Compute the sum of all the roots of $ (2x \plus{} 3)(x \minus{} 4) \plus{} (2x \plus{} 3)(x \minus{} 6) \equal{} 0$. $ \textbf{(A)}\ 7/2 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 13$

Do there exist a circle and an infinite set of points on it such that the distance between any two of the points is rational?

Baron and Peter are playing a game. They are given a simple finite graph $G$ with $n\ge 3$ vertex and $k$ edges that connects the vertices. First Peter labels two vertices A and B, and places a counter at A. Baron starts first. A move for Baron is move the counter along an edge. Peter's move is to remove an edge from the graph. Baron wins if he reaches $B$, otherwise Peter wins. Given the value of $n$, what is the largest $k$ so that Peter can always win?

Suppose the estimated $20$ billion dollar cost to send a person to the planet Mars is shared equally by the $250$ million people in the U.S. Then each person's share is $ \text{(A)}\ 40\text{ dollars}\qquad\text{(B)}\ 50\text{ dollars}\qquad\text{(C)}\ 80\text{ dollars}\qquad\text{(D)}\ 100\text{ dollars}\qquad\text{(E)}\ 125\text{ dollars} $

Prove that for any natural number $n > 1$, \[ \frac{1}{2} < \frac{1}{n^2+1} + \frac{2}{n^2 +2} + \ldots + \frac{n}{n^2 + n} < \frac{1}{2} + \frac{1}{2n}. \]

Let $x,y,z{}$ be positive real numbers. Prove that \[(xy+yz+zx)\left(\frac{1}{x^2+y^2}+\frac{1}{y^2+z^2}+\frac{1}{z^2+x^2}\right)>\frac{5}{2}.\]

Determine all functions $f: \mathbb{R} \to \mathbb{R}$ satisfying \[f(x + yf(x)) + f(xy) = f(x) + f(2019y),\] for all real numbers $x$ and $y$.

Let $P(x)$ and $Q(x)$ be polynomials with integer coefficients. Let $a_n = n! +n$. Show that if $\frac{P(a_n)}{Q(a_n)}$ is an integer for every $n$, then $\frac{P(n)}{Q(n)}$ is an integer for every integer $n$ such that $Q(n)\neq 0$.

In ZS Chess, an Ivanight attacks like a knight, except that if the attacked square is out of range, it goes through the edge and comes out from the other side of the board, and attacks that square instead. The ZS chessboard is an $8 \times 8$ board, where cells are coloured with $n$ distinct colours, where $n$ is a natural number, such that a Ivanight placed on any square attacks $ 8 $ squares that consist of all $n$ colours, and the colours appear equally many times in those $ 8 $ squares. For which values of $n$ does such a ZS chess board exist?

On a circle $O$ with radius $\overline{OA}$, points $B$ and $C$ are drawn such that $\angle AOC = \angle BOA = 30^\circ$, as shown. A second circle passing through $B$, $C$, and the midpoint of $\overline{OA}$ is drawn. The ratio of the radius of this new circle to the radius of circle $O$ can be expressed in the form $\tfrac{a \sqrt 3 - b}{c}$ where $a$, $b$, and $c$ are positive integers and $c$ is as small as possible. What is $a + b + c$? [asy] size(100); pair O,A,B,C; O = (0,0); label("$O$",O,W); A = (2,0); label("$A$",A,E); B = (sqrt(3),1); label("$B$",B,N*1.8); C = (sqrt(3),-1); label("$C$",C,S*1.8); draw(Circle(O,2)); dot((1,0)^^A^^B^^C^^O); draw(O--B); draw(O--C); draw(O--A); draw(Circle((2.04904,0),1.04904),dashed); [/asy] [center]Note: In the diagram, $A$ is not necessarily the center of the second circle.[/center] $\textbf{(A) }10\qquad\textbf{(B) }12\qquad\textbf{(C) }15\qquad\textbf{(D) }21\qquad\textbf{(E) }27$

Let $ABCD$ be a quadrilateral inscribed in a circle $\omega$, and let $P$ be a point on the extension of $AC$ such that $PB$ and $PD$ are tangent to $\omega$. The tangent at $C$ intersects $PD$ at $Q$ and the line $AD$ at $R$. Let $E$ be the second point of intersection between $AQ$ and $\omega$. Prove that $B$, $E$, $R$ are collinear.

Evaluate $ \int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \frac{x^2}{(1\plus{}x\tan x)(x\minus{}\tan x)\cos ^ 2 x}\ dx.$

Determine all natural integers $n$ for which there is no triplet $(a, b, c)$ of natural numbers such that: $$n = \frac{a \cdot \,\,lcm(b, c) + b \cdot lcm \,\,(c, a) + c \cdot lcm \,\, (a, b)}{lcm \,\,(a, b, c)}$$

Let $f$ be an injective function from ${1,2,3,\ldots}$ in itself. Prove that for any $n$ we have: $\sum_{k=1}^{n} f(k)k^{-2} \geq \sum_{k=1}^{n} k^{-1}.$

Let $O$ be the circumcenter and $H$ the orthocenter of an acute triangle $ABC$. Show that there exist points $D$, $E$, and $F$ on sides $BC$, $CA$, and $AB$ respectively such that \[ OD + DH = OE + EH = OF + FH\] and the lines $AD$, $BE$, and $CF$ are concurrent.

Exactly one number from the set $\{ -1,0,1 \}$ is written in each unit cell of a $2005 \times 2005$ table, so that the sum of all the entries is $0$. Prove that there exist two rows and two columns of the table, such that the sum of the four numbers written at the intersections of these rows and columns is equal to $0$.

The integers from $1$ to $1000000$ are divided into two groups consisting of numbers with odd or even sums of digits respectively. Which group contains more numbers?

In triangle $ABC$(with incenter $I$) let the line parallel to $BC$ from $A$ intersect circumcircle of $\triangle ABC$ at $A_1$ let $AI\cap BC=D$ and $E$ is tangency point of incircle with $BC$ let $ EA_1\cap \odot (\triangle ADE)=T$ prove that $AI=TI$.

Consider the paths from $(0,0)$ to $(6,3)$ that only take steps of unit length up and right. Compute the sum of the areas bounded by the path, the $x$-axis, and the line $x = 6$ over all such paths. (In particular, the path from $(0,0)$ to $(6,0)$ to $(6,3)$ corresponds to an area of $0.$)

Let $n$ be a positive integer. A sequence of $n$ positive integers (not necessarily distinct) is called [b]full[/b] if it satisfies the following condition: for each positive integer $k\geq2$, if the number $k$ appears in the sequence then so does the number $k-1$, and moreover the first occurrence of $k-1$ comes before the last occurrence of $k$. For each $n$, how many full sequences are there ?

If $T_{n} = 1 + 2 + 3 + \ldots + n$ and \[P_{n} = \frac{T_{2}}{T_{2} - 1} \cdot \frac{T_{3}}{T_{3} - 1} \cdot \frac{T_{4}}{T_{4} - 1} \cdot\,\, \cdots \,\,\cdot \frac{T_{n}}{T_{n} - 1}\quad\text{for }n = 2,3,4,\ldots,\] then $P_{1991}$ is closest to which of the following numbers? $ \textbf{(A)}\ 2.0\qquad\textbf{(B)}\ 2.3\qquad\textbf{(C)}\ 2.6\qquad\textbf{(D)}\ 2.9\qquad\textbf{(E)}\ 3.2 $

There are $ n$ markers, each with one side white and the other side black. In the beginning, these $ n$ markers are aligned in a row so that their white sides are all up. In each step, if possible, we choose a marker whose white side is up (but not one of the outermost markers), remove it, and reverse the closest marker to the left of it and also reverse the closest marker to the right of it. Prove that, by a finite sequence of such steps, one can achieve a state with only two markers remaining if and only if $ n \minus{} 1$ is not divisible by $ 3$. [i]Proposed by Dusan Dukic, Serbia[/i]

Find all primes $p,q,r$ and natural numbers $n$ such that $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=\frac{1}{n}$.

Let $U$ be an open subset of the complex plane $\mathbb{C}$ including $\mathbb{D}=\{z \in \mathbb{C} : |z| \le 1\}$ and $f$ be analytic over $U$. Prove that if for every $z$ with a complex norm equal to $1$($|z|=1$) we have $0<Re(\bar{z}f(z))$, then $f$ has only one root in $\mathbb{D}$ and that's simple.

In a conference $2n$ personalities take apart. Each person has at least $n$ acquaintaces among the others. Prove that it is possible to quarter the participants into two-person rooms, so that each participant would share the room with his/her acquaintace.