Navvye/Olympiad-Problems-CSV huggingface dataset:
1995 Kürschák Competition
Points $A$, $B$, $C$, $D$ are such that no three of them are collinear. Let $E=AB\cap CD$ and $F=BC\cap DA$. Let $k_1$, $k_2$ and $k_3$ denote the circles with diameter $\overline{AC}$, $\overline{BD}$ and $\overline{EF}$, respectively. Prove that either $k_1,k_2,k_3$ pass through one point, or no two of them intersect.
1994 Kürschák Competition
The ratio of the sides of a parallelogram is $\lambda>1$. Given $\lambda$, determine the maximum of the acute angle subtended by the diagonals of the parallelogram.
1994 Kürschák Competition
Prove that if we erase $n-3$ diagonals of a regular $n$-gon, then we may still choose $n-3$ of the remaining diagonals such that they don't intersect inside the $n$-gon; but it is possible to erase $n-2$ diagonals such that this statement doesn't hold.
1994 Kürschák Competition
Consider the sets $A_1,A_2,\dots,A_n$. Set $A_k$ is composed of $k$ disjoint intervals on the real axis ($k=1,2,\dots,n$). Prove that from the intervals contained by these sets, one can choose $\left\lfloor\frac{n+1}2\right\rfloor$ intervals such that they belong to pairwise different sets $A_k$, and no two of these intervals have a common point.
1993 Kürschák Competition
Let $a$ and $b$ be positive integers. Prove that the numbers $an^2+b$ and $a(n+1)^2+b$ are both perfect squares only for finitely many integers $n$.
1993 Kürschák Competition
Triangle $ABC$ is not isosceles. The incircle of $\triangle ABC$ touches the sides $BC$, $CA$, $AB$ in the points $K$, $L$, $M$. The parallel with $LM$ through $B$ meets $KL$ at $D$, the parallel with $LM$ through $C$ meets $KM$ at $E$.



Prove that $DE$ passes through the midpoint of $\overline{LM}$.
1993 Kürschák Competition
Let $n$ be a fixed positive integer. Compute over $\mathbb{R}$ the minimum of the following polynomial:



$$f(x)=\sum_{t=0}^{2n}(2n+1-t)x^t.$$
1992 Kürschák Competition
Define for $n$ given positive reals the strange mean as the sum of the squares of these numbers divided by their sum. Decide which of the following statements hold for $n=2$:



a) The strange mean is never smaller than the third power mean.



b) The strange mean is never larger than the third power mean.



c) The strange mean, depending on the given numbers, can be larger or smaller than the third power mean.



Which statement is valid for $n=3$?
1992 Kürschák Competition
For any positive integer $k$ define $f_1(k)$ as the square of the digital sum of $k$ in the decimal system, and $f_{n}(k)=f_1(f_{n-1}(k))$ $\forall n>1$. Compute $f_{1992}(2^{1991})$.
1992 Kürschák Competition
Consider finitely many points in the plane such that no three are collinear. Prove that we can paint the points with two colors such that there is no half-plane that contains exactly three points such that those three points have the same color.
1991 Kürschák Competition
Let $n$ be a positive integer, and $a,b\ge 1$, $c>0$ arbitrary real numbers. Prove that



$$\frac{(ab+c)^n-c}{(b+c)^n-c}\le a^n.$$
1991 Kürschák Competition
A convex polyhedron has two triangle and three quadrilateral faces. Connect every vertex of one of the triangle faces with the intersection point of the diagonals in the quadrilateral face opposite to it. Show that the resulting three lines are concurrent.
1991 Kürschák Competition
Consider $998$ red points on the plane with no three collinear. We select $k$ blue points in such a way that inside each triangle whose vertices are red points, there is a blue point as well. Find the smallest $k$ for which the described selection of blue points is possible for any configuration of $998$ red points.
1990 Kürschák Competition
Let $p>2$ be a prime number and $n$ a positive integer. Prove that $pn^2$ has at most one positive divisor $d$ for which $n^2+d$ is a square number.
1990 Kürschák Competition
The incenter of $\triangle A_1A_2A_3$ is $I$, and the center of the $A_i$-excircle is $J_i$ ($i=1,2,3$). Let $B_i$ be the intersection point of side $A_{i+1}A_{i+2}$ and the bisector of $\angle A_{i+1}IA_{i+2}$ ($A_{i+3}:=A_i$ $\forall i$). Prove that the three lines $B_iJ_i$ are concurrent.
1990 Kürschák Competition
We would like to give a present to one of $100$ children. We do this by throwing a biased coin $k$ times, after predetermining who wins in each possible outcome of this lottery.



Prove that we can choose the probability $p$ of throwing heads, and the value of $k$ such that, by distributing the $2^k$ different outcomes between the children in the right way, we can guarantee that each child has the same probability of winning.
1989 Kürschák Competition
In the plane, two intersecting lines $a$ and $b$ are given, along with a circle $\omega$ that has no common points with these lines. For any line $\ell||b$, define $A=\ell\cap a$, and $\{B,C\}=\ell\cap \omega$ such that $B$ is on segment $AC$. Construct the line $\ell$ such that the ratio $\frac{|BC|}{|AB|}$ is maximal.
1989 Kürschák Competition
For any positive integer $n$ denote $S(n)$ the digital sum of $n$ when represented in the decimal system. Find every positive integer $M$ for which $S(Mk)=S(M)$ holds for all integers $1\le k\le M$.
1989 Kürschák Competition
We play the following game in a Cartesian coordinate system in the plane. Given the input $(x,y)$, in one step, we may move to the point $(x,y\pm 2x)$ or to the point $(x\pm 2y,y)$. There is also an additional rule: it is not allowed to make two steps that lead back to the same point (i.e, to step backwards).



Prove that starting from the point $\left(1;\sqrt 2\right)$, we cannot return to it in finitely many steps.
1988 Kürschák Competition
Prove that if there exists a point $P$ inside the convex quadrilateral $ABCD$ such that the triangles $PAB$, $PBC$, $PCD$, $PDA$ have the same area, then one of the diagonals of $ABCD$ bisects the area of the quadrilateral.
1988 Kürschák Competition
Set $T\subset\{1,2,\dots,n\}^3$ has the property that for any two triplets $(a,b,c)$ and $(x,y,z)$ in $T$, we have $a<b<c$, and also, we know that at most one of the equalities $a=x$, $b=y$, $c=z$ holds. Maximize $|T|$.
1988 Kürschák Competition
Consider the convex lattice quadrilateral $PQRS$ whose diagonals intersect at $E$. Prove that if $\angle P+\angle Q<180^\circ$, then the $\triangle PQE$ contains inside it or on one of its sides a lattice point other than $P$ and $Q$.
1987 Kürschák Competition
Is there a set of points in space whose intersection with any plane is a finite but nonempty set of points?
1987 Kürschák Competition
Any two members of a club with $3n+1$ people plays ping-pong, tennis or chess with each other. Everyone has exactly $n$ partners who plays ping-pong, $n$ who play tennis and $n$ who play chess.



Prove that we can choose three members of the club who play three different games amongst each other.
1986 Kürschák Competition
Let $n>2$ be a positive integer. Find the largest value $h$ and the smallest value $H$ for which

$$h<{a_1\over a_1+a_2}+{a_2\over a_2+a_3}+\cdots+{a_n\over a_n+a_1}<H$$

holds for any positive reals $a_1,\dots,a_n$.
1986 Kürschák Competition
A and B plays the following game: they choose randomly $k$ integers from $\{1,2,\dots,100\}$; if their sum is even, A wins, else B wins. For what values of $k$ does A and B have the same chance of winning?
1985 Kürschák Competition
We have triangulated a convex $(n+1)$-gon $P_0P_1\dots P_n$ (i.e., divided it into $n-1$ triangles with $n-2$ non-intersecting diagonals). Prove that the resulting triangles can be labelled with the numbers $1,2,\dots,n-1$ such that for any $i\in\{1,2,\dots,n-1\}$, $P_i$ is a vertex of the triangle with label $i$.
1985 Kürschák Competition
For every $n\in\mathbb{N}$, define the power sum of $n$ as follows. For every prime divisor $p$ of $n$, consider the largest positive integer $k$ for which $p^k\le n$, and sum up all the $p^k$'s. (For instance, the power sum of $100$ is $2^6+5^2=89$.) Prove that the power sum of $n$ is larger than $n$ for infinitely many positive integers $n$.
1985 Kürschák Competition
We reflected each vertex of a triangle on the opposite side. Prove that the area of the triangle formed by these three reflection points is smaller than the area of the initial triangle multiplied by five.
2012 Kyrgyzstan National Olympiad
Prove that $ n $ must be prime in order to have only one solution to the equation $\frac{1}{x}-\frac{1}{y}=\frac{1}{n}$, $x,y\in\mathbb{N}$.
2012 Kyrgyzstan National Olympiad
Given positive real numbers $ {a_1},{a_2},...,{a_n} $ with $ {a_1}+{a_2}+...+{a_n}= 1 $. Prove that

$ \left({\frac{1}{{a_1^2}}-1}\right)\left({\frac{1}{{a_2^2}}-1}\right)...\left({\frac{1}{{a_n^2}}-1}\right)\geqslant{({n^2}-1)^n} $.
2012 Kyrgyzstan National Olympiad
Prove that if the diagonals of a convex quadrilateral are perpendicular, then the feet of perpendiculars dropped from the intersection point of diagonals on the sides of this quadrilateral lie on one circle. Is the converse true?
2012 Kyrgyzstan National Olympiad
Find all functions $ f:\mathbb{R}\to\mathbb{R} $ such that $ f(f(x)^2+f(y)) = xf(x)+y $,$ \forall x,y\in R $.
2012 Kyrgyzstan National Olympiad
The sequence of natural numbers is defined as follows: for any $ k\geq 1 $,$ a_{k+2}= a_{k+1}\cdot a_k+1 $. Prove that for $ k\geq 9 $ the number $ a_k-22 $ is composite.
2012 Kyrgyzstan National Olympiad
The numbers $ 1, 2,\ldots, 50 $ are written on a blackboard. Each minute any two numbers are erased and their positive difference is written instead. At the end one number remains. Which values can take this number?
2011 Kyrgyzstan National Olympiad
For a given chord $MN$ of a circle discussed the triangle $ABC$, whose base is the diameter $AB$ of this circle,which do not intersect the $MN$, and the sides $AC$ and $BC$ pass through the ends of $M$ and $N$ of the chord $MN$. Prove that the heights of all such triangles $ABC$ drawn from the vertex $C$ to the side $AB$, intersect at one point.
2011 Kyrgyzstan National Olympiad
In a convex $n$-gon all angles are equal from a certain point, located inside the $n$-gon, all its sides are seen under equal angles. Can we conclude that this $n$-gon is regular?
2011 Kyrgyzstan National Olympiad
Given positive numbers ${a_1},{a_2},...,{a_n}$ with ${a_1} + {a_2} + ... + {a_n} = 1$. Prove that $\left( {\frac{1}{{a_1^2}} - 1} \right)\left( {\frac{1}{{a_2^2}} - 1} \right)...\left( {\frac{1}{{a_n^2}} - 1} \right) \geqslant {({n^2} - 1)^n}$.
2011 Kyrgyzstan National Olympiad
Given equation ${a^5} - {a^3} + a = 2$, with real $a$ . Prove that $3 < {a^6} < 4$.
2011 Kyrgyzstan National Olympiad
Points $M$ and $N$ are chosen on sides $AB$ and $BC$,respectively, in a triangle $ABC$, such that point $O$ is interserction of lines $CM$ and $AN$. Given that $AM+AN=CM+CN$. Prove that $AO+AB=CO+CB$.
2011 Kyrgyzstan National Olympiad
a) Among the $21$ pairwise distances between the $7$ points of the plane, prove that  one and the same number occurs not more than $12$ times.



b) Find a maximum number of times may meet the same number among the $15$ pairwise distances between $6$ points of the plane.
2011 Kyrgyzstan National Olympiad
Given that $g(n) = \frac{1}{{2 + \frac{1}{{3 + \frac{1}{{... + \frac{1}{{n - 1}}}}}}}}$ and $k(n) = \frac{1}{{2 + \frac{1}{{3 + \frac{1}{{... + \frac{1}{{n - 1 + \frac{1}{n}}}}}}}}}$, for natural $n$. Prove that $\left| {g(n) - k(n)} \right| \le \frac{1}{{(n - 1)!n!}}$.
2011 Kyrgyzstan National Olympiad
Given a sequence $x_1,x_2,...,x_n$ of real numbers with ${x_{n + 1}}^3 = {x_n}^3 - 3{x_n}^2 + 3{x_n}$, where $(n=1,2,3,...)$. What must be value of $x_1$, so that $x_{100}$ and $x_{1000}$ becomes equal?
2010 Kyrgyzstan National Olympiad
Given that $a,b,c > 0$ and $a + b + c = 1$. Prove that $\sqrt {\frac{{ab}}{{ab + c}}}  + \sqrt {\frac{{bc}}{{bc + a}}}  + \sqrt {\frac{{ca}}{{ca + b}}}  \leqslant \frac{3}{2}$.
2010 Kyrgyzstan National Olympiad
Fifteen pairwise coprime positive integers chosen so that each of them less than 2010. Show that at least one of them is prime.
2010 Kyrgyzstan National Olympiad
At the meeting, each person is familiar with 22 people. If two persons $A$ and $B$ know each with one another, among the remaining people they do not have a common friend. For each pair individuals $A$ and $B$ are not familiar with each other, there are among the remaining six common acquaintances. How many people were at the meeting?
2010 Kyrgyzstan National Olympiad
Point $O$ is chosen in a triangle $ABC$ such that ${d_a},{d_b},{d_c}$ are distance from point $O$ to sides $BC,CA,AB$, respectively. Find position of point $O$ so that product ${d_a} \cdot {d_b} \cdot {d_c}$ becomes maximum.
2010 Kyrgyzstan National Olympiad
Let $k$ be a constant number larger than $1$. Find all polynomials $P(x)$ such that $P({x^k}) = {\left( {P(x)} \right)^k}$ for all real $x$.
2010 Kyrgyzstan National Olympiad
Let $p$ - a prime, where  $p> 11$. Prove that there exists a number $k$ such that the product $p \cdot k$ can be written in the decimal system with only ones.
2010 Kyrgyzstan National Olympiad
Find all natural triples $(a,b,c)$, such that:

$a - )\,a \le b \le c$

$b - )\,(a,b,c) = 1$

$c - )\,\left. {{a^2}b} \right|{a^3} + {b^3} + {c^3}\,,\,\left. {{b^2}c} \right|{a^3} + {b^3} + {c^3}\,,\,\left. {{c^2}a} \right|{a^3} + {b^3} + {c^3}$.
2010 Kyrgyzstan National Olympiad
Solve in none-negative integers ${x^3} + 7{x^2} + 35x + 27 = {y^3}$.
2009 Kyrgyzstan National Olympiad
$ a,b,c$ are sides of triangle $ ABC$. For any choosen triple from $ (a + 1,b,c),(a,b + 1,c),(a,b,c + 1)$ there exist a triangle which sides are choosen triple. Find all possible values of area which triangle $ ABC$ can take.
2009 Kyrgyzstan National Olympiad
$ x$ and $ y$ are real numbers.

$A)$ If it is known that $ x + y$ and $ x + y^2$ are rational numbers, can we conclude that $ x$ and $ y$ are also rational numbers.

$B)$ If it is known that $ x + y$ , $ x + y^2$ and $ x + y^3$ are rational numbers, can we conclude that $ x$ and $ y$ are also rational numbers.
2009 Kyrgyzstan National Olympiad
For function $ f: \mathbb{R} \to \mathbb{R}$ given that $ f(x^2 +x +3) +2 \cdot f(x^2 - 3x + 5) = 6x^2 - 10x +17$, calculate $ f(2009)$.
2009 Kyrgyzstan National Olympiad
Find all real $(x,y)$ such that

$x + {y^2} = {y^3}$

$y + {x^2} = {x^3}$
2009 Kyrgyzstan National Olympiad
Prove for all natural $n$ that $\left. {{{40}^n} \cdot n!} \right|(5n)!$
2009 Kyrgyzstan National Olympiad
Find all natural $a,b$ such that $\left. {a(a + b) + 1} \right|(a + b)(b + 1) - 1$.
2009 Kyrgyzstan National Olympiad
Does $ a^2 + b^2 + c^2 \leqslant 2(ab + bc + ca)$  hold for every $ a,b,c$  if it is known that $ a^4 + b^4 + c^4 \leqslant 2(a^2 b^2 + b^2 c^2 + c^2 a^2 )$.
2009 Kyrgyzstan National Olympiad
Does there exist a function $ f: {\Bbb N} \to {\Bbb N}$ such that $ f(f(n - 1)) = f(n + 1) - f(n)$ for all $ n > 2$.
2009 Kyrgyzstan National Olympiad
For any positive $ a_1 ,a_2 ,...,a_n$ prove that  $ \frac {{a_1 }} {{a_2 + a_3 }} + \frac {{a_2 }} {{a_3 + a_4 }} + ... + \frac {{a_n }} {{a_1 + a_2 }} > \frac {n} {4}$   holds.
2016 Latvia National Olympiad
Given positive integers $x$ and $y$ such that $xy^2$ is a perfect cube, prove that $x^2y$ is also a perfect cube.
2016 Latvia National Olympiad
Triangle $ABC$ has median $AF$, and $D$ is the midpoint of the median. Line $CD$ intersects $AB$ in $E$. Prove that $BD = BF$ implies $AE = DE$!
2016 Latvia National Olympiad
Is it possible to insert numbers $1, \ldots, 16$ into a table $4 \times 4$ (each cell should have a different number) so that every two adjacent cells (i.e. cells sharing a common side) have numbers $a$ and $b$ satisfying

(a) $|a-b| \geq 6$

(b) $|a-b| \geq 7$
2016 Latvia National Olympiad
Find the least prime factor of the number $\frac{2016^{2016}-3}{3}$.
2016 Latvia National Olympiad
The integer sequence $(s_i)$ "having pattern 2016'" is defined as follows:

$\circ$ The first member $s_1$ is 2.

$\circ$ The second member $s_2$ is the least positive integer exceeding $s_1$ and having digit 0 in its decimal notation.

$\circ$ The third member $s_3$ is the least positive integer exceeding $s_2$ and having digit 1 in its decimal notation.

$\circ$ The third member $s_3$ is the least positive integer exceeding $s_2$ and having digit 6 in its decimal notation.

The following members are defined in the same way. The required digits change periodically: $2 \rightarrow 0 \rightarrow 1 \rightarrow 6 \rightarrow 2 \rightarrow 0 \rightarrow \ldots$. The first members of this sequence are the following: $2; 10; 11; 16; 20; 30; 31; 36; 42; 50$. What are the 4 numbers that immediately follow $s_k = 2016$ in this sequence?
2016 Latvia National Olympiad
Given that $x$ and $y$ are positive integers such that $xy^{10}$ is perfect 33rd power of a positive integer, prove that $x^{10}y$ is also a perfect 33rd power!
2016 Latvia National Olympiad
The bisectors of the angles $\sphericalangle CAB$ and $\sphericalangle BCA$ intersect the circumcircle of $ABC$ in $P$ and $Q$ respectively. These bisectors intersect each other in point $I$. Prove that $PQ \perp BI$.
2016 Latvia National Olympiad
Assume that real numbers $x$, $y$ and $z$ satisfy $x + y + z = 3$. Prove that $xy + xz + yz \leq 3$.
2016 Latvia National Olympiad
In a Pythagorean triangle all sides are longer than 5. Is it possible that (a) all three sides are prime numbers, (b) exactly two sides are prime numbers. (Note: We call a triangle "Pythagorean", if it is a right-angled triangle where all sides are positive integers.)
2016 Latvia National Olympiad
All vertices of a regular 2016-gon are initially white. What is the least number of them that can be painted black so that:

(a) There is no right triangle

(b) There is no acute triangle

having all vertices in the vertices of the 2016-gon that are still white?
2016 Latvia National Olympiad
Given that $x$ and $y$ are positive integers such that $xy^{433}$ is a perfect 2016-power of a positive integer, prove that $x^{433}y$ is also a perfect 2016-power.
2016 Latvia National Olympiad
An acute triangle $ABC$ ($AB > AC$) has circumcenter $O$, but $D$ is the midpoint of $BC$. Circle with diameter $AD$ intersects sides $AB$ and $AC$ in $E$ and $F$ respectively. On segment $EF$ pick a point $M$ so that $DM \parallel AO$. Prove that triangles $ABD$ and $FDM$ are similar.
2016 Latvia National Olympiad
Prove that for every integer $n$ ($n > 1$) there exist two positive integers $x$ and $y$ ($x \leq y$) such that

$$\frac{1}{n} = \frac{1}{x(x+1)} + \frac{1}{(x+1)(x+2)} + \cdots + \frac{1}{y(y+1)}$$
2016 Latvia National Olympiad
The integer sequence $(s_i)$ "having pattern 2016'" is defined as follows:

$\circ$ The first member $s_1$ is 2.

$\circ$ The second member $s_2$ is the least positive integer exceeding $s_1$ and having digit 0 in its decimal notation.

$\circ$ The third member $s_3$ is the least positive integer exceeding $s_2$ and having digit 1 in its decimal notation.

$\circ$ The third member $s_3$ is the least positive integer exceeding $s_2$ and having digit 6 in its decimal notation.

The following members are defined in the same way. The required digits change periodically: $2 \rightarrow 0 \rightarrow 1 \rightarrow 6 \rightarrow 2 \rightarrow 0 \rightarrow \ldots$. The first members of this sequence are the following: $2; 10; 11; 16; 20; 30; 31; 36; 42; 50$.

Does this sequence contain a) 2001, b) 2006?
2016 Latvia National Olympiad
Prove that every triangle can be cut into three pieces so that every piece has axis of symmetry. Show how to cut it (a) using three line segments, (b) using two line segments.
2016 Latvia National Olympiad
Given that $x$, $y$ and $z$ are positive integers such that $x^3y^5z^6$ is a perfect 7th power of a positive integer, show that also $x^5y^6z^3$ is a perfect 7th power.
2016 Latvia National Olympiad
Triangle $ABC$ has incircle $\omega$ and incenter $I$. On its sides $AB$ and $BC$ we pick points $P$ and $Q$ respectively, so that $PI = QI$ and $PB > QB$. Line segment $QI$ intersects $\omega$ in $T$. Draw a tangent line to $\omega$ passing through $T$; it intersects the sides $AB$ and $BC$ in $U$ and $V$ respectively. Prove that $PU = UV + VQ$!
2016 Latvia National Olympiad
Prove that among any 18 consecutive positive 3-digit numbers, there is at least one that is divisible by the sum of its digits!
2016 Latvia National Olympiad
Two functions are defined by equations: $f(a) = a^2 + 3a + 2$ and $g(b, c) = b^2 - b + 3c^2 + 3c$. Prove that for any positive integer $a$ there exist positive integers $b$ and $c$ such that $f(a) = g(b, c)$.
2016 Latvia National Olympiad
Consider the graphs of all the functions $y = x^2 + px + q$ having 3 different intersection points with the coordinate axes. For every such graph we pick these 3 intersection points and draw a circumcircle through them. Prove that all these circles have a common point!
2010 Lithuania National Olympiad
Let $a,b$ be real numbers. Prove the inequality

$$ 2(a^4+a^2b^2+b^4)\ge 3(a^3b+ab^3).$$
2010 Lithuania National Olympiad
In trapezoid $ABCD$, $AD$ is parallel to $BC$. Knowing that $AB=AD+BC$, prove that the bisector of $\angle A$ also bisects $CD$.
2010 Lithuania National Olympiad
In an $m\times n$ rectangular chessboard,there is a stone in the lower leftmost square. Two persons A,B move the stone alternately. In each step one can move the stone upward or rightward any number of squares. The one who moves it into the upper rightmost square wins. Find all $(m,n)$ such that the first person has a winning strategy.
2010 Lithuania National Olympiad
Decimal digits $a,b,c$ satisfy

$$ 37\mid (a0a0\ldots a0b0c0c\ldots 0c)_{10} $$

where there are $1001$ a's and $1001$ c's. Prove that $b=a+c$.
2010 Lithuania National Olympiad
$a,b$ are real numbers such that:

$$ a^3+b^3=8-6ab. $$

Find the maximal and minimal value of $a+b$.
2010 Lithuania National Olympiad
Let $I$ be the incenter of a triangle $ABC$. $D,E,F$ are the symmetric points of $I$ with respect to $BC,AC,AB$ respectively. Knowing that $D,E,F,B$ are concyclic,find all possible values of $\angle B$.
2010 Lithuania National Olympiad
Arrange arbitrarily $1,2,\ldots ,25$ on a circumference. We consider all $25$ sums obtained by adding $5$ consecutive numbers. If the number of distinct residues of those sums modulo $5$ is $d$ $(0\le d\le 5)$,find all possible values of $d$.
2006 Lithuania National Olympiad
Solve the system of equations:

$\left\{ \begin{aligned} x^4+y^2-xy^3-\frac{9}{8}x = 0 \\ y^4+x^2-yx^3-\frac{9}{8}y=0 \end{aligned} \right.$
2006 Lithuania National Olympiad
Two circles are tangent externaly at a point $B$. A line tangent to one of the circles at a point $A$ intersects the other circle at points $C$ and $D$. Show that $A$ is equidistant to the lines $BC$ and $BD$.
2006 Lithuania National Olympiad
Show that if $a+b+c=0$ then $(\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b})(\frac{b-c}{a}+\frac{c-a}{b}+\frac{a-b}{c})=9$.
2006 Lithuania National Olympiad
Find the maximal cardinality $|S|$  of the subset $S \subset A=\{1, 2, 3, \dots, 9\}$ given that no two sums $a+b | a, b \in S, a \neq b$ are equal.
2021 Macedonian Mathematical Olympiad
Let $(a_n)^{+\infty}_{n=1}$ be a sequence defined recursively as follows: $a_1=1$ and $$a_{n+1}=1 + \sum\limits_{k=1}^{n}ka_k$$For every $n > 1$, prove that $\sqrt[n]{a_n} < \frac {n+1}{2}$.
2021 Macedonian Mathematical Olympiad
In the City of Islands there are $2021$ islands connected by bridges. For any given pair of islands $A$ and $B$, one can go from island $A$ to island $B$ using the bridges. Moreover, for any four islands $A_1, A_2, A_3$ and $A_4$: if there is a bridge from $A_i$ to $A_{i+1}$ for each $i \in \left \{ 1,2,3 \right \}$, then there is a bridge between $A_{j}$ and $A_{k}$ for some $j,k \in \left \{ 1,2,3,4 \right \}$ with $|j-k|=2$. Show that there is at least one island which is connected to any other island by a bridge.
2021 Macedonian Mathematical Olympiad
Let $ABCD$ be a trapezoid with $AD \parallel BC$ and $\angle BCD < \angle ABC < 90^\circ$. Let $E$ be the intersection point of the diagonals $AC$ and $BD$. The circumcircle $\omega$ of $\triangle BEC$ intersects the segment $CD$ at $X$. The lines $AX$ and $BC$ intersect at $Y$, while the lines $BX$ and $AD$ intersect at $Z$. Prove that the line $EZ$ is tangent to $\omega$ iff the line $BE$ is tangent to the circumcircle of $\triangle BXY$.