Show that $1 + 1/2 + 1/3 + ... + 1/n$ is not an integer for $n > 1$.

Prove that the volume of a tire (torus) is equal to the volume of a cylinder whose base is a meridian section of that and whose height is the length of the circumference formed by the centers of the meridian sections.

[b]7.[/b] Let $a_0$ and $a_1$ be arbitrary real numbers, and let $a_{n+1}=a_n + \frac{2}{n+1}a_{n-1}$ $(n= 1, 2, \dots)$ Show that the sequence $\left (\frac{a_n}{n^2} \right )_{n=1}^{\infty}$ is convergent and find its limit. [b](S. 10)[/b]

Positive integer $l$ and positive real numbers $a_1, a_2, ..., a_l$ are given. For every positive integer $n$ we define $$c_n=\sum_{k_1+k_2+...+k_l=n}\frac{(2n)!}{(2k_1)!(2k_2)!...(2k_l)!}a_1^{k_1}a_2^{k_2}...a_l^{k_l}.$$ Prove that for every positive integer $n$ the inequality $\sqrt[n]{c_n}\leq \sqrt[n+1]{c_{n+1}}$ holds.

Prove that for any positive integer $ m$ there exist an infinite number of pairs of integers $(x,y)$ such that $(\text{i})$ $x$ and $y$ are relatively prime; $(\text{ii})$ $x$ divides $y^2+m;$ $(\text{iii})$ $y$ divides $x^2+m.$

Investigate if there exist infinitely many natural numbers $n$ such that $n$ divides $2^n+3^n$.

Let $n$ be a positive integer, and let $a_0,\,a_1,\dots,\,a_n$ be nonnegative integers such that $a_0\ge a_1\ge \dots\ge a_n.$ Prove that \[ \sum_{i=0}^n i\binom{a_i}{2}\le\frac{1}{2}\binom{a_0+a_1+\dots+a_n}{2}. \] [i]Note:[/i] $\binom{k}{2}=\frac{k(k-1)}{2}$ for all nonnegative integers $k$.

Sabbir noticed one day that everyone in the city of BdMO has a distinct word of length $10$, where each letter is either $A$ or $B$. Sabbir saw that two citizens are friends if one of their words can be altered a few times using a special rule and transformed into the other ones word. The rule is, if somewhere in the word $ABB$ is located consecutively, then these letters can be changed to $BBA$ or if $BBA$ is located somewhere in the word consecutively, then these letters can be changed to $ABB$ (if wanted, the word can be kept as it is, without making this change.) For example $AABBA$ can be transformed into $AAABB$ (the opposite is also possible.) Now Sabbir made a team of $N$ citizens where no one is friends with anyone. What is the highest value of $N.$

Two distinct numbers a and b satisfy that the two equations $x^{2019} + ax + 2b = 0$ and $x^{2019}+ bx + 2a = 0$ have a common solution. Determine all possible values of $a + b$.

Determine whether the polynomial $P(x)=(x^2-2x+5)(x^2-4x+20)+1$ is irreducible over $\mathbb{Z}[X]$.

You are at a point $(a,b)$ and you need to reach another point $(c,d)$. Both points are below the line $x = y$ and have integer coordinates. You can move in steps of length 1, either upwards of to the right, but you may not move to a point on the line $x = y$. How many different paths are there?

Let $K$ and $M$ be points on the side $AB$ of a triangle $\triangle{ABC}$, and let $L$ and $N$ be points on the side $AC$. The point $K$ is between $M$ and $B$, and the point $L$ is between $N$ and $C$. If $\frac{BK}{KM}=\frac{CL}{LN}$, then prove that the orthocentres of the triangles $\triangle{ABC}$, $\triangle{AKL}$ and $\triangle{AMN}$ lie on one line.

Determine all polynomials P(x) with real coefficients such that [(x + 1)P(x − 1) − (x − 1)P(x)] is a constant polynomial.

Let $A_1A_2A_3 \ldots A_{21}$ be a 21-sided regular polygon inscribed in a circle with centre $O$. How many triangles $A_iA_jA_k$, $1 \leq i < j < k \leq 21$, contain the centre point $O$ in their interior?

Let $a{}$ and $b>1$ be natural numbers. Prove that there exists a natural number $n < b^2$ such that the number $a^n + n$ is divisible by $b{}$.

[b]Problem 1. [/b]In the cells of square table are written the numbers $1$, $0$ or $-1$ so that in every line there is exactly one $1$, amd exactly one $-1$. Each turn we change the places of two columns or two rows. Is it possible, from any such table, after finite number of turns to obtain its opposite table (two tables are opposite if the sum of the numbers written in any two corresponding squares is zero)? [i] Emil Kolev[/i]

Let $ \left( x_n \right)_{n\ge 1} $ be a sequence of positive real numbers, verifying the inequality $ x_n\le \frac{x_{n-1}+x_{n-2}}{2} , $ for any natural number $ n\ge 3. $ Show that $ \left( x_n \right)_{n\ge 1} $ is convergent.

Three positive integers $a,b,c$ with $a>c$ satisfy the following equations : $$ac + b+c = bc + a + 66, \; \; \; \; a+b+c=32$$ Find the value of $a$.

There is a square $ABCD$ in the plane with $|AB|=a$. Determine the smallest possible radius value of three equal circles to cover a given square.

Let $N$ be the number of ordered triples of 3 positive integers $(a,b,c)$ such that $6a$, $10b$, and $15c$ are all perfect squares and $abc = 210^{210}$. Find the number of divisors of $N$. [i]Proposed by Andy Xu[/i]

Find the maximum value of $\frac{1}{x^{2}}+\frac{2}{y^{2}}+\frac{3}{z^{2}}$, where $x, y, z$ are positive reals satisfying $\frac{1}{\sqrt{2}}\leq z <\frac{ \min(x\sqrt{2}, y\sqrt{3})}{2}, x+z\sqrt{3}\geq\sqrt{6}, y\sqrt{3}+z\sqrt{10}\geq 2\sqrt{5}.$

Find all positive integers $m,n$ and prime numbers $p$ for which $\frac{5^m+2^np}{5^m-2^np}$ is a perfect square.

The median $AM$ is drawn in the triangle $ABC$ ($AB \ne AC$). The point $P$ is the foot of the perpendicular drawn on the segment $AM$ from the point $B$. On the segment $AM$ we chose such a point $Q$ that $AQ = 2PM$. Prove that $\angle CQM = \angle BAM$.

Let $G$ be a finite group and $a\in G$ a fixed element. Define the set $$S_a=\{g\in G\mid ga\neq ag, \,ga^2=a^2g\}.$$ Show that: [list=a] [*] if $g\in S_a$, then $ag^{-1}\in S_a$; [*] $|S_a|$ is divisible by $4$.

Two sides of an integer sided triangle have lengths $18$ and $x$. If there are exactly $35$ possible integer $y$ such that $18,x,y$ are the sides of a non-degenerate triangle, find the number of possible integer values $x$ can have.