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 $\textit{captures}$.

A sequence $(a_n)$ is defined by means of the recursion \[a_1 = 1, a_{n+1} = \frac{1 + 4a_n +\sqrt{1+ 24a_n}}{16}.\] Find an explicit formula for $a_n.$

Let $C_1(O_1)$ and $ C_2(O_2)$ be two circles such that $C_1$ passes through $O_2$. Point $M$ lies on $C_1$ such that $M \notin O_1O_2$. The tangents from $M$ at $O_2$ meet again $C_1$ at $A$ and $B$. Prove that the tangents from $A$ and $B$ at $C_2$ - others than $MA$ and $MB$ - meet at a point located on $C_1$.

Is it possible to place within a square an equilateral triangle whose area is larger than $9/ 20$ of the area of the square?

There are $20$ points in the plane and no three of them are collinear. Of these points $10$ are red while the other $10$ are blue. Prove that there exists a straight line such that there are $5$ red points and $5$ blue points on either side of this line. (A Kushnirenko, Moscow)

We will say that two positive integers $a$ and $b$ form a [i]suitable pair[/i] if $a+b$ divides $ab$ (its sum divides its multiplication). Find $24$ positive integers that can be distribute into $12$ suitable pairs, and so that each integer number appears in only one pair and the largest of the $24$ numbers is as small as possible.

$$\int_1^\infty \frac{1 + 2x \ln 2}{x\sqrt{x 4^x - 1}}\,dx$$ [i]Proposed by Vlad Oleksenko[/i]

Determine the set $S$ with minimal number of points defining $7$ distinct lines

For a given odd prime number $p$, define $f(n)$ the remainder of $d$ divided by $p$, where $d$ is the biggest divisor of $n$ which is not a multiple of $p$. For example when $p=5$, $f(6)=1, f(35)=2, f(75)=3$. Define the sequence $a_1, a_2, \ldots, a_n, \ldots$ of integers as the followings: [list] [*]$a_1=1$ [*]$a_{n+1}=a_n+(-1)^{f(n)+1}$ for all positive integers $n$. [/list] Determine all integers $m$, such that there exist infinitely many positive integers $k$ such that $m=a_k$.

It is given set $A=\{1,2,3,...,2n-1\}$. From set $A$, at least $n-1$ numbers are expelled such that: $a)$ if number $a \in A$ is expelled, and if $2a \in A$ then $2a$ must be expelled $b)$ if $a,b \in A$ are expelled, and $a+b \in A$ then $a+b$ must be also expelled Which numbers must be expelled such that sum of numbers remaining in set stays minimal

If $n$ is an even positive integer, the [i]double factorial[/i] notation $n!!$ represents the product of all the even integers from $2$ to $n$. For example, $8!! = 2 \cdot 4 \cdot 6 \cdot 8$. What is the units digit of the following sum? $$2!! + 4!! + 6!! + \cdots + 2018!! + 2020!! + 2022!!$$ $\textbf{(A)} ~0\qquad\textbf{(B)} ~2\qquad\textbf{(C)} ~4\qquad\textbf{(D)} ~6\qquad\textbf{(E)} ~8\qquad$

Find all functions $f:\mathbb R^+ \to \mathbb R^+$ satisfying \begin{align*} f\left(\frac{1}{x}\right) \geq 1 - \frac{\sqrt{f(x)f\left(\frac{1}{x}\right)}}{x} \geq x^2 f(x), \end{align*} for all positive real numbers $x$.

Let $BC$ be a fixed chord of a circle $\omega$. Let $A$ be a variable point on the major arc $BC$ of $\omega$. Let $H$ be the orthocenter of $ABC$. The points $D, E$ lie on $AB, AC$ such that $H$ is the midpoint of $DE$. $O_A$ is the circumcenter of $ADE$. Prove that as $A$ varies, $O_A$ lies on a fixed circle.

On a cube, 27 points are marked in the following manner: one point in each corner, one point on the middle of each edge, one point on the middle of each face, and one in the middle the cube. The number of lines containing three out of these points is A. 33 B. 42 C. 49 D. 72 E. 81

Let $a$ and $b$ satisfy $a \ge b >0, a + b = 1$. i) Prove that if $m$ and $n$ are positive integers with $m < n$, then $a^m - a^n \ge b^m- b^n > 0$. ii) For each positive integer $n$, consider a quadratic function $f_n(x) = x^2 - b^nx- a^n$. Show that $f(x)$ has two roots that are in between $-1$ and $1$.

Let $O_1O_2O_3$ be an acute angled triangle.Let $\omega_1$, $\omega_2$, $\omega_3$ be the circles with centres $O_1$, $O_2$, $O_3$ respectively such that any of two are tangent to each other. Circumcircle of $O_1O_2O_3$ intersects $\omega_1$ at $A_1$ and $B_1$, $\omega_2$ at $A_2$ and $B_2$, $\omega_3$ at $A_3$ and $B_3$ respectively. Prove that the incenter of triangle which can be constructed by lines $A_1B_1$, $A_2B_2$, $A_3B_3$ and the incenter of $O_1O_2O_3$ are coincide.

Aaron, Darren, Karen, Maren, and Sharon rode on a small train that has five cars that seat one person each. Maren sat in the last car. Aaron sat directly behind Sharon. Darren sat in one of the cars in front of Aaron. At least one person sat between Karen and Darren. Who sat in the middle car? $\textbf{(A) }\text{Aaron}\qquad \textbf{(B) }\text{Darren}\qquad \textbf{(C) }\text{Karen}\qquad \textbf{(D) }\text{Maren}\qquad \textbf{(E) }\text{Sharon}\qquad$

A figure with area $1$ is cut out of paper. We divide this figure into $10$ parts and color them in $10$ different colors. Now, we turn around the piece of paper, divide the same figure on the other side of the paper in $10$ parts again (in some different way). Show that we can color these new parts in the same $10$ colors again (hereby, different parts should have different colors) such that the sum of the areas of all parts of the figure colored with the same color on both sides is $\geq \frac{1}{10}.$

We say that the real-valued function $ f(x)$ defined on the interval $ (0,1)$ is approximately continuous on $ (0,1)$ if for any $ x_0 \in (0,1)$ and $ \varepsilon >0$ the point $ x_0$ is a point of interior density $ 1$ of the set \[ H\equal{} \{x : \;|f(x)\minus{}f(x_0)|< \varepsilon \ \}.\] Let $ F \subset (0,1)$ be a countable closed set, and $ g(x)$ a real-valued function defined on $ F$. Prove the existence of an approximately continuous function $ f(x)$ defined on $ (0,1)$ such that \[ f(x)\equal{}g(x) \;\textrm{for all}\ \;x \in F\ .\] [i]M. Laczkovich, Gy. Petruska[/i]