Consider $2009$ cards, each having one gold side and one black side, lying on parallel on a long table. Initially all cards show their gold sides. Two player, standing by the same long side of the table, play a game with alternating moves. Each move consists of choosing a block of $50$ consecutive cards, the leftmost of which is showing gold, and turning them all over, so those which showed gold now show black and vice versa. The last player who can make a legal move wins. (a) Does the game necessarily end? (b) Does there exist a winning strategy for the starting player? [i]Proposed by Michael Albert, Richard Guy, New Zealand[/i]

Find all the functions $ f: \mathbb{N}\rightarrow \mathbb{N}$ such that \[ 3f(f(f(n))) \plus{} 2f(f(n)) \plus{} f(n) \equal{} 6n, \quad \forall n\in \mathbb{N}.\]

A $1\times 2019$ board is filled with numbers $1, 2, \dots, 2019$ in an increasing order. In each step, three consecutive tiles are selected, then one of the following operations is performed: $\text{(i)}$ the number in the middle is increased by $2$ and its neighbors are decreased by $1$, or $\text{(ii)}$ the number in the middle is decreased by $2$ and its neighbors are increased by $1$. After several such operations, the board again contains all the numbers $1, 2,\dots, 2019$. Prove that each number is in its original position.

Let $ p_1, p_2, p_3$ and $ p_4$ be four different prime numbers satisying the equations $ 2p_1 \plus{} 3p_2 \plus{} 5p_3 \plus{} 7p_4 \equal{} 162$ $ 11p_1 \plus{} 7p_2 \plus{} 5p_3 \plus{} 4p_4 \equal{} 162$ Find all possible values of the product $ p_1p_2p_3p_4$

It is known that for some integers $a_{2021},a_{2020},...,a_1,a_0$ the expression $$a_{2021}n^{2021}+a_{2020}n^{2020}+...+a_1n+a_0$$ is divisible by $2021$ for any arbitrary integer $n$. Is it required that each of the numbers $a_{2021},a_{2020},...,a_1,a_0$ also divisible by $2021$?

Prove that $[\sqrt{n}+\sqrt{n+1}]=[\sqrt{4n+1}]$ for all $n \in N$.

Prove that \[ \lim_{n \to \infty} n \left( \frac{\pi}{4} - n \int_0^1 \frac{x^n}{1+x^{2n}} \, dx \right) = \int_0^1 f(x) \, dx , \] where $f(x) = \frac{\arctan x}{x}$ if $x \in \left( 0,1 \right]$ and $f(0)=1$. [i]Dorin Andrica, Mihai Piticari[/i]

The incircle of a non-isosceles triangle $ABC$ with the center $I$ touches the sides $BC,AC,AB$ at $A_{1},B_{1},C_{1}$ . let $AI,BI,CI$ meets $BC,AC,AB$ at $A_{2},B_{2},C_{2}$. let $A'$ is a point on $AI$ such that $A_{1}A'\perp B_{2}C_{2}$ .$B',C'$ respectively. prove that two triangle $A'B'C',A_{1}B_{1}C_{1}$ are equal.

On a circumference of a circle, seven points are selected, at which different positive integers are assigned to each of them. Then fit simultaneously, each number is replaced by the least common multiple of the two neighboring numbers to it. If the same number $n$ is obtained in each of the seven points, determine the smallest possible value for $n$. [hide=original wording]Sobre una circunferencia de un círculo, se seleccionan siete puntos, a los cuales se le asignan enteros positivos distintos a cada uno de ellos. Luego, en forma simultánea, cada número se reemplaza por el mínimo común múltiplo de los dos números vecinos a él. Si se obtiene el mismo número n en cada uno de los siete puntos, determine el menor valor posible para n.[/url]

Given two non-parallel segments $AB$ and $CD$ on the plane, find the locus of points $P$ on the plane such that the area of triangle $ABP$ is equal to the area of triangle $CDP$.

Let $\triangle{ABC}$ a triangle with $\angle{CAB}=90^{\circ}$ and $L$ a point on the segment $BC$. The circumcircle of triangle $\triangle{ABL}$ intersects $AC$ at $M$ and the circumcircle of triangle $\triangle{CAL}$ intersects $AB$ at $N$. Show that $L$, $M$ and $N$ are collinear.

In triangle $ABC$, points $P$, $Q$, and $R$ are marked on the sides $AB$, $BC$, and $AC$ respectively. The lengths of the sides of triangle $PQR$ are known to be 7, 8, and 9 centimeters. Find the radii of the circles inscribed in triangles $APR$, $BPQ$, and $CQR$ given that all three circles are tangent to the incircle of triangle $PQR$. [i]Proposed by Giorgi Arabidze, Georgia[/i]

Determine all the numbers formed by three different and non-zero digits, such that the six numbers obtained by permuting these digits leaves the same remainder after the division by $4$.

Prove that one can find infinite number of distinct pairs of integers such that every digit of each number is no less than $7$ and the product of two numbers in each pair is also a number with all its digits being no less than $7$. (6)

Find all real numbers $a$ for which there exist functions $f,g: \mathbb{R} \to \mathbb{R}$, where $g$ is strictly increasing, such that $f(1) = 1$, $f(2) = a$, and \[ f(x) - f(y) \leq (x-y)(g(x) - g(y)) \] for all real numbers $x$ and $y$.

(1) A function is defined $ f(x) \equal{} \ln (x \plus{} \sqrt {1 \plus{} x^2})$ for $ x\geq 0$. Find $ f'(x)$. (2) Find the arc length of the part $ 0\leq \theta \leq \pi$ for the curve defined by the polar equation: $ r \equal{} \theta\ (\theta \geq 0)$. Remark: [color=blue]You may not directly use the integral formula of[/color] $ \frac {1}{\sqrt {1 \plus{} x^2}},\ \sqrt{1 \plus{} x^2}$ here.

Let $(G,\cdot)$ be a finite group with the identity element, $e$. The smallest positive integer $n$ with the property that $x^{n}= e$, for all $x \in G$, is called the [i]exponent[/i] of $G$. (a) For all primes $p \geq 3$, prove that the multiplicative group $\mathcal G_{p}$ of the matrices of the form $\begin{pmatrix}\hat 1 & \hat a & \hat b \\ \hat 0 & \hat 1 & \hat c \\ \hat 0 & \hat 0 & \hat 1 \end{pmatrix}$, with $\hat a, \hat b, \hat c \in \mathbb Z \slash p \mathbb Z$, is not commutative and has [i]exponent[/i] $p$. (b) Prove that if $\left( G, \circ \right)$ and $\left( H, \bullet \right)$ are finite groups of [i]exponents[/i] $m$ and $n$, respectively, then the group $\left( G \times H, \odot \right)$ with the operation given by $(g,h) \odot \left( g^\prime, h^\prime \right) = \left( g \circ g^\prime, h \bullet h^\prime \right)$, for all $\left( g,h \right), \, \left( g^\prime, h^\prime \right) \in G \times H$, has the [i]exponent[/i] equal to $\textrm{lcm}(m,n)$. (c) Prove that any $n \geq 3$ is the [i]exponent[/i] of a finite, non-commutative group. [i]Ion Savu[/i]

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that \[f(x)f(y) = (x+y+1)^2 \cdot f \left( \frac{xy-1}{x+y+1} \right)\] $\forall x,y \in \mathbb{R}$ with $x+y+1 \neq 0$ and $f(x) > 1$ $\forall x > 0.$

In the corners of triangle $ABC$ there are three circles with the same radius. Each of them is tangent to two of the triangle's sides. The vertices of triangle $MNK$ lie on different sides of triangle $ABC$, and each edge of $MNK$ is also tangent to one of the three circles. Likewise, the vertices of triangle $PQR$ lie on different sides of triangle $ABC$, and each edge of $PQR$ is also tangent to one of the three circles (see picture below). Prove that triangles $MNK,PQR$ have the same inradius. [img]https://i.imgur.com/bYuBabS.png[/img]

When Tim was Jim’s age, Kim was twice as old as Jim. When Kim was Tim’s age, Jim was 30. When Jim becomes Kim’s age, Tim will be 88. When Jim becomes Tim’s age, what will be the sum of the ages of Tim, Jim, and Kim?

Let $ABC$ be a triangle with circumcenter $O$. Additionally, $\angle BAC=20^\circ$ and $\angle BCA = 70^\circ$. Let $D, E$ be points on side $AC$ such that $BO$ bisects $\angle ABD$ and $BE$ bisects $\angle CBD$. If $P$ and $Q$ are points on line $BC$ such that $DP$ and $EQ$ are perpendicular to $AC$, what is $\angle PAQ$? [i]Proposed by Daniel Li[/i]

Let $P(z)=a_nz^n+a_{n-1}z^{n-1}+\ldots+a_mz^m$ be a polynomial with complex coefficients such that $a_m\neq 0, a_n\neq 0$ and $n>m$. Prove that \[\text{max}_{|z|=1}\{|P(z)|\}\ge\sqrt{2|a_ma_n|+\sum_{k=m}^{n} |a_k|^2}\]

A point $P$ is chosen inside a triangle $ABC$ with circumcircle $\Omega$. Let $\Gamma$ be the circle passing through the circumcenters of the triangles $APB$, $BPC$, and $CPA$. Let $\Omega$ and $\Gamma$ intersect at points $X$ and $Y$. Let $Q$ be the reflection of $P$ in the line $XY$ . Prove that $\angle BAP = \angle CAQ$.

Let $n \geq 2$ be a natural number. We consider a $(2n - 1) \times (2n - 1)$ table.Ana and Bob play the following game: starting with Ana, the two of them alternately color the vertices of the unit squares, Ana with red and Bob with blue, in $2n^2$ rounds. Then, starting with Ana, each one forms a vector with origin at a red point and ending at a blue point, resulting in $2n^2$ vectors with distinct origins and endpoints. If the sum of these vectors is zero, Ana wins. Otherwise, Bob wins. Show that Bob has a winning strategy.

Let $b$ and $c$ be real numbers not both equal to $1$ such that $1,b,c$ is an arithmetic progression and $1,c,b$ is a geometric progression. What is $100(b-c)$? [i]Proposed by Noah Kravitz[/i]