Let $ABCD$ be a cyclic quadrilateral. Let lines $AB$ and $CD$ intersect in $P,$ and lines $BC$ and $DA$ intersect in $Q.$ The feet of the perpendiculars from $P$ to $BC$ and $DA$ are $K$ and $L,$ and the feet of the perpendiculars from $Q$ to $AB$ and $CD$ are $M$ and $N.$ The midpoint of diagonal $AC$ is $F.$ Prove that the circumcircles of triangles $FKN$ and $FLM,$ and the line $PQ$ are concurrent. [i]Based on a problem by Ádám Péter Balogh, Szeged[/i]

Given real numbers $a,\alpha,\beta, \sigma \ and \ \varrho$ s.t. $\sigma, \varrho > 0$ and $\sigma \varrho = \frac{1}{16}$, prove that there exist integers $x$ and $y$ s.t. \[ - \sigma \leq (x+\alpha_(ax + y + \beta ) \leq \varrho \]

Prove that $\sqrt{x}+\sqrt{y}+\sqrt{xy}$ is equal to$ \sqrt{x}+\sqrt{y+xy+2y\sqrt{x}}$ and compare the numbers $\sqrt{3}+\sqrt{10+2\sqrt{3}}$ and $\sqrt{5+\sqrt{22}}+\sqrt{8- \sqrt{22}+2\sqrt{15-3\sqrt{22}}}$.

Altitudes $AA_1$ and $CC_1$ of an acute-angled triangle $ABC$ intersect at point $H$. A straight line passing through point $H$ parallel to line $A_1C_1$ intersects the circumscribed circles of triangles $AHC_1$ and $CHA_1$ at points $X$ and $Y$, respectively. Prove that points $X$ and $Y$ are equidistant from the midpoint of segment $BH$.

Let S be a 1990-element set and P be a set of 100-ary sequences $(a_1,a_2,...,a_{100})$ ,where $a_i's$ are distinct elements of S.An ordered pair (x,y) of elements of S is said to [i]appear[/i] in $(a_1,a_2,...,a_{100})$ if $x=a_i$ and $y=a_j$ for some i,j with $1\leq i<j\leq 100$.Assume that every ordered pair (x,y) of elements of S appears in at most one member in P.Show that $|P|\leq 800$.

David starts at the point $A$ and goes up and right along the grid lines to point $B$. At each of the points $C$, $D$, and $E$ there is a bully. Find the number of paths David can take which make him encounter exactly one bully. [asy] size(150); draw((0,0)--(4,0)--(4,3)--(0,3)--cycle); draw((0,1)--(4,1)); draw((0,2)--(4,2)); draw((1,0)--(1,3)); draw((2,0)--(2,3)); draw((3,0)--(3,3)); dot((0,0)); label("A", (0,0), W); dot((4,3)); label("B", (4,3), E); dot((1,1.5)); label("C", (1,1.5), W); dot((2,0.5)); label("D", (2,0.5), W); dot((2.5,2)); label("E", (2.5,2), N); [/asy]

The euclidean plane is divided into regions by drawing a finite number of circles. Show that it is possible to color each of these regions either red or blue in such a way that no two adjacent regions have the same color.

Given that ${a_n}$ and ${b_n}$ are two sequences of integers defined by \begin{align*} a_1=1, a_2=10, a_{n+1}=2a_n+3a_{n-1} & ~~~\text{for }n=2,3,4,\ldots, \\ b_1=1, b_2=8, b_{n+1}=3b_n+4b_{n-1} & ~~~\text{for }n=2,3,4,\ldots. \end{align*} Prove that, besides the number $1$, no two numbers in the sequences are identical.

Prove that any uncountable subset of the Euclidean $ n$-space contains an countable subset with the property that the distances between different pairs of points are different (that is, for any points $ P_1 \not\equal{} P_2$ and $ Q_1\not\equal{} Q_2$ of this subset, $ \overline{P_1P_2}\equal{}\overline{Q_1Q_2}$ implies either $ P_1\equal{}Q_1$ and $ P_2\equal{}Q_2$, or $ P_1\equal{}Q_2$ and $ P_2\equal{}Q_1$). Show that a similar statement is not valid if the Euclidean $ n$-space is replaced with a (separable) Hilbert space.

Is it possible to place $100$ balls in space so that no two of them have a common interior point and each of them touches at least one third of the others?

A bank has a set S of codes formed only with 0 and 1,each one with length n.Two codes are 'friends' if they are different on only one position.We know that each code has exactly k 'friends'.Prove that: 1)S has an even number of elements 2)S contains at least $2^k$ codes

For a non-empty set $T$ denote by $p(T)$ the product of all elements of $T$. Does there exist a set $T$ of $2021$ elements such that for any $a\in T$ one has that $P(T)-a$ is an odd integer? Consider two cases: 1) All elements of $T$ are irrational numbers. 2) At least one element of $T$ is a rational number.

The non-negative numbers $ x_1, x_2, \ldots, x_5$ satisfy $ \sum_{i \equal{} 1}^5 \frac {1}{1 \plus{} x_i} \equal{} 1$. Prove that $ \sum_{i \equal{} 1}^5 \frac {x_i}{4 \plus{} x_i^2} \leq 1$.

Let $a>0,b>0,c>0$ and $\sqrt{1987+a}+\sqrt{1987+b}=2\sqrt{1987+c}$. Prove that $\frac{1}{2} (a+b )\ge c $.

Consider on the coordinate plane all rectangles whose (i) vertices have integer coordinates; (ii) edges are parallel to coordinate axes; (iii) area is $2^k$, where $k = 0,1,2....$ Is it possible to color all points with integer coordinates in two colors so that no such rectangle has all its vertices of the same color?

$\triangle PQR$ is isosceles and right angled at $R$. Point $A$ is inside $\triangle PQR$, such that $PA=11, QA=7$, and $RA=6$. Legs $\overline{PR}$ and $\overline{QR}$ have length $s=\sqrt{a+b\sqrt{2}}$, where $a$ and $b$ are positive integers. What is $a+b$?

Let $A$ be a set of positive integers such that for any $x,y\in A$, $$x>y\implies x-y\ge\frac{xy}{25}.$$Find the maximal possible number of elements of the set $A$.

Let $ABC$ be an acute scalene triangle with circumcircle $\omega$. Let $P$ and $Q$ be interior points of the sides $AB$ and $AC$, respectively, such that $PQ$ is parallel to $BC$. Let $L$ be a point on $\omega$ such that $AL$ is parallel to $BC$. The segments $BQ$ and $CP$ intersect at $S$. The line $LS$ intersects $\omega$ at $K$. Prove that $\angle BKP = \angle CKQ$. Proposed by [i]Ervin Macić, Bosnia and Herzegovina[/i]

A square is partitioned in $n^2\geq 4$ rectanles using $2(n-1)$ lines,$n-1$ of which,are parallel to the one side of the square,$n-1$ are parallel to the other side.Prove that we can choose $2n$ rectangles of the partition,such that,for each two of them,we can place the one inside the other (possibly with rotation).

Let $X_1X_2X_3$ be a triangle with $X_1X_2 = 4, X_2X_3 = 5, X_3X_1 = 7,$ and centroid $G$. For all integers $n \ge 3$, define the set $S_n$ to be the set of $n^2$ ordered pairs $(i,j)$ such that $1\le i\le n$ and $1\le j\le n$. Then, for each integer $n\ge 3$, when given the points $X_1, X_2, \ldots , X_{n}$, randomly choose an element $(i,j)\in S_n$ and define $X_{n+1}$ to be the midpoint of $X_i$ and $X_j$. The value of \[ \sum_{i=0}^\infty \left(\mathbb{E}\left[X_{i+4}G^2\right]\left(\dfrac{3}{4}\right)^i\right) \] can be expressed in the form $p + q \ln 2 + r \ln 3$ for rational numbers $p, q, r$. Let $|p| + |q| + |r| = \dfrac mn$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$. Note: $\mathbb{E}(x)$ denotes the expected value of $x$. [i]Proposed by Yang Liu[/i]

Find all functions $f: (0, \infty) \to (0, \infty)$ such that \begin{align*} f(y(f(x))^3 + x) = x^3f(y) + f(x) \end{align*} for all $x, y>0$. [i]Proposed by Jason Prodromidis, Greece[/i]

Let $ABCD$ be a convex quadrilateral with $\angle DAB =\angle ABC =\angle BCD > 90^o$. The circle circumscribed around the triangle $ABC$ intersects the sides $AD$ and $CD$ at points $K$ and $L$, respectively, different from any vertex of the quadrilateral $ABCD$ . Segments $AL$ and $CK$ intersect at point $P$. Prove that $\angle ADB =\angle PDC$.

For which positive integers $n$ is there a permutation $(x_1,x_2,\ldots,x_n)$ of $1,2,\ldots,n$ such that all the differences $|x_k-k|$, $k = 1,2,\ldots,n$, are distinct?

Evaluate $\sum_{k=0}^{37}(-1)^k\binom{75}{2k}$.

Let $ x, y, z$ be real numbers such that $ x \plus{} y \plus{} z \geq xyz$. Find the smallest possible value of $ \frac {x^2 \plus{} y^2 \plus{} z^2}{xyz}$.