Prove that for all positive reals $a,b,c$, \[\frac{1}{a+\frac{1}{b}+1}+\frac{1}{b+\frac{1}{c}+1}+\frac{1}{c+\frac{1}{a}+1}\ge \frac{3}{\sqrt[3]{abc}+\frac{1}{\sqrt[3]{abc}}+1}. \][i]Proposed by David Stoner[/i]

Let $z$ be a complex number that satisfies the equation \[\frac{z-4}{z^2-5z+1} + \frac{2z-4}{2z^2-5z+1} + \frac{z-2}{z^2-3z+1} = \frac{3}{z}.\] Over all possible values of $z$, find the sum of the values of \[\left| \frac{1}{z^2-5z+1} + \frac{1}{2z^2-5z+1} + \frac{1}{z^2-3z+1} \right|.\] [i]Proposed by Justin Hsieh[/i]

Let $0 \le a < b$ be real numbers. Prove that there is no continuous function $f : [a, b] \to \mathbb{R}$ such that \[ \int_a^b f(x)x^{2n} \mathrm dx>0 \quad \text{and} \quad \int_a^b f(x)x^{2n+1} \mathrm dx <0 \] for every integer $n \ge 0$.

Let $n$ be an integer number greater than $2$, let $x_{1},x_{2},\ldots ,x_{n}$ be $n$ positive real numbers such that \[\sum_{i=1}^{n}\frac{1}{x_{i}+1}=1\] and let $k$ be a real number greater than $1$. Show that: \[\sum_{i=1}^{n}\frac{1}{x_{i}^{k}+1}\ge\frac{n}{(n-1)^{k}+1}\] and determine the cases of equality.

A uniform rod of length $2k$ and weight $w$ rests with the end $A$ against a vertical wall, while the lower end $B$ is fastened by a string $BC$ of length $2b$ coming from a point $C$ in the wall above $A.$ If the system is in equilibrium, determine the angle $ABC.$

Let $n$ be a positive integer. Find the maximal number of non-congruent triangles whose sides lengths are integers $\leq n.$

Mariana plays with an $8\times 8$ board with all its squares blank. She says that two houses are [i]neighbors [/i] if they have a common side or vertex, that is, two houses can be neighbors vertically, horizontally or diagonally. The game consists of filling the $64$ squares on the board, one after the other, each with a number according to the following rule: she always chooses a house blank and fill it with an integer equal to the number of neighboring houses that are still in White. Once this is done, the house is no longer considered blank. Show that the value of the sum of all $64$ numbers written on the board at the end of the game does not depend in the order of filling. Also, calculate the value of this sum. Note: A house is not neighbor to itself. [hide=original wording]Mariana brinca com um tabuleiro 8 x 8 com todas as suas casas em branco. Ela diz que duas casas s˜ao vizinhas se elas possu´ırem um lado ou um v´ertice em comum, ou seja, duas casas podem ser vizinhas verticalmente, horizontalmente ou diagonalmente. A brincadeira consiste em preencher as 64 casas do tabuleiro, uma ap´os a outra, cada uma com um n´umero de acordo com a seguinte regra: ela escolhe sempre uma casa em branco e a preenche com o n´umero inteiro igual `a quantidade de casas vizinhas desta que ainda estejam em branco. Feito isso, a casa n˜ao ´e mais considerada em branco. Demonstre que o valor da soma de todos os 64 n´umeros escritos no tabuleiro ao final da brincadeira n˜ao depende da ordem do preenchimento. Al´em disso, calcule o valor dessa soma. Observa¸c˜ao: Uma casa n˜ao ´e vizinha a si mesma[/hide]

Nonzero real numbers $x_1, x_2, \ldots, x_n$ satisfy the following condition: $$x_1 - \frac{1}{x_2} = x_2 - \frac{1}{x_3} = \ldots = x_{n-1} - \frac{1}{x_n} = x_n - \frac{1}{x_1}$$ Determine all $n$ for which $x_1, x_2, \ldots, x_n$ have to be equal. [i](Proposed by Oleksii Masalitin, Anton Trygub)[/i]

It is known that the equation$ |x - 1| + |x - 2| +... + |x - 2001| = a$ has exactly one solution. Find $a$.

Let $ABC$ be a triangle with circumcircle $\Omega$, incircle $\omega$, and $A$-excircle $\omega_A$. Let $X$ and $Y$ be the tangency points of $\omega_A$ with $AB$ and $AC$. Lines $XY$ and $BC$ intersect in $T$. The tangent from $T$ to $\omega$ different from $BC$ intersects $\omega$ at $K$. The radical axis of $\omega_A$ and $\Omega$ intersects $BC$ in $S$. The tangent from $S$ to $\omega_A$ different from $BC$ intersects $\omega_A$ at $L$. Prove that $A$, $K$ and $L$ are collinear. [i]Proposed by Ana Boiangiu[/i]

Show that the number $x$ is rational if and only if three distinct terms that form a geometric progression can be chosen from the sequence \[x, ~ x+1, ~ x+2,~ x+3,\ldots . \]

Let nonnegative real numbers $ a_{1},a_{2},a_{3},a_{4}$ satisfy $ a_{1} \plus{} a_{2} \plus{} a_{3} \plus{} a_{4} \equal{} 1.$ Prove that $ max\{\sum_{1}^4{\sqrt {a_{i}^2 \plus{} a_{i}a_{i \minus{} 1} \plus{} a_{i \minus{} 1}^2 \plus{} a_{i \minus{} 1}a_{i \minus{} 2}}},\sum_{1}^4{\sqrt {a_{i}^2 \plus{} a_{i}a_{i \plus{} 1} \plus{} a_{i \plus{} 1}^2 \plus{} a_{i \plus{} 1}a_{i \plus{} 2}}}\}\ge 2.$ Where for all integers $ i, a_{i \plus{} 4} \equal{} a_{i}$ holds.

The parallel sides of a trapezoid are $ 3$ and $ 9$. The non-parallel sides are $ 4$ and $ 6$. A line parallel to the bases divides the trapezoid into two trapezoids of equal perimeters. The ratio in which each of the non-parallel sides is divided is: [asy]defaultpen(linewidth(.8pt)); unitsize(2cm); pair A = origin; pair B = (2.25,0); pair C = (2,1); pair D = (1,1); pair E = waypoint(A--D,0.25); pair F = waypoint(B--C,0.25); draw(A--B--C--D--cycle); draw(E--F); label("6",midpoint(A--D),NW); label("3",midpoint(C--D),N); label("4",midpoint(C--B),NE); label("9",midpoint(A--B),S);[/asy]$ \textbf{(A)}\ 4: 3\qquad \textbf{(B)}\ 3: 2\qquad \textbf{(C)}\ 4: 1\qquad \textbf{(D)}\ 3: 1\qquad \textbf{(E)}\ 6: 1$

$T$ is a point on the plane of triangle $ABC$ such that $\angle ATB = \angle BTC = \angle CTA = 120^\circ$. Prove that the lines symmetric to $AT, BT$ and $CT$ with respect to $BC, CA$ and $AB$, respectively, are concurrent.

There are $2015$ distinct circles in a plane, with radius $1$. Prove that you can select $27$ circles, which form a set $C$, which satisfy the following. For two arbitrary circles in $C$, they intersect with each other or For two arbitrary circles in $C$, they don't intersect with each other.

Solve for $z\in \mathbb{C}$ the equation : \[ |z-|z+1||=|z+|z-1|| \]

Triangle $ABC$ has side lengths $13$, $14$ and $15$. Let $k, k_A,k_B,k_C$ be four circles of radius $ r$ inside the triangle such that $k_A$ is tangent to sides $AB$ and $AC$, $k_B$ is tangent to sides $BA$ and $BC$, $k_C$ is tangent to sides $CA$ and $CB$, and $k$ is externally tangent to circles $k_A$, $k_B$ and $k_C$. Let $r = m/n$ where $m$ and $n$ are coprime. Find $m + n$.

Let $\vartriangle ABC$ be a triangle in which $\angle ACB = 40^o$ and $\angle BAC = 60^o$ . Let $D$ be a point inside the segment $BC$ such that $CD =\frac{AB}{2}$ and let $M$ be the midpoint of the segment $AC$. How much is the angle $\angle CMD$ in degrees?

In a sequence $P_n$ of quadratic trinomials each trinomial, starting with the third, is the sum of the two preceding trinomials. The first two trinomials do not have common roots. Is it possible that $P_n$ has an integral root for each $n$?

Determine all pairs of integers $(x,y)$ satisfying the equation $$x^3 +x^2y+xy^2 +y^3 = 8(x^2 +xy+y^2 +1).$$

Let $ d$ be the sum of the lengths of all the diagonals of a plane convex polygon with $ n$ vertices (where $ n>3$). Let $ p$ be its perimeter. Prove that: \[ n\minus{}3<{2d\over p}<\Bigl[{n\over2}\Bigr]\cdot\Bigl[{n\plus{}1\over 2}\Bigr]\minus{}2,\] where $ [x]$ denotes the greatest integer not exceeding $ x$.

How many numbers between $ 1$ and $ 2005$ are integer multiples of $ 3$ or $ 4$ but not $ 12$? $ \textbf{(A)}\ 501\qquad \textbf{(B)}\ 668\qquad \textbf{(C)}\ 835\qquad \textbf{(D)}\ 1002\qquad \textbf{(E)}\ 1169$

If $N$ is the number of triangles of different shapes (i.e., not similar) whose angles are all integers (in degrees), what is $\frac{N}{100}$?

The distance light travels in one year is approximately $ 5,870,000,000,000$ miles. The distance light travels in $ 100$ years is: $ \textbf{(A)}\ 587 * 10^8 \text{ miles} \qquad\textbf{(B)}\ 587 * 10^{10} \text{ miles} \qquad\textbf{(C)}\ 587*10^{ \minus{} 10} \text{ miles}$ $ \textbf{(D)}\ 587 * 10^{12} \text{ miles} \qquad\textbf{(E)}\ 587* 10^{ \minus{} 12} \text{ miles}$

Find the smallest positive integer $n$ with the following property: if we write down all positive integers from $1$ to $10^n$ and add together the reciprocals of every non-zero digit written down, we obtain an integer.