In triangle $ABC,$ the points $D$ and $E$ are the intersections of the angular bisectors from $C$ and $B$ with the sides $AB$ and $AC,$ respectively. Points $F$ and $G$ on the extensions of $AB$ and $AC$ beyond $B$ and $C,$ respectively, satisfy $BF = CG = BC.$ Prove that $F G \parallel DE.$

Denote the number of different prime divisors of the number $n$ by $\omega (n)$, where $n$ is an integer greater than $1$. Prove that there exist infinitely many numbers $n$ for which $\omega (n)< \omega (n+1)<\omega (n+2)$ holds.

Three (not necessarily distinct) points in the plane which have integer coordinates between $ 1$ and $2020$, inclusive, are chosen uniformly at random. The probability that the area of the triangle with these three vertices is an integer is $a/b$ in lowest terms. If the three points are collinear, the area of the degenerate triangle is $0$. Find $a + b$.

The numbers -2, 4, 6, 9 and 12 are rearranged according to these rules: 1. The largest isn't first, but it is in one of the first three places. 2. The smallest isn't last, but it is in one of the last three places. 3. The median isn't first or last. What is the average of the first and last numbers? $\textbf{(A)}\: 3.5 \qquad\textbf{(B)} \:5 \qquad\textbf{(C)} \:6.5 \qquad\textbf{(D)} \:7.5 \qquad\textbf{(E)} \:8$

The sum of the squares of the roots of the equation $x^2+2hx=3$ is $10$. The absolute value of $h$ is equal to $\textbf{(A) }-1\qquad\textbf{(B) }\textstyle\frac{1}{2}\qquad\textbf{(C) }\textstyle\frac{3}{2}\qquad\textbf{(D) }2\qquad \textbf{(E) }\text{None of these}$

Let $P$ and $P^{\prime }$ be two parallelograms with equal area, and let their sidelengths be $a,$ $b$ and $a^{\prime },$ $b^{\prime }.$ Assume that $a^{\prime }\leq a\leq b\leq b^{\prime },$ and moreover, it is possible to place the segment $b^{\prime }$ such that it completely lies in the interior of the parallelogram $P.$ Show that the parallelogram $P$ can be partitioned into four polygons such that these four polygons can be composed again to form the parallelogram $% P^{\prime }.$

Consider the polynomial f(x) = cx(x - 2) where $c$ is a positive real number. For any $n \in Z^+$, the notation $g_n(x)$ is a composite function $n$ times of $f$ and assume that the equation $g_n(x) = 0$ has all of the $2^n$ solutions are real numbers. 1. For $c = 5$, find in terms of $n$, the sum of all the solutions of $g_n(x)$, of which each multiple (if any) is counted only once. 2. Prove that $c\ge 1$.

Let $m$ and $n$ be positive integers. A sequence of points $(A_0,A_1,\ldots,A_n)$ on the Cartesian plane is called [i]interesting[/i] if $A_i$ are all lattice points, the slopes of $OA_0,OA_1,\cdots,OA_n$ are strictly increasing ($O$ is the origin) and the area of triangle $OA_iA_{i+1}$ is equal to $\frac{1}{2}$ for $i=0,1,\ldots,n-1$. Let $(B_0,B_1,\cdots,B_n)$ be a sequence of points. We may insert a point $B$ between $B_i$ and $B_{i+1}$ if $\overrightarrow{OB}=\overrightarrow{OB_i}+\overrightarrow{OB_{i+1}}$, and the resulting sequence $(B_0,B_1,\ldots,B_i,B,B_{i+1},\ldots,B_n)$ is called an [i]extension[/i] of the original sequence. Given two [i]interesting[/i] sequences $(C_0,C_1,\ldots,C_n)$ and $(D_0,D_1,\ldots,D_m)$, prove that if $C_0=D_0$ and $C_n=D_m$, then we may perform finitely many [i]extensions[/i] on each sequence until the resulting two sequences become identical.

We place randomly the numbers $1,2, \dots ,2006$ around a circle. A move consists of changing two neighbouring numbers. After a limited numbers of moves all the numbers are diametrically opposite to their starting position. Show that we changed at least once two numbers which had the sum $2007$.

Let $ ABC$ an acute triangle with circumcircle center $ O.$ The line $ (BO)$ intersects the circumcircle again in $ D,$ and the extension of the altitude from $ A$ intersects the circle in $ E.$ Prove that the quadrilateral $ BECD$ and the triangle $ ABC$ have the same area.

Consider in a plane $ P$ the points $ O,A_1,A_2,A_3,A_4$ such that \[ \sigma(OA_iA_j) \geq 1 \quad \forall i, j \equal{} 1, 2, 3, 4, i \neq j.\] where $ \sigma(OA_iA_j)$ is the area of triangle $ OA_iA_j.$ Prove that there exists at least one pair $ i_0, j_0 \in \{1, 2, 3, 4\}$ such that \[ \sigma(OA_iA_j) \geq \sqrt{2}.\]

In the following, a [i]word[/i] will mean a finite sequence of letters "$a$" and "$b$". The [i]length[/i] of a word will mean the number of the letters of the word. For instance, $abaab$ is a word of length $5$. There exists exactly one word of length $0$, namely the empty word. A word $w$ of length $\ell$ consisting of the letters $x_1$, $x_2$, ..., $x_{\ell}$ in this order is called a [i]palindrome[/i] if and only if $x_j=x_{\ell+1-j}$ holds for every $j$ such that $1\leq j\leq\ell$. For instance, $baaab$ is a palindrome; so is the empty word. For two words $w_1$ and $w_2$, let $w_1w_2$ denote the word formed by writing the word $w_2$ directly after the word $w_1$. For instance, if $w_1=baa$ and $w_2=bb$, then $w_1w_2=baabb$. Let $r$, $s$, $t$ be nonnegative integers satisfying $r + s = t + 2$. Prove that there exist palindromes $A$, $B$, $C$ with lengths $r$, $s$, $t$, respectively, such that $AB=Cab$, if and only if the integers $r + 2$ and $s - 2$ are coprime.

The numbers from $1$ to $2000$ are placed on the vertices of a regular polygon with $2000$ sides, one on each vertex, so that the following is true: If four integers $A, B, C, D$ satisfy that $1 \leq A<B<C<D \leq 2000$, then the segment that joins the vertices of the numbers $A$ and $B$ and the segment that joins the vertices of $C$ and $D$ do not intersect inside the polygon. Prove that there exists a perfect square such that the number diametrically opposite to it is not a perfect square.

Given the rebus: $$AB \cdot AC \cdot BC = BBBCCC $$ where different letters correspond to different digits and the same letters to the same digits, find the sum $AB + AC + BC.$ [i]Proposed by Giorgi Arabidze, Georgia [/i]

Let $ABC$ be an acute-angled triangle. Let $A_1$ be the midpoint of the arc $BC$ which contain the point $A$. Let $A_2$ and $A_3$ be the foot(s) of the perpendicular(s) of the point $A_1$ to the lines $AB$ and $AC$, respectively. Define $B_2,B_3,C_2,C_3$ analogously. a) Prove that the line $A_2A_3$ cuts $BC$ in the midpoint. b) Prove that the lines $A_2A_3,B_2B_3$ and $C_2C_3$ are concurrents.

Given $m+n$ numbers: $a_i,$ $i = 1,2, \ldots, m,$ $b_j$, $j = 1,2, \ldots, n,$ determine the number of pairs $(a_i,b_j)$ for which $|i-j| \geq k,$ where $k$ is a non-negative integer.

Given is a triangle $ABC$ and a point $X$ inside its circumcircle. If $I_B, I_C$ denote the $B, C$ excenters, then prove that $XB \cdot XC <XI_B \cdot XI_C$.

Evaluate $\textstyle\sum_{n=0}^\infty \mathrm{Arccot}(n^2+n+1)$, where $\mathrm{Arccot}\,t$ for $t \geq 0$ denotes the number $\theta$ in the interval $0 < \theta \leq \pi/2$ with $\cot \theta = t$.

Triangle $ABC$ circumscribed $(O)$ has $A$-excircle $(J)$ that touches $AB,\ BC,\ AC$ at $F,\ D,\ E$, resp. a. $L$ is the midpoint of $BC$. Circle with diameter $LJ$ cuts $DE,\ DF$ at $K,\ H$. Prove that $(BDK),\ (CDH)$ has an intersecting point on $(J)$. b. Let $EF\cap BC =\{G\}$ and $GJ$ cuts $AB,\ AC$ at $M,\ N$, resp. $P\in JB$ and $Q\in JC$ such that $$\angle PAB=\angle QAC=90{}^\circ .$$ $PM\cap QN=\{T\}$ and $S$ is the midpoint of the larger $BC$-arc of $(O)$. $(I)$ is the incircle of $ABC$. Prove that $SI\cap AT\in (O)$.

Let $m,n$ be positive integers such that the equation (in respect of $x$) \[(x+m)(x+n)=x+m+n\] has at least one integer root. Prove that $\frac{1}{2}n<m<2n$.

\[\int_1^\infty \frac{\sec^{-1}(x^{2})-\sec^{-1}(\sqrt x)}{x\log(x)}\mathrm dx\] [i]Proposed by Connor Gordon[/i]

Sequences $(a_n), (b_n)$ are defined by $ a_1 = 1, b_1 = 2$, $a_{n+1} = \frac{ 1 + a_n + a_nb_n}{b_n}$, $ b_{n+1} = \frac{ 1 +b_n+ a_nb_n}{a_n}$ for all positive integers $n$. Prove that $a_{2020} < 5$.

The negative real numbers $x, y, z, t$ satisfy simultaneously equalities, $$x + y + z = t, \,\,\,\,\frac{1}{x}+ \frac{1}{y}+\frac{1}{z}= \frac{1}{t}, \\,\,\,\, x^3 + y^3 + z^3 = 1000^3$$ Compute $x + y + z + t$.

Determine all pairs $(a,b)$ of natural numbers such that the number $$\frac{a^2(b-a)}{b+a}$$ is the square of a prime number.

Let $N$ be the number of functions $f:\{1,2,3,4,5,6,7,8,9,10\} \rightarrow \{1,2,3,4,5\}$ that have the property that for $1\leq x\leq 5$ it is true that $f(f(x))=x$. Given that $N$ can be written in the form $5^a\cdot b$ for positive integers $a$ and $b$ with $b$ not divisible by $5$, find $a+b$. [i]Proposed by Nathan Ramesh