Prove that for all positive integers $ n$: $ \frac{2n}{3n\plus{}1} \le \displaystyle\sum_{k\equal{}n\plus{}1}^{2n}\frac{1}{k} \le \frac{3n\plus{}1}{4(n\plus{}1)}$.

Given is a convex polyhedron with $ k $ faces $ S_1, \ldots, S_k $. Let us denote the vector of length 1 perpendicular to the wall $ S_i $ ($ i = 1, \ldots, k $) directed outside the given polyhedron by $ \overrightarrow{n_i} $, and the surface area of this wall by $ P_i $. Prove that $$ \sum_{i=1}^k P_i \cdot \overrightarrow{n_i} = \overrightarrow{0}.$$

For a positive integer $n$, denote $A_n=\{(x,y)\in\mathbb Z^2|x^2+xy+y^2=n\}$. (a) Prove that the set $A_n$ is always finite. (b) Prove that the number of elements of $A_n$ is divisible by $6$ for all $n$. (c) For which $n$ is the number of elements of $A_n$ divisible by $12$?

Let $p, u, m, a, c$ be positive real numbers satisfying $5p^5+4u^5+3m^5+2a^5+c^5=91$. What is the maximum possible value of: \[18pumac + 2(2 + p)^2 + 23(1 + ua)^2 + 15(3 + mc)^2?\]

Let $c(O, R)$ be a circle, $AB$ a diameter and $C$ an arbitrary point on the circle different than $A$ and $B$ such that $\angle AOC > 90^o$. On the radius $OC$ we consider point $K$ and the circle $c_1(K, KC)$. The extension of the segment $KB$ meets the circle $(c)$ at point $E$. From $E$ we consider the tangents $ES$ and $ET$ to the circle $(c_1)$. Prove that the lines $BE, ST$ and $AC$ are concurrent.

In each unit square of square $100*100$ write any natural number. Called rectangle with sides parallel sides of square $good$ if sum of number inside rectangle divided by $17$. We can painted all unit squares in $good$ rectangle. One unit square cannot painted twice or more. Find maximum $d$ for which we can guaranteed paint at least $d$ points.

Given $15$ coins of the same appearance. It is known that one of them weighs $1$g, two coins weigh $2$g each, three coins weigh $3$g each, four coins weigh $4$g each, and five coins weigh $5$g each. There are inscriptions on the coins, indicating their weight. It is allowed to perform two weighings on a balance without additional weights. Find a way to check that there are no wrong inscriptions. (It is not required to check which inscriptions are wrong and which ones are correct.) (8 marks)

On an international conference there are 4 official languages. Each two of the attendees can have a conversation on one of the languages. Prove that at least 60% of the attendees can speak the same language.

Let $k$ be a positive integer. A natural number $m$ is called [i]$k$-typical[/i] if each divisor of $m$ leaves the remainder $1$ when being divided by $k$. Prove: [b]a)[/b] If the number of all divisors of a positive integer $n$ (including the divisors $1$ and $n$) is $k$-typical, then $n$ is the $k$-th power of an integer. [b]b)[/b] If $k > 2$, then the converse of the assertion [b]a)[/b] is not true.

Find an arithmetic progression of $2016$ natural numbers such that neither is a perfect power but its multiplication is a perfect power. Clarification: A perfect power is a number of the form $n^k$ where $n$ and $k$ are both natural numbers greater than or equal to $2$.

We call a positive integer [i]interesting [/i] if the number and the number with its digits written in reverse order both leave remainder $2$ in division by $4$. a) Determine if $2018$ is an interesting number. b) For every positive integer $n$, find how many interesting $n$-digit numbers there are.

Given any integer $ n \geq 2,$ assume that the integers $ a_1, a_2, \ldots, a_n$ are not divisible by $ n$ and, moreover, that $ n$ does not divide $ \sum^n_{i\equal{}1} a_i.$ Prove that there exist at least $ n$ different sequences $ (e_1, e_2, \ldots, e_n)$ consisting of zeros or ones such $ \sum^n_{i\equal{}1} e_i \cdot a_i$ is divisible by $ n.$

Prove that the application $ \mathbb{R}\ni x\mapsto 2x+ \{ x\} $ and its inverse are bijective and continuous.

If $xy = a, xz =b,$ and $yz = c$, and none of these quantities is zero, then $x^2+y^2+z^2$ equals: $ \textbf{(A)}\ \frac{ab+ac+bc}{abc}\qquad\textbf{(B)}\ \frac{a^2+b^2+c^2}{abc}\qquad\textbf{(C)}\ \frac{(a+b+c)^2}{abc}\qquad\textbf{(D)}\ \frac{(ab+ac+bc)^2}{abc}\qquad\textbf{(E)}\ \frac{(ab)^2+(ac)^2+(bc)^2}{abc} $

A strategical video game consists of a map of finitely many towns. In each town there are $k$ directions, labelled from $1$ through $k$. One of the towns is designated as initial, and one – as terminal. Starting from the initial town the hero of the game makes a finite sequence of moves. At each move the hero selects a direction from the current town. This determines the next town he visits and a certain positive amount of points he receives. Two strategical video games are equivalent if for every sequence of directions the hero can reach the terminal town from the initial in one game, he can do so in the other game, and, in addition, he accumulates the same amount of points in both games. For his birthday John receives two strategical video games – one with $N$ towns and one with $M$ towns. He claims they are equivalent. Marry is convinced they are not. Marry is right. Prove that she can provide a sequence of at most $N +M$ directions that shows the two games are indeed not equivalent. [i]Stefan Gerdjikov, Bulgaria[/i]

Let $AC$ be a line segment in the plane and $B$ a points between $A$ and $C$. Construct isosceles triangles $PAB$ and $QAC$ on one side of the segment $AC$ such that $\angle APB = \angle BQC = 120^{\circ}$ and an isosceles triangle $RAC$ on the other side of $AC$ such that $\angle ARC = 120^{\circ}.$ Show that $PQR$ is an equilateral triangle.

A function $ f:\mathbb{R}^2\longrightarrow\mathbb{R} $ is [i]olympic[/i] if, any finite number of pairwise distinct elements of $ \mathbb{R}^2 $ at which the function takes the same value represent in the plane the vertices of a convex polygon. Prove that if $ p $ if a complex polynom of degree at least $ 1, $ then the function $ \mathbb{R}^2\ni (x,y)\mapsto |p(x+iy)| $ is olympic if and only if the roots of $ p $ are all equal.

Let $(I_b)$, $(I_c)$ are excircles of a triangle $ABC$. Given a circle $ \omega $ passes through $A$ and externally tangents to the circles $(I_b)$ and $(I_c)$ such that it intersects with $BC$ at points $M$, $N$. Prove that $ \angle BAM=\angle CAN $. A. Smirnov

At a conference, six people place their name badges in a hat, which is shaken up; one badge is then distributed to each person such that each distribution is equally likely. Each turn, every person who does not yet have their own badge finds the person whose badge they have and takes that person's badge. For example, if Alice has Bob's badge and Bob has Charlie's badge, Alice would have Charlie's badge after a turn. Compute the probability that everyone will eventually end up with their own badge.

The identifier of a book is an n-tuple of numbers 0, 1, .... , 9, followed by a checksum. The checksum is computed by a fixed rule that satisfies the following property: whenever one increases a single number in the n-tuple (without modifying the other numbers), the checksum also increases. Find the smallest possible number of required checksums if all possible n-tuples are in use.

Could the three-dimensional space be expressed as the union of disjoint circumferences?

Let $k>1$ be an integer, set $n=2^{k+1}$. Prove that for any positive integers $a_1<a_2<\cdots<a_n$, the number $\prod_{1\leq i<j\leq n}(a_i+a_j)$ has at least $k+1$ different prime divisors.

The width of a lane in a circular running track is $1.22$ meters. One loop in the first lane (shortest lane) is $400$ meters. Thus $12.5$ loops makes it a $5{,}000$ meter distance. Which lane should an athlete run in if they want to make $12$ loops as close to the $5{,}000$ meter distance as possible? \[\rm a. ~second\qquad \mathrm b. ~third \qquad \mathrm c. ~fourth \qquad\mathrm d. ~fifth \qquad\mathrm e. ~sixth\]

Find all pairs $(x,y)$ of positive integers that satisfy $$2^x+17=y^4$$.

Two walls and the ceiling of a room meet at right angles at point $P$. A fly is in the air one meter from one wall, eight meters from the other wall, and $9$ meters from point $P$. How many meters is the fly from the ceiling? $\textbf{(A) }\sqrt{13}\qquad\textbf{(B) }\sqrt{14}\qquad\textbf{(C) }\sqrt{15}\qquad\textbf{(D) }4\qquad\textbf{(E) }\sqrt{17}$