This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

AND:
OR:
NO:

Found problems: 36

2019 Kosovo Team Selection Test, 2
Tags: functional equation , algebra , function , bounded , Reals , TST , Kosovo
Determine all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that for every $x,y \in \mathbb{R}$ $$f(x^{4}-y^{4})+4f(xy)^{2}=f(x^{4}+y^{4})$$

2020 Kosovo National Mathematical Olympiad, 2
Tags: Kosovo , national olympiad , Olympiad , number theory , divisible
Let $a_1,a_2,...,a_n$ be integers such that $a_1^{20}+a_2^{20}+...+a_n^{20}$ is divisible by $2020$. Show that $a_1^{2020}+a_2^{2020}+...+a_n^{2020}$ is divisible by $2020$.

2025 Kosovo National Mathematical Olympiad`, P3
Tags: number theory , Kosovo , national olympiad , prime , remainder
Let $m$ and $n$ be natural numbers such that $m^3-n^3$ is a prime number. What is the remainder of the number $m^3-n^3$ when divided by $6$?

2020 Kosovo National Mathematical Olympiad, 3
Tags: algebra , minimum value , logarithms , national olympiad , Olympiad , Kosovo
Let $a$ and $b$ be real numbers such that $a+b=\log_2( \log_2 3)$. What is the minimum value of $2^a + 3^b$ ?

2024 Kosovo Team Selection Test, P2
Tags: TST , geometry , circle , concurrent lines , Tangents , beautiful geometry , Kosovo
Let $\omega$ be a circle and let $A$ be a point lying outside of $\omega$. The tangents from $A$ to $\omega$ touch $\omega$ at points $B$ and $C$. Let $M$ be the midpoint of $BC$ and let $D$ a point on the side $BC$ different from $M$. The circle with diameter $AD$ intersects $\omega$ at points $X$ and $Y$ and the circumcircle of $\bigtriangleup ABC$ again at $E$. Prove that $AD$, $EM$, and $XY$ are concurrent.

2025 Kosovo EGMO Team Selection Test, P2
Tags: nt , Diophantine equation , number theory , Kosovo , Egmo tst , 2025
Find all natural numbers $m$ and $n$ such that $3^m+n!-1$ is the square of a natural number.

2025 Kosovo National Mathematical Olympiad`, P1
Tags: algebra , system of equations , Kosovo , national olympiad , nice
Find all real numbers $a$, $b$ and $c$ that satisfy the following system of equations: $$\begin{cases} ab-c = 3 \\ a+bc = 4 \\ a^2+c^2 = 5\end{cases}$$

2025 Kosovo National Mathematical Olympiad`, P3
Tags: Kosovo , national olympiad , number theory , Sets
A subset $S$ of the natural numbers is called [i]dense [/i] for every $7$ consecutive natural numbers, at least $5$ of them are in $S$. Show that there exists a dense subset for which the equation $a^2+b^2=c^2$ has no solution for $a,b,c \in S$.

2025 Kosovo National Mathematical Olympiad`, P4
Tags: geometry , national olympiad , Kosovo
Let $D$ be a point on the side $AC$ of triangle $\triangle ABC$ such that $AB=AD=DC$ and let $E$ be a point on the side $BC$ such that $BE=2CE$. Prove that $\angle BDE = 90 ^{\circ}$.

2020 Kosovo National Mathematical Olympiad, 4
Tags: national olympiad , Olympiad , Kosovo , algebra , number theory , Sequence
Let $a_0$ be a fixed positive integer. We define an infinite sequence of positive integers $\{a_n\}_{n\ge 1}$ in an inductive way as follows: if we are given the terms $a_0,a_1,...a_{n-1}$ , then $a_n$ is the smallest positive integer such that $\sqrt[n]{a_0\cdot a_1\cdot ...\cdot a_n}$ is a positive integer. Show that the sequence $\{a_n\}_{n\ge 1}$ is eventually constant. [b]Note:[/b] The sequence $\{a_n\}_{n\ge 1}$ is eventually constant if there exists a positive integer $k$ such that $a_n=c$, for every $n\ge k$.

2020 Kosovo National Mathematical Olympiad, 1
Tags: Sequence , terms , national olympiad , Olympiad , Kosovo , algebra , combinatorics
Some positive integers, sum of which is $23$, are written in sequential form. Neither one of the terms nor the sum of some consecutive terms in the sequence is equal to $3$. [b]a) [/b]Is it possible that the sequence contains exactly $11$ terms? [b]b)[/b]Is it possible that the sequence contains exactly $12$ terms?

2020 Kosovo National Mathematical Olympiad, 1
Tags: algebra , maximum , Kosovo
Let $x\in\mathbb{R}$. What is the maximum value of the following expression: $\sqrt{x-2018} + \sqrt{2020-x}$ ?

2025 Kosovo National Mathematical Olympiad`, P4
Tags: remainder , number theory , Kosovo , national olympiad
When a number is divided by $2$ it has quotient $x$ and remainder $1$. Whereas, when the same number is divided by $3$ it has quotient $y$ and remainder $2$. What is the remainder when $x+y$ is divided by $5$?

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?

2022 Kosovo Team Selection Test, 1
Tags: function , functional equation , Kosovo , TST , 2022
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all real numbers $x$ and $y$, $$f(x^2)+2f(xy)=xf(x+y)+yf(x).$$ [i]Proposed by Dorlir Ahmeti, Kosovo[/i]

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.

2016 Kosovo Team Selection Test, 2
Tags: number theory , TST , Kosovo , 2016
Show that for all positive integers $n\geq 2$ the last digit of the number $2^{2^n}+1$ is $7$ .

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 $2020$?

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.

2021 Kosovo National Mathematical Olympiad, 3
Tags: inequalities , algebra , Kosovo
Let $a,b$ and $c$ be positive real numbers such that $a^5+b^5+c^5=ab^2+bc^2+ca^2$. Prove the inequality: $$\frac{a^2+b^2}{b}+\frac{b^2+c^2}{c}+\frac{c^2+a^2}{a}\geq 2(ab+bc+ca).$$

2022 Kosovo National Mathematical Olympiad, 3
Tags: geometry , Kosovo , simson , parallelism
Let $\bigtriangleup ABC$ be a triangle and $D$ be a point in line $BC$ such that $AD$ bisects $\angle BAC$. Furthermore, let $F$ and $G$ be points on the circumcircle of $\bigtriangleup ABC$ and $E\neq D$ point in line $BC$ such that $AF=AE=AD=AG$. If $X$ and $Y$ are the feet of perpendiculars from $D$ to $EF$ and $EG,$ respectively. Prove that $XY\parallel AD$.

2025 Kosovo National Mathematical Olympiad`, P2
Tags: functional equation , Reals , Kosovo , algebra
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ with the property that for every real numbers $x$ and $y$ it holds that $$f(x+yf(x+y))=f(x)+f(xy)+y^2.$$

2024 Kosovo Team Selection Test, P3
Tags: Kosovo , functional equation , algebra
Find all functions $f:\mathbb R\to\mathbb R$ such that $$(x-y)f(x+y) - (x+y)f(x-y) = 2y(f(x)-f(y) - 1)$$for all $x, y\in\mathbb R$.

2020 Kosovo National Mathematical Olympiad, 2
Tags: number theory , Olympiad , Kosovo , Perfect Squares
Find all positive integers $x$, $y$ such that $2^x+5^y+2$ is a perfect square.