Olympiad problems

2003 German National Olympiad, 5

$n$ is a positive integer. Let $a(n)$ be the smallest number for which $n\mid a(n)!$ Find all solutions of:$$\frac{a(n)}{n}=\frac{2}{3}$$

2023 Ecuador NMO (OMEC), 4

Tags: national olympiad , combinatorics

A number is [i]additive[/i] if it has three digits, all of them are different and the sum of two of the digits is equal to the remaining one. (For example, $123 (1+2=3), 945 (4+5=9)$). Find the sum of all additive numbers.

2020 Kosovo National Mathematical Olympiad, 4

Tags: number theory , prime numbers , national olympiad , Olympiad , compos

Let $p$ and $q$ be prime numbers. Show that $p^2+q^2+2020$ is composite.

2021 Ecuador NMO (OMEC), 5

Tags: geometry , national olympiad

Find an acutangle triangle such that its sides and altitudes have integer length.

2019 Czech and Slovak Olympiad III A, 6

Tags: Combinatorial Number Theory , numbers in a table , national olympiad , number theory

Assume we can fill a table $n\times n$ with all numbers $1,2,\ldots,n^2-1,n^2$ in such way that arithmetic means of numbers in every row and every column is an integer. Determine all such positive integers $n$.

2021 Ecuador NMO (OMEC), 2

Tags: algebra , polynomial , national olympiad

Let $P(x)$ a grade 3 polynomial such that: $$P(1)=1, P(2)=4, P(3)=9$$ Find the value of $P(10)+P(-6)$

2018 Belarusian National Olympiad, 11.3

Tags: number theory , national olympiad

For all pairs $(m, n)$ of positive integers that have the same number $k$ of divisors we define the operation $\circ$. Write all their divisors in an ascending order: $1=m_1<\ldots<m_k=m$, $1=n_1<\ldots<n_k=n$ and set $$ m\circ n= m_1\cdot n_1+\ldots+m_k\cdot n_k. $$ Find all pairs of numbers $(m, n)$, $m\geqslant n$, such that $m\circ n=497$.

2022 Bosnia and Herzegovina Junior BMO TST, 1

Tags: JBMO TST , algebra , national olympiad

Let $a,b,c$ be real numbers such that $$a^2-bc=b^2-ca=c^2-ab=2$$. Find the value of $$ab+bc+ca$$ and find at least one triplet $(a,b,c)$ that satisfy those conditions.

2018 Czech and Slovak Olympiad III A, 3

Tags: geometry , national olympiad , circumcircle

In triangle $ABC$ let be $D$ an intersection of $BC$ and the $A$-angle bisector. Denote $E,F$ the circumcenters of $ABD$ and $ACD$ respectively. Assuming that the circumcenter of $AEF$ lies on the line $BC$ what is the possible size of the angle $BAC$ ?

2019 Czech and Slovak Olympiad III A, 2

Tags: constructive geometry , rectangle , geometry , national olympiad

Let be $ABCD$ a rectangle with $|AB|=a\ge b=|BC|$. Find points $P,Q$ on the line $BD$ such that $|AP|=|PQ|=|QC|$. Discuss the solvability with respect to the lengths $a,b$.

2025 Kosovo National Mathematical Olympiad`, P2

Tags: geometry , similar triangles , national olympiad , Kosovo

Let $h_a$, $h_b$ and $h_c$ be the altitudes of a triangle $\triangle ABC$ ejected from the vertices $A$,$B$ and $C$, respectively. Similarly, let $h_x$, $h_y$ and $h_z$ be the altitudes of an another triangle $\triangle XYZ$. Show that if $$h_a : h_b : h_c = h_x : h_y : h_z, $$ then the triangles $\triangle ABC$ and $\triangle XYZ$ are similar.

2020 Kosovo National Mathematical Olympiad, 1

Tags: combinatorics , national olympiad , Olympiad , Kosovo , game strategy , Combinatorial games , Game Theory

Two players, Agon and Besa, choose a number from the set $\{1,2,3,4,5,6,7,8\}$, in turns, until no number is left. Then, each player sums all the numbers that he has chosen. We say that a player wins if the sum of his chosen numbers is a prime and the sum of the numbers that his opponent has chosen is composite. In the contrary, the game ends in a draw. Agon starts first. Does there exist a winning strategy for any of the players?

1983 Czech and Slovak Olympiad III A, 5

Tags: Inequality , algebra , national olympiad , inequalities

Find all pair $(x,y)$ of positive integers satisfying $$\left|\frac{x}{y}-\sqrt2\right|<\frac{1}{y^3}.$$

2015 Czech and Slovak Olympiad III A, 1

Tags: number theory , national olympiad , Diophantine equation

Find all 4-digit numbers $n$, such that $n=pqr$, where $p<q<r$ are distinct primes, such that $p+q=r-q$ and $p+q+r=s^2$, where $s$ is a prime number.

1983 Czech and Slovak Olympiad III A, 4

Tags: algebra , Binomial coefficient identity , Arithmetic Progression , national olympiad , arithmetic sequence , binomial coefficients

Consider an arithmetic progression $a_0,\ldots,a_n$ with $n\ge2$. Prove that $$\sum_{k=0}^n(-1)^k\binom{n}{k}a_k=0.$$

1957 Czech and Slovak Olympiad III A, 1

Tags: equation , parametric equation , algebra , national olympiad , parameterization

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 $p$.

2020 Kosovo National Mathematical Olympiad, 3

Tags: number theory , prime numbers , national olympiad , OLYM

Find all prime numbers $p$ such that $3^p + 5^p -1$ is a prime number.

2025 Kosovo National Mathematical Olympiad`, P2

Tags: Kosovo , national olympiad , number theory

Find all natural numbers $n$ such that $\frac{\sqrt{n}}{2}+\frac{10}{\sqrt{n}}$ is a natural number.

2022 Ecuador NMO (OMEC), 6

Tags: number theory , primes , national olympiad

Prove that for all prime $p \ge 5$, there exist an odd prime $q \not= p$ such that $q$ divides $(p-1)^p + 1$

2017 Spain Mathematical Olympiad, 6

Tags: national olympiad , Spain , geometry

In the triangle $ABC$, the respective mid points of the sides $BC$, $AB$ and $AC$ are $D$, $E$ and $F$. Let $M$ be the point where the internal bisector of the angle $\angle ADB$ intersects the side $AB$, and $N$ the point where the internal bisector of the angle $\angle ADC$ intersects the side $AC$. Also, let $O$ be the intersection point of $AD$ and $MN$, $P$ the intersection point of $AB$ and $FO$, and $R$ the intersection point of $AC$ and $EO$. Prove that $PR=AD$.

1987 Czech and Slovak Olympiad III A, 1

Tags: geometry , constructive geometry , quadrilateral , cyclic quadrilateral , trapezoid , circumcircle , national olympiad

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.

2025 Israel National Olympiad (Gillis), P5

Tags: combinatorics , graph theory , national olympiad

$2024$ otters live in the river. Some are friends with each other. Is it possible that, for any collection of $1012$ otters, there is exactly one additional otter that is friends with all $1012$ otters?

2020 Kosovo National Mathematical Olympiad, 2

Tags: combinatorics , national olympiad , olympi , Kosovo

A natural number $n$ is written on the board. Ben plays a game as follows: in every step, he deletes the number written on the board, and writes either the number which is three greater or two less than the number he has deleted. Is it possible that for every value of $n$, at some time, he will get to the number $2020$?

2016 Spain Mathematical Olympiad, 2

Tags: number theory , Quadratic Residues , national olympiad , Olympiad , prime numbers

Given a positive prime number $p$. Prove that there exist a positive integer $\alpha$ such that $p|\alpha(\alpha-1)+3$, if and only if there exist a positive integer $\beta$ such that $p|\beta(\beta-1)+25$.

2025 Kosovo National Mathematical Olympiad`, P3

Tags: geometry , similar triangles , national olympiad , Kosovo

Let $g_a$, $g_b$ and $g_c$ be the medians of a triangle $\triangle ABC$ erected from the vertices $A$, $B$ and $C$, respectively. Similarly, let $g_x$, $g_y$ and $g_z$ be the medians of an another triangle $\triangle XYZ$. Show that if $$g_a : g_b : g_c = g_x : g_y : g_z, $$ then the triangles $\triangle ABC$ and $\triangle XYZ$ are similar.