Determine $ Q \in \mathbb{R}$ which is so big that a sequence with non-negative reals elements $ a_1 ,a_2, \ldots$ which satisfies the following two conditions: [b](i)[/b] $ \forall m,n \geq 1$ we have $ a_{m \plus{} n} \leq 2 \left(a_m \plus{} a_n \right)$ [b](ii)[/b] $ \forall k \geq 0$ we have $ a_{2^k} \leq \frac {1}{(k \plus{} 1)^{2008}}$ such that for each sequence element we have the inequality $ a_n \leq Q.$

A sequence of integers $a_1$, $a_2$, $a_3$, $\ldots$ is chosen so that $a_n = a_{n - 1} - a_{n - 2}$ for each $n \ge 3$. What is the sum of the first 2001 terms of this sequence if the sum of the first 1492 terms is 1985, and the sum of the first 1985 terms is 1492?

A sequence of $X$s and $O$s is given, such that no three consecutive characters in the sequence are all the same, and let $N$ be the number of characters in this sequence. Maia may swap two consecutive characters in the sequence. After each swap, any consecutive block of three or more of the same character will be erased (if there are multiple consecutive blocks of three or more characters after a swap, then they will be erased at the same time), until there are no more consecutive blocks of three or more of the same character. For example, if the original sequence were $XXOOXOXO$ and Maia swaps the fifth and sixth character, the end result will be $$XXOOOXXO \to XXXXO \to O.$$ Find the maximum value $N$ for which Maia can’t necessarily erase all the characters after a series of swaps. Partial credit will be awarded for correct proofs of lower and upper bounds on $N$.

Given a natural number $n$, for which we can find a prime number less than $\sqrt{n}$ that is not a divisor of $n$. The sequence $a_1, a_2,... ,a_n$ is the numbers $1, 2,... ,n$ arranged in some order. For this sequence, we will find the longest ascending subsequense $a_{i_1} < a_{i_2} < ... < a_{i_k}$, ($i_1 <...< i_k$) and the longest decreasing substring $a_{j_1} > ... > a_{j_l}$, ($j_1 < ... < j_l$) . Prove that at least one of these two subsequnsces $a_{i_1} , . . . , a_{i_k}$ and $a_{j_1} > ... > a_{j_l}$ contains a number that is not a divisor of $n$.

The numbers $1, 2,\ldots, 2006$ are written around the circumference of a circle. A [i]move[/i] consists of exchanging two adjacent numbers. After a sequence of such moves, each number ends up $13$ positions to the right of its initial position. lf the numbers $1, 2,\ldots, 2006$ are partitioned into $1003$ distinct pairs, then show that in at least one of the moves, the two numbers of one of the pairs were exchanged.

For a positive integer $ n>2$, let $ \left(a_{1}, a_{2}, \ldots, a_{n}\right)$ be a sequence of $ n$ positive reals which is either non-decreasing (this means, we have $ a_{1}\leq a_{2}\leq \ldots \leq a_{n}$) or non-increasing (this means, we have $ a_{1}\geq a_{2}\geq \ldots \geq a_{n}$), and which satisfies $ a_{1}\neq a_{n}$. Let $ x$ and $ y$ be positive reals satisfying $ \frac{x}{y}\geq \frac{a_{1}-a_{2}}{a_{1}-a_{n}}$. Show that: \[ \frac{a_{1}}{a_{2}\cdot x+a_{3}\cdot y}+\frac{a_{2}}{a_{3}\cdot x+a_{4}\cdot y}+\ldots+\frac{a_{n-1}}{a_{n}\cdot x+a_{1}\cdot y}+\frac{a_{n}}{a_{1}\cdot x+a_{2}\cdot y}\geq \frac{n}{x+y}. \]

Players Aino and Eino take turns choosing numbers from the set $\{0,..., n\}$ with $n\in \Bbb{N}$ being fixed in advance. The game ends when the numbers picked by one of the players include an arithmetic progression of length $4.$ The one who obtains the progression wins. Prove that for some $n,$ the starter of the game wins. Find the smallest such $n.$

Let \( a_1 \) be an integer greater than or equal to 2. Consider the sequence such that its first term is \( a_1 \), and for \( a_n \), the \( n \)-th term of the sequence, we have \[ a_{n+1} = \frac{a_n}{p_k^{e_k - 1}} + 1, \] where \( p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k} \) is the prime factorization of \( a_n \), with \( 1 < p_1 < p_2 < \cdots < p_k \), and \( e_1, e_2, \dots, e_k \) positive integers. For example, if \( a_1 = 2024 = 2^3 \cdot 11 \cdot 23 \), the next two terms of the sequence are \[ a_2 = \frac{a_1}{23^{1-1}} + 1 = \frac{2024}{1} + 1 = 2025 = 3^4 \cdot 5^2; \] \[ a_3 = \frac{a_2}{5^{2-1}} + 1 = \frac{2025}{5} + 1 = 406. \] Determine for which values of \( a_1 \) the sequence is eventually periodic and what all the possible periods are. [b]Note:[/b] Let \( p \) be a positive integer. A sequence \( x_1, x_2, \dots \) is eventually periodic with period \( p \) if \( p \) is the smallest positive integer such that there exists an \( N \geq 0 \) satisfying \( x_{n+p} = x_n \) for all \( n > N \).

Let $a_1, a_2, \cdots, a_n$ be a sequence of nonnegative integers. For $k=1,2,\cdots,n$ denote \[ m_k = \max_{1 \le l \le k} \frac{a_{k-l+1} + a_{k-l+2} + \cdots + a_k}{l}. \] Prove that for every $\alpha > 0$ the number of values of $k$ for which $m_k > \alpha$ is less than $\frac{a_1+a_2+ \cdots +a_n}{\alpha}.$

There are $n^2$ empty boxes, each with a square base. The height and width of each box are integers between $1$ and $n$ inclusive, and no two boxes are identical. One box [i]fits inside[/i] another if its height and width are both smaller, and additionally, one of its dimensions is at least $2$ units smaller. In this way, we can form sequences of boxes (the first inside the second, the second inside the third, and so on). We place each of these sequences on a different shelf. How many shelves are needed to store all the boxes, with certainty?

Each of the small squares of a $50\times 50$ table is coloured in red or blue. Initially all squares are red. A [i]step[/i] means changing the colour of all squares on a row or on a column. a) Prove that there exists no sequence of steps, such that at the end there are exactly $2011$ blue squares. b) Describe a sequence of steps, such that at the end exactly $2010$ squares are blue. [i]Adriana & Lucian Dragomir[/i]

Find all sequences of positive integers $(a_n)_{n\ge 1}$ which satisfy $$a_{n+2}(a_{n+1}-1)=a_n(a_{n+1}+1)$$ for all $n\in \mathbb{Z}_{\ge 1}$. [i](Bogdan Blaga)[/i]

Let $M$ be an arbitrary positive real number greater than $1$, and let $a_1,a_2,...$ be an infinite sequence of real numbers with $a_n\in [1,M]$ for any $n\in \mathbb{N}$. Show that for any $\epsilon\ge 0$, there exists a positive integer $n$ such that $$\frac{a_n}{a_{n+1}}+\frac{a_{n+1}}{a_{n+2}}+\cdots+\frac{a_{n+t-1}}{a_{n+t}}\ge t-\epsilon$$ holds for any positive integer $t$.

Let $a_n$ be a sequence of reals. Suppose $\sum a_n$ converges. Do these sums converge aswell? (a) $a_1+a_2+(a_4+a_3)+(a_8+...+a_5)+(a_{16}+...+a_9)+...$ (b) ${a_1+a_2+(a_3)+(a_4)+(a_5+a_7)+(a_6+a_8)+(a_9+a_{11}+a_{13}+a_{15})+(a_{10}+a_{12}+a_{14}+a_{16})+(a_{17}+a_{19}+...}$

The sequence $a_1, a_2,..., a_{2000}$ of real numbers satisfies the condition \[a_1^3+a_2^3+...+a_n^3=(a_1+a_2+...+a_n)^2\] for all $n$, $1\leq n \leq 2000$. Prove that every element of the sequence is an integer.

Let $S_n$ be the number of sequences $(a_1, a_2, \ldots, a_n),$ where $a_i \in \{0,1\},$ in which no six consecutive blocks are equal. Prove that $S_n \rightarrow \infty$ when $n \rightarrow \infty.$