A mouse is sitting in a toy car on a negligibly small turntable. The car cannot turn on its own, but the mouse can control when the car is launched and when the car stops (the car has brakes). When the mouse chooses to launch, the car will immediately leave the turntable on a straight trajectory at $1$ meter per second. Suddenly someone turns on the turntable; it spins at $30$ rpm. Consider the set $S$ of points the mouse can reach in his car within $1$ second after the turntable is set in motion. What is the area of $S$, in square meters?

Let $M$ be the midpoint of $\overline{AB}$ in regular tetrahedron $ABCD$. What is $\cos({\angle CMD})$? $\textbf{(A)} ~\frac{1}{4} \qquad\textbf{(B)} ~\frac{1}{3} \qquad\textbf{(C)} ~\frac{2}{5} \qquad\textbf{(D)} ~\frac{1}{2} \qquad\textbf{(E)} ~\frac{\sqrt{3}}{2} $

Given positive integers $a,b,c,d$ such that $a\mid c^d$ and $b\mid d^c$. Prove that \[ ab\mid (cd)^{max(a,b)} \]

Given a triangle $ABC$, on the side $BC$ which marked the point $E$ such that $BE \ge CE$. Construct on the sides $AB$ and $AC$ the points $D$ and $F$, respectively, such that $\angle DEF = 90 {} ^ \circ$ and the segment $BF$ is bisected by the segment $DE $. (Black Maxim)

The integer $n$ is the smallest positive multiple of 15 such that every digit of $n$ is either 8 or 0. Compute $\frac{n}{15}$.

A graph $G$ has $n + k$ vertices. Let $A$ be a subset of $n$ vertices of the graph $G$, and $B$ be a subset of other $k$ vertices. Each vertex of $A$ is joined to at least $k - p$ vertices of $B$. Prove that if $np < k$ then there is a vertex in $B$ that can be joined to all vertices of $A$.

A sphere $S$ divides every edge of a convex polyhedron $P$ into three equal parts. Show that there exists a sphere tangent to all the edges of $P$.

Compute $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}.$

For a positive integer $n$, let $A(n)$ be the equal to the remainder when $n$ is divided by $11$ and let $T(n)=A(1)+A(2)+A(3)+ \dots + A(n)$. Find the value of $$A(T(2021))$$

Vero had an isosceles triangle made of paper. Using scissors, he divided it into three smaller triangles and painted them blue, red and green. Having done so, he observed that: $\bullet$ with the blue triangle and the red triangle an isosceles triangle can be formed, $\bullet$ with the blue triangle and the green triangle an isosceles triangle can be formed, $\bullet$ with the red triangle and the green triangle an isosceles triangle can be formed. Show what Vero's triangle looked like and how he might have made the cuts to make this situation be possible.

Let $Z=\{ z_1,\ldots, z_{n-1}\}$, $n\ge 2$, be a set of different complex numbers such that $Z$ contains the conjugate of any its element. a) Show that there exists a constant $C$, depending on $Z$, such that for any $\varepsilon\in (0,1)$ there exists an algebraic integer $x_0$ of degree $n$, whose algebraic conjugates $x_1, x_2, \ldots, x_{n-1}$ satisfy $|x_1-z_1|\le \varepsilon, \ldots, |x_{n-1}-z_{n-1}|\le \varepsilon$ and $|x_0|\le \frac{C}{\varepsilon}$. b) Show that there exists a set $Z=\{ z_1, \ldots, z_{n-1}\}$ and a positive number $c_n$ such that for any algebraic integer $x_0$ of degree $n$, whose algebraic conjugates satisfy $|x_1-z_1|\le \varepsilon,\ldots, |x_{n-1}-z_{n-1}|\le \varepsilon$, it also holds that $|x_0|>\frac{c_n}{\varepsilon}$. (translated by L. Erdős)

The entries in a $ 3\times3$ array include all the digits from 1 through 9, arranged so that the entries in every row and column are in increasing order. How many such arrays are there? $ \textbf{(A)}\ 18\qquad\textbf{(B)}\ 24\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 42\qquad\textbf{(E)}\ 60$

An ordered triple $(p, q, r)$ of prime numbers is called [i]parcera[/i] if $p$ divides $q^2-4$, $q$ divides $r^2-4$ and $r$ divides $p^2-4$. Find all parcera triples.

A function $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfies the conditions \[\begin{cases}f(x+24)\le f(x)+24\\ f(x+77)\ge f(x)+77\end{cases}\quad\text{for all}\ x\in\mathbb{R}\] Prove that $f(x+1)=f(x)+1$ for all real $x$.

In tetrahedron $SABC$, $\angle ASB=\frac{\pi}{2}, \angle ASC=\alpha(0<\alpha<\frac{\pi}{2}), \angle BSC=\beta(0<\beta<\frac{\pi}{2})$. Let $\theta=A-SC-B$, prove that $\theta=-\arccos(\cot\alpha\cdot\cot\beta)$.

Determine the largest possible number of groups one can compose from the integers $1,2,3,..., 19,20$, so that the product of the numbers in each group is a perfect square. (The group may contain exactly one number, in that case the product equals this number, each number must be in exactly one group.) (E. Barabanov, I. Voronovich)

$f$ is a function on the set of complex numbers such that $f(z)=1/(z*)$, where $z*$ is the complex conjugate of $z$. $S$ is the set of complex numbers $z$ such that the real part of $f(z)$ lies between $1/2020$ and $1/2018$. If $S$ is treated as a subset of the complex plane, the area of $S$ can be expressed as $m× \pi$ where $m$ is an integer. What is the value of $m$?

Let $a,b$ be two integers and suppose that $n$ is a positive integer for which the set $\mathbb{Z} \backslash \{ax^n + by^n \mid x,y \in \mathbb{Z}\}$ is finite. Prove that $n=1$.

A $k$-set is a set with exactly $k$ elements. For a $6$-set $A$ and any collection $\mathcal{F}$ of $4$-sets, we say that $A$ is [i]$\mathcal{F}$-good[/i] if there are exactly three elements $B_1, B_2, B_3$ in $\mathcal{F}$ that are subsets of $A$, and they furthermore satisfy $$(A \backslash B_1) \cup (A \backslash B_2) \cup (A \backslash B_3) = A.$$ Find all $n \geq 6$ so that there exists a collection $\mathcal{F}$ of $4$-subsets of $\{1, 2, \ldots , n\}$ such that every $6$-set $A \subseteq \{1, 2, \ldots , n\}$ is $\mathcal{F}$-good. [i] Proposed by usjl[/i]

Find all functions $f$ which are defined on all non-negative real numbers, take nonnegative real values only, and satisfy the condition $x \cdot f(y) + y\cdot f(x) = f(x) \cdot f(y) \cdot (f(x) + f(y))$ for all non-negative real numbers $x, y$.

We consider the non-zero natural numbers $(m, n)$ such that the numbers $$\frac{m^2 + 2n}{n^2 - 2m} \,\,\,\, and \,\,\, \frac{n^2 + 2m}{m^2-2n}$$ are integers. a) Show that $|m - n| \le 2$: b) Find all the pairs $(m, n)$ with the property from hypothesis $a$.

We shall call a positive integer [i]ascending [/i] if its digits read from left to right they are in strictly increasing order. For example, $458$ is ascending and $2339$ is not. Find the largest ascending number that is a multiple of $56$.

[b]8.[/b] Among all point lattices on the plane intersecting every closed convex region of unit width, which on's fundamental parallelogram has the largest area? ([b]G.36[/b]) [L. Fejes-Tóth]

A rectangular box measures $a \times b \times c$, where $a,$ $b,$ and $c$ are integers and $1 \leq a \leq b \leq c$. The volume and surface area of the box are numerically equal. How many ordered triples $(a,b,c)$ are possible? $ \textbf{(A) }4\qquad\textbf{(B) }10\qquad\textbf{(C) }12\qquad\textbf{(D) }21\qquad\textbf{(E) }26 $

$ABCD$ is a cyclic quadrilateral in which $AB=3$, $BC=5$, $CD=6$, and $AD=10$. $M$, $I$, and $T$ are the feet of the perpendiculars from $D$ to lines $AB$, $AC$, and $BC$ respectively. Determine the value of $MI/IT$.