to show that these numbers do in fact appear on the list by giving a colouring for each of these numbers, and furthermore prove that the numbers $7$, $ 8$, $10$, $11$ and all numbers bigger than $ 12$ do not appear.)



(b) What are the three smallest (most negative) numbers on his list?



(c) What is the smallest positive number (so, greater than $0$) on his list?
2021 Dutch Mathematical Olympiad
We consider sports tournaments with $n \ge 4$ participating teams and where every pair of teams plays against one another at most one time. We call such a tournament balanced  if any four participating teams play exactly three matches between themselves. So, not all teams play against one another.

Determine the largest value of $n$ for which a balanced tournament with $n$ teams exists.
2021 Dutch Mathematical Olympiad
A frog jumps around on the grid points in the plane, from one grid point to another. The frog starts at the point $(0, 0)$. Then it makes, successively, a jump of one step horizontally, a jump of $2$ steps vertically, a jump of $3$ steps horizontally, a jump of $4$ steps vertically, et cetera. Determine all $n > 0$ such that the frog can be back in $(0, 0)$ after $n$ jumps.
2021 Dutch Mathematical Olympiad
In triangle ABC we have $\angle ACB = 90^o$. The point $M$ is the middle of $AB$. The line through $M$ parallel to $BC$ intersects $AC$ in $D$. The midpoint of line segment $CD$ is $E$. The lines $BD$ and $CM$ are perpendicular.

(a) Prove that triangles $CME$ and $ABD$ are similar.

(b) Prove that $EM$ and $AB$ are perpendicular.







Be aware: the figure is not drawn to scale.
2021 Dutch Mathematical Olympiad
We consider an integer $n > 1$ with the following property: for every positive divisor $d$ of $n$ we have that $d + 1$ is a divisor of$ n + 1$. Prove that $n$ is a prime number.
2020 Dutch Mathematical Olympiad
Daan distributes the numbers $1$ to $9$ over the nine squares of a $3\times 3$-table (each square receives exactly one number). Then, in each row, Daan circles the median number (the number that is neither the smallest nor the largest of the three). For example, if the numbers $8, 1$, and $2$ are in one row, he circles the number $2$. He does the same for each column and each of the two diagonals. If a number is already circled, he does not circle it again. He calls the result of this process a median table. Above, you can see a median table that has $5$ circled numbers.

(a) What is the smallest  possible number of circled numbers in a median table?

Prove that a smaller number is not possible and give an example in which a minimum number of numbers is circled.

(b) What is the largest possible number of circled numbers in a median table?

Prove that a larger number is not possible and give an example in which a maximum number of numbers is circled.
2020 Dutch Mathematical Olympiad
For a given value $t$, we consider number sequences $a_1, a_2, a_3,...$ such that $a_{n+1} =\frac{a_n + t}{a_n + 1}$ for all $n \ge 1$.

(a) Suppose that $t = 2$. Determine all starting values $a_1 > 0$ such that $\frac43 \le a_n \le \frac32$ holds for all $n \ge 2$.

(b) Suppose that $t = -3$. Investigate whether $a_{2020} = a_1$ for all starting values $a_1$ different from $-1$ and $1$.
2020 Dutch Mathematical Olympiad
Given is a parallelogram $ABCD$ with $\angle A < 90^o$ and $|AB| < |BC|$. The angular bisector of angle $A$ intersects side $BC$ in $M$ and intersects the extension of $DC$ in $N$. Point $O$ is the centre of the circle through $M, C$, and $N$. Prove that $\angle OBC = \angle ODC$.

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2020 Dutch Mathematical Olympiad
Determine all pairs of integers $(x, y)$ such that $2xy$ is a perfect square and $x^2 + y^2$ is a prime number.
2020 Dutch Mathematical Olympiad
Sabine has a very large collection of shells. She decides to give part of her collection to her sister.

On the first day, she lines up all her shells. She takes the shells that are in a position that is a perfect square (the first, fourth, ninth, sixteenth, etc. shell), and gives them to her sister. On the second day, she lines up her remaining shells. Again, she takes the shells that are in a position that is a perfect square, and gives them to her sister. She repeats this process every day.

The $27$th day is the first day that she ends up with fewer than $1000$ shells. The $28$th day she ends up with a number of shells that is a perfect square for the tenth time.

What are the possible numbers of shells that Sabine could have had in the very beginning?
2019 Dutch Mathematical Olympiad
A complete  number is a $9$ digit number that contains each of the digits $1$ to $9$ exactly once. The difference  number of a number $N$ is the number you get by taking the differences of consecutive digits in $N$ and then stringing these digits together. For instance, the difference  number of $25143$ is equal to $3431$. The complete  number $124356879$ has the additional property that its difference  number, $12121212$, consists of digits alternating between $1$ and $2$.

Determine all $a$ with $3 \le a \le 9$ for which there exists a complete  number $N$ with the additional property that the digits of its difference number alternate between $1 $ and $a$.
2019 Dutch Mathematical Olympiad
There are $n$ guests at a party. Any two guests are either friends or not friends. Every guest is friends with exactly four of the other guests. Whenever a guest is not friends with two other guests, those two other guests cannot be friends with each other either.

What are the possible values of $n$?
2019 Dutch Mathematical Olympiad
Points $A, B$, and $C$ lie on a circle with centre $M$. The reflection of point $M$ in the line $AB$ lies inside triangle $ABC$ and is the intersection of the angle bisectors of angles $A$ and $B$.  Line $AM$ intersects the circle again in point $D$.

Show that $|CA| \cdot |CD| = |AB| \cdot |AM|$.
2019 Dutch Mathematical Olympiad
The sequence of Fibonacci numbers $F_0, F_1, F_2, . . .$  is defined by $F_0 = F_1 = 1 $ and $F_{n+2} = F_n+F_{n+1}$ for all $n > 0$. For example, we have $F_2 = F_0 + F_1 = 2, F_3 = F_1 + F_2 = 3, F_4 = F_2 + F_3 = 5$, and $F_5 = F_3 + F_4 = 8$. The sequence $a_0, a_1, a_2, ...$ is defined by $a_n =\frac{1}{F_nF_{n+2}}$ for all $n \ge 0$.

Prove that for all $m \ge 0$ we have: $a_0 + a_1 + a_2 + ... + a_m < 1$.
2019 Dutch Mathematical Olympiad
Thomas and Nils are playing a game. They have a number of cards, numbered $1, 2, 3$, et cetera.

At the start, all cards are lying face up on the table. They take alternate turns. The person whose turn it is, chooses a card that is still lying on the table and decides to either keep the card himself or to give it to the other player. When all cards are gone, each of them calculates the sum of the numbers on his own cards. If the difference between these two outcomes is divisible by $3$, then Thomas wins. If not, then Nils wins.

(a) Suppose they are playing with $2018$ cards (numbered from $1$ to $2018$) and that Thomas starts. Prove that Nils can play in such a way that he will win the game with certainty.

(b) Suppose they are playing with $2020 $cards (numbered from $1$ to $2020$) and that Nils starts. Which of the two players can play in such a way that he wins with certainty?
2018 Dutch Mathematical Olympiad
We call a positive integer a shuffle number if the following hold:

(1) All digits are nonzero.

(2) The number is divisible by $11$.

(3) The number is divisible by $12$. If you put the digits in any other order, you again have a number that is divisible by $12$.

How many $10$-digit  shuffle numbers are there?
2018 Dutch Mathematical Olympiad
The numbers $1$ to $15$ are each coloured blue or red. Determine all possible colourings that satisfy the following rules:

• The number $15$ is red.

• If numbers $x$ and $y$ have different colours and $x + y \le 15$, then $x + y$ is blue.

• If numbers $x$ and $y$ have different colours and $x \cdot  y \le 15$, then $x \cdot y$ is red.
2018 Dutch Mathematical Olympiad
Determine all triples $(x, y,z)$ consisting of three distinct real numbers, that satisfy the following system of equations:

$\begin {cases}x^2 + y^2 = -x + 3y + z \\
y^2 + z^2 = x + 3y - z \\
x^2 + z^2 = 2x + 2y - z \end {cases}$
2018 Dutch Mathematical Olympiad
In triangle $ABC, \angle A$ is smaller than $\angle C$. Point $D$ lies on the (extended) line $BC$ (with $B$ between $C$ and $D$) such that $|BD| = |AB|$. Point $E$ lies on the bisector of $\angle ABC$ such that $\angle BAE = \angle ACB$. Line segment $BE$ intersects line segment $AC$ in point $F$. Point $G$ lies on line segment $AD$ such that $EG$ and $BC$ are parallel. Prove that $|AG| =|BF|$.

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2018 Dutch Mathematical Olympiad
At a quiz show there are three doors. Behind exactly one of the doors, a prize is hidden. You may ask the quizmaster whether the prize is behind the left-hand door. You may also ask whether the prize is behind the right-hand door. You may ask each of these two questions multiple times, in any order that you like. Each time, the quizmaster will answer ‘yes’ or ‘no’. The quizmaster is allowed to lie at most $10$ times. You have to announce in advance how many questions you will be asking (but which questions you will ask may depend on the answers of the quizmaster).

What is the smallest number you can announce, such that you can still determine with absolute certainty the door behind which the prize is hidden?
2017 Dutch Mathematical Olympiad
We consider positive integers written down in the (usual) decimal system. Within such an integer, we number the positions of the digits from left to right, so the leftmost digit (which is never a $0$) is at position $1$.

An integer is called even-steven if each digit at an even position (if there is one) is greater than or equal to its neighbouring digits (if these exist).

An integer is called oddball if each digit at an odd position is greater than or equal to its neighbouring digits (if these exist).

For example, $3122$ is oddball but not even-steven, $7$ is both even-steven and oddball, and $123$ is neither even-steven nor oddball.

(a) Prove: every oddball integer greater than $9$ can be obtained by adding two oddball  integers.

(b) Prove: there exists an oddball integer greater than $9$ that cannot be obtained by adding two even-steven integers.
2017 Dutch Mathematical Olympiad
A parallelogram $ABCD$ with $|AD| =|BD|$ has been given. A point $E$ lies on line segment $|BD|$ in such a way that $|AE| = |DE|$. The (extended) line $AE$ intersects line segment $BC$ in $F$. Line $DF$ is the angle bisector of angle $CDE$. Determine the size of angle $ABD$.

//cdn.artofproblemsolving.com/images/5/f/7/5f7dbc59833b5137688ed1b0cb9cda4daffe962a.png
2017 Dutch Mathematical Olympiad
Six teams participate in a hockey tournament. Each team plays exactly once against each other team. A team is awarded $3$ points for each game they win, $1$ point for each draw, and $0$ points for each game they lose. After the tournament, a ranking is made. There are no ties in the list. Moreover, it turns out that each team (except the very last team) has exactly $2$ points more than the team ranking one place lower.

Prove that the team that finished fourth won exactly two games.
2017 Dutch Mathematical Olympiad
If we divide the number $13$ by the three numbers $5, 7$, and $9$, then these divisions leave remainders: when dividing by $5$ the remainder is $3$, when dividing by $7$ the remainder is $6$, and when dividing by $9$ the remainder is 4. If we add these remainders, we obtain $3 + 6 + 4 = 13$, the original number.

(a) Let $n$ be a positive integer and let $a$ and $b$ be two positive integers smaller than $n$. Prove: if you divide $n$ by $a$ and $b$, then the sum of the two remainders never equals $n$.

(b) Determine all integers $n > 229$ having the property that if you divide $n$ by $99, 132$, and $229$, the sum of the three remainders is $n$.
2017 Dutch Mathematical Olympiad
The eight points below are the vertices and the midpoints of the sides of a square. We would like to draw a number of circles through the points, in such a way that each pair of points lie on (at least) one of the circles.

Determine the smallest number of circles needed to do this.