Find all positive integers $a$ and $b$ for which there are three consecutive integers at which the polynomial \[ P(n) = \frac{n^5+a}{b} \] takes integer values.

A sequence $\{an\}$ of positive integers is defined by \[a_n=\left[ n +\sqrt n + \frac 12 \right] , \qquad \forall n \in \mathbb N\] Determine the positive integers that occur in the sequence.

If $x=14$ and $y=6$, then compute $\tfrac{x^2-y^2}{x-y}$.

Let $ABC$ be a right triangle, right at $B$, and let $M$ be the midpoint of the side $BC$. Let $P$ be the point in bisector of the angle $ \angle BAC$ such that $PM$ is perpendicular to $BC (P$ is outside the triangle $ABC$). Determine the triangle area $ABC$ if $PM = 1$ and $MC = 5$.

Let $f(x)=x^3 +17$. Prove that for each natural number $n \ge 2$, there is a natural number $x$ for which $f(x)$ is divisible by $3^n$ but not $3^{n+1}$.

From the sequence $1,\frac12, \frac13, ...$ can one choose (a) a subsequence of $100$ different numbers, (b) an infinite subsequence such that each number (beginning from the third) is equal to the difference between the two preceding numbers ($a_k=a_{k-2}-a_{k-1}$)? (SI Tokarev)

Each time you click a toggle switch, the switch either turns from [i]off[/i] to [i]on[/i] or from [i]on[/i] to [i]off[/i]. Suppose that you start with three toggle switches with one of them [i]on[/i] and two of them [i]off[/i]. On each move you randomly select one of the three switches and click it. Let $m$ and $n$ be relatively prime positive integers so that $\frac{m}{n}$ is the probability that after four such clicks, one switch will be [i]on[/i] and two of them will be [i]off[/i]. Find $m+n$.

Consider an integer $n \geq 1$, $a_1,a_2, \ldots , a_n$ real numbers in $[-1,1]$ satisfying \begin{align*}a_1+a_2+\ldots +a_n=0 \end{align*} and a function $f: [-1,1] \mapsto \mathbb{R}$ such \begin{align*} \mid f(x)-f(y) \mid \le \mid x-y \mid \end{align*} for every $x,y \in [-1,1]$. Prove \begin{align*} \left| f(x) - \frac{f(a_1) +f(a_2) + \ldots + f(a_n)}{n} \right| \le 1 \end{align*} for every $x$ $\in [-1,1]$. For a given sequence $a_1,a_2, \ldots ,a_n$, Find $f$ and $x$ so hat the equality holds.

In a sequence of natural numbers, the first number is $a$, and each subsequent number is the smallest number coprime to all the previous ones and greater than all of them. Prove that in this sequence from some place all numbers will be primes.

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 $0$.