Which statement is correct? $ \textbf{(A)}\ \text{If } x<0, \text{ then } x^2 > x. \qquad \textbf{(B)}\ \text{If } x^2 > 0, \text{ then } x > 0. \qquad$ $ \textbf{(C)}\ \text{If } x^2 > x, \text{ then } x>0. \qquad \textbf{(D)}\ \text{If } x^2 > x, \text{ then } x<0. \qquad$ $ \textbf{(E)}\ \text{If } x<1, \text{ then } x^2<x.$

For real variables $0 \leq x, y, z, w \leq 1$, find the maximum value of $$x(1-y)+2y(1-z)+3z(1-w)+4w(1-x)$$

Let $ \mathbb{Z}$ denote the set of all integers. Prove that for any integers $ A$ and $ B,$ one can find an integer $ C$ for which $ M_1 \equal{} \{x^2 \plus{} Ax \plus{} B : x \in \mathbb{Z}\}$ and $ M_2 \equal{} {2x^2 \plus{} 2x \plus{} C : x \in \mathbb{Z}}$ do not intersect.

Prove that in any parallelepiped the sum of the lengths of the edges is less than or equal to twice the sum of the lengths of the four diagonals.

How many of the twelve pentominoes pictured below have at least one line of symmetry? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 7$ [asy]unitsize(5mm); defaultpen(linewidth(1pt)); draw(shift(2,0)*unitsquare); draw(shift(2,1)*unitsquare); draw(shift(2,2)*unitsquare); draw(shift(1,2)*unitsquare); draw(shift(0,2)*unitsquare); draw(shift(2,4)*unitsquare); draw(shift(2,5)*unitsquare); draw(shift(2,6)*unitsquare); draw(shift(1,5)*unitsquare); draw(shift(0,5)*unitsquare); draw(shift(4,8)*unitsquare); draw(shift(3,8)*unitsquare); draw(shift(2,8)*unitsquare); draw(shift(1,8)*unitsquare); draw(shift(0,8)*unitsquare); draw(shift(6,8)*unitsquare); draw(shift(7,8)*unitsquare); draw(shift(8,8)*unitsquare); draw(shift(9,8)*unitsquare); draw(shift(9,9)*unitsquare); draw(shift(6,5)*unitsquare); draw(shift(7,5)*unitsquare); draw(shift(8,5)*unitsquare); draw(shift(7,6)*unitsquare); draw(shift(7,4)*unitsquare); draw(shift(6,1)*unitsquare); draw(shift(7,1)*unitsquare); draw(shift(8,1)*unitsquare); draw(shift(6,0)*unitsquare); draw(shift(7,2)*unitsquare); draw(shift(11,8)*unitsquare); draw(shift(12,8)*unitsquare); draw(shift(13,8)*unitsquare); draw(shift(14,8)*unitsquare); draw(shift(13,9)*unitsquare); draw(shift(11,5)*unitsquare); draw(shift(12,5)*unitsquare); draw(shift(13,5)*unitsquare); draw(shift(11,6)*unitsquare); draw(shift(13,4)*unitsquare); draw(shift(11,1)*unitsquare); draw(shift(12,1)*unitsquare); draw(shift(13,1)*unitsquare); draw(shift(13,2)*unitsquare); draw(shift(14,2)*unitsquare); draw(shift(16,8)*unitsquare); draw(shift(17,8)*unitsquare); draw(shift(18,8)*unitsquare); draw(shift(17,9)*unitsquare); draw(shift(18,9)*unitsquare); draw(shift(16,5)*unitsquare); draw(shift(17,6)*unitsquare); draw(shift(18,5)*unitsquare); draw(shift(16,6)*unitsquare); draw(shift(18,6)*unitsquare); draw(shift(16,0)*unitsquare); draw(shift(17,0)*unitsquare); draw(shift(17,1)*unitsquare); draw(shift(18,1)*unitsquare); draw(shift(18,2)*unitsquare);[/asy]

In a school there are $40$ different clubs, each of them contains exactly $30$ children. For every $i$ from $1$ to $30$ define $n_i$ as a number of children who attend exactly $i$ clubs. Prove that it is possible to organize $40$ new clubs with $30$ children in each of them such, that the analogical numbers $n_1, n_2,..., n_{30}$ will be the same for them.

Let $x, y$ be two non-negative integers. Prove that $47$ divides $3^x - 2^y$ if and only if $23$ divides $4x + y$.

The schoolchildren sat down on chairs located in transverse and longitudinal rows. The tallest student was chosen from each transverse row, and the lowest was chosen among them. Then the lowest student was selected from each longitudinal row, and the tallest was chosen among them. Which of these two students is higher?

Determine all positive integers $n > 2$ , such that \[ \frac{1}{2} \varphi(n) \equiv 1 ( \bmod 6) \]

Leo and Smilla find $2020$ gold nuggets with masses $1,2,\dots,2020$ gram, which they distribute to a red and a blue treasure chest according to the following rule: First, Leo chooses one of the chests and tells its colour to Smilla. Then Smilla chooses one of the not yet distributed nuggets and puts it into this chest. This is repeated until all the nuggets are distributed. Finally, Smilla chooses one of the chests and wins all the nuggets from this chest. How many gram of gold can Smilla make sure to win?

In an election between $\text{A}$ and $\text{B}$, during the counting of the votes, neither candidate was more than $2$ votes ahead, and the vote ended in a tie, $6$ votes to $6$ votes. Two votes for the same candidate are indistinguishable. In how many orders could the votes have been counted? One possibility is $\text{AABBABBABABA}$.

Triangle $ABC$ is isosceles with base $AC$. Points $P$ and $Q$ are respectively in $CB$ and $AB$ and such that $AC=AP=PQ=QB$. The number of degrees in angle $B$ is: ${{ \textbf{(A)}\ 25 \frac{5}{7} \qquad\textbf{(B)}\ 26 \frac{1}{3} \qquad\textbf{(C)}\ 30\qquad\textbf{(D)}\ 40}\qquad\textbf{(E)}\ \text{Not determined by the information given} } $

$1999-999+99$ equals $\textbf{(A)}\ 901 \qquad \textbf{(B)}\ 1099 \qquad \textbf{(C)}\ 1000 \qquad \textbf{(D)}\ 199 \qquad \textbf{(E)}\ 99$

The plane is divided into strips of width $1$ by parallel lines (a strip - the region between two parallel lines). The points from the interior of each strip are coloured with red or white, such that in each strip only one color is used (the points of a strip are coloured with the same color). The points on the lines are not coloured. Show that there is an equilateral triangle of side-length $100$, with all vertices of the same colour.

The quadrilateral $ABCD$ is a square of sidelength $1$, and the points $E, F, G, H$ are the midpoints of the sides. Determine the area of quadrilateral $PQRS$. [img]https://1.bp.blogspot.com/--fMGH2lX6Go/XzcDqhgGKfI/AAAAAAAAMXo/x4NATcMDJ2MeUe-O0xBGKZ_B4l_QzROjACLcBGAsYHQ/s0/2000%2BMohr%2Bp1.png[/img]

An integer is given $N> 1$. Arne and Britt play the following game: (1) Arne says a positive integer $A$. (2) Britt says an integer $B> 1$ that is either a divisor of $A$ or a multiple of $A$. ($A$ itself is a possibility.) (3) Arne says a new number $A$ that is either $B - 1, B$ or $B + 1$. The game continues by repeating steps 2 and 3. Britt wins if she is okay with being told the number $N$ before the $50$th has been said. Otherwise, Arne wins. a) Show that Arne has a winning strategy if $N = 10$. b) Show that Britt has a winning strategy if $N = 24$. c) For which $N$ does Britt have a winning strategy?

A rectangular pool table has vertices at $(0, 0) (12, 0) (0, 10),$ and $(12, 10)$. There are pockets only in the four corners. A ball is hit from $(0, 0)$ along the line $y = x$ and bounces off several walls before eventually entering a pocket. Find the number of walls that the ball bounces off of before entering a pocket.

How many integer solutions exist that satisfy this equation? $$x+4y-343\sqrt{x}-686\sqrt{y}+4\sqrt{xy}+2022=0$$.

Let $\theta_i \in \left ( 0,\frac{\pi}{4} \right ]$ for $i=1,2,3,4$. Prove that: $\tan \theta _1 \tan \theta _2 \tan \theta _3 \tan \theta _4 \le (\frac{\sin^8 \theta _1+\sin^8 \theta _2+\sin^8 \theta _3+\sin^8 \theta _4}{\cos^8 \theta _1+\cos^8 \theta _2+\cos^8 \theta _3+\cos^8 \theta _4})^\frac{1}{2}$ [hide=edit]@below, fixed now. There were some problems (weird characters) so aops couldn't send it.[/hide]

Two circles $C_1$ and $C_2$ are tangent at a point $P$. The straight line at $D$ is tangent at $A$ to one of the circles and cuts the other circle at the points $B$ and $C$. Prove that the straight line $PA$ is a bisector (interior or exterior) of the angle $BPC$.

Let $S$ be a set of real numbers. We say that $S$ is [i]strong[/i] if for any two distinct $a$ and $b$ from $S$, the number $a^2 + b\sqrt{2023}$ is rational. We say that $S$ is [i]very strong[/i] if for every $a$ from $S$, the number $a\sqrt{2023}$ is rational. a) Prove that if $S$ is a very strong set, then it is also strong. b) Find the smallest natural number $k$ such that every strong set of $k$ distinct real numbers is very strong.

Find the sum of all integers $n$ such that $1 < n < 30$ and $n$ divides $$1+\sum^{n-1}_{k=1}k^{2k}.$$

Given are two circles which intersect at points $P$ and $Q$. Consider an arbitrary line $\ell$ through $Q$, let the second points of intersection of this line with the circles be $A$ and $B$ respectively. Let $C$ be the point of intersection of the tangents to the circles in those points. Let $D$ be the intersection of the line $AB$ and the bisector of the angle $CPQ$. Prove that all possible $D$ for any choice of $\ell$ lie on a single circle. Alexey Zaslavsky

Let $n \geq 3$ be an integer. Find the minimum degree of one algebraic (polynomial) equation that defines the set of vertices of the correct $n$-gon on plane $R^2$.

The length of a rectangle is $ 5$ inches and its width is less than $ 4$ inches. The rectangle is folded so that two diagonally opposite vertices coincide. If the length of the crease is $ \sqrt {6}$, then the width is: $ \textbf{(A)}\ \sqrt {2} \qquad \textbf{(B)}\ \sqrt {3} \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ \sqrt {5} \qquad \textbf{(E)}\ \sqrt {\frac {11}{2}}$