For the positive real numbers $ a,b,c$ prove the inequality \[ \frac {c}{a \plus{} 2b} \plus{} \frac {a}{b \plus{} 2c} \plus{} \frac {b}{c \plus{} 2a}\ge1. \]

Let $m$ circles intersect in points $A$ and $B$. We write numbers using the following algorithm: we write $1$ in points $A$ and $B$, in every midpoint of the open arc $AB$ we write $2$, then between every two numbers written in the midpoint we write their sum and so on repeating $n$ times. Let $r(n,m)$ be the number of appearances of the number $n$ writing all of them on our $m$ circles. a) Determine $r(n,m)$; b) For $n=2006$, find the smallest $m$ for which $r(n,m)$ is a perfect square. Example for half arc: $1-1$; $1-2-1$; $1-3-2-3-1$; $1-4-3-5-2-5-3-4-1$; $1-5-4-7-3-8-5-7-2-7-5-8-3-7-4-5-1$...

Find, with proof, all positive integers $n$ for which $2^n + 12^n + 2011^n$ is a perfect square.

For a real number $a$, denote by $(a]$ the smallest integer that is not less than $a$. Find all real values of $x$ for which holds the equality $$(\sin x]^2 + (\cos x]^2 =|tg x| +|ctg x|.$$

The parallelepiped contains a sphere of radius $r$ and is contained within a sphere of radius $R$. Prove that $ \frac{R}{r} \geq \sqrt{3} $.

Prove that the line joining the centroid and the incenter of a non-isosceles triangle is perpendicular to the base if and only if the sum of the other two sides is thrice the base.

Consider a triangle $XYZ$ and a point $O$ in its interior. Three lines through $O$ are drawn, parallel to the respective sides of the triangle. The intersections with the sides of the triangle determine six line segments from $O$ to the sides of the triangle. The lengths of these segments are integer numbers $a, b, c, d, e$ and $f$ (see figure). Prove that the product $a \cdot b \cdot c\cdot d \cdot e \cdot f$ is a perfect square. [asy] unitsize(1 cm); pair A, B, C, D, E, F, O, X, Y, Z; X = (1,4); Y = (0,0); Z = (5,1.5); O = (1.8,2.2); A = extension(O, O + Z - X, X, Y); B = extension(O, O + Y - Z, X, Y); C = extension(O, O + X - Y, Y, Z); D = extension(O, O + Z - X, Y, Z); E = extension(O, O + Y - Z, Z, X); F = extension(O, O + X - Y, Z, X); draw(X--Y--Z--cycle); draw(A--D); draw(B--E); draw(C--F); dot("$A$", A, NW); dot("$B$", B, NW); dot("$C$", C, SE); dot("$D$", D, SE); dot("$E$", E, NE); dot("$F$", F, NE); dot("$O$", O, S); dot("$X$", X, N); dot("$Y$", Y, SW); dot("$Z$", Z, dir(0)); label("$a$", (A + O)/2, SW); label("$b$", (B + O)/2, SE); label("$c$", (C + O)/2, SE); label("$d$", (D + O)/2, SW); label("$e$", (E + O)/2, SE); label("$f$", (F + O)/2, NW); [/asy]

Let $B$ and $C$ be two fixed points in the plane. For each point $A$ of the plane, outside of the line $BC$, let $G$ be the barycenter of the triangle $ABC$. Determine the locus of points $A$ such that $\angle BAC + \angle BGC = 180^{\circ}$. Note: The locus is the set of all points of the plane that satisfies the property.

Several distinct real numbers are written on a blackboard. Peter wants to make an expression such that its values are exactly these numbers. To make such an expression, he may use any real numbers, brackets, and usual signs $+$ , $-$ and $\times$. He may also use a special sign $\pm$: computing the values of the resulting expression, he chooses values $+$ or $-$ for every $\pm$ in all possible combinations. For instance, the expression $5 \pm 1$ results in $\{4, 6 \}$, and $(2 \pm 0.5) \pm 0.5$ results in $\{1, 2, 3 \}$. Can Pete construct such an expression: $a)$ If the numbers on the blackboard are $1, 2, 4$; $b)$ For any collection of $100$ distinct real numbers on a blackboard?

A rectangle with perimeter $ 176$ is divided into five congruent rectangles as shown in the diagram. What is the perimeter of one of the five congruent rectangles? [asy]defaultpen(linewidth(.8pt)); draw(origin--(0,3)--(4,3)--(4,0)--cycle); draw((0,1)--(4,1)); draw((2,0)--midpoint((0,1)--(4,1))); real r = 4/3; draw((r,3)--foot((r,3),(0,1),(4,1))); draw((2r,3)--foot((2r,3),(0,1),(4,1)));[/asy]$ \textbf{(A)}\ 35.2\qquad \textbf{(B)}\ 76\qquad \textbf{(C)}\ 80\qquad \textbf{(D)}\ 84\qquad \textbf{(E)}\ 86$

Let $ABC$ be a triangle with sides $BC = 13$, $CA = 14$ and $AB = 15$. We denote by $I$ the intersection point of the angle bisectors and $M$ to the midpoint of $AB$. The line $IM$ cuts at $P$ at the altitude drawn from $C$. Find the length of $CP$.

Let $ a,b,c,d,e,f,g$ and $ h$ be distinct elements in the set \[ \{ \minus{} 7, \minus{} 5, \minus{} 3, \minus{} 2,2,4,6,13\}. \]What is the minimum possible value of \[ (a \plus{} b \plus{} c \plus{} d)^2 \plus{} (e \plus{} f \plus{} g \plus{} h)^2 \]$ \textbf{(A)}\ 30\qquad \textbf{(B)}\ 32\qquad \textbf{(C)}\ 34\qquad \textbf{(D)}\ 40\qquad \textbf{(E)}\ 50$

Let $P(x) = a_{1998}X^{1998} + a_{1997}X^{1997} +...+a_1X + a_0$ be a polynomial with real coefficients such that $P(0) \ne P (-1)$, and let $a, b$ be real numbers. Let $Q(x) = b_{1998}X^{1998} + b_{1997}X^{1997} +...+b_1X + b_0$ be the polynomial with real coefficients obtained by taking $b_k = aa_k + b$ ,$\forall k = 0, 1,2,..., 1998$. Show that if $Q(0) = Q (-1) \ne 0$ , then the polynomial $Q$ has no real roots.

Suppose that $p$ and $q$ are two different positive integers and $x$ is a real number. Form the product $(x+p)(x+q).$ Find the sum $S(x,n) = \sum (x+p)(x+q),$ where $p$ and $q$ take values from 1 to $n.$ Does there exist integer values of $x$ for which $S(x,n) = 0.$

Discuss $ \lim_{x\to 0}\frac{\lambda +\sin\frac{1}{x} \pm\cos\frac{1}{x}}{x} . $

$2020*N$ is a perfect cube. If $N$ can be expressed as $2^a*5^b*101^c,$ find the least possible value of $a+b+c$ such that $a,b,c$ are all positive integers and not necessarily distinct. [i]Proposed by Ephram Chun[/i]

There are$n$ balls numbered $1,2,\cdots,n$, respectively. They are painted with $4$ colours, red, yellow, blue, and green, according to the following rules: First, randomly line them on a circle. Then let any three clockwise consecutive balls numbered $i, j, k$, in order. 1) If $i>j>k$, then the ball $j$ is painted in red; 2) If $i<j<k$, then the ball $j$ is painted in yellow; 3) If $i<j, k<j$, then the ball $j$ is painted in blue; 4) If $i>j, k>j$, then the ball $j$ is painted in green. And now each permutation of the balls determine a painting method. We call two painting methods distinct, if there exists a ball, which is painted with two different colours in that two methods. Find out the number of all distinct painting methods.

A game is played between two players on a $10 \times 10$ checkerboard. They move alternately, the first player marking $X$s on vacant cells and the second $O$s. When all $100$ cells have been marked, they calculate two numbers $C$ and $Z$. $C$ is the total number of five consecutive $X$s in a row, a column or a diagonal, so that $6$ consecutive $X$s contribute a count of $2$ to $C$, $7$ consecutive $X$s contribute $3$, and so on. Similarly, $Z$ is the total number of five consecutive Os. The first player wins if $C > Z$, loses if $C < Z$ and draws if $C = Z$. Does the first player have a strategy which guarantees (a) a draw or a win (b) a win regardless of the counter-strategy of the second player? (A Belov)

Determine all sets of real numbers $S$ such that: [list] [*] $1$ is the smallest element of $S$, [*] for all $x,y\in S$ such that $x>y$, $\sqrt{x^2-y^2}\in S$ [/list] [i]Adian Anibal Santos Sepcic[/i]

The circles $(R, r)$ and $(P, \rho)$, where $r > \rho$, touch externally at $A$. Their direct common tangent touches $(R, r)$ at B and $(P, \rho)$ at $C$. The line $RP$ meets the circle $(P, \rho)$ again at $D$ and the line $BC$ at $E$. If $|BC| = 6|DE|$, prove that: [b](a)[/b] the lengths of the sides of the triangle $RBE$ are in an arithmetic progression, and [b](b)[/b] $|AB| = 2|AC|.$

Is the infinite series $$\sum_{n=1}^{\infty} \frac{1}{n^{1+\frac{1}{n}}}$$ convergent?

If $x=1+2^p$ and $y=1+2^{-p}$, then $y$ in terms of $x$ is: $\textbf{(A) }\dfrac{x+1}{x-1}\qquad\textbf{(B) }\dfrac{x+2}{x-1}\qquad\textbf{(C) }\dfrac{x}{x-1}\qquad\textbf{(D) }2-x\qquad \textbf{(E) }\dfrac{x-1}{x}$

In the triangle $\triangle ABC$, let $G$ be the centroid, and let $I$ be the center of the inscribed circle.  Let $\alpha$ and $\beta$ be the angles at the vertices $A$ and $B$, respectively.  Suppose that the segment $IG$ is parallel to $AB$ and that $\beta = 2\tan^{-1}(1/3)$.  Find $\alpha$.

Find all integer solutions of the equation $a^3+3ab^2+7b^3=2011$.

We call a function $f: \mathbb{Q}^+ \to \mathbb{Q}^+$ [i]good[/i] if for all $x,y \in \mathbb{Q}^+$ we have: $$f(x)+f(y)\geq 4f(x+y).$$ a) Prove that for all good functions $f: \mathbb{Q}^+ \to \mathbb{Q}^+$ and $x,y,z \in \mathbb{Q}^+$ $$f(x)+f(y)+f(z) \geq 8f(x+y+z)$$ b) Does there exists a good functions $f: \mathbb{Q}^+ \to \mathbb{Q}^+$ and $x,y,z \in \mathbb{Q}^+$ such that $$f(x)+f(y)+f(z) < 9f(x+y+z) ?$$