Two nonhorizontal, non vertical lines in the $xy$-coordinate plane intersect to form a $45^{\circ}$ angle. One line has slope equal to $6$ times the slope of the other line. What is the greatest possible value of the product of the slopes of the two lines? $\textbf{(A)}\ \frac16 \qquad\textbf{(B)}\ \frac23 \qquad\textbf{(C)}\ \frac32 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 6$

Let $ABC$ be a triangle, having no right angles, and let $D$ be a point on the side $BC$. Let $E$ and $F$ be the feet of the perpendiculars drawn from the point $D$ to the lines $AB$ and $AC$ respectively. Let $P$ be the point of intersection of the lines $BF$ and $CE$. Prove that the line $AP$ is the altitude of the triangle $ABC$ from the vertex $A$ if and only if the line $AD$ is the angle bisector of the angle $CAB$.

A [b] bijective[/b] mapping from a plane to itself maps every circle to a circle. Prove that it maps every line to a line.

What's the largest number of integers from $1$ to $2022$ that you can choose so that no sum of any two different chosen integers is divisible by any difference of two different chosen integers? [i](Proposed by Oleksii Masalitin)[/i]

Define a sequence {$a_n$}$^{\infty}_{n=1}$ by $a_1 = 4, a_2 = a_3 = (a^2 - 2)^2$ and $a_n = a_{n-1}.a_{n-2} - 2(a_{n-1} + a_{n-2}) - a_{n-3} + 8, n \ge 4$, where $a > 2$ is a natural number. Prove that for all $n$ the number $2 + \sqrt{a_n}$ is a perfect square.

A point $ P$ is chosen at random in the interior of a unit square $ S$. Let $ d(P)$ denote the distance from $ P$ to the closest side of $ S$. The probability that $ \frac15\le d(P)\le\frac13$ is equal to $ \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m\plus{}n$.

In the array of $13$ squares shown below, $8$ squares are colored red, and the remaining $5$ squares are colored blue. If one of all possible such colorings is chosen at random, the probability that the chosen colored array appears the same when rotated $90^{\circ}$ around the central square is $\tfrac{1}{n}$, where $n$ is a positive integer. Find $n$. [asy] draw((0,0)--(1,0)--(1,1)--(0,1)--(0,0)); draw((2,0)--(2,2)--(3,2)--(3,0)--(3,1)--(2,1)--(4,1)--(4,0)--(2,0)); draw((1,2)--(1,4)--(0,4)--(0,2)--(0,3)--(1,3)--(-1,3)--(-1,2)--(1,2)); draw((-1,1)--(-3,1)--(-3,0)--(-1,0)--(-2,0)--(-2,1)--(-2,-1)--(-1,-1)--(-1,1)); draw((0,-1)--(0,-3)--(1,-3)--(1,-1)--(1,-2)--(0,-2)--(2,-2)--(2,-1)--(0,-1)); size(100); [/asy]

Consider the convex quadrilateral $ABCD$. The point $P$ is in the interior of $ABCD$. The following ratio equalities hold: \[\angle PAD:\angle PBA:\angle DPA=1:2:3=\angle CBP:\angle BAP:\angle BPC\] Prove that the following three lines meet in a point: the internal bisectors of angles $\angle ADP$ and $\angle PCB$ and the perpendicular bisector of segment $AB$. [i]Proposed by Dominik Burek, Poland[/i]

There are $n\geq 3$ pupils standing in a circle, and always facing the teacher that stands at the centre of the circle. Each time the teacher whistles, two arbitrary pupils that stand next to each other switch their seats, while the others stands still. Find the least number $M$ such that after $M$ times of whistling, by appropriate switchings, the pupils stand in such a way that any two pupils, initially standing beside each other, will finally also stand beside each other; call these two pupils $ A$ and $ B$, and if $ A$ initially stands on the left side of $ B$ then $ A$ will finally stand on the right side of $ B$.

Let $ N$ denote the set of natural numbers. Let $ \phi: N\rightarrow N$ be a bijective function and assume that there exists a finite limit \[ \lim_{n\rightarrow\infty}\frac{\phi(n)}{n}\equal{}L. \] What are the possible values of $ L$?

Determine all positive real numbers $x$ and $y$ satisfying the equation \[x+y+\frac{1}{x}+\frac{1}{y}+4=2\cdot (\sqrt{2x+1}+\sqrt{2y+1})\]

The ratio of the sides of a parallelogram is $\lambda>1$. Given $\lambda$, determine the maximum of the acute angle subtended by the diagonals of the parallelogram.

Air Michael and Air Patrick operate direct flights connecting Belfast, Cork, Dublin, Galway, Limerick, and Waterord. For each pair of cities exactly one of the airlines operates the route (in both directions) connecting the cities. Prove that there are four cities for which one of the airlines operates a round trip. (Note that a round trip of four cities $ P,Q,R,$ and $ S$, is a journey that follows the path $ P \rightarrow Q \rightarrow R \rightarrow S \rightarrow P$.)

Given is the sequence $(a_n)_{n\geq 0}$ which is defined as follows:$a_0=3$ and $a_{n+1}-a_n=n(a_n-1) \ , \ \forall n\geq 0$. Determine all positive integers $m$ such that $\gcd (m,a_n)=1 \ , \ \forall n\geq 0$.

Victor writes down all $7-$digit numbers using the digits $1, 2, 3, 4, 5, 6,$ and $7$ exactly once. Prove that there are no two numbers among them where one is a multiple of the other.

Let $x, y, z$ be non-negative real numbers satisfying \[x^2 + y^2 + z^2 = 5 \quad \text{ and } \quad yz + zx + xy = 2.\] Which values can the greatest of the numbers $x^2 -yz, y^2 - xz$ and $z^2 - xy$ have?

$ \left|AC\right| \equal{} 8 \sqrt {2}$, $ B$ is the midpoint of $ \left[AC\right]$, $ E$ is the midpoint of arc $ AB$ of a circle having chord $ \left[AB\right]$, and $ D$ is the point of tangency drawing from $ C$.($ D$ lies on the opposite side of line $ AB$ to $ E$). If $ \left[DE\right]\bigcap \left[AB\right] \equal{} \left\{F\right\}$, $ \left|CF\right| \equal{} ?$ $\textbf{(A)}\ 5\sqrt {2} \qquad\textbf{(B)}\ 4\sqrt {2} \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 4\sqrt {3}$

Let $P_1$, $P_2$, $P_3$ be pairwise distinct parabolas in the plane. Find the maximum possible number of intersections between two or more of the $P_i$. In other words, find the maximum number of points that can lie on two or more of the parabolas $P_1$, $P_2$, $P_3$ .

Given an integer $n>1$. The board $n\times n$ is colored white and black in a chess-like manner. We call any non-empty set of different cells of the board as a [i]figure[/i]. We call figures $F_1$ and $F_2$ [i]similar[/i], if $F_1$ can be obtained from $F_2$ by a rotation with respect to the center of the board by an angle multiple of $90^\circ$ and a parallel transfer. (Any figure is similar to itself.) We call a figure $F$ [i]connected[/i] if for any cells $a,b\in F$ there is a sequence of cells $c_1,\ldots,c_m\in F$ such that $c_1 = a$, $c_m = b$, and also $c_i$ and $c_{i+1}$ have a common side for each $1\le i\le m - 1$. Find the largest possible value of $k$ such that for any connected figure $F$ consisting of $k$ cells, there are figures $F_1,F_2$ similar to $F$ such that $F_1$ has more white cells than black cells and $F_2$ has more black cells than white cells in it.

$10$ people attended a party. For every $3$ people, there exist at least $2$ people who don’t know each other. Prove that there exist $4$ people who don’t know each other.

Apples cost $2$ dollars. Bananas cost $3$ dollars. Oranges cost $5$ dollars. Compute the number of distinct baskets of fruit such that there are $100$ pieces of fruit and the basket costs $300$ dollars. Two baskets are distinct if and only if, for some type of fruit, the two baskets have differing amounts of that fruit.

$m$ boys and $n$ girls ($m>n$) sat across a round table, supervised by a teacher, and they did a game, which went like this. At first, the teacher pointed a boy to start the game. The chosen boy put a coin on the table. Then, consecutively in a clockwise order, everyone did his turn. If the next person is a boy, he will put a coin to the existing pile of coins. If the next person is a girl, she will take a coin from the existing pile of coins. If there is no coin on the table, the game ends. Notice that depending on the chosen boy, the game could end early, or it could go for a full turn. If the teacher wants the game to go for at least a full turn, how many possible boys could be chosen? [i]Hendrata Dharmawan, Boston, USA[/i]

Two triangles are circumscribed to a circumference. Show that if a circumference containing five of their vertices exists, then it will contain the sixth vertex too.

Let be two matrices $ A,B\in\mathcal{M}_2\left( \mathbb{R} \right) $ that don’t commute. [b]a)[/b] If $ A^3=B^3, $ then $ \text{tr} \left( A^n \right) =\text{tr} \left( B^n \right) , $ for all natural numbers $ n. $ [b]b)[/b] If $ A^n\neq B^n $ and $ \text{tr} \left( A^n \right) =\text{tr} \left( B^n \right) , $ for all natural numbers $ n, $ then find some of the matrices $ A,B. $

Is there a positive integer $m$ such that the equation \[ {1\over a}+{1\over b}+{1\over c}+{1\over abc}={m\over a+b+c} \] has infinitely many solutions in positive integers $a,b,c$?