Find $x$ and $y$ ($a$ and $b$ parameters): $$\begin{cases} \dfrac{x-y\sqrt{x^2-y^2}}{\sqrt{1-x^2+y^2}} = a\\ \\ \dfrac{y-x\sqrt{x^2-y^2}}{\sqrt{1-x^2+y^2}} = b\end{cases}$$

It is given regular $n$-sided polygon, $n \geq 6$. How many triangles they are inside the polygon such that all of their sides are formed by diagonals of polygon and their vertices are vertices of polygon?

Let $X \neq \varnothing$ be a finite set and let $f: X \to X$ be a function such that for every $x \in X$ and a fixed prime $p$ we have $f^p(x)=x.$ Let $Y=\{x \in X | f(x) \neq x\}.$ Prove that the number of the members of the set $Y$ is divisible by $p.$ [i]Note.[/i] ${f^p(x)=x = \underbrace{f(f(f(\cdots ((f}_{ p \text{ times}}(x) ) \cdots )))} .$

A uniform disk rotates at a fixed angular velocity on an axis through its center normal to the plane of the disk, and has kinetic energy $E$. If the same disk rotates at the same angular velocity about an axis on the edge of the disk (still normal to the plane of the disk), what is its kinetic energy? (a) $\frac{1}{2}E$ (b) $\frac{3}{2}E$ (c) $2E$ (d) $3E$ (e) $4E$

Show that there exists a positive integer $N$ such that the decimal representation of $2000^N$ starts with the digits $200120012001.$

Given the following system of equations $a_1 + a_2 + a_3 = 1$ $a_2 + a_3 + a_4 = 2$ $a_3 + a_4 + a_5 = 3$ $...$ $a_{12} + a_{13} + a_{14} = 12$ $a_{13} + a_{14} + a_1 = 13$ $a_{14 }+ a_1 + a_2 = 14$ find the value of $a_{14}$.

Is it possible for an isosceles triangle with all its sides of positive integer lengths to have an angle of $36^o$? [i] (Adapted from Archimedes 2011, Traian Preda)[/i]

By $h(n)$, where $n$ is an integer greater than $1$, let us denote the greatest prime divisor of the number $n$. Are there infinitely many numbers $n$ for which $h(n) < h(n+1)< h(n+2)$ holds?

We have a rectangle with integer sides $m, n$ that is subdivided into $mn$ squares of side $1$. Find the number of little squares that are crossed by the diagonal (without counting those that are touched only in one vertex)

Suppose for some positive integers, that $\frac{p+\frac{1}{q}}{q+\frac{1}{p}}= 17$. What is the greatest integer $n$ such that $\frac{p+q}{n}$ is always an integer?

We are given an infinite sequence of $ 1$'s and $ 2$'s with the following properties: (1) The first element of the sequence is $ 1$. (2) There are no two consecutive $ 2$'s or three consecutive $ 1$'s. (3) If we replace consecutive $ 1$'s by a single $ 2$, leave the single $ 1$'s alone, and delete the original $ 2$'s, then we recover the original sequence. How many $ 2$'s are there among the first $ n$ elements of the sequence? [i]P. P. Palfy[/i]

There is a polynomial $P$ such that for every real number $x$, \[ x^{512} + x^{256} + 1 = (x^2 + x + 1) P(x). \] When $P$ is written in standard polynomial form, how many of its coefficients are nonzero?

The pentagon $ABCDE$ is inscribed in the circle $\omega$. Let $T$ be the intersection point of the diagonals $BE$ and $AD$. A line is drawn through the point $T$ parallel to $CD$, which intersects $AB$ and $CE$ at points $X$ and $Y$, respectively. Prove that the circumscribed circle of the triangle $AXY$ is tangent to $\omega$.

Find minimal number of divisors that can number $|2016^m-36^n|$ have,where $m$ and $n$ are natural numbers.

A computer science teacher has asked his students to write a program that, given a list of $n$ numbers $a_1, a_2, ..., a_n$, calculates the list $b_1, b_2, ..., b_n$ where $b_k$ is the number of times the number $a_k$ appears in the list. So, for example, for the list $1,2,3,1$, the program returns the list $2,1,1,2$. Next, the teacher asked Alexandre to run the program for a list of $2025$ numbers. Then he asked him to apply the program to the resulting list, and so on, until a number greater than or equal to $k$ appears in the list. Find the largest value of $k$ for which, whatever the initial list of $2025$ positive integers $a_1, a_2, ..., a_{2025}$, it is possible for Alexander to do what the teacher asked him to do.

Given arbitrary positive integer $ a$ larger than $ 1$, show that for any positive integer $ n$, there always exists a n-degree integral coefficient polynomial $ p(x)$, such that $ p(0)$, $ p(1)$, $ \cdots$, $ p(n)$ are pairwise distinct positive integers, and all have the form of $ 2a^k\plus{}3$, where $ k$ is also an integer.

Show that for all $n\ge 3$ there are $n$ different positive integers $x_1,x_2, ...,x_n$ such that $$\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}= 1.$$