Find all triangles such that its angles form an arithmetic sequence and the corresponding sides form a geometric sequence.

Suppose that $a_1,...,a_{15}$ are prime numbers forming an arithmetic progression with common difference $d > 0$ if $a_1 > 15$ show that $d > 30000$

Prove that if a sequence of Mikusinski operators of the form $ \mu e^{\minus{}\lambda s}$ ( $ \lambda$ and $ \mu$ nonnegative real numbers, $ s$ the differentiation operator) is convergent in the sense of Mikusinski, then its limit is also of this form. [i]E. Geaztelyi[/i]

Prove that there exists a sequence of unbounded positive integers $a_1\leq a_2\leq a_3\leq\cdots$, such that there exists a positive integer $M$ with the following property: for any integer $n\geq M$, if $n+1$ is not prime, then any prime divisor of $n!+1$ is greater than $n+a_n$.

Let $p$ be an odd prime $p$ such that $2h \neq 1 \; \pmod{p}$ for all $h \in \mathbb{N}$ with $h< p-1$, and let $a$ be an even integer with $a \in] \tfrac{p}{2}, p [$. The sequence $\{a_n\}_{n \ge 0}$ is defined by $a_{0}=a$, $a_{n+1}=p -b_{n}$ \; $(n \ge 0)$, where $b_{n}$ is the greatest odd divisor of $a_n$. Show that the sequence $\{a_n\}_{n \ge 0}$ is periodic and find its minimal (positive) period.

The sequence $a_{n}$ is defined by $a_{1}\geq 2$ and the recurrence formula \[a_{n+1}=a_{n}\sqrt{\frac{a_{n}^3+2}{2(a_{n}^3+1)}}\] for $n\geq 1$. Prove that for every integer $n$, the inequality $a_{n}>\sqrt{\frac{3}{n}}$ holds.

For every triangle $ABC$, denote by $D(ABC)$ the triangle whose vertices are the tangency points of the incircle of $\vartriangle ABC$ with the sides. Assume that $\vartriangle ABC$ is not equilateral. (a) Prove that $D(ABC)$ is also not equilateral. (b) Find in the sequence $T_1 = \vartriangle ABC, T_{k+1} = D(T_k)$ for $k \in N$ a triangle whose largest angle $\alpha$ satisfies $0 < \alpha -60^o < 0.0001^o$

In a sequence of $71$ nonzero real numbers, each number (apart from the fitrst one and the last one) is one less than the product of its two neighbors. Prove that the first and the last number are equal. [i](Josef Tkadlec)[/i]

Sequence $(a_n)$ is defined as $a_{n+1}-2a_n+a_{n-1}=7$ for every $n\geq 2$, where $a_1 = 1, a_2=5$. What is $a_{17}$? $ \textbf{(A)}\ 895 \qquad\textbf{(B)}\ 900 \qquad\textbf{(C)}\ 905 \qquad\textbf{(D)}\ 910 \qquad\textbf{(E)}\ \text{None of the above} $

Let $n \ge 2$ be a positive integer. For every number from $1$ to $n$, there is a card with this number and which is either black or white. A magician can repeatedly perform the following move: For any two tiles with different number and different colour, he can replace the card with the smaller number by one identical to the other card. For instance, when $n=5$ and the initial configuration is $(1B, 2B, 3W, 4B,5B)$, the magician can choose $1B, 3W$ on the first move to obtain $(3W, 2B, 3W, 4B, 5B)$ and then $3W, 4B$ on the second move to obtain $(4B, 2B, 3W, 4B, 5B)$. Determine in terms of $n$ all possible lengths of sequences of moves from any possible initial configuration to any configuration in which no more move is possible.

On a coordinate plane are placed four counters, each of whose centers has integer coordinates. One can displace any counter by the vector joining the centers of two of the other counters. Prove that any two preselected counters can be made to coincide by a finite sequence of moves. [i]Р. Sadykov[/i]

Let $ n > 1$ be an integer. Find all sequences $ a_1, a_2, \ldots a_{n^2 \plus{} n}$ satisfying the following conditions: \[ \text{ (a) } a_i \in \left\{0,1\right\} \text{ for all } 1 \leq i \leq n^2 \plus{} n; \] \[ \text{ (b) } a_{i \plus{} 1} \plus{} a_{i \plus{} 2} \plus{} \ldots \plus{} a_{i \plus{} n} < a_{i \plus{} n \plus{} 1} \plus{} a_{i \plus{} n \plus{} 2} \plus{} \ldots \plus{} a_{i \plus{} 2n} \text{ for all } 0 \leq i \leq n^2 \minus{} n. \] [i]Author: Dusan Dukic, Serbia[/i]

Let $m$ and $n$ are fixed natural numbers and $Oxy$ is a coordinate system in the plane. Find the total count of all possible situations of $n+m-1$ points $P_1(x_1,y_1),P_2(x_2,y_2),\ldots,P_{n+m-1}(x_{n+m-1},y_{n+m-1})$ in the plane for which the following conditions are satisfied: (i) The numbers $x_i$ and $y_i~(i=1,2,\ldots,n+m-1)$ are integers and $1\le x_i\le n,1\le y_i\le m$. (ii) Every one of the numbers $1,2,\ldots,n$ can be found in the sequence $x_1,x_2,\ldots,x_{n+m-1}$ and every one of the numbers $1,2,\ldots,m$ can be found in the sequence $y_1,y_2,\ldots,y_{n+m-1}$. (iii) For every $i=1,2,\ldots,n+m-2$ the line $P_iP_{i+1}$ is parallel to one of the coordinate axes. [i](Ivan Gochev, Hristo Minchev)[/i]

Prove that the sequence of the first digits of the numbers in the form $2^n+3^n$ is nonperiodic.

Let $(a_n)_{n\ge 1}$ be a sequence of positive integers defined as $a_1,a_2>0$ and $a_{n+1}$ is the least prime divisor of $a_{n-1}+a_{n}$, for all $n\ge 2$. Prove that a real number $x$ whose decimals are digits of the numbers $a_1,a_2,\ldots a_n,\ldots $ written in order, is a rational number. [i]Laurentiu Panaitopol[/i]