For two arbitrary reals $x, y$ which are larger than $0$ and less than $1.$ Prove that$$\frac{x^2}{x+y}+\frac{y^2}{1-x}+\frac{(1-x-y)^2}{1-y}\geq\frac{1}{2}.$$

Let $P_0(x) = x^3 + 313x^2 - 77x - 8$. For integers $n \ge 1$, define $P_n(x) = P_{n - 1}(x - n)$. What is the coefficient of $x$ in $P_{20}(x)$?

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that $$\displaystyle{f(f(x)) = \frac{x^2 - x}{2}\cdot f(x) + 2-x,}$$ for all $x \in \mathbb{R}.$ Find all possible values of $f(2).$

How many solutions, depending on the value of the parameter $a$, has the equation $$\sqrt{x^2-4}+\sqrt{2x^2-7x+5}=a ?$$

Show that a sequence $(a_n)$ of $+1$ and $-1$ is periodic with period a power of $2$ if and only if $a_n=(-1)^{P(n)}$, where $P$ is an integer-valued polynomial with rational coefficients.

Starting from an equilateral triangle with perimeter $P_0$, we carry out the following iterations: the first iteration consists of dividing each side of the triangle into three segments of equal length, construct an exterior equilateral triangle on each of the middle segments, and then remove these segments (bases of each new equilateral triangle formed). The second iteration consists of apply the same process of the first iteration on each segment of the resulting figure after the first iteration. Successively, follow the other iterations. Let $A_n$ be the area of the figure after the $n$- th iteration, and let $P_n$ the perimeter of the same figure. If $A_n = P_n$, find the value of $P_0$ (in its simplest form).

Find the tangents of the angles of a triangle knowing that they are positive integers.

Find all real numbers $x,y,z$ so that \begin{align*} x^2 y + y^2 z + z^2 &= 0 \\ z^3 + z^2 y + z y^3 + x^2 y &= \frac{1}{4}(x^4 + y^4). \end{align*}

Given integer $n$, let $W_n$ be the set of complex numbers of the form $re^{2qi\pi}$, where $q$ is a rational number so that $q_n \in Z$ and $r$ is a real number. Suppose that p is a polynomial of degree $ \ge 2$ such that there exists a non-constant function $f : W_n \to C$ so that $p(f(x))p(f(y)) = f(xy)$ for all $x, y \in W_n$. If $p$ is the unique monic polynomial of lowest degree for which such an $f$ exists for $n = 65$, find $p(10)$.

[list=a][*] Find infinitely many pairs of integers $a$ and $b$ with $1<a<b$, so that $ab$ exactly divides $a^{2}+b^{2}-1$. [*] With $a$ and $b$ as above, what are the possible values of \[\frac{a^{2}+b^{2}-1}{ab}?\] [/list]

Let's define [i]almost mean[/i] of numbers $a_1, a_2, \ldots, a_n$ as $\frac{a_1 + a_2 + \ldots + a_n}{n+1}$. Oleksiy has positive real numbers $b_1, b_2, \ldots, b_{2023}$, not necessarily distinct. For each pair $(i, j)$ with $1 \leq i, j \leq 2023$, Oleksiy wrote on a board [i]almost mean[/i] of numbers $b_i, b_{i+1}, \ldots, b_j$. Prove that there are at least $45$ distinct numbers on the board. [i]Proposed by Anton Trygub[/i]

The distance between the city and the house of Borya is 2km. Once Borya went from the city to his house with speed 4km/h. Simultaneously with that a dog Sharik started running out of house in the direction to city, and whenever Sharik meets Borya or the house, it starts running back (so the dog runs between Borya and the house), and when the dog runs to the house, its speed is 8km/h, and when it runs from the house, its speed is 12km/h. What distance will Sharik run until Borya comes to the house? [i]Yauheni Barabanau[/i]

The first four terms of an arithmetic sequence are $a, x, b, 2x$. The ratio of $a$ to $b$ is $ \textbf{(A)}\ \frac{1}{4} \qquad\textbf{(B)}\ \frac{1}{3} \qquad\textbf{(C)}\ \frac{1}{2} \qquad\textbf{(D)}\ \frac{2}{3} \qquad\textbf{(E)}\ 2 $

Determine the minimum value of $\dfrac{m^m}{1\cdot 3\cdot 5\cdot \ldots \cdot(2m-1)}$ for positive integers $m$.

Show that the solution set of the inequality \[ \sum^{70}_{k \equal{} 1} \frac {k}{x \minus{} k} \geq \frac {5}{4} \] is a union of disjoint intervals, the sum of whose length is 1988.

For every positive integer $n$ determine the least possible value of the expression \[|x_{1}|+|x_{1}-x_{2}|+|x_{1}+x_{2}-x_{3}|+\dots +|x_{1}+x_{2}+\dots +x_{n-1}-x_{n}|\] given that $x_{1}, x_{2}, \dots , x_{n}$ are real numbers satisfying $|x_{1}|+|x_{2}|+\dots+|x_{n}| = 1$.

Two pirates divide the loot, consisting of two bags of coins and a diamond, according to the following rules. First the first pirate takes take a few coins from any bag and transfer them from this bag in the other the same number of coins. Then the second pirate does the same (choosing the bag from which he takes the coins at his discretion) and etc. until you can take coins according to these rules. The pirate who takes the coins last gets the diamond. Who will get the diamond if is each of the pirates trying to get it? Give your answer depending on the initial number of coins in the bags.

Determine the maximum number $ h$ satisfying the following condition: for every $ a\in [0,h]$ and every polynomial $ P(x)$ of degree 99 such that $ P(0)\equal{}P(1)\equal{}0$, there exist $ x_1,x_2\in [0,1]$ such that $ P(x_1)\equal{}P(x_2)$ and $ x_2\minus{}x_1\equal{}a$. [i]Proposed by F. Petrov, D. Rostovsky, A. Khrabrov[/i]

Calculate the following $$P=(4\sin^2{0} -3)(4\sin^2\frac{\pi}{2^{2005}} -3)(4\sin^2\frac{2\pi}{2^{2005}} -3)(4\sin^2\frac{3\pi}{2^{2005}} -3)...$$ $$...\,\,\,\,(4\sin^2\frac{(2^{2004}-1)\pi}{2^{2005}} -3)(4\sin^2\frac{\pi}{2} -3)$$

Adamu and Afaafa choose, each in his turn, positive integers as coefficients of a polynomial of degree $n$. Adamu wins if the polynomial obtained has an integer root; otherwise, Afaafa wins. Afaafa plays first if $n$ is odd; otherwise Adamu plays first. Prove that: [list] [*] Adamu has a winning strategy if $n$ is odd. [*] Afaafa has a winning strategy if $n$ is even. [/list]

Let $x_1, x_2, ... , x_n$ be a real numbers such that\\ 1) $1 \le x_1, x_2, ... , x_n \le 160$ 2) $x^{2}_{i} + x^{2}_{j} + x^{2}_{k} \ge 2(x_ix_j + x_jx_k + x_kx_i)$ for all $1\le i < j < k \le n$ Find the largest possible $n$.

Let $f(0), f(1), f(2), \dots$ be a sequence satisfying \[ f(0) = 0 \quad \text{and} \quad f(n) = n - f(f(n-1)) \] for $n=1,2,3,\dots$. Give a formula for $f(n)$ such that its value can be immediately computed using $n$ without having to compute the previous terms.

If $ a$, $ b$, $ x$, $ y$ are integers greater than 1 such that $ a$ and $ b$ have no common factor except 1 and $ x^a \equal{} y^b$ show that $ x \equal{} n^b$, $ y \equal{} n^a$ for some integer $ n$ greater than 1.

Which is greater: \[\frac{1^{-3}-2^{-3}}{1^{-2}-2^{-2}}-\frac{2^{-3}-3^{-3}}{2^{-2}-3^{-2}}+\frac{3^{-3}-4^{-3}}{3^{-2}-4^{-2}}-\cdots +\frac{2019^{-3}-2020^{-3}}{2019^{-2}-2020^{-2}}\] or \[1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\cdots +\frac{1}{5781}?\]

Given a polynomial $P$ and a positive integer $k$, we denote the $k$-fold composition of $P$ by $P^{\circ k}$. A polynomial $P$ with real coefficients is called [b]perfect[/b] if for each integer $n$ there is a positive integer $k$ so that $P^{\circ k}(n)$ is an integer. Is it true that for each perfect polynomial $P$, there exists a positive $m$ so that for each integer $n$ there is $0<k\leq m$ for which $P^{\circ k}(n)$ is an integer?