Suppose that $ ABCD$ is a cyclic quadrilateral. Let $ E \equal{} AC\cap BD$ and $ F \equal{} AB\cap CD$. Denote by $ H_{1}$ and $ H_{2}$ the orthocenters of triangles $ EAD$ and $ EBC$, respectively. Prove that the points $ F$, $ H_{1}$, $ H_{2}$ are collinear. Original formulation: Let $ ABC$ be a triangle. A circle passing through $ B$ and $ C$ intersects the sides $ AB$ and $ AC$ again at $ C'$ and $ B',$ respectively. Prove that $ BB'$, $CC'$ and $ HH'$ are concurrent, where $ H$ and $ H'$ are the orthocentres of triangles $ ABC$ and $ AB'C'$ respectively.

The inner angles of the pentagon inscribed in the circle are equal to each other. Prove that this pentagon is regular.

One day in a room there were several inhabitants of an island where only truth-tellers and liars live. Three of them made the following statements: [list] [*] There are no more than three of us here. We are all liars. [*] There are no more than four of us here. Not all of us are liars. [*] There are five of us here. At least three of us are liars. [/list] How many people are in the room and how many of them are liars?

Let $K$ and $M$ be points on the side $AB$ of a triangle $\triangle{ABC}$, and let $L$ and $N$ be points on the side $AC$. The point $K$ is between $M$ and $B$, and the point $L$ is between $N$ and $C$. If $\frac{BK}{KM}=\frac{CL}{LN}$, then prove that the orthocentres of the triangles $\triangle{ABC}$, $\triangle{AKL}$ and $\triangle{AMN}$ lie on one line.

Let $ABCD$ be a quadrilateral with $\angle BAD = \angle ABC = 90^{\circ}$, and suppose $AB=BC=1$, $AD=2$. The circumcircle of $ABC$ meets $\overline{AD}$ and $\overline{BD}$ at point $E$ and $F$, respectively. If lines $AF$ and $CD$ meet at $K$, compute $EK$.

Let $n$ be a positive integer. For $i$ and $j$ in $\{1,2, \ldots, n\}$, let $s(i, j)$ be the number of pairs $(a, b)$ of nonnegative integers satisfying $a i+b j=n$. Let $S$ be the $n$-by-n matrix whose $(i, j)$-entry is $s(i, j)$. For example, when $n=5$, we have $S=\left[\begin{array}{lllll}6 & 3 & 2 & 2 & 2 \\ 3 & 0 & 1 & 0 & 1 \\ 2 & 1 & 0 & 0 & 1 \\ 2 & 0 & 0 & 0 & 1 \\ 2 & 1 & 1 & 1 & 2\end{array}\right]$. Compute the determinant of $S$.

A committee of 5 is to be chosen from a group of 9 people. How many ways can it be chosen, if Bill and Karl must serve together or not at all, and Alice and Jane refuse to serve with each other?

Solve the equation $ \sin^6x \plus{} \cos^6x \equal{} \frac{1}{4}$.

Determine all functions $f:\mathbb{R}_+\to\mathbb{R}_+$ which satisfy \[f\left(\frac{y}{f(x)}\right)+x=f(xy)+f(f(x)),\]for any positive real numbers $x$ and $y$. [i]Proposed by Pavel Ciurea[/i]

Let $\sigma_k(n)$ be the sum of the $k^{th}$ powers of the divisors of $n$. For all $k \ge 2$ and all $n \ge 3$, we have that $$\frac{\sigma_k(n)}{n^{k+2}} (2020n + 2019)^2 > m.$$ Find the largest possible value of $m$.

Show that $\log_{10} 2$ is irrational.

Let $n \ge 2$ be a positive integer, and write in a digit form \[\frac{1}{n}=0.a_1a_2\dots.\] Suppose that $n = a_1 + a_2 + \cdots$. Determine all possible values of $n$.

We consider all $14$-digit positive integers, divisible by $18$, whose digits are exclusively $ 1$ and $2$, but there are no consecutive digits $2$. How many of these numbers are there?

In the picture below, the circles are tangent to each other and to the edges of the rectangle. The larger circle's radius equals 1. Determine the area of the rectangle. [img]https://i.imgur.com/g3GUg4Z.png[/img]

Without overlapping, hexagonal tiles are placed inside an isosceles right triangle of area $1$ whose hypotenuse is horizontal. The tiles are similar to the figure below, but are not necessarily all the same size.[asy] unitsize(.85cm); draw((0,0)--(1,0)--(1,1)--(2,2)--(-1,2)--(0,1)--(0,0),linewidth(1)); draw((0,2)--(0,1)--(1,1)--(1,2),dashed); label("\footnotesize $a$",(0.5,0),S); label("\footnotesize $a$",(0,0.5),W); label("\footnotesize $a$",(1,0.5),E); label("\footnotesize $a$",(0,1.5),E); label("\footnotesize $a$",(1,1.5),W); label("\footnotesize $a$",(-0.5,2),N); label("\footnotesize $a$",(0.5,2),N); label("\footnotesize $a$",(1.5,2),N); [/asy] The longest side of each tile is parallel to the hypotenuse of the triangle, and the horizontal side of length $a$ of each tile lies between this longest side of the tile and the hypotenuse of the triangle. Furthermore, if the longest side of a tile is farther from the hypotenuse than the longest side of another tile, then the size of the first tile is larger or equal to the size of the second tile. Find the smallest value of $\lambda$ such that every such configuration of tiles has a total area less than $\lambda$.

Let $H$ be the orthocenter of acute-angled $\vartriangle ABC$, and $X, Y$ points on the ray $AB, AC$. ($B$ lies between $X, A$, and $C$ lies between $Y, A$.) Lines $HX, HY$ intersect $BC$ at $D, E$ respectively. Let the line through $D$ parallel to $AC$ intersect $XY$ at $Z$. Prove that $\angle XHY = 90^o$ if and only if $ZE \parallel AB$.

Prove that the systems of hyperbolas \begin{align*}x^2-y^2&=a\\xy&=b\end{align*}are orthogonal.

Prove that for a nowhere dense, compact set $K\subset \mathbb{R}^2$ the following are equivalent: (i) $K=\bigcup_{i=1}^{\infty}K_n$ where $K_n$ is a compact set with connected complement for all $n$. (ii) $K$ does not have a nonempty closed subset $S\subseteq K$ such that any neighborhood of any point in $S$ contains a connected component of $\mathbb{R}^2 \setminus S$.

Let $ABC$ be a triangle with $\angle{C} = 90^{\circ}$, and let $H$ be the foot of the altitude from $C$. A point $D$ is chosen inside the triangle $CBH$ so that $CH$ bisects $AD$. Let $P$ be the intersection point of the lines $BD$ and $CH$. Let $\omega$ be the semicircle with diameter $BD$ that meets the segment $CB$ at an interior point. A line through $P$ is tangent to $\omega$ at $Q$. Prove that the lines $CQ$ and $AD$ meet on $\omega$.

Let $ 1<a_1<a_2< \ldots <a_n<x$ be positive integers such that $ \sum_{i\equal{}1}^n 1/a_i \leq 1$. Let $ y$ denote the number of positive integers smaller that $ x$ not divisible by any of the $ a_i$. Prove that \[ y > \frac{cx}{\log x}\] with a suitable positive constant $ c$ (independent of $ x$ and the numbers $ a_i$). [i]I. Z. Ruzsa[/i]

Prove that the sum of the squares of two consecutive positive integers cannot be equal to a sum of the fourth powers of two consecutive positive integers.

Prove that for the sides $a, b, c$, the angles $A, B, C$ and the area $S$ of the triangle holds $$\cot A+ \cot B + \cot C = \frac{a^2+b^2+c^2}{4S}.$$

Let $ABCD$ be a rectangle of centre $O$, such that $\angle DAC=60^{\circ}$. The angle bisector of $\angle DAC$ meets $DC$ at $S$. Lines $OS$ and $AD$ meet at $L$, and lines $BL$ and $AC$ meet at $M$. Prove that lines $SM$ and $CL$ are parallel.

Suppose that $ f(x\plus{}3)\equal{}3x^2\plus{}7x\plus{}4$ and $ f(x)\equal{}ax^2\plus{}bx\plus{}c$. What is $ a\plus{}b\plus{}c$? $ \textbf{(A)}\minus{}\!1 \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 3$

The expression below has six empty boxes. Each box is to be filled in with a number from $1$ to $6$, where all six numbers are used exactly once, and then the expression is evaluated. What is the maximum possible final result that can be achieved? $$\dfrac{\frac{\square}{\square}+\frac{\square}{\square}}{\frac{\square}{\square}}$$