Let $f(x)$ is such function, that $f(x)=1$ for integer $x$ and $f(x)=0$ for non integer $x$. Build such function using only variable $x$, integer numbers, and operations $+,-,*,/,[.]$(plus, minus, multiply,divide and integer part)

Let $f(x) = 1 + 2x + 3x^2 + 4x^3 + 5x^4$ and let $\zeta = e^{2\pi i/5} = \cos \frac{2\pi}{5} + i \sin \frac{2\pi}{5}$. Find the value of the following expression: $$f(\zeta)f(\zeta^2)f(\zeta^3)f(\zeta^4).$$

There are $ n$ markers, each with one side white and the other side black. In the beginning, these $ n$ markers are aligned in a row so that their white sides are all up. In each step, if possible, we choose a marker whose white side is up (but not one of the outermost markers), remove it, and reverse the closest marker to the left of it and also reverse the closest marker to the right of it. Prove that, by a finite sequence of such steps, one can achieve a state with only two markers remaining if and only if $ n \minus{} 1$ is not divisible by $ 3$. [i]Proposed by Dusan Dukic, Serbia[/i]

There are $15$ boys and $15$ girls in the class. The first girl is friends with $4$ boys, the second with $5$, the third with $6$, . . . , the $11$th with $14$, and each of the other four girls is friends with all the boys. It turned out that there are exactly $3 \cdot 2^{25}$ ways to split the entire class into pairs, so that each pair has a boy and a girl who are friends. Prove that any of the friends of the first girl are friends with all the other girls too. [i]G.M.Sharafetdinova[/i]

Bisectors of $\angle BAC$, $\angle CAD$ in a rectangle $ABCD$ , intersect the sides $BC$, $CD$ at points $M$ and $N$ resp. Prove that $\frac{(MB)}{(MC)}+\frac{(ND)}{(NC)}>1$

find all positive integer $a\geq 2 $ and $b\geq2$ such that $a$ is even and all the digits of $a^b+1$ are equals.

An equilateral triangle with side $3$ is divided into $9$ small equal equilateral triangles with sides of length $1$. Each vertex of a triangle small (bold dots) is numbered with a different number than the $1$ to $10$. Inside each small triangle, write the sum of the numbers corresponding to its three vertices. Prove that there are three small triangles for which it is verified that the sum of the numbers written inside is at least $48$. [img]https://cdn.artofproblemsolving.com/attachments/2/1/b2f58b6d59cb26e2fe29d0df59c1a42639a496.png[/img]

[b]Version 1.[/b] Find all primes $p$ satisfying the following conditions: (i) $\frac{p+1}{2}$ is a prime number. (ii) There are at least three distinct positive integers $n$ for which $\frac{p^2+n}{p+n^2}$ is an integer. [b]Version 2.[/b] Let $p \neq 5$ be a prime number such that $\frac{p+1}{2}$ is also a prime. Suppose there exist positive integers $a <b$ such that $\frac{p^2+a}{p+a^2}$ and $\frac{p^2+b}{p+b^2}$ are integers. Show that $b=(a-1)^2+1$.

Let $k$ be a positive integer. Find the maximum value of \[a^{3k-1}b+b^{3k-1}c+c^{3k-1}a+k^2a^kb^kc^k,\] where $a$, $b$, $c$ are non-negative reals such that $a+b+c=3k$.

[color=darkred]Prove that if $n\ge 2$ is a natural number and $x_1,x_2,\ldots,x_n$ are positive real numbers, then: \[4\left(\frac {x_1^3-x_2^3}{x_1+x_2}+\frac {x_2^3-x_3^3}{x_2+x_3}+\ldots+\frac {x_{n-1}^3-x_n^3}{x_{n-1}+x_n}+\frac {x_n^3-x_1^3}{x_n+x_1}\right)\le \\ \\ \le(x_1-x_2)^2+(x_2-x_3)^2+\ldots+(x_{n-1}-x_n)^2+(x_n-x_1)^2\, .\][/color]

$n$ is a natural number. Given $3n \cdot 3n$ table, the unit cells are colored white and black such that starting from the left up corner diagonals are colored in pure white or black in ratio of 2:1 respectively. ( See the picture below). In one step any chosen $2 \cdot 2$ square's white cells are colored orange, orange are colored black and black are colored white. Find all $n$ such that with finite steps, all the white cells in the table turns to black, and all black cells in the table turns to white. ( From starting point)

A subset $A$ of the natural numbers $\mathbb{N} = \{0, 1, 2,\dots\}$ is called [i]good[/i] if every integer $n>0$ has at most one prime divisor $p$ such that $n-p\in A$. (a) Show that the set $S = \{0, 1, 4, 9,\dots\}$ of perfect squares is good. (b) Find an infinite good set disjoint from $S$. (Two sets are [i]disjoint[/i] if they have no common elements.)

You have five segments of lengths $a_1, a_2, a_3, a_4,$ and $a_5$ such that it is possible to form a triangle with any three of them. Demonstrate that at least one of those triangles has angles that are all acute.

A square and a triangle have equal perimeters. The lengths of the three sides of the triangle are $ 6.2$ cm, $ 8.3$ cm, and $ 9.5$ cm. The area of the square is \[ \textbf{(A)}\ 24 \text{ cm}^2 \qquad \textbf{(B)}\ 36 \text{ cm}^2 \qquad \textbf{(C)}\ 48 \text{ cm}^2 \qquad \textbf{(D)}\ 64 \text{ cm}^2 \qquad \textbf{(E)}\ 144 \text{ cm}^2 \]

In triangle $ABC$, $K$ and $L$ are points on the side $BC$ ($K$ being closer to $B$ than $L$) such that $BC \cdot KL = BK \cdot CL$ and $AL$ bisects $\angle KAC$. Show that $AL \perp AB.$

Jake, Jonathan, and Joe are playing a dice game involving polyhedron dice. The dice are as follows: 4 sides, 6 sides, 12 sides, and 20 sides. An n-sided dice has the numbers 1 through n labeled on the sides. Jake starts by selecting a 4-sided die and a 20-sided die. The amount of points that a player gets is the sum of the numbers on the rolled dice. Jonathan then selects a 12-sided die and an 20-sided die. Finally, Joe selects a 20-sided die and a 6-sided die. [list=a] [*]What is the probability that Joe places last? [*]What is the probability that Joe places second? [*]What is the probability that Joe places first? [*]What is the probability that there is a three-way tie? [/list] [i]Proposed by beanielove2[/i]

Two fixed circles $\omega_1$ and $\omega_2$ pass through point $O$. A circle of an arbitrary radius $R$ centered at $O$ meets $\omega_1$ at points $A$ and $B$, and meets $\omega_2$ at points $C$ and $D$. Let $X$ be the common point of lines $AC$ and $BD$. Prove that all the points X are collinear as $R$ changes.