What is $ (20\minus{}(2010\minus{}201)) \plus{} (2010\minus{}(201\minus{}20))$? $ \textbf{(A)}\ \minus{}4020\qquad \textbf{(B)}\ 0\qquad \textbf{(C)}\ 40\qquad \textbf{(D)}\ 401\qquad \textbf{(E)}\ 4020$

Let $ ABCD$ be an isosceles trapezium with $ AB \parallel{} CD$ and $ \bar{BC} \equal{} \bar{AD}.$ The parallel to $ AD$ through $ B$ meets the perpendicular to $ AD$ through $ D$ in point $ X.$ The line through $ A$ drawn which is parallel to $ BD$ meets the perpendicular to $ BD$ through $ D$ in point $ Y.$ Prove that points $ C,X,D$ and $ Y$ lie on a common circle.

Let $ n \in \mathbb{Z}^\plus{}$ and let $ a, b \in \mathbb{R}.$ Determine the range of $ x_0$ for which \[ \sum^n_{i\equal{}0} x_i \equal{} a \text{ and } \sum^n_{i\equal{}0} x^2_i \equal{} b,\] where $ x_0, x_1, \ldots , x_n$ are real variables.

A triangle with side lengths in the ratio $ 3: 4: 5$ is inscribed in a circle of radius $ 3.$ What is the area of the triangle? $ \textbf{(A)}\ 8.64 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 5\pi \qquad \textbf{(D)}\ 17.28 \qquad \textbf{(E)}\ 18$

There are $n$ vertices and $m > n$ edges in a graph. Each edge is colored either red or blue. In each year, we are allowed to choose a vertex and flip the color of all edges incident to it. Prove that there is a way to color the edges (initially) so that they will never all have the same color

Each person stands on a whole number on the number line from $0$ to $2022$ . In each turn, two people are selected by a distance of at least $2$. These go towards each other by $1$. When no more such moves are possible, the process ends. Show that this process always ends after a finite number of moves, and determine all possible configurations where people can end up standing. (whereby is for each configuration is only of interest how many people stand at each number.) [i](Birgit Vera Schmidt)[/i] [hide=original wording]Bei jeder ganzen Zahl auf dem Zahlenstrahl von 0 bis 2022 steht zu Beginn eine Person. In jedem Zug werden zwei Personen mit Abstand mindestens 2 ausgewählt. Diese gehen jeweils um 1 aufeinander zu. Wenn kein solcher Zug mehr möglich ist, endet der Vorgang. Man zeige, dass dieser Vorgang immer nach endlich vielen Zügen endet, und bestimme alle möglichen Konfigurationen, wo die Personen am Ende stehen können. (Dabei ist für jede Konfiguration nur von Interesse, wie viele Personen bei jeder Zahl stehen.)[/hide]

Determine if there are triplets ($x,y,z)$ of real numbers such that $$\begin{cases} x+y+z=7 \\ xy+yz+zx=11\end{cases}$$ If the answer is affirmative, find the minimum and maximum values ​​of $z$ in such a triplet.

[b]a.)[/b] Let $m>1$ be a positive integer. Prove there exist finite number of positive integers $n$ such that $m+n|mn+1$. [b]b.)[/b] For positive integers $m,n>2$, prove that there exists a sequence $a_0,a_1,\cdots,a_k$ from positive integers greater than $2$ that $a_0=m$, $a_k=n$ and $a_i+a_{i+1}|a_ia_{i+1}+1$ for $i=0,1,\cdots,k-1$.

Let $a$ and $b$ be two positive integers. Prove that the integer \[a^2+\left\lceil\frac{4a^2}b\right\rceil\] is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.) [i]Russia[/i]

Find $ \lim_{a\rightarrow{\infty}} \frac{1}{a^2}\int_0^a \ln (1\plus{}e^x)dx$.

Let $\Gamma$ be the circumcircle of a triangle $ABC,$ and let $D$ and $E$ be two points different from the vertices on the sides $AB$ and $AC,$ respectively. Let $A'$ be the second point where $\Gamma$ intersects the bisector of the angle $BAC,$ and let $P$ and $Q$ be the second points where $\Gamma$ intersects the lines $A'D$ and $A'E,$ respectively. Let $R$ and $S$ be the second points of intersection of the lines $AA'$ and the circumcircles of the triangles $APD$ and $AQE,$ respectively. Show that the lines $DS, \: ER$ and the tangent line to $\Gamma$ through $A$ are concurrent.

Let real numbers $x_1, x_2, \cdots , x_n$ satisfy $0 < x_1 < x_2 < \cdots< x_n < 1$ and set $x_0 = 0, x_{n+1} = 1$. Suppose that these numbers satisfy the following system of equations: \[\sum_{j=0, j \neq i}^{n+1} \frac{1}{x_i-x_j}=0 \quad \text{where } i = 1, 2, . . ., n.\] Prove that $x_{n+1-i} = 1- x_i$ for $i = 1, 2, . . . , n.$

Find the number of ordered integer triplets $ x, y, z $ with absolute value less than or equal to 100 such that $ 2x^2 + 3y^2 + 3z^2 + 2xy + 2xz - 4yz < 5 $.

Given three points $A, B$ and $C$ (not collinear) construct the equilateral triangle of greater perimeter such that each of its sides passes through one of the given points.

Given two 2's, "plus" can be changed to "times" without changing the result: 2+2=2·2. The solution with three numbers is easy too: 1+2+3=1·2·3. There are three answers for the five-number case. Which five numbers with this property has the largest sum?

Let $A_1A_2 . . .A_{100}$ be the vertices of a regular $100$-gon. Let $\pi$ be a randomly chosen permutation of the numbers from $1$ through $100$. The segments $A_{\pi (1)}A_{\pi (2)}$, $A_{\pi (2)}A_{\pi (3)}$, $...$ ,$A_{\pi (99)}A_{\pi (100)}, A_{\pi (100)}A_{\pi (1)}$ are drawn. Find the expected number of pairs of line segments that intersect at a point in the interior of the $100$-gon.

A group of $25$ friends were discussing a large positive integer. "It can be divided by $1$," said the first friend. "It can be divided by $2$," said the second friend. "And by $3$," said the third friend. "And by $4$," added the fourth friend. This continued until everyone had made such a comment. If exactly $2$ friends were incorrect, and those two friends said consecutive numbers, what was the least possible integer they were discussing?

Determine whether there exist an odd positive integer $n$ and $n\times n$ matrices $A$ and $B$ with integer entries, that satisfy the following conditions: [list=1] [*]$\det (B)=1$;[/*] [*]$AB=BA$;[/*] [*]$A^4+4A^2B^2+16B^4=2019I$.[/*] [/list] (Here $I$ denotes the $n\times n$ identity matrix.) [i]Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan[/i]

Compute the sum of the real solutions to $\lfloor x \rfloor \{x\} = 2020x$. Here, $\lfloor x \rfloor$ is defined as the greatest integer less than or equal to $x$, and$ \{x\} = x -\lfloor x \rfloor$.

In the right triangle $ABC$ ($m ( \angle A) = 90^o$) $D$ is the foot of the altitude from $A$. The bisectors of the angles $ABD$ and $ADB$ intersect in $I_1$ and the bisectors of the angles $ACD$ and $ADC$ in $I_2$. Find the angles of the triangle if the sum of distances from $I_1$ and $I_2$ to $AD$ is equal to $\frac14$ of the length of $BC$.

A cube $ABCDA_1B_1C_1D_1$ is given. Find the locus of points $E$ on the face $A_1B_1C_1D_1$ for which there exists a line intersecting the lines $AB$, $A_1D_1$, $B_1D$, and $EC$.

$\definecolor{A}{RGB}{190,0,60}\color{A}\fbox{A1.}$ Find all $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $$\definecolor{A}{RGB}{80,0,200}\color{A} x^4+y^4+z^4\ge f(xy)+f(yz)+f(zx)\ge xyz(x+y+z)$$holds for all $a,b,c\in\mathbb{R}$. [i]Proposed by [/i][b][color=#FFFF00]usjl[/color][/b]. [color=#B6D7A8]#1733[/color]

For every $n$ non-negative integer let $S(n)$ denote a subset of the positive integers, for which $i$ is an element of $S(n)$ if and only if the $i$-th digit (from the right) in the base two representation of $n$ is a digit $1$. Two players, $A$ and $B$ play the following game: first, $A$ chooses a positive integer $k$, then $B$ chooses a positive integer $n$ for which $2^n\geqslant k$. Let $X$ denote the set of integers $\{ 0,1,\dotsc ,2^n-1\}$, let $Y$ denote the set of integers $\{ 0,1,\dotsc ,2^{n+1}-1\}$. The game consists of $k$ rounds, and in each round player $A$ chooses an element of set $X$ or $Y$, then player $B$ chooses an element from the other set. For $1\leqslant i\leqslant k$ let $x_i$ denote the element chosen from set $X$, let $y_i$ denote the element chosen from set $Y$. Player $B$ wins the game, if for every $1\leqslant i\leqslant k$ and $1\leqslant j\leqslant k$, $x_i<x_j$ if and only if $y_i<y_j$ and $S(x_i)\subset S(x_j)$ if and only if $S(y_i)\subset S(y_j)$. Which player has a winning strategy? [i]Proposed by Levente Bodnár, Cambridge[/i]

For a positive integer $n$, prove that $$\sum_{n \le p \le n^4} \frac{1}{p} < 4$$ where the sum is taken across primes $p$ in the range $[n, n^4]$ [i]N. Filonov[/i]

Based on a city's rules, the buildings of a street may not have more than $9$ stories. Moreover, if the number of stories of two buildings is the same, no matter how far they are from each other, there must be a building with a higher number of stories between them. What is the maximum number of buildings that can be built on one side of a street in this city?