A $9\times 1$ rectangle is divided into unit squares. A broken line, from the lower left to the upper right corner, goes through all $20$ vertices of the unit squares and consists of $19$ line segments. How many such lines are there?

Find all real values $x$ that satisfy the following equation: $$\frac{\sin 3x cos \left(\frac{\pi}{3}-4x \right)+ 1}{\sin \left(\frac{\pi}{3}-7x \right) - cos\left(\frac{\pi}{6}+x \right)+m}= 0$$ where $m$ is a given real number.

Parallelogram $ABCD$ has $AB=CD=6$ and $BC=AD=10$, where $\angle ABC$ is obtuse. The circumcircle of $\triangle ABD$ intersects $BC$ at $E$ such that $CE=4$. Compute $BD$.

Find all real-valued functions $f$ defined on pairs of real numbers, having the following property: for all real numbers $a, b, c$, the median of $f(a,b), f(b,c), f(c,a)$ equals the median of $a, b, c$. (The [i]median[/i] of three real numbers, not necessarily distinct, is the number that is in the middle when the three numbers are arranged in nondecreasing order.)

Let $ ABC$ be a triangle where $ \angle ACB\equal{}90^{\circ}$. Let $ D$ be the midpoint of $ BC$ and let $ E$, and $ F$ be points on $ AC$ such that $ CF\equal{}FE\equal{}EA$. The altitude from $ C$ to the hypotenuse $ AB$ is $ CG$, and the circumcentre of triangle $ AEG$ is $ H$. Prove that the triangles $ ABC$ and $ HDF$ are similar.

The quadrilateral $ ABCD$ inscribed in a circle wich has diameter $ BD$. Let $ A',B'$ are symmetric to $ A,B$ with respect to the line $ BD$ and $ AC$ respectively. If $ A'C \cap BD \equal{} P$ and $ AC\cap B'D \equal{} Q$ then prove that $ PQ \perp AC$

Tom has a collection of $13$ snakes, $4$ of which are purple and $5$ of which are happy. He observes that $\bullet$ all of his happy snakes can add $\bullet$ none of his purple snakes can subtract, and $\bullet$ all of his snakes that can’t subtract also can’t add Which of these conclusions can be drawn about Tom’s snakes? $\textbf{(A)}$ Purple snakes can add. $\textbf{(B)}$ Purple snakes are happy. $\textbf{(C)}$ Snakes that can add are purple. $\textbf{(D)}$ Happy snakes are not purple. $\textbf{(E)}$ Happy snakes can't subtract.

Let $ ABCD$ be a trapezoid with parallel sides $ AB > CD$. Points $ K$ and $ L$ lie on the line segments $ AB$ and $ CD$, respectively, so that $AK/KB=DL/LC$. Suppose that there are points $ P$ and $ Q$ on the line segment $ KL$ satisfying \[\angle{APB} \equal{} \angle{BCD}\qquad\text{and}\qquad \angle{CQD} \equal{} \angle{ABC}.\] Prove that the points $ P$, $ Q$, $ B$ and $ C$ are concyclic. [i]Proposed by Vyacheslev Yasinskiy, Ukraine[/i]

Somewhere in the universe, $n$ students are taking a $10$-question math competition. Their collective performance is called [i]laughable[/i] if, for some pair of questions, there exist $57$ students such that either all of them answered both questions correctly or none of them answered both questions correctly. Compute the smallest $n$ such that the performance is necessarily laughable.

The circle $C_1$ and the circle $C_2$ passing through the center of $C_1$ intersect each other at $A$ and $B$. The line tangent to $C_2$ at $B$ meets $C_1$ at $B$ and $D$. If the radius of $C_1$ is $\sqrt 3$ and the radius of $C_2$ is $2$, find $\dfrac{|AB|}{|BD|}$. $ \textbf{(A)}\ \dfrac 12 \qquad\textbf{(B)}\ \dfrac {\sqrt 3}2 \qquad\textbf{(C)}\ \dfrac {2\sqrt 3}2 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ \dfrac {\sqrt 5}2 $

Given triangle $ABC$. Find the locus of points $M$ such that there is a rotation with center at $M$ that takes $C$ to a certain point on side $AB$.

From a square board $11$ squares long and $11$ squares wide, the central square is removed. Prove that the remaining $120$ squares cannot be covered by $15$ strips each $8$ units long and one unit wide.

Prove that there exist subfields of $\mathbb R$ that are a) non-measurable and b) of measure zero and continuum cardinality. (translated by Miklós Maróti)

Given that $x$, $y$ are positive integers with $x(x+1)|y(y+1)$, but neither $x$ nor $x+1$ divides either of $y$ or $y+1$, and $x^2 + y^2$ as small as possible, find $x^2 + y^2$.

Among the $n$ inhabitants of an island, every two are either friends or enemies. Some day, the chief of the island orders that each inhabitant (including himself) makes and wears a necklace consisting of marbles, in such a way that the necklaces of two friends have at least one marble of the same type and that the necklaces of two enemies differ at all marbles. (A necklace may also be marbleless). Show that the chief’s order can be achieved by using $\left\lfloor\frac{n^2}4\right\rfloor$ different types of stones, but not necessarily by using fewer types.

Given positive integers $x$ and $y$ such that $xy^2$ is a perfect cube, prove that $x^2y$ is also a perfect cube.

If $x$, $y$, $z$ are positive numbers satisfying \[x+\frac{y}{z}=y+\frac{z}{x}=z+\frac{x}{y}=2.\] Find all the possible values of $x+y+z$.

Let $ABC$ be a triangle such that $|BC|=7$ and $|AB|=9$. If $m(\widehat{ABC}) = 2m(\widehat{BCA})$, then what is the area of the triangle? $ \textbf{(A)}\ 14\sqrt 5 \qquad\textbf{(B)}\ 30 \qquad\textbf{(C)}\ 10\sqrt 6 \qquad\textbf{(D)}\ 20 \sqrt 2 \qquad\textbf{(E)}\ 12 \sqrt 3 $

Let be the set $ G=\{ (u,v)\in \mathbb{C}^2| u\neq 0 \} $ and a function $ \varphi :\mathbb{C}\setminus\{ 0\}\longrightarrow\mathbb{C}\setminus\{ 0\} $ having the property that the operation $ *:G^2\longrightarrow G $ defined as $$ (a,b)*(c,d)=(ac,bc+d\varphi (a)) $$ is associative. [b]a)[/b] Show that $ (G,*) $ is a group. [b]b)[/b] Describe $ \varphi , $ knowing that $(G,*) $ is a commutative group. [i]Marius Perianu[/i]

You are given $20$ weights such that any object of integer weight $m$, $1 \le m \le1997$, can be balanced by placing it on one pan of a balance and a subset of the weights on the other pan. What is the minimal value of largest of the $20$ weights if the weights are (a) all integers; (b) not necessarily integers? (M Rasin)

[i](8 pts)[/i] Let $n$ be a positive integer, and let $a, b, c$ be real numbers. (a) [i](2 pts)[/i] Given that $a\cos x+b\cos 2x +c\cos 3x \geq -1$ for all reals $x$, find, with proof, the maximum possible value of $a+b+c$. (b) [i](6 pts)[/i] Let $f$ be a degree $n$ polynomial with real coefficients. Suppose that $|f(z)| \leq \left|f(z)+\frac{2}{z}\right|$ for all complex $z$ lying on the unit circle. Find, with proof, the maximum possible value of $f(1)$.

a) Determine if there exists a convex hexagon $ABCDEF$ with $$\angle ABD + \angle AED > 180^{\circ},$$ $$\angle BCE + \angle BFE > 180^{\circ},$$ $$\angle CDF + \angle CAF > 180^{\circ}.$$ b) The same question, with additional condition, that diagonals $AD, BE,$ and $CF$ are concurrent.

Find all polynomials $ p(x)$ satisfying the equation: $ (x\minus{}16)p(2x)\equal{}16(x\minus{}1)p(x)$ for all $ x$.

For some complex number $\omega$ with $|\omega| = 2016$, there is some real $\lambda>1$ such that $\omega, \omega^{2},$ and $\lambda \omega$ form an equilateral triangle in the complex plane. Then, $\lambda$ can be written in the form $\tfrac{a + \sqrt{b}}{c}$, where $a,b,$ and $c$ are positive integers and $b$ is squarefree. Compute $\sqrt{a+b+c}$.

Evan is a $n$-dimensional being that lives in a house formed by the points of $\mathbb{Z}_{\geq 0}^n$. His room is the set of points in which coordinates are all less than or equal to $2021$. Evan's room has been infested with bees, so he decides to flush them out through $\textit{captures}$. A $\textit{capture}$ can be performed by eliminating a bee from point $ (a_1, a_2, \ldots, a_n) $ and replacing it with $ n $ bees, one in each of the points: $$ (a_1 + 1, a_2 , \ldots, a_n), (a_1, a_2 + 1, \ldots, a_n), \ldots, (a_1, a_2, \ldots, a_n + 1) $$ However, two bees can never occupy the same point in the house. Determine, for every $ n $, the greatest value $ A (n) $ of bees such that, for some initial arrangement of these bees in Evan's room, he is able to accomplish his goal with a finite amount of $\textit{captures}$.