Navvye/Olympiad-Problems-CSV huggingface dataset:
2005 Korea National Olympiad
For triangle $ABC$, $P$ and $Q$ satisfy $\angle BPA + \angle AQC=90^{\circ}$. It is provided that the vertices of the triangle $BAP$ and $ACQ$ are ordered counterclockwise(or clockwise). Let the intersection of the circumcircles of the two triangles be $N$ ($A \neq N$, however if $A$ is the only intersection $A=N$), and the midpoint of segment $BC$ be $M$. Show that the length of $MN$ does not depend on $P$ and $Q$.
2005 Korea National Olympiad
For a positive integer $K$, define a sequence, $\{a_n\}_n$, as following $a_1=K$, $$ a_{n+1} = \{ \begin{array} {cc} a_n-1 , & \mbox{ if } a_n \mbox{ is even} \\ \frac{a_n-1}2 , & \mbox{ if } a_n \mbox{ is odd} \end{array},  $$ for all $n\geq 1$.



Find the smallest value of $K$, which makes $a_{2005}$ the first term equal to 0.
2005 Korea National Olympiad
Find all $f: \mathbb R \to\mathbb R$ such that for all real numbers $x$, $f(x) \geq 0$ and for all real numbers $x$ and $y$, $$ f(x+y)+f(x-y)-2f(x)-2y^2=0.  $$
2005 Korea National Olympiad
Let $P$ be a point that lies outside of circle $O$. A line passes through $P$ and meets the circle at $A$ and $B$, and another line passes through $P$ and meets the circle at $C$ and $D$. The point $A$ is between $P$ and $B$, $C$ is between $P$ and $D$. Let the intersection of segment $AD$ and $BC$ be $L$ and construct $E$ on ray $(PA$ so that $BL \cdot PE = DL \cdot PD$.



Show that $M$ is the midpoint of the segment $DE$, where $M$ is the intersection of lines $PL$ and $DE$.
2005 Korea National Olympiad
Real numbers $x_1, x_2, x_3, \cdots , x_n$ satisfy $x_1^2 + x_2^2 + x_3^2 + \cdots + x_n^2 = 1$. Show that  $$ \frac{x_1}{1+x_1^2}+\frac{x_2}{1+x_1^2+x_2^2}+\cdots+\frac{x_n}{1+ x_1^2 + x_2^2 + x_3^2 + \cdots + x_n^2} < \sqrt{\frac n2} .  $$
2005 Korea National Olympiad
For a positive integer $n$, let $f(n)$ be the number of factors of $n^2+n+1$. Show that there are infinitely many integers $n$ which satisfy $f(n) \geq f(n+1)$.
2005 Korea National Olympiad
A group of 6 students decided to make study groups and service activity groups according to the following principle:



Each group must have exactly 3 members. For any pair of students, there are same number of study groups and service activity groups that both of the students are members.



Supposing there are at least one group and no three students belong to the same study group and service activity group, find the minimum number of groups.
2004 Korea National Olympiad
For arbitrary real number $x$, the function $f : \mathbb R \to \mathbb R$ satisfies $f(f(x))-x^2+x+3=0$. Show that the function $f$ does not exist.
2004 Korea National Olympiad
$x$ and $y$ are positive and relatively prime and $z$ is an integer. They satisfy $(5z-4x)(5z-4y)=25xy$. Show that at least one of $10z+x+y$ or quotient of this number divided by $3$ is a square number (i.e. prove that $10z+x+y$ or integer part of $\frac{10z+x+y}{3}$ is a square number).
2004 Korea National Olympiad
Positive real numbers, $a_1, .. ,a_6$ satisfy $a_1^2+..+a_6^2=2$. Think six squares that has side length of $a_i$ ($i=1,2,\ldots,6$). Show that the squares can be packed inside a square of length $2$, without overlapping.
2004 Korea National Olympiad
Let $k$ and $N$ be positive real numbers which satisfy $k\leq N$. For $1\leq i \leq k$, there are subsets $A_i$ of $\{1,2,3,\ldots,N\}$ that satisfy the following property.



For arbitrary subset of $\{ i_1, i_2, \ldots , i_s \} \subset \{ 1, 2, 3, \ldots, k \} $, $A_{i_1} \triangle A_{i_2}   \triangle ... \triangle A_{i_s}$ is not an empty set.



Show that a subset $\{ j_1, j_2, .. ,j_t \} \subset \{ 1, 2, ... ,k \} $ exist that satisfies $n(A_{j_1} \triangle A_{j_2} \triangle \cdots \triangle A_{j_t})  \geq  k$. ($A \triangle B=A \cup B-A \cap B$)
2004 Korea National Olympiad
$A, B, C$, and $D$ are the four different points on the circle $O$ in the order. Let the centre of the scribed circle of triangle $ABC$, which is tangent to $BC$, be $O_1$. Let the centre of the scribed circle of triangle $ACD$, which is tangent to $CD$, be $O_2$.



(1) Show that the circumcentre of triangle $ABO_1$ is on the circle $O$.



(2) Show that the circumcircle of triangle $CO_1O_2$ always pass through a fixed point on the circle $O$, when $C$ is moving along arc $BD$.
1997 Korea National Olympiad
Let $f(n)$ be the number of ways to express positive integer $n$ as a sum of positive odd integers.

Compute $f(n).$

(If the order of odd numbers are different,  then it is considered as different expression.)
1997 Korea National Olympiad
For positive integer $n,$ let $a_n=\sum_{k=0}^{[\frac{n}{2}]}\binom{n-2}{k}(-\frac{1}{4})^k.$

Find $a_{1997}.$ (For real $x,$ $[x]$ is defined as largest integer that does not exceeds $x.$)
1997 Korea National Olympiad
Let $ABCDEF$ be a convex hexagon such that $AB=BC,CD=DE, EF=FA.$

Prove that $\frac{BC}{BE}+\frac{DE}{DA}+\frac{FA}{FC}\ge\frac{3}{2}$ and find when equality holds.
1997 Korea National Olympiad
For any prime number $p>2,$ and an integer $a$ and $b,$ if $1+\frac{1}{2^3}+\frac{1}{3^3}+\cdots+\frac{1}{(p-1)^3}=\frac{a}{b},$ prove that $a$ is divisible by $p.$
1997 Korea National Olympiad
Let $a,b,c$ be the side lengths of any triangle $\triangle ABC$ opposite to $A,B$ and $C,$ respectively. Let $x,y,z$ be the length of medians from $A,B$ and $C,$ respectively.

If $T$ is the area of $\triangle ABC$, prove that $\frac{a^2}{x}+\frac{b^2}{y}+\frac{c^2}{z}\ge\sqrt{\sqrt{3}T}$
1997 Korea National Olympiad
Find all polynomial $P(x,y)$ for any reals $x,y$ such that

(i) $x^{100}+y^{100}\le P(x,y)\le 101(x^{100}+y^{100})$

(ii) $(x-y)P(x,y)=(x-1)P(x,1)+(1-y)P(1,y).$
1997 Korea National Olympiad
Let $X,Y,Z$ be the points outside the $\triangle ABC$ such that $\angle BAZ=\angle CAY,\angle CBX=\angle ABZ,\angle ACY=\angle BCX.$ Prove that the lines $AX, BY, CZ$ are concurrent.
1997 Korea National Olympiad
For any positive integers $x,y,z$ and $w,$ prove that $x^2,y^2,z^2$ and $w^2$ cannot be four consecutive terms of arithmetic sequence.
1996 Korea National Olympiad
If you draw $4$ points on the unit circle, prove that you can always find two points where their distance between is less than $\sqrt{2}.$
1996 Korea National Olympiad
Let the $f:\mathbb{N}\rightarrow\mathbb{N}$ be the function such that

(i) For all positive integers $n,$ $f(n+f(n))=f(n)$

(ii) $f(n_o)=1$ for some $n_0$



Prove that $f(n)\equiv 1.$
1996 Korea National Olympiad
Let $a=\lfloor \sqrt{n} \rfloor$ for given positive integer $n.$

Express the summation $\sum_{k=1}^{n}\lfloor \sqrt{k} \rfloor$ in terms of $n$ and $a.$
1996 Korea National Olympiad
Circle $C$(the center is $C$.) is inside the $\angle XOY$ and it is tangent to the two sides of the angle. Let $C_1$ be the circle that passes through the center of $C$ and tangent to two sides of angle and let $A$ be one of the endpoint of diameter of $C_1$ that passes through $C$ and $B$ be the intersection of this diameter and circle $C.$ Prove that the cirlce that $A$ is the center and $AB$ is the radius is also tangent to the two sides of $\angle XOY.$
1996 Korea National Olympiad
Find all integer solution triple $(x,y,z)$ such that $x^2+y^2+z^2-2xyz=0.$
1996 Korea National Olympiad
Find the minimum value of $k$ such that there exists two sequence ${a_i},{b_i}$ for $i=1,2,\cdots ,k$ that satisfies the following conditions.



(i) For all $i=1,2,\cdots ,k,$ $a_i,b_i$ is the element of $S=\{1996^n|n=0,1,2,\cdots\}.$

(ii) For all $i=1,2,\cdots, k, a_i\ne b_i.$

(iii) For all $i=1,2,\cdots, k, a_i\le a_{i+1}$ and $b_i\le b_{i+1}.$

(iv) $\sum_{i=1}^{k} a_i=\sum_{i=1}^{k} b_i.$
1996 Korea National Olympiad
Let $A_n$ be the set of real numbers such that each element of $A_n$ can be expressed as $1+\frac{a_1}{\sqrt{2}}+\frac{a_2}{(\sqrt{2})^2}+\cdots +\frac{a_n}{(\sqrt{n})^n}$ for given $n.$ Find both $|A_n|$ and sum of the products of two distinct elements of $A_n$ where each $a_i$ is either $1$ or $-1.$
1996 Korea National Olympiad
Let $\triangle ABC$ be the acute triangle such that $AB\ne AC.$ Let $V$ be the intersection of $BC$ and angle bisector of $\angle A.$ Let $D$ be the foot of altitude from $A$ to $BC.$ Let $E,F$ be the intersection of circumcircle of $\triangle AVD$ and $CA,AB$ respectively. Prove that the lines $AD, BE,CF$ is concurrent.
1995 Korea National Olympiad
For any positive integer $m$,show that there exist integers $a,b$ satisfying

$\left | a \right |\leq m$, $ \left | b \right |\leq m$, $0< a+b\sqrt{2}\leq \frac{1+\sqrt{2}}{m+2}$
1995 Korea National Olympiad
find all functions from the nonegative integers into themselves, such that: $2f(m^2+n^2)=f^2(m)+f^2(n)$ and for $m\geq n$ $f(m^2)\geq f(n^2)$.
1995 Korea National Olympiad
Let $ABC$ be an equilateral triangle of side $1$, $D$ be a point on $BC$, and $r_1, r_2$ be the inradii of triangles $ABD$ and $ADC$. Express $r_1r_2$ in terms of $p = BD$ and find the maximum of $r_1r_2$.
1995 Korea National Olympiad
Let $O$ and $R$ be the circumcenter and circumradius of a triangle $ABC$, and let $P$ be any point in the plane of the triangle. The perpendiculars $PA_1,PB_1,PC_1$ are dropped from $P$ to $BC,CA,AB$. Express $S_{A_1B_1C_1}/S_{ABC}$ in terms of $R$ and $d = OP$, where $S_{XYZ}$ is the area of $\triangle XYZ$.
1995 Korea National Olympiad
Let $a,b$ be integers and $p$ be a prime number such that:

(i) $p$ is the greatest common divisor of $a$ and $b$;

(ii) $p^2$ divides $a$.

Prove that the polynomial $x^{n+2}+ax^{n+1}+bx^{n}+a+b$ cannot be decomposed into the product of two polynomials with integer coefficients and degree greater than $1$.
1995 Korea National Olympiad
Let $m,n$ be positive integers with $1 \le  n < m$. A box is locked with several padlocks which must all be opened to open the box, and which all have different keys. The keys are distributed among $m$ people. Suppose that among these people, no $n$ can open the box, but any $n+1$ can open it. Find the smallest possible number $l$ of locks and then the total number of keys for which this is possible.
1994 Korea National Olympiad
Let $ S$ be the set of nonnegative integers. Find all functions $ f,g,h: S\rightarrow S$ such that



$ f(m+n)=g(m)+h(n),$ for all $ m,n\in S$, and



$ g(1)=h(1)=1$.
1994 Korea National Olympiad
Let $ \alpha,\beta,\gamma$ be the angles of a triangle. Prove that

$csc^2\frac{\alpha}{2}+csc^2\frac{\beta}{2}+csc^2\frac{\gamma}{2} \ge 12$

and find the conditions for equality.
1994 Korea National Olympiad
In a triangle $ABC$, $I$ and $O$ are the incenter and circumcenter respectively, $A',B',C'$ the excenters, and $O'$ the circumcenter of $\triangle A'B'C'$. If $R$ and $R'$ are the circumradii of triangles $ABC$ and $A'B'C'$, respectively, prove that:

(i) $R'= 2R $

(ii) $IO' = 2IO$
1994 Korea National Olympiad
Consider the equation $ y^2-k=x^3$, where $ k$ is an integer.



Prove that the equation cannot have five integer solutions of the form

$ (x_1,y_1),(x_2,y_1-1),(x_3,y_1-2),(x_4,y_1-3),(x_5,y_1-4)$.



Also show that if it has the first four of these pairs as solutions, then $ 63|k-17$.
1994 Korea National Olympiad
Given a set $S \subset N$ and a positive integer n, let $S\oplus \{n\} = \{s+n / s \in  S\}$. The sequence $S_k$ of sets is defined inductively as follows: $S_1 = {1}$, $S_k=(S_{k-1} \oplus \{k\})  \cup \{2k-1\}$ for $k = 2,3,4, ...$

(a) Determine $N - \cup _{k=1}^{\infty}  S_k$.

(b) Find all $n$ for which $1994 \in S_n$.
1994 Korea National Olympiad
Let $\alpha,\beta ,\gamma$ be the angles of $\triangle ABC$.

a) Show that $cos^2\alpha +cos^2\beta +cos^2 \gamma =1-2cos\alpha cos\beta cos\gamma$ .

b) Given that $cos\alpha : cos\beta : cos\gamma = 39 : 33 : 25$, find $sin\alpha : sin\beta : sin\gamma$ .
2009 Kosovo National Mathematical Olympiad
Find the graph of the function $y=x-|x+x^2|$
2009 Kosovo National Mathematical Olympiad
If $x_1$ and $x_2$ are the solutions of the equation

$x^2-(m+3)x+m+2=0$

Find all real values of $m$ such that the following inequations are valid

$\frac {1}{x_1}+\frac {1}{x_2}>\frac{1}{2}$

and

$x_1^2+x_2^2<5$
2009 Kosovo National Mathematical Olympiad
Prove that $\sqrt 2$ is irrational.
2009 Kosovo National Mathematical Olympiad
Prove that if in the product of four consequtive natural numbers we add $1$, we get a perfect square.
2009 Kosovo National Mathematical Olympiad
In a circle four distinct points are fixed and each of them is assigned with a real number. Let those numbers be $x_1,x_2,x_3,x_4$ such that $x_1+x_2+x_3+x_4>0$. Now we define a game with these numbers: If one of them, i.e. $x_i$, is a negative number, the player makes a move by adding the number $x_i$ to his neighbors and changes the sign of the chosen number. The game ends when all the numbers are negative. Prove that this game ends in a finite number of steps.
2009 Kosovo National Mathematical Olympiad
Find the graph of the function $y=1-|1-sin x|$.
2009 Kosovo National Mathematical Olympiad
Solve the equation:

$x^2+2xcos(x-y)+1=0$
2009 Kosovo National Mathematical Olympiad
Let $n\geq2$ be an integer. $n$ is a prime if it is only divisible by $1$ and $n$. Prove that there are infinitely many prime numbers.
2009 Kosovo National Mathematical Olympiad
Prove that $n^{11}-n$ is divisible by $11$.
2009 Kosovo National Mathematical Olympiad
Find the graph of the function $y=x+|1-x^3|$.
2009 Kosovo National Mathematical Olympiad
Let $p$ be a prime number and $n$ a natural one. How many natural numbers are between $1$ and $p^n$ that are relatively prime with $p^n$?
2009 Kosovo National Mathematical Olympiad
Let $a,b$ and $c$ be the sides of a triangle, prove that

$\frac {a}{b+c}+\frac {b}{c+a}+\frac {c}{a+b}<2$.
2009 Kosovo National Mathematical Olympiad
$(a)$ Let $a_1,a_2,a_3$ be three real numbers. Prove that

$(a_1-a_2)(a_1-a_3)+(a_2-a_1)(a_2-a_3)+(a_3-a_1)(a_2-a_2)\geq 0$.

$(b)$ Prove that the inequality above doesn't hold if we use four number instead of three.
2010 Kosovo National Mathematical Olympiad
Solve the equation

$|x+1|-|x-1|=2$.
2010 Kosovo National Mathematical Olympiad
Someones age is equal to the sum of the digits of his year of birth. How old is he and when was he born, if it is known that he is older than $11$.

P.s. the current year in the problem is $2010$.
2010 Kosovo National Mathematical Olympiad
Prove that in any polygon, there exist two sides whose radio is less than $2$.(Essentialy if $a_1\geq a_2\geq...\geq a_n$ are the sides of a polygon prove that there exist $i,j\in\{1,2,..,n\}$ so that $i<j$ and $\frac {a_i}{a_j}<2$).
2010 Kosovo National Mathematical Olympiad
Prove that $\sqrt 3$ is irrational.
2010 Kosovo National Mathematical Olympiad
Let $x,y$ be positive real numbers such that $x+y=1$. Prove that

$\left(1+\frac {1}{x}\right)\left(1+\frac {1}{y}\right)\geq 9$.
2010 Kosovo National Mathematical Olympiad
Find the graph of the function $y=|2^{|x|}-1|$.
2010 Kosovo National Mathematical Olympiad
The equation is given

$x^2-(m+3)x+m+2=0$.

If $x_1$ and $x_2$ are its solutions find all $m$ such that

$\frac{x_1}{x_1+1}+\frac{x_2}{x_2+1}=\frac{13}{10}$.
2010 Kosovo National Mathematical Olympiad
Let $a,b,c$ be non negative integers. Suppose that $c$ is even and $a^5+4b^5=c^5$. Prove that $b=0$.
2010 Kosovo National Mathematical Olympiad
Solve the inequation

$\sqrt {3-x}-\sqrt {x+1}>\frac {1}{2}$.
2010 Kosovo National Mathematical Olympiad
Let $a_1,a_2,...,a_n$ be an arithmetic progression of positive real numbers. Prove that

$\tfrac {1}{\sqrt a_1+\sqrt a_2}+\tfrac {1}{\sqrt a_2+\sqrt a_3}+...+\tfrac {1}{\sqrt a_{n-1}+\sqrt a_n}=\tfrac{n-1}{\sqrt {a_1}+\sqrt{a_n}}$.
2010 Kosovo National Mathematical Olympiad
Arrange the numbers $\cos 2, \cos 4, \cos 6$ and $\cos 8$ from the biggest to the smallest where $2,4,6,8$ are given as radians.
2010 Kosovo National Mathematical Olympiad
Prove that $\sqrt[3]{5}$ is irrational.
2010 Kosovo National Mathematical Olympiad
If the real function

$f(x)=\cos x+\sum_{i=1}^{n}\cos(a_ix)$

is periodic, prove that $a_i,i\in\{1,2,...,n\}$, are rational numbers.
2010 Kosovo National Mathematical Olympiad
The set $S\subseteq \mathbb{R}$ is given with the properties:

$(a) \mathbb{Z}\subset S$,

$(b) (\sqrt 2 +\sqrt 3)\in S$,

$(c)$ If $x,y\in S$ then $x+y\in S$, and

$(d)$ If $x,y\in S$ then $x\cdot y\in S$.

Prove that $(\sqrt 2+\sqrt 3)^{-1}\in S$.
2010 Kosovo National Mathematical Olympiad
Let $n\in \mathbb{N}$. Prove that the polynom

$p(x)=x^{2n}-2x^{2n-1}+3x^{2n-2}-...-2nx+2n+1$

doesn't have real roots.
2010 Kosovo National Mathematical Olympiad
Let $(p_1,p_2,..., p_n)$ be a random permutation of the set $\{1,2,...,n)$. If $n$ is odd, prove that the product

$(p_1-1)\cdot (p_2-2)\cdot ...\cdot (p_n-n)$

is an even number.

@below fixed.
2011 Kosovo National Mathematical Olympiad
Let  $x = \left( 1 + \frac{1}{n}\right)^n$ and $y = \left( 1 + \frac{1}{n}\right)^{n+1}$ where $n \in \mathbb{N}$. Which one of the numbers  $x^y$, $y^x$ is bigger ?
2011 Kosovo National Mathematical Olympiad
It is given the function $f: \left(\mathbb{R} - \{0\}\right)  \to \mathbb{R}$ such that $f(x)=x+\frac{1}{x}$.

Is this function injective ? Justify your answer.
2011 Kosovo National Mathematical Olympiad
A little boy wrote  the numbers $1,2,\cdots,2011$ on a blackboard. He picks any two numbers $x,y$, erases them with a sponge and writes the number $|x-y|$. This process continues until only one number is left. Prove that the number left is even.
2011 Kosovo National Mathematical Olympiad
In triangle $ABC$ medians of triangle $BE$ and $AD$ are perpendicular to each other. Find the length of $\overline{AB}$, if $\overline{BC}=6$ and $\overline{AC}=8$
2011 Kosovo National Mathematical Olympiad
Let $n>1$ be an integer and $S_n$ the set of all permutations $\pi : \{1,2,\cdots,n \} \to \{1,2,\cdots,n \}$ where $\pi$ is bijective function. For every permutation $\pi \in S_n$ we define:



$$ F(\pi)= \sum_{k=1}^n  |k-\pi(k)|  \ \ \text{and} \ \  M_{n}=\frac{1}{n!}\sum_{\pi \in S_n} F(\pi) $$

where $M_n$ is taken with all permutations $\pi \in S_n$. Calculate the sum $M_n$.
2011 Kosovo National Mathematical Olympiad
Suppose that the roots $p,q$ of the equation $x^2-x+c=0$ where $c \in \mathbb{R}$, are rational numbers. Prove that the roots of the equation $x^2+px-q=0$ are also rational numbers.
2011 Kosovo National Mathematical Olympiad
Find all solutions to the equation:



$$ \left(\left\lfloor x+\frac{7}{3} \right\rfloor \right)^2-\left\lfloor x-\frac{9}{4} \right\rfloor = 16 $$
2011 Kosovo National Mathematical Olympiad
Prove that the following inequality holds:

$$ \left( \log_{24}48 \right)^2+ \left( \log_{12}54 \right)^2>4$$
2011 Kosovo National Mathematical Olympiad
Let $ a$, $ b$, $ c$ be the sides of a triangle, and $ S$ its area. Prove:

$$ a^{2} + b^{2} + c^{2}\geq 4S \sqrt {3}

$$

In what case does equality hold?
2011 Kosovo National Mathematical Olympiad
It is given the function $f:\mathbb{R} \to \mathbb{R}$ such that it holds $f(\sin x)=\sin (2011x)$. Find the value of $f(\cos x)$.
2011 Kosovo National Mathematical Olympiad
Is it possible that by using the following transformations:

$$ f(x) \mapsto x^2 \cdot f \left(\frac{1}{x}+1 \right) \ \ \ \text{or} \ \ \ f(x) \mapsto (x-1)^2 \cdot f\left(\frac{1}{x-1} \right)$$

the function $f(x)=x^2+5x+4$ is sent to the function $g(x)=x^2+10x+8$ ?
2011 Kosovo National Mathematical Olympiad
Find maximal value of the function $f(x)=8-3\sin^2 (3x)+6 \sin (6x)$
2011 Kosovo National Mathematical Olympiad
A point $P$ is given in the square $ABCD$ such that $\overline{PA}=3$, $\overline{PB}=7$ and $\overline{PD}=5$. Find the area of the square.
2011 Kosovo National Mathematical Olympiad
The complex numbers $z_1$ and $z_2$ are given such that $z_1=-1+i$ and $z_2=2+4i$. Find the complex number $z_3$ such that $z_1,z_2,z_3$ are the points of an equilateral triangle. How many solutions do we have ?
2011 Kosovo National Mathematical Olympiad
It is given the function $f:\left( \mathbb{R} - \{0\} \right) \times \left( \mathbb{R}-\{0\} \right) \to \mathbb{R}$ such that $f(a,b)= \left| \frac{|b-a|}{|ab|}+\frac{b+a}{ab}-1 \right|+ \frac{|b-a|}{|ab|}+ \frac{b+a}{ab}+1$ where $a,b \not=0$. Prove that:

$$ f(a,b)=4 \cdot \text{max} \left\{\frac{1}{a},\frac{1}{b},\frac{1}{2} \right\}$$
2011 Kosovo National Mathematical Olympiad
If $a,b,c$ are real positive numbers prove that the inequality holds:



$$ \frac{\sqrt{a^3+b^3}}{a^2+b^2}+\frac{\sqrt{b^3+c^3}}{b^2+c^2}+\frac{\sqrt{c^3+a^3}}{c^2+a^2} \ge \frac{6(ab+bc+ac)}{(a+b+c)\sqrt{(a+b)(b+c)(c+a)}} $$
2011 Kosovo National Mathematical Olympiad
It is given a convex hexagon $A_1A_2 \cdots A_6$ such that all its interior angles are same valued (congruent). Denote  by $a_1= \overline{A_1A_2},\ \ a_2=\overline{A_2A_3},\ \cdots , a_6=\overline{A_6A_1}.$



$a)$ Prove that holds: $ a_1-a_4=a_2-a_5=a_3-a_6 $

$b)$ Prove that if $a_1,a_2,a_3,...,a_6$ satisfy the above equation, we can construct a convex hexagon with its same-valued (congruent)  interior angles.
2012 Kosovo National Mathematical Olympiad
Find the value of

$(1+2)(1+2^2)(1+2^4)(1+2^8)...(1+2^{2048})$.
2012 Kosovo National Mathematical Olympiad
If $a>1,b>1$ are the legths of the catheti of an right triangle and $c$ the length of its hypotenuse, prove that

$a+b\leq c\sqrt 2$
2012 Kosovo National Mathematical Olympiad
The integers $a_1,a_2,...,a_{2012}$ are given. Exactly $29$ of them are divisible by $3$. Prove that the sum $a_1^2+a_2^2+...+a_{2012}^2$   is divisible by $3$.
2012 Kosovo National Mathematical Olympiad
Inside of the square $ABCD$ the point $P$ is given such that $|PA|:|PB|:|PC|=1:2:3$. Find $\angle APB$.
2012 Kosovo National Mathematical Olympiad
The following square table is given with seven raws and seven columns:

$a_{11},a_{12},a_{13},a_{14},a_{15},a_{16},a_{17}$

$a_{21},a_{22},a_{23},a_{24},a_{25},a_{26},a_{27}$

$a_{31},a_{32},a_{33},a_{34},a_{35},a_{36},a_{37}$

$a_{41},a_{42},a_{43},a_{44},a_{45},a_{46},a_{47}$

$a_{51},a_{52},a_{53},a_{54},a_{55},a_{56},a_{57}$

$a_{61},a_{62},a_{63},a_{64},a_{65},a_{66},a_{67}$

$a_{71},a_{72},a_{73},a_{74},a_{75},a_{76},a_{77}$

Suppose $a_{ij}\in\{0,1\},\forall i,j\in\{1,...,7\}$. Prove that there exists at least one combination of the numbers $1$ and $0$ so that the following conditions hold:

$(i)$ Each raw and each column has exactly three $1$'s.

$(ii)$$\sum_{j=1}^7a_{lj}a_{ij}=1,\forall l,i\in\{1,...,7\}$ and $l\neq i$.(so for any two distinct raws there is exactly one $r$ so that the both raws have $1$ in the $r$-th place).

$(iii)$$\sum_{i=1}^7a_{ij}a_{ik}=1,\forall j,k\in\{1,...,7\}$ and $j\neq k$.(so for any two distinct columns there is exactly one $s$ so that the both columns have $1$ in the $s$-th place).
2012 Kosovo National Mathematical Olympiad
Find the two last digits of $2012^{2012}$.
2012 Kosovo National Mathematical Olympiad
Let $a,b,c$ be the lengths of the sides of a triangle. Prove that,

$\left|\frac {a}{b}+\frac {b}{c}+\frac {c}{a}-\frac {b}{a}-\frac {c}{b}-\frac {a}{c}\right|<1$
2012 Kosovo National Mathematical Olympiad
Let $n\not\equiv 2\pmod{3}$. Is $\sqrt{\lfloor n+\tfrac {2n}{3}\rfloor+7},\forall n \in \mathbb {N}$, a natural number?
2012 Kosovo National Mathematical Olympiad
The right triangle $ABC$ with a right angle at $C$. From all the rectangles $CA_1MB_1$, where $A_1\in BC, M\in AB$ and $B_1\in AC$ which one has the biggest area?
2012 Kosovo National Mathematical Olympiad
If

$(x^2-x-1)^n=a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+...+a_1x+a_0$,

where $a_i,i\in\{0,1,2,..,2n\}$, find $a_1+a_3+...+a_{2n-1}$ and $a_0+a_2+a_4+...+a_{2n}$.
2012 Kosovo National Mathematical Olympiad
The bisector of acute angle $\alpha$ of the right triangle $ABC$  splits the side $a$ in two segments with lengths $m$ and $n$. Find the acute angle $\beta$ and the lenths of the other two sides by knowing $m$ and $n$.
2012 Kosovo National Mathematical Olympiad
Prove that for any integer $n\geq 2$ it holds that

$\dbinom {2n}{n}>\frac {4^n}{2n}$.
2012 Kosovo National Mathematical Olympiad
Find the set of solutions to the equation

$\log_{\lfloor x\rfloor}(x^2-1)=2$
2012 Kosovo National Mathematical Olympiad
Prove that for all $n\in\mathbb{N}$ we have

$\sum_{k=0}^n\dbinom {n}{k}^2=\dbinom {2n}{n}$.
2012 Kosovo National Mathematical Olympiad
In a sphere $S_0$ we radius $r$ a cube $K_0$ has been inscribed. Then in the cube $K_0$ another sphere $S_1$ has been inscribed and so on to infinity. Calculate the volume of all spheres created in this way.