Let $G$ be a finite graph. Prove that one can partition $G$ into two graphs $A \cup B=G$ such that if we erase all edges conecting a vertex from $A$ to a vertex from $B$, each vertex of the new graph has even degree.

Let $A$ be a finite set of positive integers. Prove that there exists a finite set $B$ of positive integers such that $A \subseteq B$ and \[\prod_{x\in B} x = \sum_{x\in B} x^2.\]

There are $2018$ peoples. We call the group of people as "club" if all members of same "club" are all friends, but not friends with a nonmember of "club". Prove, that we can divide peoples for $90$ rooms, such that no one room has all members of some "club".

Decide whether there is an integer $n > 1$ with the following properties: (a) $n$ is not a prime number. (b) For all integers $a$, $a^n - a$ is divisible by $n$

For each positive integer $n$, determine the smallest possible value of the polynomial $$ W_n(x)=x^{2n}+2x^{2n-1}+3x^{2n-2}+\ldots + (2n-1)x^2+2nx. $$

A positive integer $n$ is called [i]naughty[/i] if it can be written in the form $n=a^b+b$ with integers $a,b \geq 2$. Is there a sequence of $102$ consecutive positive integers such that exactly $100$ of those numbers are naughty?

Given a scalene triangle $ \triangle ABC $. $ B', C' $ are points lie on the rays $ \overrightarrow{AB}, \overrightarrow{AC} $ such that $ \overline{AB'} = \overline{AC}, \overline{AC'} = \overline{AB} $. Now, for an arbitrary point $ P $ in the plane. Let $ Q $ be the reflection point of $ P $ w.r.t $ \overline{BC} $. The intersections of $ \odot{\left(BB'P\right)} $ and $ \odot{\left(CC'P\right)} $ is $ P' $ and the intersections of $ \odot{\left(BB'Q\right)} $ and $ \odot{\left(CC'Q\right)} $ is $ Q' $. Suppose that $ O, O' $ are circumcenters of $ \triangle{ABC}, \triangle{AB'C'} $ Show that 1. $ O', P', Q' $ are colinear 2. $ \overline{O'P'} \cdot \overline{O'Q'} = \overline{OA}^{2} $

If $ \log_{k}{x}\cdot \log_{5}{k} \equal{} 3$, then $ x$ equals: $ \textbf{(A)}\ k^6\qquad \textbf{(B)}\ 5k^3\qquad \textbf{(C)}\ k^3\qquad \textbf{(D)}\ 243\qquad \textbf{(E)}\ 125$

For a positive integer $n$, and a non empty subset $A$ of $\{1,2,...,2n\}$, call $A$ good if the set $\{u\pm v|u,v\in A\}$ does not contain the set $\{1,2,...,n\}$. Find the smallest real number $c$, such that for any positive integer $n$, and any good subset $A$ of $\{1,2,...,2n\}$, $|A|\leq cn$.

Let $n$ be a positive integer and $a_1, a_2, \dots, a_n$ non-zero real numbers. What is the least number of non-zero coefficients that the polynomial $P(x) = (x - a_1)(x - a_2)\cdots(x - a_n)$ can have?

Ayase randomly picks a number $x \in (0, 1]$ with uniform probability. He then draws the six points $(0, 0, 0),(x, 0, 0),(2x, 3x, 0),(5, 5, 2),(7, 3, 0),(9, 1, 4)$. If the expected value of the volume of the convex polyhedron formed by these six points can be written as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, find $m + n$

Given equilateral triangle with the side $l$. What is the minimal length $d$ of a brush (segment), that will paint all the triangle, if its ends are moving along the sides of the triangle.

[color=indigo]Let $ABC$ is an acute angled triangle and let $A_{1},\, B_{1},\, C_{1}$ are points respectively on $BC,\,CA,\,AB$ such that $\triangle ABC$ is similar to $\triangle A_{1}B_{1}C_{1}.$ Prove that orthocenter of $A_{1}B_{1}C_{1}$ coincides with circumcenter of $ABC$.[/color]

Lyuba, Tanya, Lena and Ira ran across a flat field. At some point it turned out that among the pairwise distances between them there are distances of $1, 2, 3, 4$ and $5$ meters, and there are no other distances. Give an example of how this could be.

Find the biggest real number $ k$ such that for each right-angled triangle with sides $ a$, $ b$, $ c$, we have \[ a^{3}\plus{}b^{3}\plus{}c^{3}\geq k\left(a\plus{}b\plus{}c\right)^{3}.\]

If the perimeter of rectangle $ABCD$ is $20$ inches, the least value of diagonal $\overline{AC}$, in inches, is: $\textbf{(A)}\ 0\qquad \textbf{(B)}\ \sqrt{50}\qquad \textbf{(C)}\ 10\qquad \textbf{(D)}\ \sqrt{200}\qquad \textbf{(E)}\ \text{none of these}$

A circle $\omega$ is inscribed in triangle $ABC$, tangent to the side $BC$ at point $K$. Circle $\omega'$ is symmetrical to the circle $\omega$ with respect to point $A$. The point $A_0$ is chosen so that the segments $BA_0$ and $CA_0$ touch $\omega'$. Let $M$ be the midpoint of side $BC$. Prove that the line $AM$ bisects the segment $KA_0$.

Does there exist a polynomial $P(x)$ with integer coefficients such that $P(1+\sqrt[3]{2})=1+\sqrt[3]{2}$ and $P(1+\sqrt5)=2+3\sqrt5$?

Line $\ell$ intersects sides $BC$ and $AD$ of cyclic quadrilateral $ABCD$ in its interior points $R$ and $S$, respectively, and intersects ray $DC$ beyond point $C$ at $Q$, and ray $BA$ beyond point $A$ at $P$. Circumcircles of the triangles $QCR$ and $QDS$ intersect at $N \neq Q$, while circumcircles of the triangles $PAS$ and $PBR$ intersect at $M\neq P$. Let lines $MP$ and $NQ$ meet at point $X$, lines $AB$ and $CD$ meet at point $K$ and lines $BC$ and $AD$ meet at point $L$. Prove that point $X$ lies on line $KL$.

Consider a $9 \times 9$ grid of squares. Haruki fills each square in this grid with an integer between 1 and 9, inclusive. The grid is called a $\textit{super-sudoku}$ if each of the following three conditions hold: [list] [*] Each column in the grid contains each of the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ exactly once. [*] Each row in the grid contains each of the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ exactly once. [*] Each $3 \times 3$ subsquare in the grid contains each of the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ exactly once. [/list] How many possible super-sudoku grids are there?

The figure $ABCDEF$ is a regular hexagon. Find all points $M$ belonging to the hexagon such that Area of triangle $MAC =$ Area of triangle $MCD$.

In a triangle $T$, all the angles are less than $90^o$, and the longest side has length $s$. Show that for every point $p$ in $T$ we can pick a corner $h$ in $T$ such that the distance from $p$ to $h$ is less than or equal to $s/\sqrt3$.

Compute $\mathop {\lim }\limits_{n \to \infty } \left\{ {{{\left( {2 + \sqrt 3 } \right)}^n}} \right\}$

Find the sum of all integers $n$ for which $n - 3$ and $n^2 + 4$ are both perfect cubes.

We define a sequence $ \left(a_{1},a_{2},a_{3},\ldots \right)$ by \[ a_{n} \equal{} \frac {1}{n}\left(\left\lfloor\frac {n}{1}\right\rfloor \plus{} \left\lfloor\frac {n}{2}\right\rfloor \plus{} \cdots \plus{} \left\lfloor\frac {n}{n}\right\rfloor\right), \] where $\lfloor x\rfloor$ denotes the integer part of $x$. [b]a)[/b] Prove that $a_{n+1}>a_n$ infinitely often. [b]b)[/b] Prove that $a_{n+1}<a_n$ infinitely often. [i]Proposed by Johan Meyer, South Africa[/i]