Navvye/Olympiad-Problems-CSV huggingface dataset:
2012 Kosovo National Mathematical Olympiad
Solve the recurrence $R_0=1, R_n=nR_{n-1}+2^n\cdot n!$.
2012 Kosovo National Mathematical Olympiad
Let $x,y$ be positive real numbers such that $x+y+xy=3$. Prove that $x+y\geq 2$. For what values of $x$ and $y$ do we have $x+y=2$?
2013 Kosovo National Mathematical Olympiad
Prove that:



$\sqrt{10+\sqrt{24}+\sqrt{40}+\sqrt{60}}=\sqrt{2}+\sqrt3+\sqrt5$
2013 Kosovo National Mathematical Olympiad
Find all integer $n$ such that $n-5$ divide $n^2+n-27$.
2013 Kosovo National Mathematical Olympiad
For all real numbers $a$ prove that $3(a^4+a^2+1)\geq (a^2+a+1)^2$
2013 Kosovo National Mathematical Olympiad
Find all value of parameter $a$ such that equations $x^2-ax+1=0$ and $x^2-x+a=0$ have at least one same solution.

For this value $a$ find same solution of this equations(real or imaginary).
2013 Kosovo National Mathematical Olympiad
Let $ABC$ be an equilateral triangle, with sidelength equal to $a$. Let $P$ be a point in the interior of triangle $ABC$, and let $D,E$ and $F$ be the feet of the altitudes from $P$ on $AB, BC$ and $CA$, respectively. Prove that $\frac{|PD|+|PE|+|PF|}{3a}=\frac{\sqrt{3}}{6}$
2013 Kosovo National Mathematical Olympiad
Let be $a,b$ real numbers such that $|a|\neq |b|$ and $\frac{a+b}{a-b}+\frac{a-b}{a+b}=6$ .



Calculate:



$\frac{a^3+b^3}{a^3-b^3}+\frac{a^3-b^3}{a^3+b^3}$
2013 Kosovo National Mathematical Olympiad
Three numbers have sum $k$ (where $k\in \mathbb{R}$) such that the numbers are arethmetic progression.If First of two numbers remain the same and to the third number we add $\frac{k}{6}$ than we have geometry progression.

Find those numbers?
2013 Kosovo National Mathematical Olympiad
How many positive integers which are less or equal with $2013$ such that $3$ or $5$ divide the number.
2013 Kosovo National Mathematical Olympiad
Let be $n$ positive integer than calculate:



$1\cdot 1!+2\cdot2!+...+n\cdot n!$
2013 Kosovo National Mathematical Olympiad
Let $P$ be a point inside or outside (but not on) of a triangle $ABC$. Prove that $PA +PB +PC$ is greater than half of the perimeter of the triangle
2013 Kosovo National Mathematical Olympiad
Let be $z_1$ and $z_2$ two complex numbers such that $|z_1+2z_2|=|2z_1+z_2|$.Prove that for all real numbers $a$ is true $|z_1+az_2|=|az_1+z_2|$
2013 Kosovo National Mathematical Olympiad
Solve equation $27\cdot3^{3\sin x}=9^{\cos^2x}$ where $x\in [0,2\pi )$
2013 Kosovo National Mathematical Olympiad
Prove that solution of equation $y=x^2+ax+b$ and $x=y^2+cy+d$ it belong a circle.
2013 Kosovo National Mathematical Olympiad
Let be $a,b,c$ three positive integer.Prove that $4$ divide $a^2+b^2+c^2$ only and only if $a,b,c$ are even.
2013 Kosovo National Mathematical Olympiad
Let $ABCD$ be a convex quadrilateral with perpendicular diagonals. . Assume that $ABCD$ has been inscribed in the circle with center $O$. Prove that $AOC$ separates $ABCD$ into two quadrilaterals of equal area
2013 Kosovo National Mathematical Olympiad
Which number is bigger $\sqrt[2012]{2012!}$ or $\sqrt[2013]{2013!}$.
2013 Kosovo National Mathematical Olympiad
Math teacher wrote in a table a polynomial $P(x)$ with integer coefficients and he said:

"Today my daughter have a birthday.If in polynomial $P(x)$ we have $x=a$ where $a$ is the age of my daughter we have $P(a)=a$ and $P(0)=p$ where $p$ is a prime number such that $p>a$."

How old is the daughter of math teacher?
2013 Kosovo National Mathematical Olympiad
Find all numbers $x$ such that:



$1+2\cdot2^x+3\cdot3^x<6^x$
2013 Kosovo National Mathematical Olympiad
Calculate:

$\sqrt{3\sqrt{5\sqrt{3\sqrt{5...}}}}$
2013 Kosovo National Mathematical Olympiad
A trapezium has parallel sides of length equal to $a$ and $b$ ($a <b$), and the distance between the parallel sides is the altitude $h$. The extensions of the non-parallel lines intersect at a point that is a vertex of two triangles that have as sides the parallel sides of the trapezium. Express the areas of the triangles as functions of $a,b$ and $h$.
2014 Kosovo National Mathematical Olympiad
Prove that for any integer the number $2n^3+3n^2+7n$ is divisible by $6$.
2014 Kosovo National Mathematical Olympiad
Solve $|x-1|-2|x+5|>3+x$.
2014 Kosovo National Mathematical Olympiad
A student read the book with $480$ pages two times. If he in the second time for every day read $16$ pages more than in the first time and he finished it $5$ days earlier than in the first time. For how many days did he read the book in the first time?
2014 Kosovo National Mathematical Olympiad
The number $2015$ has been written in the table. Two friends play this game: In the table they write the difference of the number in the table and one of its factors. The game is lost by the one who reaches $0$. Which of the two can secure victory?
2014 Kosovo National Mathematical Olympiad
A square $ABCD$ with sude length 1 is given and a circle with diameter $AD$. Find the radius of the circumcircle of this figure.
2014 Kosovo National Mathematical Olympiad
Let $a$ and $b$ be the solutions to $x^2-x+q=0$, find

$a^3+b^3+3(a^3b+ab^3)+6(a^3b^2+a^2b^3)$.
2016 Kosovo National Mathematical Olympiad
If $a,b\neq 0$ are real numbers such that $a^2b^2(a^2b^2+4)=2(a^6+b^6)$  , then show that $a,b$ can’t be both of them rational .
2016 Kosovo National Mathematical Olympiad
Find all real numbers $x$ which satisfied $|2x+1|+|x-1|=2-x$ .
2016 Kosovo National Mathematical Olympiad
Show that the sum $S=5+5^2+5^3+…+5^{2016}$ is divisible by $31$
2016 Kosovo National Mathematical Olympiad
In a planet $Papella$ year has $400$ days with days coundting from $1-400$ . A holiday would be that day which is divisible by $6$ . The new  gonverment decide to reform a new calendar and split in $10$ months with $40$ day each month , and they decide that day of month which is divisible by $6$ to be holiday . Show that after reform the number of holidays after one year decreased less then $ 10  $ percent .
2016 Kosovo National Mathematical Olympiad
It is given rectangle $ABCD$ with length $|AB|=15cm$ and with length of altitude $|BE|=12cm$ where $BC$ is altitude of triangle $ABC$ . Find perimeter and area of rectangle $ABCD$ .
2016 Kosovo National Mathematical Olympiad
Find all triples $(x,y,z)$ of integers such that satisfied:



$x^2+y^2+z^2+xy+yz+zx=6$
2016 Kosovo National Mathematical Olympiad
Show that the number $2017^{2016}-2016^{2017}$ is divisible by $5$ .
2016 Kosovo National Mathematical Olympiad
The distance from $A$ to $B$ is $408km$ . From $A$ in direction of $B$ move motorcyclist , and from $B$ in direction of $A$ move a bicyclist . If a motorcyclist start to move $2$ hours earlier then byciclist , then they will meet $7$ hours after bicyclist start to move . If a bicyclist start to move $2$ hours earlier then motorcyclist , then they will meet $8$ hours after after motorcyclist start to move . Find the velocity of motorcyclist and bicyclist if we now that the velocity of them was constant all the time .
2016 Kosovo National Mathematical Olympiad
Let be $f: (0,+\infty)\rightarrow \mathbb{R}$ monoton-decreasing .



If $f(2a^2+a+1)<f(3a^2-4a+1)$ find interval of $a$ .
2016 Kosovo National Mathematical Olympiad
If $a,b,c$ are sides of right triangle with $c$ hypothenuse then show that for every positive integer $n>2$ we have $c^n>a^n+b^n$ .
2016 Kosovo National Mathematical Olympiad
Find all couples $(m,n)$ of positive integers such that satisfied $m^2+1=n^2+2016$ .
2016 Kosovo National Mathematical Olympiad
Evaluate the sum of all three digits number which are not divisible by $13$ .
2016 Kosovo National Mathematical Olympiad
If $\alpha $ is an acute angle and $a,b\geq 0$ then show that:



$\left( a+\frac{b}{\sin \alpha}\right)\left(b+\frac{a}{\cos \alpha}\right)\geq a^2+b^2+3ab$
2016 Kosovo National Mathematical Olympiad
Solve equation in real numbers $\log_{2}(4^x+4)=x+\log_{2}(2^{x+1}-3)$
2016 Kosovo National Mathematical Olympiad
In angle $\angle AOB=60^{\circ}$ are two circle which circumscribed and tangjent to each other . If we write with $r$ and $R$ the radius of smaller and bigger circle respectively and if $r=1$ find $R$ .
2016 Kosovo National Mathematical Olympiad
Find all three digit numbers such that the square of that number is equal to the sum of  their digits in power of $5$ .
2016 Kosovo National Mathematical Olympiad
Sum of all coefficients of polynomial $P(x)$ is equal with $2$ . Also the sum of coefficients which are at odd exponential in $x^k$ are equal to sum of coefficients which are at even exponential in $x^k$  . Find the residue of polynomial $P(x)$ when it is divide by $x^2-1$ .
2016 Kosovo National Mathematical Olympiad
Let be $a,b,c$ complex numbers such that $|a|=|b|=|c|=r$ then show that



$\left | \frac{ab+bc+ca}{a+b+c}\right|=r$
2016 Kosovo National Mathematical Olympiad
In all rectangles with same diagonal $d$ find that one with bigger area .
2016 Kosovo National Mathematical Olympiad
In trapezoid $ABCD$ with $AB$ parallel to $CD$ show that :



$\frac{|AB|^2-|BC|^2+|AC|^2}{|CD|^2-|AD|^2+|AC|^2}=\frac{|AB|}{|CD|}=\frac{|AB|^2-|AD|^2+|BD|^2}{|CD|^2-|BC|^2+|BD|^2}$
2017 Kosovo National Mathematical Olympiad
1. Find all primes of the form $n^3-1$ .
2017 Kosovo National Mathematical Olympiad
2 .Solve the inequality $|x-1|-2|x-4|>3+2x$
2017 Kosovo National Mathematical Olympiad
3.

3 red birds for 4 days eat 36 grams of seed, 5 blue birds for 3 days eat 60 gram of seed.

For how many days could be feed 2 red birds and 4 blue birds with 88 gr seed?
2017 Kosovo National Mathematical Olympiad
4. Find all triples of consecutive numbers ,whose sum of squares is equal to some fourdigit number with all four digits being  equal.
2017 Kosovo National Mathematical Olympiad
5.

Given the point T in rectangle ABCD, the distances from T to A,B,C is 15,20,25.

Find the distance from T to D.
2017 Kosovo National Mathematical Olympiad
Find all ordered pairs $(a,b)$, of natural numbers, where $1<a,b\leq 100$, such that

$\frac{1}{\log_{a}{10}}+\frac{1}{\log_{b}{10}}$ is a natural number.
2017 Kosovo National Mathematical Olympiad
Prove that for every positive real $a,b,c$ the inequality holds :

$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+1 \geq \frac{2\sqrt2}{3} (\sqrt{\frac{a+b}{c}}+\sqrt{\frac{b+c}{a}}+\sqrt{\frac{c+a}{b}})$

When does the equality hold?
2017 Kosovo National Mathematical Olympiad
$n$ teams participated in a basketball tournament. Each team has played with each team exactly one game. There was no tie. If in the end of the tournament the $i$-th team has $x_{i}$ wins and $y_{i}$ loses $(1\leq i \leq n)$ prove that:

$\sum_{i=1}^{n} {x_{i}}^2=\sum_{i=1}^{n} {y_{i}}^2$
2017 Kosovo National Mathematical Olympiad
Prove that :

$\cos36-\sin18=\frac{1}{2}$
2017 Kosovo National Mathematical Olympiad
A sphere with ray $R$ is cut by two parallel planes. such that the center of the sphere is outside the region determined by these planes. Let $S_{1}$ and $S_{2}$ be the areas of the intersections, and $d$ the distance between these planes. Find the area of the intersection of the sphere with the plane parallel with these two planes, with equal distance from them.
2017 Kosovo National Mathematical Olympiad
The sequence ${a_{n}}$ $n\in \mathbb{N}$ is given in a recursive way with $a_{1}=1$,

$a_{n}=\prod_{i=1}^{n-1} a_{i}+1$, for all $n\geq 2$.

Determine the least number $M$, such that $\sum_{n=1}^{m} \frac{1}{a_{n}} <M$ for all $m\in \mathbb{N}$
2017 Kosovo National Mathematical Olympiad
Solve the system of equations

$x+y+z=\pi$

$\tan x\tan z=2$

$\tan y\tan z=18$
2017 Kosovo National Mathematical Olympiad
Let $a\geq 2$ a fixed natural number, and let $a_{n}$ be the sequence $a_{n}=a^{a^{.^{.^{a}}}}$ (e.g $a_{1}=a$, $a_{2}=a^a$, etc.). Prove that $(a_{n+1}-a_{n})|(a_{n+2}-a_{n+1})$ for every natural number $n$.
2017 Kosovo National Mathematical Olympiad
Prove the identity

$\sum_{k=2}^{n} k(k-1)\binom{n}{k} =\binom{n}{2} 2^{n-1}$ for all $n=2,3,4,...$
2017 Kosovo National Mathematical Olympiad
Lines determined by sides $AB$ and $CD$ of the convex quadrilateral $ABCD$ intersect at point $P$. Prove that $\alpha +\gamma =\beta +\delta$ if and only if $PA\cdot PB=PC\cdot PD$, where $\alpha ,\beta ,\gamma ,\delta$ are the measures of the internal angles of vertices $A, B, C, D$ respectively.
2019 Kosovo National Mathematical Olympiad
Calculate $1^2-2^2+3^2-4^2+...-2018^2+2019^2$.
2019 Kosovo National Mathematical Olympiad
Show that when the product of three conscutive numbers we add arithmetic mean of them it is a perfect cube.
2019 Kosovo National Mathematical Olympiad
Let $ABCD$ be a rectangle with $AB>BC$. Let points $E,F$ be on side $CD$ such that $CE=ED$ and $BC=CF$. Show that if $AC$ is prependicular to $BE$ then $AB=BF$.
2019 Kosovo National Mathematical Olympiad
Find all sequence of consecutive positive numbers which the sum of them is equal with $2019$.
2019 Kosovo National Mathematical Olympiad
There are given in a table numbers $1,2,...,18$. What is minimal number of numbers we should erase such that the sum of every two remaining numbers is not perfect square of a positive integer.
2019 Kosovo National Mathematical Olympiad
Find last three digits of the number $\frac{2019!}{2^{1009}}$ .
2019 Kosovo National Mathematical Olympiad
Show that for any positive real numbers $a,b,c$ the following inequality is true:

$$4(a^3+b^3+c^3+3)\geq 3(a+1)(b+1)(c+1)$$When does equality hold?
2019 Kosovo National Mathematical Olympiad
The doctor instructed a person to take $48$ pills for next $30$ days. Every day he take at least $1$ pill and at most $6$ pills. Show that exist the numbers of conscutive days such that the total numbers of pills he take is equal with $11$.
2019 Kosovo National Mathematical Olympiad
Find all real numbers $x,y,z$ such that satisfied the following equalities at same time:



$\sqrt{x^3-y}=z-1 \wedge \sqrt{y^3-z}=x-1\wedge \sqrt{z^3-x}=y-1$
2019 Kosovo National Mathematical Olympiad
Let $ABCDE$ be a regular pentagon. Let point $F$ be intersection of segments $AC$ and $BD$. Let point $G$ be in segment $AD$ such that $2AD=3AG$. Let point $H$ be the midpoint of side $DE$. Show that the points $F,G,H$ lie on a line.
2019 Kosovo National Mathematical Olympiad
Let $a,b$ be real numbers grater then $4$. Show that at least one of the trinomials $x^2+ax+b$ or $x^2+bx+a$ has two different real zeros.
2019 Kosovo National Mathematical Olympiad
Find all positive integers $n$ such that $6^n+1$ it has all the same digits when it is writen in decimal representation.
2019 Kosovo National Mathematical Olympiad
Let $ABC$ be a triangle with $\angle CAB=60^{\circ}$ and with incenter $I$. Let points $D,E$ be on sides $AC,AB$, respectively, such that $BD$ and $CE$ are angle bisectors of angles $\angle ABC$ and $\angle BCA$, respectively. Show that $ID=IE$.
2019 Kosovo National Mathematical Olympiad
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that:

$$f(xy+f(x))=xf(y)$$for all $x,y\in\mathbb{R}$.
2019 Kosovo National Mathematical Olympiad
There are given points with integer coordinate $(m,n)$ such that $1\leq m,n\leq 4$. Two players, Ana and Ben, are playing a game: First Ana color one of the coordinates with red one, then she pass the turn to Ben who color one of the remaining coordinates with yellow one, then this process they repeate again one after other. The game win the first player who can create a rectangle with same color of vertices and the length of sides are positive integer numbers, otherwise the game is a tie. Does there exist a strategy for any of the player to win the game?
2019 Kosovo National Mathematical Olympiad
Does there exist a triangle with length $a,b,c$ such that:



a) $\begin{cases} a+b+c=6 \\ a^2+b^2+c^2=13 \\ a^3+b^3+c^3=28 \end{cases}$



b) $\begin{cases} a+b+c=6 \\ a^2+b^2+c^2=13 \\ a^3+b^3+c^3=30 \end{cases}$
2019 Kosovo National Mathematical Olympiad
Suppose that each point on a plane is colored with one of the colors red or yellow. Show that exist a convex pentagon with three right angles and all vertices are with same color.
2019 Kosovo National Mathematical Olympiad
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?
2019 Kosovo National Mathematical Olympiad
Let $ABC$ be an acute triagnle with its circumcircle $\omega$. Let point $D$ be the foot of triangle $ABC$ from point $A$. Let points $E,F$ be midpoints of sides $AB,AC$, respectively. Let points $P$ and $Q$ be the second intersections of of circle $\omega$ with circumcircle of triangles $BDE$ and $CDF$, respectively. Suppose that $A,P,B,Q$ and $C$ be on a circle in this order. Show that the lines $EF,BQ$ and $CP$ are concurrent.
2019 Kosovo National Mathematical Olympiad
Find all positive integers $x,y$ such that $2^x+19^y$ is a perfect cube.
2020 Kosovo National Mathematical Olympiad
Compare the following two numbers: $2^{2^{2^{2^{2}}}}$ and $3^{3^{3^{3}}}$.
2020 Kosovo National Mathematical Olympiad
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 $2020$?
2020 Kosovo National Mathematical Olympiad
Let $\triangle ABC$ be a triangle. Let $O$ be the circumcenter of triangle $\triangle ABC$ and $P$ a variable point in line segment $BC$. The circle with center $P$ and radius $PA$ intersects the circumcircle of triangle $\triangle ABC$ again at another point $R$ and $RP$ intersects the circumcircle of triangle $\triangle ABC$ again at another point $Q$. Show that points $A$, $O$, $P$ and $Q$ are concyclic.
2020 Kosovo National Mathematical Olympiad
Let $p$ and $q$ be prime numbers. Show that $p^2+q^2+2020$ is composite.
2020 Kosovo National Mathematical Olympiad
Let $x\in\mathbb{R}$. What is the maximum value of the following expression: $\sqrt{x-2018}  + \sqrt{2020-x}$ ?
2020 Kosovo National Mathematical 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?
2020 Kosovo National Mathematical Olympiad
Find all prime numbers $p$ such that $3^p + 5^p -1$ is a prime number.
2020 Kosovo National Mathematical Olympiad
Let $B'$ and $C'$ be points in the circumcircle of triangle $\triangle ABC$ such that $AB=AB'$ and $AC=AC'$. Let $E$ and $F$ be the foot of altitudes from $B$ and $C$ to $AC$ and $AB$, respectively. Show that $B'E$ and $C'F$ intersect on the circumcircle of triangle $\triangle ABC$.
2020 Kosovo National Mathematical Olympiad
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 $3$.

a) Is it possible that the sequence contains exactly $11$ terms?

b)Is it possible that the sequence contains exactly $12$ terms?
2020 Kosovo National Mathematical Olympiad
Find all positive integers $x$, $y$ such that $2^x+5^y+2$ is a perfect square.
2020 Kosovo National Mathematical Olympiad
Let $a$ and $b$ be real numbers such that $a+b=\log_2( \log_2 3)$. What is the minimum value of $2^a + 3^b$ ?
2020 Kosovo National Mathematical Olympiad
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
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?
2020 Kosovo National Mathematical Olympiad
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 $2020$.
2020 Kosovo National Mathematical Olympiad
Let $ABC$ be a triangle with incenter $I$. The points $D$ and $E$ lie on the segments $CA$

and $BC$ respectively, such that $CD = CE$. Let $F$ be a point on the segment $CD$. Prove

that the quadrilateral $ABEF$ is circumscribable if and only if the quadrilateral $DIEF$

is cyclic.



Proposed by Dorlir Ahmeti, Albania
2020 Kosovo National Mathematical Olympiad
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.

Note: The sequence $\{a_n\}_{n\ge 1}$ is eventually constant if there exists a positive integer $k$ such that $a_n=c$, for every $n\ge k$.
2021 Kosovo National Mathematical Olympiad
Find all natural two digit numbers such that when you substract by seven times the sum of its digit

from the number you get a prime number.
2021 Kosovo National Mathematical Olympiad
Dua has all the odd natural numbers less than 20. Asija has all the even numbers less than 21. They play the following game. In each round, they take a number from each other and after every round, they may fix two or more consecutive numbers so that their opponent cannot take these fixed numbers in the next round. The game is won by the player who attains 10 consecutive numbers first. Does either player have a winning strategy?
2021 Kosovo National Mathematical Olympiad
Find all real numbers $a,b,c$ and $d$ such that:

$a^2+b^2+c^2+d^2=a+b+c+d-ab=3.$
2021 Kosovo National Mathematical Olympiad
Let $ABCDE$ be a convex pentagon such that:

$\angle ABC=90,\angle BCD=135,\angle DEA=60$ and $AB=BC=CD=DE$. Find angle $\angle DAE$.
2021 Kosovo National Mathematical Olympiad
Nine weights are placed in a scale with the respective values $1kg,2kg,...,9kg$. In how many ways can we place six weights in the left side and three weights in the right side such that the right side is heavier than the left one?
2021 Kosovo National Mathematical Olympiad
Find all functions $f:\mathbb R\rightarrow \mathbb R$ such that for all real numbers $x,y$:

$f(x)f(y)+f(xy)\leq x+y$.
2021 Kosovo National Mathematical Olympiad
Prove that for any natural numbers $a,b,c$ and $d$ there exist infinetly natural numbers $n$ such that $a^n+b^n+c^n+d^n$ is composite.
2021 Kosovo National Mathematical Olympiad
Let $M$ be the midpoint of segment $BC$ of $\triangle ABC$. Let $D$ be a point such that $AD=AB$, $AD\perp AB$ and points $C$ and $D$ are on different sides of $AB$. Prove that: $$\sqrt{AB\cdot AC+BC\cdot AM}\geq\frac{\sqrt{2}}{2}CD.$$
2021 Kosovo National Mathematical Olympiad
There are $9$ point in the Cartezian plane with coordinates

$(0,0),(0,1),(0,2),(1,0),(1,1),(1,2),(2,0),(2,1),(2,2).$

Some points are coloured in red and the others in blue. Prove that for any colouring of the points we can always find a right isosceles triangle whose vertexes have the same colour.
2021 Kosovo National Mathematical Olympiad
Does there exist a natural number $n$ such that $n!$ ends with exactly $2021$ zeros?
2021 Kosovo National Mathematical Olympiad
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:

$$\frac{a^2+b^2}{b}+\frac{b^2+c^2}{c}+\frac{c^2+a^2}{a}\geq 2(ab+bc+ca).$$