Let $n$ be an integer greater than $1$. Let $x\in\mathbb R$. (a) Evaluate $S(x,n)=\sum(x+p)(x+q)$, where the summation is over all pairs $(p,q)$ of different numbers from $\{1,2,\ldots,n\}$. (b) Do there exist integers $x,n$ for which $S(x,n)=0$?

Let $OABC$ be a tetrahedon with $\angle BOC=\alpha,\angle COA =\beta$ and $\angle AOB =\gamma$. The angle between the faces $OAB$ and $OAC$ is $\sigma$ and the angle between the faces $OAB$ and $OBC$ is $\rho$. Show that $\gamma > \beta \cos\sigma + \alpha \cos\rho$.

In a $2 \times 3$ rectangle there is a polyline of length $36$, which can have self-intersections. Show that there exists a line parallel to two sides of the rectangle, which intersects the other two sides in their interior points and intersects the polyline in fewer than $10$ points. [i]Proposed by Josef Tkadlec, Czechia and Vojtech Bálint, Slovakia[/i]

Let $(a_n)_{n\ge 1}$ be an increasing sequence of positive integers. Assume that there is a constant $M>0$ satisfying$$0<a_{n+1}-a_n<M.a_n^{5/8},\forall n\ge 1.$$ Prove that: there exists a real number $A$ such that for each $k\in \mathbb{Z}^+,[A^{3^k}]$ is an element of $(a_n)_{n\ge 1}.$

The polynomial $P (x)$ is such that the polynomials $P (P (x))$ and $P (P (P (x)))$ are strictly monotone on the whole real axis. Prove that $P (x)$ is also strictly monotone on the whole real axis.

a) Find an integer $a$ for which $(x - a)(x - 10) + 1$ factors in the product $(x + b)(x + c)$ with integers $b$ and $c$. b) Find nonzero and nonequal integers $a, b, c$ so that $x(x - a)(x - b)(x - c) + 1$ factors into the product of two polynomials with integer coefficients.

A number is called [i]winning[/i] if it can be expressed in the form $\frac{a}{20}+\frac{b}{23}$ where $a$ and $b$ are positive integers. How many [i]winning[/i] numbers are less than 1? [i]Proposed by Andy Xu[/i]

For which real $a$ are there distinct reals $x$, $y$ such that $$\begin{cases} x = a - y^2 \\ y = a - x^2 \,\,\, ? \end {cases}$$

Suppose that $x$ and $y$ are nonzero real numbers such that \[\frac{3x+y}{x-3y}= -2.\] What is the value of \[\frac{x+3y}{3x-y}?\] $\textbf{(A) } {-3} \qquad \textbf{(B) } {-1} \qquad \textbf{(C) } 1 \qquad \textbf{(D) }2 \qquad \textbf{(E) } 3$

Let $ABCD$ be a convex quadrilateral. $DA$ and $CB$ meet at $F$ and $AB$ and $DC$ meet at $E$. The bisectors of the angles $DFC$ and $AED$ are perpendicular. Prove that these angle bisectors are parallel to the bisectors of the angles between the lines $AC$ and $BD.$

A point $K$ is chosen on the side $CD$ of a rectangle $ABCD$. From the vertex $B$, the perpendicular $BH$ is dropped to the segment $AK$. The segments $AK$ and $BH$ divide the rectangle into three parts such that each of them has the inscribed circle (see figure). Prove that if the circles tangent to $CD$ are equal then the third circle is also equal to them.

The figure below shows a regular $ABCDE$ pentagon inscribed in an equilateral triangle $MNP$ . Determine the measure of the angle $CMD$. [img]http://4.bp.blogspot.com/-LLT7hB7QwiA/Xp9fXOsihLI/AAAAAAAAL14/5lPsjXeKfYwIr5DyRAKRy0TbrX_zx1xHQCK4BGAYYCw/s200/2012%2Bobm%2Bl2.png[/img]

Let $a$ and $b$ be real numbers for which the following is true: $acscx + b cot x \ge 1$, for all $0 <x < \pi$ Find the least value of $a^2 + b$.

Eight points are chosen on the circumference of a circle, labelled $P_1$, $P_2$, ..., $P_8$ in clockwise order. A route is a sequence of at least two points $P_{a_1}$, $P_{a_2}$, $...$, $P_{a_n}$ such that if an ant were to visit these points in their given order, starting at $P_{a_1}$ and ending at $P_{a_n}$, by following $n-1$ straight line segments (each connecting each $P_{a_i}$ and $P_{a_{i+1}}$), it would never visit a point twice or cross its own path. Find the number of routes.

A function $ f: R^3\rightarrow R$ for all reals $ a,b,c,d,e$ satisfies a condition: \[ f(a,b,c)\plus{}f(b,c,d)\plus{}f(c,d,e)\plus{}f(d,e,a)\plus{}f(e,a,b)\equal{}a\plus{}b\plus{}c\plus{}d\plus{}e\] Show that for all reals $ x_1,x_2,\ldots,x_n$ ($ n\geq 5$) equality holds: \[ f(x_1,x_2,x_3)\plus{}f(x_2,x_3,x_4)\plus{}\ldots \plus{}f(x_{n\minus{}1},x_n,x_1)\plus{}f(x_n,x_1,x_2)\equal{}x_1\plus{}x_2\plus{}\ldots\plus{}x_n\]

Three faces of a regular tetrahedron are painted in white and the remaining one in black. Initially, the tetrahedron is positioned on a plane with the black face down. It is then tilted several times over its edges. After a while it returns to its original position. Can it now have a white face down?

Let $m$ and $n$ be two positive integers greater than $1$. Prove that there are $m$ positive integers $N_1$ , $\ldots$ , $N_m$ (some of them may be equal) such that \[\sqrt{m}=\sum_{i=1}^m{(\sqrt{N_i}-\sqrt{N_i-1})^{\frac{1}{n}}.}\]

Let $n$ be a positive integer. We start with $n$ piles of pebbles, each initially containing a single pebble. One can perform moves of the following form: choose two piles, take an equal number of pebbles from each pile and form a new pile out of these pebbles. Find (in terms of $n$) the smallest number of nonempty piles that one can obtain by performing a finite sequence of moves of this form.

Let $S$ S be a point chosen at random from the interior of the square $ABCD$, which has side $AB$ and diagonal $AC$. Let $P$ be the probability that the segments $AS$, $SB$, and $AC$ are congruent to the sides of a triangle. Then $P$ can be written as $\dfrac{a-\pi\sqrt{b}-\sqrt{c}}{d}$ where $a,b,c,$ and $d$ are all positive integers and $d$ is as small as possible. Find $ab+cd$.

The six children listed below are from two families of three siblings each. Each child has blue or brown eyes and black or blond hair. Children from the same family have at least one of these characteristics in common. Which two children are Jim's siblings? \[ \begin{array}{c|c|c}\text{Child}&\text{Eye Color}&\text{Hair Color}\\ \hline \text{Benjamin}& \text{Blue} & \text{Black} \\ \hline \text{Jim} & \text{Brown} & \text{Blonde} \\ \hline \text{Nadeen} & \text{Brown} & \text{Black}\\ \hline \text{Austin}& \text{Blue} & \text{Blonde}\\ \hline \text{Tevyn} & \text{Blue} & \text{Black} \\ \hline \text{Sue} & \text{Blue} & \text{Blonde} \\ \hline \end{array} \] $\textbf{(A)}\ \text{Nadeen and Austin} \qquad \textbf{(B)}\ \text{Benjamin and Sue}\qquad \textbf{(C)}\ \text{Benjamin and Austin}\qquad$ $\textbf{(D)}\ \text{Nadeen and Tevyn} \qquad \textbf{(E)}\ \text{Austin and Sue} $

John's digital clock is broken. It scrambles the digits of the time and displays them in a random order. For example, if the current time is $4:21$, it could display $4:12$, $2:14$, or any other reordering of $4$, $1$, and $2$. If his clock reads $6:71$ one morning, how many possibilities are there for the correct time? $\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }6$

Triangle $ABC$ has circumcenter $O$ and circumcircle $\omega$. Let $A_{\omega}$ be the point diametrically opposite $A$ on $\omega$, and let $H$ be the foot of the altitude from $A$ onto $BC$. Let $H_B$ and $H_C$ be the reflections of $H$ over $B$ and $C$, respectively. Point $P$ is the intersection of line $A_{\omega}B$ and the perpendicular of $BC$ at point $H_B$, and point $Q$ is the intersection of line $A_{\omega}C$ and the perpendicular of $CB$ at point $H_C$. The circles $\omega_1$ and $\omega_2$ have the respective centers $P$ and $Q$ and respective radii $PA$ and $QA$. Suppose that $\omega$, $\omega_1$, and $\omega_2$ intersect at another common point $X$. If $AO = \frac{\sqrt{105}}{5}$ and $AX = 4$, then $|AB - CA|^2$ can be written as $m - n\sqrt{p}$ for positive integers $m$ and $n$ and squarefree positive integer $p$. Find $m + n + p$. [i]Note: the reflection of a point $P$ over another point $Q \neq P$ is the point $P'$ such that $Q$ is the midpoint of $P$ and $P'$.[/i]

$ABC$ is an acute-angled triangle. Let $D$ be the point on $BC$ such that $AD$ is the bisector of $\angle A$. Let $E, F$ be the feet of perpendiculars from $D$ to $AC,AB$ respectively. Suppose the lines $BE$ and $CF$ meet at $H$. The circumcircle of triangle $AFH$ meets $BE$ at $G$ (apart from $H$). Prove that the triangle constructed from $BG$, $GE$ and $BF$ is right-angled.

Prove that if $t$ is a natural number then there exists a natural number $n>1$ such that $(n,t)=1$ and none of the numbers $n+t,n^2+t,n^3+t,....$ are perfect powers.

Given any $22$ points in the plane, no three collinear. Show that the points can be divided into $11$ pairs, so that the $11$ line segments defined by the pairs have at least five different intersections