Let $ ABCD$ be a quadrilateral which is a cyclic quadrilateral and a tangent quadrilateral simultaneously. (By a [i]tangent quadrilateral[/i], we mean a quadrilateral that has an incircle.) Let the incircle of the quadrilateral $ ABCD$ touch its sides $ AB$, $ BC$, $ CD$, and $ DA$ in the points $ K$, $ L$, $ M$, and $ N$, respectively. The exterior angle bisectors of the angles $ DAB$ and $ ABC$ intersect each other at a point $ K^{\prime}$. The exterior angle bisectors of the angles $ ABC$ and $ BCD$ intersect each other at a point $ L^{\prime}$. The exterior angle bisectors of the angles $ BCD$ and $ CDA$ intersect each other at a point $ M^{\prime}$. The exterior angle bisectors of the angles $ CDA$ and $ DAB$ intersect each other at a point $ N^{\prime}$. Prove that the straight lines $ KK^{\prime}$, $ LL^{\prime}$, $ MM^{\prime}$, and $ NN^{\prime}$ are concurrent.

How many real numbers $x$ satisfy the equation $3^{2x+2}-3^{x+3}-3^{x}+3=0$? $\textbf {(A) } 0 \qquad \textbf {(B) } 1 \qquad \textbf {(C) } 2 \qquad \textbf {(D) } 3 \qquad \textbf {(E) } 4$

There is a board with the shape of an equilateral triangle with side $n$ divided into triangular cells with the shape of equilateral triangles with side $ 1$ (the figure below shows the board when $n = 4$). Each and every triangular cell is colored either red or blue. What is the least number of cells that can be colored blue without two red cells sharing one side? [img]https://cdn.artofproblemsolving.com/attachments/0/1/d1f034258966b319dc87297bdb311f134497b5.png[/img]

Cassidy has string of $n$ bits, where $n$ is a positive integer, which initially are all $0$s or $1$s. Every second, Cassidy may choose to do one of two things: 1. Change the first bit (so the first bit changes from a $0$ to a $1$, or vice versa) 2. Change the first bit after the first $1$. Let $M$ be the minimum number of such moves it takes to get from $1\dots 1$ to $0 \dots 0$ (both of length $12$), and $N$ the number of starting sequences with $12$ bits that Cassidy can turn into all $0$s. Find $M + N$.

Find all functions $f:\mathbb{N}\setminus\{1\} \rightarrow\mathbb{N}$ such that for all distinct $x,y\in \mathbb{N}$ with $y\ge 2018$, $$\gcd(f(x),y)\cdot \mathrm{lcm}(x,f(y))=f(x)f(y).$$

The Young Scientist and the Old Scientist play the following game, taking turns in an alternating fashion, with the Young Scientist starting first. The player on turn fills in one of the stars in the equation \[ x^4 + *x^3 + *x^2 + *x + * = 0 \] with a positive real number. Who has a winning strategy if the goals of the players are: a) the Young Scientist - to make the resulting equation have no real roots, and the Old Scientist -- to make it have real roots? b) the Young Scientist - to make the resulting equation have real roots, and the Old Scientist -- to make it have none?

Consider a polinomial $p \in \mathbb{R}[x]$ of degree $n$ and with no real roots. Prove that $$\int_{-\infty}^{\infty}\frac{(p'(x))^2}{(p(x))^2+(p'(x))^2}dx$$ converges, and is less or equal than $n^{3/2}\pi.$

The acute angles of a right triangle are $a^{\circ}$ and $b^{\circ}$, where $a>b$ and both $a$ and $b$ are prime numbers. What is the least possible value of $b$? $\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }5\qquad\textbf{(D) }7\qquad\textbf{(E) }11$

Four positive integers $x,y,z$ and $t$ satisfy the relations \[ xy - zt = x + y = z + t. \] Is it possible that both $xy$ and $zt$ are perfect squares?

Let $E$ be a finite set of points in space such that $E$ is not contained in a plane and no three points of $E$ are collinear. Show that $E$ contains the vertices of a tetrahedron $T = ABCD$ such that $T \cap E = \{A,B,C,D\}$ (including interior points of $T$ ) and such that the projection of $A$ onto the plane $BCD$ is inside a triangle that is similar to the triangle $BCD$ and whose sides have midpoints $B,C,D.$

Let $A=(a_{jk})$ be a $10\times 10$ array of positive real numbers such that the sum of numbers in row as well as in each column is $1$. Show that there exists $j<k$ and $l<m$ such that \[a_{jl}a_{km}+a_{jm}a_{kl}\ge \frac{1}{50}\]

Let $n\ge 1$ be a positive integer. A square of side length $n$ is divided by lines parallel to each side into $n^2$ squares of side length $1$. Find the number of parallelograms which have vertices among the vertices of the $n^2$ squares of side length $1$, with both sides smaller or equal to $2$, and which have tha area equal to $2$. (Greece)

Let $a,b,c,d$ be odd integers such that $0<a<b<c<d$ and $ad=bc$. Prove that if $a+d=2^k$ and $b+c=2^m$ for some integers $k$ and $m$, then $a=1$.

Does there exist a surjective function $f:R \to R$ such that the expression $f(x + y) - f(x) - f(y)$ takes exactly two values $0$ and $1$ for various real $x$ and $y$ ? (E. Barabanov)

Let $ABC$ be an acute triangle ($AB < BC < AC$) with circumcircle $\Gamma$. Assume there exists $X \in AC$ satisfying $AB=BX$ and $AX=BC$. Points $D, E \in \Gamma$ are taken such that $\angle ADB<90^{\circ}$, $DA=DB$ and $BC=CE$. Let $P$ be the intersection point of $AE$ with the tangent line to $\Gamma$ at $B$, and let $Q$ be the intersection point of $AB$ with tangent line to $\Gamma$ at $C$. Show that the projection of $D$ onto $PQ$ lies on the circumcircle of $\triangle PAB$.

Consider the integral lattice $\mathbb{Z}^n$, $n \geq 2$, in the Euclidean $n$-space. Define a [i]line[/i] in $\mathbb{Z}^n$ to be a set of the form $a_1 \times \cdots \times a_{k-1} \times \mathbb{Z} \times a_{k+1} \times \cdots \times a_n$ where $k$ is an integer in the range $1,2,\ldots,n$, and the $a_i$ are arbitrary integers. A subset $A$ of $\mathbb{Z}^n$ is called [i]admissible[/i] if it is non-empty, finite, and every [i]line[/i] in $\mathbb{Z}^{n}$ which intersects $A$ contains at least two points from $A$. A subset $N$ of $\mathbb{Z}^n$ is called [i]null[/i] if it is non-empty, and every [i]line[/i] in $\mathbb{Z}^n$ intersects $N$ in an even number of points (possibly zero). [b](a)[/b] Prove that every [i]admissible[/i] set in $\mathbb{Z}^2$ contains a [i]null[/i] set. [b](b)[/b] Exhibit an [i]admissible[/i] set in $\mathbb{Z}^3$ no subset of which is a [i]null[/i] set .

Two circles touch in $M$, and lie inside a rectangle $ABCD$. One of them touches the sides $AB$ and $AD$, and the other one touches $AD,BC,CD$. The radius of the second circle is four times that of the first circle. Find the ratio in which the common tangent of the circles in $M$ divides $AB$ and $CD$.

Let $a,b,c$ be the positive real numbers satisfying $a^2+b^2+c^2=3$. Prove that: $$\frac{a}{b(a+c)}+\frac{b}{c(b+a)}+\frac{c}{a(c+b)}\geq \frac{3}{2}.$$

In a triangle $ABC$ points $A_1,B_1,C_1$ are the midpoints of $BC,CA,AB$ respectively,and $A_2,B_2,C_2$ are the midpoints of the altitudes from $A,B,C$ respectively. Show that the lines $A_1A_2,B_1B_2,C_1,C_2$ are concurrent.

$P\in A_n(\mathbb R)=\{M_{n\times n}|M^2=M\}$. Which of the following are true? $\textbf{(A)}~P^T=P,\forall P\in A_n(\mathbb R)$ $\textbf{(B)}~\exists P\ne0,P\in A_n(\mathbb R)\text{ with }\operatorname{tr}(P)=0$ $\textbf{(C)}~\exists X_{n\times r}\text{ such that }Px=X\text{ for }r=\operatorname{rank}(P)$

[b]Q10.[/b] Solve the following equation \[\frac{1}{(x+29)^2}+ \frac{1}{(x+30)^2}= \frac{13}{36}.\]

$300$ people took part in the drawing for the main prize of the television lottery. They lined up in a circle, then, starting with someone who received number $1$, they began to count them. Moreover, every third person dropped out every time. (So, in the first round, everyone with numbers divisible by $3$ dropped out). The counting continued until there was only one person left. (It is clear that more than one circle was made). This person received the main prize. (It “accidentally” turned out to be the TV director’s mother-in-law). What number did this person have in the initial lineup?

A man has a seven-candle chandellier. The first evening he lighted one candle for one hour, the second evening he lighted two candles, also for one hour, and so on. After one hour the seventh evening, all seven candles simultaneously finished. How did the man choose the candles to light every evening?

What is the last digit of ${17^{17^{17^{17}}}}$?

Given the tetrahedron $ABCD$, whose opposite edges are equal, that is, $AB=CD, AC=BD$ and $BC=AD$. Prove that exist exactly $6$ planes intersecting the triangular angles of the tetrahedron and dividing the total surface and volume of this tetrahedron in half.