Set $A$ consists of $7$ consecutive positive integers less than $2011$, while set $B$ consists of $11$ consecutive positive integers. If the sum of the numbers in $A$ is equal to the sum of the numbers in $B$ , what is the maximum possible element that $A$ could contain?

Find all triples $(a, b, c)$ of integers such that $a+ b + c = 2010 \cdot 2011 $ and the solutions to the equation $$2011x^3 +ax^2 +bx+c = 0$$are all nonzero integers.

Find all positive integers $n$ such that $27^n- 2^n$ is a perfect square.

Let $ABC$ be a triangle with $AB=AC$ and $\angle BAC = 40^o$. Points $S$ and $T$ lie on the sides $AB$ and $BC$, such that $\angle BAT = \angle BCS = 10^o$. Lines $AT$ and $CS$ meet at $P$. Prove that $BT = 2PT$.

The shape of a military base is an equilateral triangle of side $10$ kilometers. Security constraints make cellular phone com­munication possible only within $2.5$ kilometers. Each of $17$ soldiers patrols the base randomly and tries to contact all others. Prove that at each moment at least two soldiers can communicate.

Prove that for any positive real numbers $a, b, c$, $$2(a^3 + b^3 + c^3 + abc) \ge (a+b)(b + c)(c + a)$$.

Let $x, y$ be distinct positive integers. Prove that the number $$\frac{(x+y)^2}{ x^3 + xy^2 - x^2y - y^3}$$is not an integer

Let $ABCD$ be a rectangle of center $O$, such that $\angle DAC = 60^o$. The angle bisector of $\angle DAC$ meets $DC$ at $S$. Lines $OS$ and $AD$ meet at $L$ and lines $BL$ and $AC$ meet at $M$. Prove that lines $SM$ and $CL$ are parallel.

Prove that $$\frac{\sin^3 a}{\sin b} +\frac{\cos^3 a}{\cos b} \ge \frac{1}{\cos(a - b)}$$for all $a$ and $b$ in the interval $(0, \pi/2)$ .

Prove that for each $n \ge 4$ a parallelogram can be dissected in $n$ cyclic quadrilaterals.

In the isosceles triangle $ABC$, with $AB = AC$, the angle bisector of $\angle B$ intersects side $AC$ at $B'$. Suppose that $ B B' + B'A = BC$. Find the angles of the triangle.

Find all quadruples $(x,y,z,w)$ of integers satisfying the sys­tem of equations $$x + y + z + w = xy + yz + zx + w^2 - w = xyz - w^3 = - 1$$

A Geostationary Earth Orbit is situated directly above the equator and has a period equal to the Earth’s rotational pe­riod. It is at the precise distance of $22,236$ miles above the Earth that a satellite can maintain an orbit with a period of rotation around the Earth exactly equal to $24$ hours. Be­ cause the satellites revolve at the same rotational speed of the Earth, they appear stationary from the Earth surface. That is why most station antennas (satellite dishes) do not need to move once they have been properly aimed at a tar­ get satellite in the sky. In an international project, a total of ten stations were equally spaced on this orbit (at the precise distance of $22,236$ miles above the equator). Given that the radius of the Earth is $3960$ miles, find the exact straight dis­tance between two neighboring stations. Write your answer in the form $a + b\sqrt{c}$, where $a, b, c$ are integers and $c > 0$ is square-free.

Pentagon $ABCDE$ is inscribed in a circle. Distances from point $E$ to lines $AB$ , $BC$ and $CD$ are equal to $a, b$ and $c$, respectively. Find the distance from point $E$ to line $AD$.

Let $n \ge 2$ be a positive integer and let $x_n$ be a positive real root to the equation $x(x+1)...(x + n) = 1$. Prove that $$x_n <\frac{1}{\sqrt{n! H_n}}$$where $H_n = 1+\frac12+...+\frac{1}{n}$.

Let $a, b, c, d$ be positive integers such that $a+b+c+d = 2011$. Prove that $2011$ is not a divisor of $ab - cd$.

Let $ABC$ be a triangle with $\angle A = 90^o$ and let $P$ be a point on the hypotenuse $BC$. Prove that $$\frac{AB^2}{PC}+\frac{AC^2}{PB} \ge \frac{BC^3}{PA^2 + PB \cdot PC}$$

Find all integers $n \ge 2$ for which $\sqrt[n]{3^n+ 4^n+5^n+8^n+10^n}$ is an integer.

Find all positive integers $x$ and $y$ such that $${x \choose y} = 1432$$

Points $A ,B ,C ,D$ lie on a line in this order. Draw parallel lines $a$ and $b$ through $A$ and $B$, respectively, and parallel lines $c$ and $d$ through $C$ and $D$, respectively, such that their points of intersection are vertices of a square. Prove that the side length of this square does not depend on the length of segment $BC$.

Let $a, b, c$ be positive real numbers. Prove that $$8(a+b+c) \left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a} \right) \le 9 \left(1+\frac{a}{b} \right)\left(1+\frac{b}{c} \right)\left(1+\frac{c}{a} \right)$$

Find all primes $q_1, q_2, q_3, q_4, q_5$ such that $q_1^4+q_2^4+q_3^4+q_4^4+q_5^4$ is the product of two consecutive even integers.

The quadrilateral $ABCD$ has $AD = DC = CB < AB$ and $AB \parallel CD$. Points $E$ and $F$ lie on the sides $CD$ and $BC$ such that $\angle ADE = \angle AEF$. Prove that: (a) $4CF \le CB$. (b) If $4CF = CB$, then $AE$ is the angle bisector of $\angle DAF$.

Let $f_n = 2^{2^n}+ 1$, $n = 1,2,3,...$, be the Fermat’s numbers. Find the least real number $C$ such that $$\frac{1}{f_1}+\frac{2}{f_2}+\frac{2^2}{f_3}+...+\frac{2^{n-1}}{f_n} <C$$for all positive integers $n$

Let $n$ be a positive integer. Prove that the interval $I_n= \left( \frac{1+\sqrt{8n+1}}{2}, \frac{1+\sqrt{8n+9}}{2}\right)$ does not contain any integer.

Consider the sequence $x_n = 2^n-n$, $n = 0,1 ,2 ,...$. Find all integers $m \ge 0$ such that $s_m = x_0 + x_1 + x_2 + ... + x_m$ is a power of $2$.

Let $f = aX^2 + bX+ c \in Z[X]$ be a polynomial such that for every positive integer $n$,$ f(n )$ is a perfect square. Prove that $f = g^2$ for some polynomial $g \in Z[X]$.

Let $ABC$ be a triangle with medians $m_a$ , $m_b$, $m_c$. Prove that: (a) There is a triangle with side lengths $m_a$ ,$m_b$, $m_c$. (b) This triangle is similar to $ABC$ if and only if the squares of the side lengths of triangle $ABC$ form an arithmetical sequence.

Let $n$ be a positive integer such that $2011^{2011}$ divides $n!$. Prove that $2011^{2012} $divides $n!$ .

Find all pairs of nonnegative integers $(a, b)$ such that $a+2b-b^2=\sqrt{2a+a^2+|2a+1-2b|}$.

Let $P$ be a point in the interior of triangle $ABC$. Lines $AP$, $BP$, $CP$ intersect sides $BC$, $CA$, $AB$ at $L$, $M$, $N$, respec­tively. Prove that $$AP \cdot BP \cdot CP \ge 8PL \cdot PM \cdot PN.$$

Find all positive integers $n$ for which the equation $$x^3 + y^3 = n! + 4$$has solutions in integers.

On a semicircle of diameter $AB$ and center $C$, consider vari­able points $M$ and $N$ such that $MC \perp NC$. The circumcircle of triangle $MNC$ intersects $AB$ for the second time at $P$. Prove that $\frac{|PM-PN|}{PC}$ constant and find its value.

Find positive integers $a_1 < a_2<... <a_{2010}$ such that $$a_1(1!)^{2010} + a_2(2!)^{2010} + ... + a_{2010}(2010!)^{2010} = (2011 !)^{2010}. $$

Let $x_1,x_2,...,x_n$ be positive real numbers for which $$\frac{1}{1+x_1}+\frac{1}{1+x_2}+...+\frac{1}{1+x_n}=1$$Prove that $x_1x_2...x_n \ge (n -1)^n$.

In a triangle $ABC$, let $O$ be the circumcenter, $H$ the ortho­center, and $M$ the midpoint of the segment $AH$. The perpendicular at $M$ onto $OM$ intersects lines $AB$ and $AC$ at $P$ and $Q$, respectively. Prove that $MP = MQ$.