Let $P$ be a regular $2006$-gon. A diagonal is called [i]good[/i] if its endpoints divide the boundary of $P$ into two parts, each composed of an odd number of sides of $P$. The sides of $P$ are also called [i]good[/i]. Suppose $P$ has been dissected into triangles by $2003$ diagonals, no two of which have a common point in the interior of $P$. Find the maximum number of isosceles triangles having two good sides that could appear in such a configuration.

a) Let $\{a_n\}_{n=1}^\infty$ is sequence of integers bigger than 1. Proove that if $x>0$ is irrational, then $\ds x_n>\frac{1}{a_{n+1}}$ for infinitely many $n$, where $x_n$ is fractional part of $a_na_{n-1}\dots a_1x$. b)Find all sequences $\{a_n\}_{n=1}^\infty$ of positive integers, for which exist infinitely many $x\in(0,1)$ such that $\ds x_n>\frac{1}{a_{n+1}}$ for all $n$. [i]Nikolai Nikolov, Emil Kolev[/i]

In the plane of a triangle $ABC,$ in its exterior$,$ we draw the triangles $ABR, BCP, CAQ$ so that $\angle PBC = \angle CAQ = 45^{\circ}$, $\angle BCP = \angle QCA = 30^{\circ}$, $\angle ABR = \angle RAB = 15^{\circ}$. Prove that [b]a.)[/b] $\angle QRP = 90\,^{\circ},$ and [b]b.)[/b] $QR = RP.$

$M$ is a postive real and $f:[0,\infty)\to[0,M]$ is a continuous function such that $$\int_0^\infty (1+x)f(x) dx<\infty$$ Then, prove the following inequality. $$\left(\int_0^\infty f(x) dx \right)^2 \leq 4M \int_0^\infty x f(x) dx$$ (@below, Thank you. I fixed.)

We have $n$ guinea pigs placed on the vertices of a regular polygon with $n$ sides inscribed in a circumference, one guinea pig in each vertex. Each guinea pig has a direction assigned, such direction is either "clockwise" or "anti-clockwise", and a velocity between $1 km/h$, $2km/h$,..., and $n km/h$, each one with a distinct velocity, and each guinea pig has a counter starting from $0$. They start moving along the circumference with the assigned direction and velocity, everyone at the same time, when 2 or more guinea pigs meet a point, all of the guinea pigs at that point follow the same direction of the fastest guinea pig and they keep moving (with the same velocity as before); each time 2 guinea pigs meet for the first time in the same point, the fastest guinea pig adds 1 to its counter. Prove that, at some moment, for each $1\leq i\leq n$ we have that the $i-$th guinea pig has $i-1$ in its counter.

Find all functions $f : \mathbb{N} \to \mathbb{N}$ such that $f(a) + f(b) \mid af(a) - bf(b)$, for all $a, b \in \mathbb{N}$. (Here, $\mathbb{N}$ is a set of positive integers.) [i]Proposed by Vukašin Pantelić[/i]

If distinct digits $D,E,L,M,Q$ (between $0$ and $9$ inclusive) satisfy \begin{tabular}{c@{\,}c@{\,}c@{\,}c} & & $E$ & $L$ \\ + & $M$ & $E$ & $M$ \\\hline & $Q$ & $E$ & $D$ \\ \end{tabular} what is the maximum possible value of the three digit integer $QED$? [i]2019 CCA Math Bonanza Individual Round #6[/i]

(F.Nilov, A.Zaslavsky) Let $ CC_0$ be a median of triangle $ ABC$; the perpendicular bisectors to $ AC$ and $ BC$ intersect $ CC_0$ in points $ A_c$, $ B_c$; $ C_1$ is the common point of $ AA_c$ and $ BB_c$. Points $ A_1$, $ B_1$ are defined similarly. Prove that circle $ A_1B_1C_1$ passes through the circumcenter of triangle $ ABC$.

A [i]permutation[/i] of a finite set $S$ is a one-to-one function from $S$ to $S$. Given a permutation $f$ of the set $\{ 1, 2, \ldots, 100 \}$, define the [i]displacement[/i] of $f$ to be the sum \[ \sum_{i = 1}^{100} \left\lvert f(i) - i \right\rvert . \] How many permutations of $\{ 1, 2, \ldots, 100 \}$ have displacement 4?

Let $p_n$ be the $n$-th prime number. Find the remainder when $\Pi_{n=1}^{2549} 2006^{p^2_{n-1}}$ is divided by $13$

Let $a$ be an integer. Prove that for any real number $x, x^3 < 3$, both the numbers $\sqrt{3 -x^2}$ and $\sqrt{a - x^3}$ cannot be rational.

We call a positive integer "special" if the number is divided by all of its digits (except the $0$s). At most how many consequtive special numbers are there? $ \textbf{(A)}\ 9 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 12 \qquad\textbf{(D)}\ 13 \qquad\textbf{(E)}\ 14 $

Which of the following is the negation of the statement: For all $x$ of a certain set, $x^2>0$? $ \textbf{(A)}\ \text{For all x}, x^2 < 0\qquad$ $\textbf{(B)}\ \text{For all x}, x^2 \le 0\qquad$ $\textbf{(C)}\ \text{For no x}, x^2>0\qquad$ ${\textbf{(D)}\ \text{For some x}, x^2>0 }\qquad$ ${{\textbf{(E)}\ \text{For some x}, x^2 \le 0}} $

Prove that $\frac{5^{125}-1}{5^{25}-1}$ is a composite number.

The center cell of a $5 \times 5$ square grid is removed. Determine the number of ways to color the remaining 24 cells one of four colors (cyan, magenta, yellow, and black) such that any $2 \times 2$ square of cells not containing the center cell contains cells of all four colors.

For each positive integer $n$, let $a_n$ be the number of permutations $\tau$ of $\{1, 2, ... , n\}$ such that $\tau (\tau (\tau (x))) = x$ for $x = 1, 2, ..., n$. The first few values are $a_1 = 1, a_2 = 1, a_3 = 3, a_4 = 9$. Prove that $3^{334}$ divides $a_{2001}$. (A permutation of $\{1, 2, ... , n\}$ is a rearrangement of the numbers $\{1, 2, ... , n\}$ or equivalently, a one-to-one and onto function from $\{1, 2, ... , n\}$ to $\{1, 2, ... , n\}$. For example, one permutation of $\{1, 2, 3\}$ is the rearrangement $\{2, 1, 3\}$, which is equivalent to the function $\sigma : \{1, 2, 3\} \to \{1, 2, 3\}$ defined by $\sigma (1) = 2, \sigma (2) = 1, \sigma (3) = 3$.)