Olympiad problems

2007 Flanders Math Olympiad, 2

Given is a half circle with midpoint $O$ and diameter $AB$. Let $Z$ be a random point inside the half circle, and let $X$ be the intersection of $OZ$ and the half circle, and $Y$ the intersection of $AZ$ and the half circle. If $P$ is the intersection of $BY$ with the tangent line in $X$ to the half circle, show that $PZ \perp BX$.

2007 Flanders Math Olympiad, 1

Tags: belgium , national olympiad

1. The numbers $1,2, \ldots$ are placed in a triangle as following: \[ \begin{matrix} 1 & & & \\ 2 & 3 & & \\ 4 & 5 & 6 & \\ 7 & 8 & 9 & 10 \\ \ldots \end{matrix} \] What is the sum of the numbers on the $n$-th row?

2025 Israel National Olympiad (Gillis), P1

Tags: combinatorics , national olympiad , easy

Let $n$ be a positive integer. $n$ letters are written around a circle, each $A$, $B$, or $C$. When the letters are read in clockwise order, the sequence $AB$ appears $100$ times, the sequence $BA$ appears $99$ times, and the sequence $BC$ appears $17$ times. How many times does the sequence $CB$ appear?

2017 Spain Mathematical Olympiad, 2

Tags: algebra , national olympiad , Spain

A midpoint plotter is an instrument which draws the exact mid point of two point previously drawn. Starting off two points $1$ unit of distance apart and using only the midpoint plotter, you have to get two point which are strictly at a distance between $\frac{1}{2017}$ and $\frac{1}{2016}$ units, drawing the minimum amount of points. ¿Which is the minimum number of times you will need to use the midpoint plotter and what strategy should you follow to achieve it?

2017 Spain Mathematical Olympiad, 4

Tags: Combinatorial games , combinatorics , national olympiad , Spain

You are given a row made by $2018$ squares, numbered consecutively from $0$ to $2017$. Initially, there is a coin in the square $0$. Two players $A$ and $B$ play alternatively, starting with $A$, on the following way: In his turn, each player can either make his coin advance $53$ squares or make the coin go back $2$ squares. On each move the coin can never go to a number less than $0$ or greater than $2017$. The player who puts the coin on the square $2017$ wins. ¿Who is the one with the wining strategy and how should he play to win?

2016 Kazakhstan National Olympiad, 1

Tags: number theory , national olympiad , 2016

Prove that one can arrange all positive divisors of any given positive integer around a circle so that for any two neighboring numbers one is divisible by another.

2020 Kosovo National Mathematical Olympiad, 4

Tags: geometry , angle bisector , bisection , national olympiad , Olympiad , Kosovo

Let $\triangle ABC$ be a triangle and $\omega$ its circumcircle. The exterior angle bisector of $\angle BAC$ intersects $\omega$ at point $D$. Let $X$ be the foot of the altitude from $C$ to $AD$ and let $F$ be the intersection of the internal angle bisector of $\angle BAC$ and $BC$. Show that $BX$ bisects segment $AF$.

1983 Czech and Slovak Olympiad III A, 2

Tags: geometric inequality , geometry , Inequality , national olympiad , inequalities

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.

2023 Ecuador NMO (OMEC), 6

Tags: geometry , national olympiad

Let $DE$ the diameter of a circunference $\Gamma$. Let $B, C$ on $\Gamma$ such that $BC$ is perpendicular to $DE$, and let $Q$ the intersection of $BC$ with $DE$. Let $P$ a point on segment $BC$ such that $BP=4PQ$. Let $A$ the second intersection of $PE$ with $\Gamma$. If $DE=2$ and $EQ=\frac{1}{2}$, find all possible values of the sides of triangle $ABC$.

2020 Israel National Olympiad, 6

Tags: combinatorics , national olympiad

On a circle the numbers from 1 to 6 are written in order, as depicted in the picture. In each move, Lior picks a number $a$ on the circle whose neighbors are $b$ and $c$ and replaces it by the number $\frac{bc}{a}$. Can Lior reach a state in which the product of the numbers on the circle is greater than $10^{100}$ in [b]a)[/b] at most 100 moves [b]b)[/b] at most 110 moves

2022 Bosnia and Herzegovina Junior BMO TST, 2

Tags: JBMO TST , number theory , national olympiad

Let $a,b,c$ be positive integers greater than $1$ such that $$p=ab+bc+ac$$ is prime. A) Prove that $a^2, b^2, c^2$ all have different reminder $mod\ p$. B) Prove that $a^3, b^3, c^3$ all have different reminder $mod\ p$.

1956 Czech and Slovak Olympiad III A, 3

Tags: system of equations , absolute value , algebra , national olympiad

Find all real pairs $x,y$ such that \begin{align*} x-|y+1|&=1, \\ x^2+y&=10. \end{align*}

2019 Czech and Slovak Olympiad III A, 1

Tags: algebra , national olympiad , system of equations

Find all triplets $(x,y,z)\in\mathbb{R}^3$ such that \begin{align*} x^2-yz &= |y-z|+1, \\ y^2-zx &= |z-x|+1, \\ z^2-xy &= |x-y|+1. \end{align*}

2022 Bosnia and Herzegovina Junior BMO TST, 4

Tags: JBMO TST , combinatorics , national olympiad

Some people know each other in a group of people, where "knowing" is a symmetric relation. For a person, we say that it is $social$ if it knows at least $20$ other persons and at least $2$ of those $20$ know each other. For a person, we say that it is $shy$ if it doesn't know at least $20$ other persons and at least $2$ of those $20$ don't know each other. Find the maximal number of people in that group, if we know that group doesn't have any $social$ nor $shy$ persons.

2016 Mexico National Olmypiad, 3

Tags: inequalities , floor function , algebra , national olympiad

Find the minimum real $x$ that satisfies $$\lfloor x \rfloor <\lfloor x^2 \rfloor <\lfloor x^3 \rfloor < \cdots < \lfloor x^n \rfloor < \lfloor x^{n+1} \rfloor < \cdots$$

1993 Bundeswettbewerb Mathematik, 4

Tags: number theory , factorial , number theory unsolved , national olympiad

Does there exist a non-negative integer n, such that the first four digits of n! is 1993?

2020 Kosovo National Mathematical Olympiad, 2

Tags: Kosovo , combinatorics , national olympiad , Olympiad

Ana baked 15 pasties. She placed them on a round plate in a circular way: 7 with cabbage, 7 with meat and 1 with cherries in that exact order and put the plate into a microwave. She doesn’t know how the plate has been rotated in the microwave. She wants to eat a pasty with cherries. Is it possible for Ana, by trying no more than three pasties, to find exactly where the pasty with cherries is?

2023 Israel National Olympiad, P1

Tags: combinatorics , national olympiad

2000 people are sitting around a round table. Each one of them is either a truth-sayer (who always tells the truth) or a liar (who always lies). Each person said: "At least two of the three people next to me to the right are liars". How many truth-sayers are there in the circle?

2016 Spain Mathematical Olympiad, 6

Tags: national olympiad , inequalities , Jensen , rearrangement inequality , Spain , ESP

Let $n\geq 2$ an integer. Find the least value of $\gamma$ such that for any positive real numbers $x_1,x_2,...,x_n$ with $x_1+x_2+...+x_n=1$ and any real $y_1+y_2+...+y_n=1$ and $0\leq y_1,y_2,...,y_n\leq \frac{1}{2}$ the following inequality holds: $$x_1x_2...x_n\leq \gamma \left(x_1y_1+x_2y_2+...+x_ny_n\right)$$

1987 Czech and Slovak Olympiad III A, 4

Tags: algebra , number theory , inequalities , Inequality , prime numbers , factorial , national olympiad

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 $$\frac{x_1\cdots x_n}{p^r}>n!.$$

2024 Israel National Olympiad (Gillis), P5

Tags: number theory , smallest prime divisor , Integer sequence , national olympiad

For positive integral $k>1$, we let $p(k)$ be its smallest prime divisor. Given an integer $a_1>2$, we define an infinite sequence $a_n$ by $a_{n+1}=a_n^n-1$ for each $n\geq 1$. For which values of $a_1$ is the sequence $p(a_n)$ bounded?

2021 Ecuador NMO (OMEC), 4

Tags: combinatorics , national olympiad

In a board $8$x$8$, the unit squares have numbers $1-64$, as shown. The unit square with a multiple of $3$ on it are red. Find the minimum number of chess' bishops that we need to put on the board such that any red unit square either has a bishop on it or is attacked by at least one bishop. Note: A bishops moves diagonally. [img]https://i.imgur.com/03baBwp.jpeg[/img]

2025 Israel National Olympiad (Gillis), P4

Tags: geometry , rectangle , national olympiad , combinatorial geometry , combinatorics

A $100\times \sqrt{3}$ rectangular table is given. What is the minimum number of disk-shaped napkins of radius $1$ required to cover the table completely? [i]Remark:[/i] The napkins are allowed to overlap and protrude the table's edges.

2025 Kosovo National Mathematical Olympiad`, P3

Tags: geometry , Kosovo , national olympiad , areas

On the side $AB$ of the parallelogram $ABCD$ we take the points $X$ and $Y$ such that the points $A$, $X$, $Y$ and $B$ appear in this order. The lines $DX$ and $CY$ intersect at the point $Z$. Suppose that the area of the triangle $\triangle XYZ$ is equal to the sum of the areas of the triangles $\triangle AXD$ and $\triangle CYB$. Prove that the area of the quadrilateral $XYCD$ is equal to $3$ times the area of the triangle $\triangle XYZ$.

2023 Macedonian Mathematical Olympiad, Problem 1

Tags: algebra , functional equation , Macedonia , national olympiad

Determine all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$ we have: $$xf(x+y)+yf(y-x) = f(x^2+y^2)\,.$$ [i]Authored by Nikola Velov[/i]