Olympiad problems

2025 Kosovo National Mathematical Olympiad`, P2

Find the smallest natural number $k$ such that the system of equations $$x+y+z=x^2+y^2+z^2=\dots=x^k+y^k+z^k $$ has only one solution for positive real numbers $x$, $y$ and $z$.

2024 Kosovo Team Selection Test, P1

Tags: number theory , prime , primes , Kosovo

Find all prime numbers $p$ and $q$ such that $p^q + 5q - 2$ is also a prime number.

2025 Kosovo National Mathematical Olympiad`, P3

Tags: number theory , decimal representation , powers , Kosovo

Find all pairs of natural numbers $(m,n)$ such that the number $5^m+6^n$ has all same digits when written in decimal representation.

2019 Kosovo National Mathematical Olympiad, 3

Tags: inequalities , algebra , Kosovo

Show that for any non-negative real numbers $a,b,c,d$ such that $a^2+b^2+c^2+d^2=1$ the following inequality hold: $$a+b+c+d-1\geq 16abcd$$ When does equality hold?

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 $AF$.

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?

2024 Kosovo EGMO Team Selection Test, P3

Tags: geometry , right triangle , Kosovo , Egmo tst , 2024

Let $\triangle ABC$ be a right triangle at the vertex $A$ such that the side $AB$ is shorter than the side $AC$. Let $D$ be the foot of the altitude from $A$ to $BC$ and $M$ the midpoint of $BC$. Let $E$ be a point on the ray $AB$, outside of the segment $AB$. Line $ED$ intersects the segment $AM$ at the point $F$. Point $H$ is on the side $AC$ such that $\angle EFH=90^{\circ}$. Suppose that $ED=FH$. Find the measure of the angle $\angle AED$.

2019 Kosovo National Mathematical Olympiad, 5

Tags: Kosovo , number theory , Diophantine equation

Find all positive integers $x,y$ such that $2^x+19^y$ is a perfect cube.

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 $\triangle XYZ$.

2025 Kosovo National Mathematical Olympiad`, P1

Tags: Kosovo , national olympiad , geometry , easy , pentagon

The pentagon $ABCDE$ below is such that the quadrilateral $ABCD$ is a square and $BC=DE$. What is the measure of the angle $\angle AEC$?

2025 Kosovo National Mathematical Olympiad`, P2

Tags: Inequality , Kosovo , algebra , inequalities

Let $x$ and $y$ be real numbers where at least one of them is bigger than $2$ and $xy+4 > 2(x+y)$ holds. Show that $xy>x+y$.