In trapezoid $ABCD$, the bisectors of angles $A$ and $D$ intersect at point $E$ lying on the side of $BC$. These bisectors divide the trapezoid into three triangles into which the circles are inscribed. One of these circles touches the base $AB$ at the point $K$, and two others touch the bisector $DE$ at points $M$ and $N$. Prove that $BK = MN$.

If $a$ and $b$ are coprime positive integers and $n$ an integer, prove that the greatest common divisor of $a^2+b^2-nab$ and $a+b$ divides $n+2$.

[i]Alice[/i], who works for the [i]Graph County Electric Works[/i], is commissioned to wire the newly erected utility poles in $k$ days. Each day she either chooses a pole and runs wires from it to as many poles as she wishes, or chooses at most $17$ pairs of poles and runs wires between each pair. [i]Bob[/i], who works for the [i]Graph County Paint Works[/i], claims that, no matter how many poles there are and how [i]Alice[/i] connects them, all the poles can be painted using not more than $2009$ colors in such a way that no pair of poles connected by a wire is the same color. Determine the greatest value of $k$ for which [i]Bob[/i]'s claim is valid.

A circular disc is divided into $12$ equal sectors and one of $6$ different colours is used to colour each sector. No two adjacent sectors can have the same colour. Find the number of such distinct colorings possible.

Find all nonnegative integer solutions $(x,y,z)$ of the equation $\frac{1}{x+2}+\frac{1}{y+2}=\frac{1}{2} +\frac{1}{z+2}$

Given is an acute-angled triangle $ABC$ with angles $\alpha$, $\beta$ and $\gamma$. On side $AB$ lies a point $P$ such that the line connecting the feet of the perpendiculars from $P$ on $AC$ and $BC$ is parallel to $AB$. Express the ratio $\frac{AP}{BP}$ in terms of $\alpha$ and $\beta$.

Let $a_1,a_2,a_3, \ldots$ be a sequence of positive integers and let $b_1,b_2,b_3,\ldots$ be the sequence of real numbers given by $$b_n = \dfrac{a_1a_2\cdots a_n}{a_1+a_2+\cdots + a_n},\ \mbox{for}\ n\geq 1$$ Show that, if there exists at least one term among every million consecutive terms of the sequence $b_1,b_2,b_3,\ldots$ that is an integer, then there exists some $k$ such that $b_k > 2021^{2021}$.

Show that $(k^3)!$ is divisible by $(k!)^{k^2+k+1}$.

Calculate \[\frac{\int_{0}^{\pi}e^{-x}\sin^{n}x\ dx}{\int_{0}^{\pi}e^{x}\sin^{n}x \ dx}\ (n=1,\ 2,\ \cdots). \]

Find the least positive integer $m$ such that $105| 9^{(p^2)} -29^p +m$ for all prime numbers $p > 3$.

Segments connect vertices $A, B, C$ of $\vartriangle ABC$ with respective points $A_1, B_1, C_1$ on the opposite sides of the triangle. Prove that the midpoints of segments $AA_1, BB_1, CC_1$ do not belong to one straight line.

Find the smallest possible number of divisors a positive integer $n$ may have, which satisfies the following conditions: 1. $24 \mid n+1$; 2. The sum of the squares of all divisors of $n$ is divisible by $48$ ($1$ and $n$ are included).

In the tetrahedron $ABCD$ , the dihedral angles at the $BC, CD$, and $DA$ edges are equal to $\alpha$, and for the remaining edges equal to $\beta$. Find the ratio $AB / CD$.

In a chess tournament $ 2n\plus{}3$ players take part. Every two play exactly one match. The schedule is such that no two matches are played at the same time, and each player, after taking part in a match, is free in at least $ n$ next (consecutive) matches. Prove that one of the players who play in the opening match will also play in the closing match.

Find the largest $c > 0$ such that for all $n \ge 1$ and $a_1,\dots,a_n, b_1,\dots, b_n > 0$ we have $$\sum_{j=1}^n a_j^4 \ge c\sum_{k = 1}^n \frac{\left(\sum_{j=1}^k a_jb_{k+1-j}\right)^4}{\left(\sum_{j=1}^k b_j^2j!\right)^2}$$ [i]Proposed by Grant Yu[/i]

The two tangents to the incircle of a right-angled triangle $ABC$ that are perpendicular to the hypotenuse $AB$ intersect it at the points $P$ and $Q$. Find $\angle PCQ$. (M Evdokimov,)

In the city of Flensburg there is a single, infinitely long, street with housesnumbered $2, 3, \ldots$. The police in Flensburg is trying to catch a thief who every night moves from the house where she is currently hiding to one of its neighbouring houses. To taunt the local law enforcement the thief reveals every morning the highest prime divisor of the number of the house she has moved to. Every Sunday afternoon the police searches a single house, and they catch the thief if they search the house she is currently occupying. Does the police have a strategy to catch the thief in finite time?

Let each of $m$ distinct points on the positive part of the $x$-axis be joined to $n$ distinct points on the positive part of the $y$-axis. Obtain a formula for the number of intersection points of these segments, assuming that no three of the segments are concurrent.

Let $f(x, y) = \left\lfloor \frac{5x}{2y} \right\rfloor + \left\lceil \frac{5y}{2x} \right\rceil$. Suppose $x, y$ are chosen independently uniformly at random from the interval $(0, 1]$. Let $p$ be the probability that $f(x, y) < 6$. If $p$ can be expressed in the form $m/n$ for relatively prime positive integers $m$ and $n$, compute $m + n$. (Note: $\lfloor x\rfloor $ is defined as the greatest integer less than or equal to $x$ and $\lceil x \rceil$ is defined as the least integer greater than or equal to$ x$.)

Let $n$ be a positive integer relatively prime to $6$. We paint the vertices of a regular $n$-gon with three colours so that there is an odd number of vertices of each colour. Show that there exists an isosceles triangle whose three vertices are of different colours.

Margie's winning art design is shown. The smallest circle has radius 2 inches, with each successive circle's radius increasing by 2 inches. Approximately what percent of the design is black? [asy] real d=320; pair O=origin; pair P=O+8*dir(d); pair A0 = origin; pair A1 = O+1*dir(d); pair A2 = O+2*dir(d); pair A3 = O+3*dir(d); pair A4 = O+4*dir(d); pair A5 = O+5*dir(d); filldraw(Circle(A0, 6), white, black); filldraw(circle(A1, 5), black, black); filldraw(circle(A2, 4), white, black); filldraw(circle(A3, 3), black, black); filldraw(circle(A4, 2), white, black); filldraw(circle(A5, 1), black, black); [/asy] $ \textbf{(A)}\ 42\qquad \textbf{(B)}\ 44\qquad \textbf{(C)}\ 45\qquad \textbf{(D)}\ 46\qquad \textbf{(E)}\ 48\qquad$

Define the functions $f, F : \mathbb N \to \mathbb N$, by \[f(n)=\left[ \frac{3-\sqrt 5}{2} n \right] , F(k) =\min \{n \in \mathbb N|f^k(n) > 0 \},\] where $f^k = f \circ \cdots \circ f$ is $f$ iterated $n$ times. Prove that $F(k + 2) = 3F(k + 1) - F(k)$ for all $k \in \mathbb N.$

Given is an isosceles triangle $ABC$ with $AB = 2$ and $AC = BC = 3$. We consider squares where $A, B$ and $C$ lie on the sides of the square (so not on the extension of such a side). Determine the maximum and minimum value of the area of such a square. Justify the answer.

There are several dominoes on a board such that each domino occupies two adjacent cells and none of the dominoes are adjacent by side or vertex. The bottom left and top right cells of the board are free. A token starts at the bottom left cell and can move to a cell adjacent by side: one step to the right or upwards at each turn. Is it always possible to move from the bottom left to the top right cell without passing through dominoes if the size of the board is a) $100 \times 101$ cells and b) $100 \times 100$ cells? [i]Nikolay Chernyatiev[/i]

For all prime $p>3$ with reminder $1$ or $3$ modulo $8$ prove that the number triples $(a,b,c), p=a^2+bc, 0<b<c<\sqrt{p}$ is odd. [i]Proposed by Navid Safaie[/i]