Let $ABCD$ be a cyclic quadrilateral with center $O$ such that $BD$ bisects $AC.$ Suppose that the angle bisector of $\angle ABC$ intersects the angle bisector of $\angle ADC$ at a single point $X$ different than $B$ and $D.$ Prove that the line passing through the circumcenters of triangles $XAC$ and $XBD$ bisects the segment $OX.$ [i]Proposed by Viktor Ahmeti and Leart Ajvazaj, Kosovo[/i]

The $100$ vertices of a prism, whose base is a $50$-gon, are labeled with numbers $1, 2, 3, \ldots, 100$ in any order. Prove that there are two vertices, which are connected by an edge of the prism, with labels differing by not more than $48$. Note: In all the triangles the three vertices do not lie on a straight line.

Every cell in a $5\times5$ grid of paper is to be painted either red or white with equal probability. An edge of the paper is said to have a "tree" if the set of cells depicted in the diagram below are all painted red when the paper is rotated so that the edge lies at the bottom. Given that at least one edge of the paper has a tree, what is the expected number of edges that have a tree? [center][img]https://cdn.artofproblemsolving.com/attachments/1/2/f81d8da53d7bc6819fc1dfe4acb9567d545856.png[/img][/center]

Find the sum of all integers between $-\sqrt {1442}$ and $\sqrt{2020}$.

Is it possible for the least common multiple of five consecutive positive integers to be a perfect square?

Let $a$, $b$, $c$, $d$ be real numbers such that $b-d \ge 5$ and all zeros $x_1, x_2, x_3,$ and $x_4$ of the polynomial $P(x)=x^4+ax^3+bx^2+cx+d$ are real. Find the smallest value the product $(x_1^2+1)(x_2^2+1)(x_3^2+1)(x_4^2+1)$ can take.

Find all real $ a$, such that there exist a function $ f: \mathbb{R}\rightarrow\mathbb{R}$ satisfying the following inequality: \[ x\plus{}af(y)\leq y\plus{}f(f(x)) \] for all $ x,y\in\mathbb{R}$

On a $999\times 999$ board a [i]limp rook[/i] can move in the following way: From any square it can move to any of its adjacent squares, i.e. a square having a common side with it, and every move must be a turn, i.e. the directions of any two consecutive moves must be perpendicular. A [i]non-intersecting route[/i] of the limp rook consists of a sequence of pairwise different squares that the limp rook can visit in that order by an admissible sequence of moves. Such a non-intersecting route is called [i]cyclic[/i], if the limp rook can, after reaching the last square of the route, move directly to the first square of the route and start over. How many squares does the longest possible cyclic, non-intersecting route of a limp rook visit? [i]Proposed by Nikolay Beluhov, Bulgaria[/i]

Let $BB_1$ and $CC_1$ be the altitudes of an acute-angled triangle $ABC$, which intersect its angle bisector $AL$ at two different points $P$ and $Q$, respectively. Denote by $F$ such a point that $PF\parallel AB$ and $QF\parallel AC$, and by $T$ the intersection point of the tangents drawn at points $B$ and $C$ to the circumscribed circle of the triangle $ABC$. Prove that the points $A, F$ and $T$ lie on the same line.

[b]2.[/b] Place 32 white and 32 black chessmen on the chessboard. Two chessmen of different colours will be said to form a "related pair" if they are placed either in the same row or in the same column. Determine the maximum and minimum number of related pairs (over all possible arrangements of the 64 chessmen considered. [b](C. 2)[/b]

Find the positive integer $n$ such that \[ \underbrace{f(f(\cdots f}_{2013 \ f\text{'s}}(n)\cdots ))=2014^2+1 \] where $f(n)$ denotes the $n$th positive integer which is not a perfect square. [i]Proposed by David Stoner[/i]

Determine all natural numbers $n$ for which there is a partition of $\{1,2,...,3n\}$ in $n$ pairwise disjoint subsets of the form $\{a,b,c\}$, such that numbers $b-a$ and $c-b$ are different numbers from the set $\{n-1, n, n+1\}$.

Consider a $2021$-by-$2021$ board of unit squares. For some integer $k$, we say the board is tiled by $k$-by-$k$ squares if it is completely covered by (possibly overlapping) $k$-by-$k$ squares with their corners on the corners of the unit squares. What is the largest integer k such that the minimum number of $k$-by-$k$ squares needed to tile the $2021$-by-$2021$ board is exactly equal to $100$?

In triangle $ABC$, $|AB| <|BC|$ holds. Point $I$ is the center of the circle inscribed in that triangle. Let $M$ be the midpoint of the side $AC$, and $N$ be the midpoint of the arc $AC$ of the circumcircle of that triangle containing point $B$. Prove that $\angle IMA = \angle INB$.

Prove that for every $n \in N$ there exists exactly one sequence of $2n + 1$ consecutive numbers, such that the sum of the squares of the first $n+1$ numbers is equal to the sum of the squares of the last $n$ numbers. Also express the smallest number of that sequence in terms of $n$.

Danielle divides a $30 \times30$ board into $100$ regions that are $3 \times 3$ squares squares each and then paint some squares black and the rest white. Then to each region assigns it the color that has the most squares painted with that color. a) If there are more black regions than white, what is the minimum number $N$ of cells that Danielle can paint black? b) In how many ways can Danielle paint the board if there are more black regions than white and she uses the minimum number $N$ of black squares?

Find all positive integers $m, n$ such that $\frac{1}{m} + \frac{1}{n} - \frac{1}{mn} =\frac{2}{5}$.

What is the value of the following expression? $$\frac{100^2-7^2}{70^2-11^2} \cdot \frac{(70-11)(70+11)}{(100-7)(100+7)}$$ $\textbf{(A) } 1 \qquad \textbf{(B) } \frac{9951}{9950} \qquad \textbf{(C) } \frac{4780}{4779} \qquad \textbf{(D) } \frac{108}{107} \qquad \textbf{(E) } \frac{81}{80} $

For any $a>0$,set $\mathcal{S}(a)=\{\lfloor{na}\rfloor|n\in \mathbb{N}\}$. Show that there are no three positive reals $a,b,c$ such that \[ \mathcal{S}(a)\cap \mathcal{S}(b)=\mathcal{S}(b)\cap \mathcal{S}(c)=\mathcal{S}(c)\cap \mathcal{S}(a)=\emptyset \] \[ \mathcal{S}(a)\cup \mathcal{S}(b)\cup \mathcal{S}(c)=\mathbb{N} \]

Which is largest positive integer n satisfying the following inequality: $n^{2007} > (2007)^n$.

On a plane, two equilateral triangles (of side length $1$) share a side, and a circle is drawn with the common side as a diameter. Find the area of the set of all points that lie inside exactly one of these shapes. [i]Proposed by Howard Halim[/i]

We define a sequence with $a_1=850$ and $$a_{n+1}=\frac{a_n^2}{a_n-1}$$ for $n\geq 1$. Find all values of $n$ for which $\lfloor a_n\rfloor =2024$.

Determine the locus of points, from which the tangent segments to two given circles are equal.

a) Prove that there is not an infinte sequence $(x_n)$, $n=1,2,...$ of positive real numbers satisfying the relation $x_{n+2}=\sqrt{x_{n+1}}-\sqrt{x_{n}}$, $\forall n \in N$ (*) b) Do there exist sequences satisfying (*) and containing arbitrary many terms? I.Voronovich

A village is constructed in the form of a square, consisting of $9$ blocks , each of side length $\ell$, in a $3 \times 3$ formation . Each block is bounded by a bitumen road . If we commence at a corner of the village, what is the smallest distance we must travel along bitumen roads , if we are to pass along each section of bitumen road at least once and finish at the same corner? (Muscovite folklore)