Find the number of solutions to the equation: $6\{x\}^3 + \{x\}^2 + \{x\} + 2x = 2018. $ With {x} we denote the fractional part of the number x.

A square is divided into 169 identical small squares and in every small square is written 0 or 1. It isn’t allowed in one row or column to have the following arrangements of adjacent digits in this order: 101, 111 or 1001. What is the the biggest possible number of 1’s in the table?

We will call one of the cells of a rectangle 11 x 13 “peculiar” , if after removing it the remaining figure can be cut into squares 2 x 2 and 3 x 3. How many of the 143 cells are “peculiar”?

The towns in one country are connected with bidirectional airlines, which are paid in at least one of the two directions. In a trip from town A to town B there are exactly 22 routes that are free. Find the least possible number of towns in the country.

Find the solutions in prime numbers of the following equation $p^4 + q^4 + r^4 + 119 = s^2 .$

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$, such that $f(x+y) = f(y) f(x f(y))$ for every two real numbers $x$ and $y$.

For a non-isosceles $ABC$ we have that $2AC = AB + BC$. Point $I$ is the center of the circle inscribed in $\triangle ABC$, point $K$ is the middle of the arc $\widehat{AC}$ that includes point $B$, and point $T$ is from the line $AC$, such that $\angle TIB = 90^\circ$. Prove that the line $TB$ is tangent to the circumscribed circle of $\triangle KBI$.

The row $x_1, x_2,…$ is defined by the following recursion $x_1=1$ and $x_{n+1}=x_n+\sqrt{x_n}$ Prove that $\sum_{n=1}^{2018}{\frac{1}{x_n}}<3$.

Let $n > 4$ be an integer. A square is divided into $n^2$ smaller identical squares, in some of which were 1’s and in the other – 0's. It is not allowed in one row or column to have the following arrangements of adjacent digits in this order: $101$, $111$ or $1001$. What is the biggest number of 1’s in the table? (The answer depends on $n$.)

$n > 1$ is an odd number and $a_1, a_2, . . . , a_n$ are positive integers such that $gcd(a_1, a_2, . . . , a_n) = 1$. If $d = gcd (a_1^n + a_1.a_2. . . a_n, a_2^n + a_1.a_2. . . a_n, . . . , a_n^n + a_1.a_2. . . a_n) $ find all possible values of $d$.

The number 1 is a solution of the equation $(x + a)(x + b)(x + c)(x + d) = 16$, where $a, b, c, d$ are positive real numbers. Find the largest value of $abcd$.

$x \geq 0$ and $y$ are real numbers for which $y^2 \geq x(x + 1)$. Prove that: $(y - 1)^2 \geq x(x-1)$.

Point $X$ lies in a right-angled isosceles $\triangle ABC$ ($\angle ABC = 90^\circ$). Prove that $AX+BX+\sqrt{2}CX \geq \sqrt{5}AB$ and find for which points $X$ the equality is met.

Prove that there exist infinitely many positive integers $n$, for which at least one of the numbers $2^{2^n}+1$ and $2018^{2^n}+1$ is composite.

On the sides $AC$ and $AB$ of an acute $\triangle ABC$ are chosen points $M$ and $N$ respectively. Point $P$ is an intersection point of the segments $BM$ and $CN$ and point $Q$ is an inner point for the quadrilateral $ANPM$, for which $\angle BQC = 90^\circ$ and $\angle BQP = \angle BMQ$. If the quadrilateral $ANPM$ is inscribed in a circle, prove that $\angle QNC = \angle PQC$.

Find all positive integers $n$ for which a square n x n can be covered with rectangles k x 1 and one square 1 x 1 when: a) $k = 4$ b) $k = 8$

Find all prime numbers $p$ and all positive integers $n$, such that $n^8 - n^2 = p^5 + p^2$

The set of numbers $(p, a, b, c)$ of positive integers is called Sozopolian when: * p is an odd prime number * $a$, $b$ and $c$ are different and * $ab + 1$, $bc + 1$ and $ca + 1$ are a multiple of $p$. a) Prove that each Sozopolian set satisfies the inequality $p+2 \leq \frac{a+b+c}{3}$ b) Find all numbers $p$ for which there exist a Sozopolian set for which the equality of the upper inequation is met.

The points $A$, $B$, $C$, $D$, and $E$ lie in one plane and have the following properties: $AB = 12, BC = 50, CD = 38, AD = 100, BE = 30, CE = 40$. Find the length of the segment $ED$.

The real numbers $a$, $b$, $c$ are such that $a+b+c+ab+bc+ca+abc \geq 7$. Prove that $\sqrt{a^2+b^2+2}+\sqrt{b^2+c^2+2}+\sqrt{c^2+a^2+2} \geq 6$

On the sides $AB$,$BC$, and $CA$ of $\triangle ABC$ are chosen points $C_1$, $A_1$, and $B_1$ respectively, in such way that $AA_1$, $BB_1$, and $CC_1$ intersect in one point $X$. If $\angle A_1C_1B = \angle B_1C_1A$, prove that $CC_1$ is perpendicular to $AB$.

Let $S$ be a real number. It is known that however we choose several numbers from the interval $(0, 1]$ with sum equal to $S$, these numbers can be separated into two subsets with the following property: The sum of the numbers in one of the subsets doesn’t exceed 1 and the sum of the numbers in the other subset doesn’t exceed 5. Find the greatest possible value of $S$.

$n$ points were chosen on a circle. Two players are playing the following game: On every move a point is chosen and it is connected with an edge to an adjacent point or with the center of the circle. The winner is the player, after whose move each point can be reached by any other (including the center) by moving on the constructed edges. Find who of the two players has a winning strategy.

Some of the towns in a country are connected with bidirectional paths, where each town can be reached by any other by going through these paths. From each town there are at least $n \geq 3$ paths. In the country there is no such route that includes all towns exactly once. Find the least possible number of towns in this country (Answer depends from $n$).

In a quadrilateral $ABCD$ diagonal $AC$ is a bisector of $\angle BAD$ and $\angle ADC = \angle ACB$. The points $X$ and $Y$ are the feet of the perpendiculars from $A$ to $BC$ and $CD$ respectively. Prove that the orthocenter of $\triangle AXY$ lies on the line $BD$.

$x$, $y$, and $z$ are positive real numbers satisfying the equation $x+y+z=\frac{1}{x} + \frac{1}{y} + \frac{1}{z}$. Prove the following inequality: $xy + yz + zx \geq 3$.

Find all positive integers $n$ for which the number $\frac{n^{3n-2}-3n+1}{3n-2}$ is whole. EDIT:In the original problem instead of whole we search for integers, so with this change $n=1$ will be a solution.

The cells of a table m x n, $m \geq 5$, $n \geq 5$ are colored in 3 colors where: (i) Each cell has an equal number of adjacent (by side) cells from the other two colors; (ii) Each of the cells in the 4 corners of the table doesn’t have an adjacent cell in the same color. Find all possible values for $m$ and $n$.

Find all functions $f :[0, +\infty) \rightarrow [0, +\infty)$ for which $f(f(x)+f(y)) = xy f (x+y)$ for every two non-negative real numbers $x$ and $y$.

Find all sets $(a, b, c)$ of different positive integers $a$, $b$, $c$, for which: * $2a - 1$ is a multiple of $b$; * $2b - 1$ is a multiple of $c$; * $2c - 1$ is a multiple of $a$.

The rows $x_n$ and $y_n$ of positive real numbers are such that: $x_{n+1}=x_n+\frac{1}{2y_n}$ and $y_{n+1}=y_n+\frac{1}{2x_n}$ for each positive integer $n$. Prove that at least one of the numbers $x_{2018}$ and $y_{2018}$ is bigger than 44,9

Are there infinitely many positive integers that can’t be presented as a sum of no more than fifteen fourth degrees of positive integers. (For example 15 isn’t such number as it can be presented as the sum of $15.1^4$)

$A = \{a_1, a_2, . . . , a_k\}$ is a set of positive integers for which the sum of some (we can have only one number too) different numbers from the set is equal to a different number i.e. there $2^k - 1$ different sums of different numbers from $A$. Prove that the following inequality holds: $\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_k}<2$

a) The real number $a$ and the continuous function $f : [a, \infty) \rightarrow [a, \infty)$ are such that $|f(x)-f(y)| < |x–y|$ for every two different $x, y \in [a, \infty)$. Is it always true that the equation $f(x)=x$ has only one solution in the interval $[a, \infty)$? b) The real numbers $a$ and $b$ and the continuous function $f : [a, b] \rightarrow [a, b]$ are such that $|f(x)-f(y)| < |x–y|$, for every two different $x, y \in [a, b]$. Is it always true that the equation $f(x)=x$ has only one solution in the interval $[a, b]$?

Let $p$ be some prime number. a) Prove that there exist positive integers $a$ and $b$ such that $a^2 + b^2 + 2018$ is multiple of $p$. b) Find all $p$ for which the $a$ and $b$ from a) can be chosen in such way that both these numbers aren’t multiples of $p$.

Find all real numbers $k$ for which the inequality $(1+t)^k (1-t)^{1-k} \leq 1$ is true for every real number $t \in (-1, 1)$.

On the extension of the heights $AH_1$ and $BH_2$ of an acute $\triangle ABC$, after points $H_1$ and $H_2$, are chosen points $M$ and $N$ in such way that $\angle MCB = \angle NCA = 30^\circ$. We denote with $C_1$ the intersection point of the lines $MB$ and $NA$. Analogously we define $A_1$ and $B_1$. Prove that the straight lines $AA_1$, $BB_1$, and $CC_1$ intersect in one point.

There are $a$ straight lines in a plane, no two of which are parallel to each other and no three intersect in one point. a) Prove that there exist a straight line for which each of the two Half-Planes defined by it contains at least $\lfloor \frac{(a-1)(a-2)}{10} \rfloor$ intersection points. b) Find all $a$ for which the evaluation in a) is the best possible.

Let $x$ and $y$ be odd positive integers. A table $x$ x $y$ is given in which the squares with coordinates $(2,1)$, $(x - 2, y)$, and $(x, y)$ are cut. The remaining part of the table is covered in dominoes and squares 2 x 2. Prove that the dominoes in a valid covering of the table are at least $\frac{3}{2}(x+y)-6$

Prove that for every positive integer $n \geq 2$ the following inequality holds: $e^{n-1}n!<n^{n+\frac{1}{2}}$