Let $ G$ be a finite non-commutative group of order $ t \equal{} 2^nm$, where $ n, m$ are positive and $ m$ is odd. Prove, that if the group contains an element of order $ 2^n$, then (i) $ G$ is not simple; (ii) $ G$ contains a normal subgroup of order $ m$.

Let $A$ be the $n\times n$ matrix whose entry in the $i$-th row and $j$-th column is \[\frac1{\min(i,j)}\] for $1\le i,j\le n.$ Compute $\det(A).$

Let $a$ and $b$ be positive integers with $a>1$ and $b>2$. Prove that $a^b+1\ge b(a+1)$ and determine when there is inequality.

Find the largest prime that divides $1\cdot 2\cdot 3+2\cdot 3\cdot 4+\cdots +44\cdot 45\cdot 46$

We call a subset $B$ of natural numbers [i]loyal[/i] if there exists natural numbers $i\le j$ such that $B=\{i,i+1,\ldots,j\}$. Let $Q$ be the set of all [i]loyal[/i] sets. For every subset $A=\{a_1<a_2<\ldots<a_k\}$ of $\{1,2,\ldots,n\}$ we set \[f(A)=\max_{1\le i \le k-1}{a_{i+1}-a_i}\qquad\text{and}\qquad g(A)=\max_{B\subseteq A, B\in Q} |B|.\] Furthermore, we define \[F(n)=\sum_{A\subseteq \{1,2,\ldots,n\}} f(A)\qquad\text{and}\qquad G(n)=\sum_{A\subseteq \{1,2,\ldots,n\}} g(A).\] Prove that there exists $m\in \mathbb N$ such that for each natural number $n>m$ we have $F(n)>G(n)$. (By $|A|$ we mean the number of elements of $A$, and if $|A|\le 1$, we define $f(A)$ to be zero). [i]Proposed by Javad Abedi[/i]

Let $N_0=\{0, 1, 2 \cdots \}$. Find all functions: $N_0 \to N_0$ such that: (1) $f(n) < f(n+1)$, all $n \in N_0$; (2) $f(2)=2$; (3) $f(mn)=f(m)f(n)$, all $m, n \in N_0$.

A brick has the shape of a cube of size $2$ with one corner unit cube removed. Given a cube of side $2^{n}$ divided into unit cubes from which an arbitrary unit cube is removed, show that the remaining figure can be built using the described bricks.

Prove that the sequence $ \{y_{n}\}_{n \ge 1}$ defined by \[ y_{0}=1, \; y_{n+1}= \frac{1}{2}\left( 3y_{n}+\sqrt{5y_{n}^{2}-4}\right) \] consists only of integers.

Find all natural numbers a, b such that $ a^{n}\plus{} b^{n} \equal{} c^{n\plus{}1}$ where c and n are naturals.

$F(0)=3$ and $F(n)=F(n-1)+4$ when $n$ is positive. Find $F(F(F(5)))$.

There are$n$ balls numbered $1,2,\cdots,n$, respectively. They are painted with $4$ colours, red, yellow, blue, and green, according to the following rules: First, randomly line them on a circle. Then let any three clockwise consecutive balls numbered $i, j, k$, in order. 1) If $i>j>k$, then the ball $j$ is painted in red; 2) If $i<j<k$, then the ball $j$ is painted in yellow; 3) If $i<j, k<j$, then the ball $j$ is painted in blue; 4) If $i>j, k>j$, then the ball $j$ is painted in green. And now each permutation of the balls determine a painting method. We call two painting methods distinct, if there exists a ball, which is painted with two different colours in that two methods. Find out the number of all distinct painting methods.

The jury of an Olympiad has $30$ members in the beginning. Each member of the jury thinks that some of his colleagues are competent, while all the others are not, and these opinions do not change. At the beginning of every session a voting takes place, and those members who are not competent in the opinion of more than one half of the voters are excluded from the jury for the rest of the olympiad. Prove that after at most $15$ sessions there will be no more exclusions. (Note that nobody votes about his own competence.)

Each rational number is painted either white or red. Call such a coloring of the rationals [i]sanferminera[/i] if for any distinct rationals numbers $x$ and $y$ satisfying one of the following three conditions: [list=1][*]$xy=1$, [*]$x+y=0$, [*]$x+y=1$,[/list]we have $x$ and $y$ painted different colors. How many sanferminera colorings are there?

Let $n$ be a positive integer. Let $P_n=\{2^n,2^{n-1}\cdot 3, 2^{n-2}\cdot 3^2, \dots, 3^n \}.$ For each subset $X$ of $P_n$, we write $S_X$ for the sum of all elements of $X$, with the convention that $S_{\emptyset}=0$ where $\emptyset$ is the empty set. Suppose that $y$ is a real number with $0 \leq y \leq 3^{n+1}-2^{n+1}.$ Prove that there is a subset $Y$ of $P_n$ such that $0 \leq y-S_Y < 2^n$

Prove $ \frac{1}{10\sqrt2}<\frac{1}{2}\frac{3}{4}\frac{5}{6}...\frac{99}{100}<\frac{1}{10} $

Two numbers are written on each vertex of a convex $100$-gon. Prove that it is possible to remove a number from each vertex so that the remaining numbers on any two adjacent vertices are different. [i]F. Petrov [/i]

In a party among any four persons there are three people who are mutual acquaintances or mutual strangers. Prove that all the people can be separated into two groups $A$ and $B$ such that in $A$ everybody knows everybody else and in $B$ nobody knows anybody else.

Let $n$ be an integer, and let $X$ be a set of $n+2$ integers each of absolute value at most $n$. Show that there exist three distinct numbers $a, b, c \in X$ such that $c=a+b$.

In a country there are several cities; some of these cities are connected by airlines, so that an airline connects exactly two cities in each case and both flight directions are possible. Each airline belongs to one of $k$ flight companies; two airlines of the same flight company have always a common final point. Show that one can partition all cities in $k+2$ groups in such a way that two cities from exactly the same group are never connected by an airline with each other.

Let $ f(x) \equal{} c_m x^m \plus{} c_{m\minus{}1} x^{m\minus{}1} \plus{}...\plus{} c_1 x \plus{} c_0$, where each $ c_i$ is a non-zero integer. Define a sequence $ \{ a_n \}$ by $ a_1 \equal{} 0$ and $ a_{n\plus{}1} \equal{} f(a_n)$ for all positive integers $ n$. (a) Let $ i$ and $ j$ be positive integers with $ i<j$. Show that $ a_{j\plus{}1} \minus{} a_j$ is a multiple of $ a_{i\plus{}1} \minus{} a_i$. (b) Show that $ a_{2008} \neq 0$

Find the number of ten-digit natural numbers (which do not start with zero) containing no block $ 1991$.

In Lineland there are $n\geq1$ towns, arranged along a road running from left to right. Each town has a [i]left bulldozer[/i] (put to the left of the town and facing left) and a [i]right bulldozer[/i] (put to the right of the town and facing right). The sizes of the $2n$ bulldozers are distinct. Every time when a left and right bulldozer confront each other, the larger bulldozer pushes the smaller one off the road. On the other hand, bulldozers are quite unprotected at their rears; so, if a bulldozer reaches the rear-end of another one, the first one pushes the second one off the road, regardless of their sizes. Let $A$ and $B$ be two towns, with $B$ to the right of $A$. We say that town $A$ can [i]sweep[/i] town $B$ [i]away[/i] if the right bulldozer of $A$ can move over to $B$ pushing off all bulldozers it meets. Similarly town $B$ can sweep town $A$ away if the left bulldozer of $B$ can move over to $A$ pushing off all bulldozers of all towns on its way. Prove that there is exactly one town that cannot be swept away by any other one.

Prove that for every nonnegative integer $n$, the number $7^{7^{n}}+1$ is the product of at least $2n+3$ (not necessarily distinct) primes.

We are given $n$ intervals $[l_1,r_1],[l_2,r_2],[l_3,r_3],\dots, [l_n,r_n]$ in the number line. We can divide the intervals into two sets such that no two intervals in the same set have overlaps. Prove that there are at most $n-1$ pairs of overlapping intervals.

Sequence of real numbers $a_0,a_1,\dots,a_{1389}$ are called concave if for each $0<i<1389$, $a_i\geq\frac{a_{i-1}+a_{i+1}}2$. Find the largest $c$ such that for every concave sequence of non-negative real numbers: \[\sum_{i=0}^{1389}ia_i^2\geq c\sum_{i=0}^{1389}a_i^2\]