Found problems: 1782
Let $x_1, x_2, ..., x_{2004}$ be a sequence of integer numbers such that $x_{k+3}=x_{k+2}+x_{k}x_{k+1}$, $\forall 1 \le k \le 2001$. Is it possible that more than half of the elements are negative?
Find the least non-negative integer $n$ such that exists a non-negative integer $k$ such that the last 2012 decimal digits of $n^k$ are all $1$'s.
Let $a_1,a_2,a_3,\dots$ be a sequence of positive real numbers such that $a_ka_{k+2}=a_{k+1}+1$ for all positive integers $k$. If $a_1$ and $a_2$ are positive integers, find the maximum possible value of $a_{2014}$.
Let $r_1=2$ and $r_n = \prod^{n-1}_{k=1} r_i + 1$, $n \geq 2.$ Prove that among all sets of positive integers such that $\sum^{n}_{k=1} \frac{1}{a_i} < 1,$ the partial sequences $r_1,r_2, ... , r_n$ are the one that gets nearer to 1.
Assume that $k$ and $n$ are two positive integers. Prove that there exist positive integers $m_1 , \dots , m_k$ such that \[1+\frac{2^k-1}{n}=\left(1+\frac1{m_1}\right)\cdots \left(1+\frac1{m_k}\right).\]
[i]Proposed by Japan[/i]
For a positive integer $n$, consider the set
\[S = \{0, 1, 1 + 2, 1 + 2 + 3, \ldots, 1 + 2 + 3 +\ldots + (n - 1)\}\]
Prove that the elements of $S$ are mutually incongruent modulo $n$ if and only if $n$ is a power of $2$.
Define a [i]beautiful number[/i] to be an integer of the form $a^n$, where $a\in\{3,4,5,6\}$ and $n$ is a positive integer.
Prove that each integer greater than $2$ can be expressed as the sum of pairwise distinct beautiful numbers.
[i]Proposed by Matthew Babbitt[/i]
Let $\{b_n\}_{n\geq 1}^{\infty}$ be a sequence of positive integers. The sequence $\{a_n\}_{n\geq 1}^{\infty}$ is defined as follows: $a_1$ is a fixed positive integer and
\[a_{n+1}=a_n^{b_n}+1 ,\qquad \forall n\geq 1.\]
Find all positive integers $m\geq 3$ with the following property: If the sequence $\{a_n\mod m\}_{n\geq 1 }^{\infty}$ is eventually periodic, then there exist positive integers $q,u,v$ with $2\leq q\leq m-1$, such that the sequence $\{b_{v+ut}\mod q\}_{t\geq 1}^{\infty}$ is purely periodic.
Find all integers $b$ such that there exists a positive real number $x$ with \[ \dfrac {1}{b} = \dfrac {1}{\lfloor 2x \rfloor} + \dfrac {1}{\lfloor 5x \rfloor} \] Here, $\lfloor y \rfloor$ denotes the greatest integer that is less than or equal to $y$.
Let $d(n)$ be the number of positive divisors of a positive integer $n$ (including $1$ and $n$). Find all values of $n$ such that $n + d(n) = d(n)^2$.
Let $n\geq 0$ be an integer and let $p \equiv 7 \pmod 8$ be a prime number. Prove that
\[ \sum^{p-1}_{k=1} \left \{ \frac {k^{2^n}}p - \frac 12 \right\} = \frac {p-1}2 . \]
[i]Călin Popescu[/i]
An elephant writes a sequence of numbers on a board starting with 1. Each minute, it doubles the sum of all the numbers on the board so far, and without erasing anything, writes the result on the board. It stops after writing a number greater than one billion. How many distinct prime factors does the largest number on the board have?
[i]Ray Li.[/i]
Archipelago consists of $ n$ islands : $ I_1,I_2,...,I_n$ and $ a_1,a_2,...,a_n$ - number of the roads on each island. $ a_1 \equal{} 55$, $ a_k \equal{} a_{k \minus{} 1} \plus{} (k \minus{} 1)$, ($ k \equal{} 2,3,...,n$)
a) Does there exist an island with 2008 roads?
b) Calculate $ a_1 \plus{} a_2 \plus{} ... \plus{} a_n.$
Find all functions $ f: \mathbb{Z}\setminus\{0\}\to \mathbb{Q}$ such that for all $ x,y \in \mathbb{Z}\setminus\{0\}$:
\[ f \left( \frac{x+y}{3}\right) =\frac{f(x)+f(y)}{2}, \; \; x, y \in \mathbb{Z}\setminus\{0\}\]
All the grids of a $m\times n$ chess board ($m,n\geq 3$), are colored either with red or with blue. Two adjacent grids (having a common side) are called a "good couple" if they have different colors. Suppose there are $S$ "good couples". Explain how to determine whether $S$ is odd or even. Is it prescribed by
some specific color grids? Justify your answers.
Let $N$ be a positive integer. Alberto and Barbara write numbers on a blackboard taking turns, according to the following rules. Alberto starts writing $1$, and thereafter if a player has written $n$ on a certain move, his adversary is allowed to write $n+1$ or $2n$ as long as he/she does not obtain a number greater than $N$. The player who writes $N$ wins.
$(a)$ Determine which player has a winning strategy for $N=2005$.
$(b)$ Determine which player has a winning strategy for $N=2004$.
$(c)$ Find for how many integers $N\le 2005$ Barbara has a winning strategy.
In a meeting, there are $2011$ scientists attending. We know that, every scientist know at least $1509$ other ones. Prove that a group of five scientists can be formed so that each one in this group knows $4$ people in his group.
Let $\mathbb{N}_0$ and $\mathbb{Z}$ be the set of all non-negative integers and the set of all integers, respectively. Let $f:\mathbb{N}_0\rightarrow\mathbb{Z}$ be a function defined as
\[f(n)=-f\left(\left\lfloor\frac{n}{3}\right\rfloor \right)-3\left\{\frac{n}{3}\right\} \]
where $\lfloor x \rfloor$ is the greatest integer smaller than or equal to $x$ and $\{ x\}=x-\lfloor x \rfloor$. Find the smallest integer $n$ such that $f(n)=2010$.
Consider the power series expansion \[\dfrac{1}{1-2x-x^2}=\sum_{n=0}^\infty a_nx^n.\] Prove that, for each integer $n\geq 0$, there is an integer $m$ such that \[a_n^2+a_{n+1}^2=a_m.\]
How many functions $\{f : 1,2, \cdots, 2013\} \rightarrow \{1,2, \cdots, 2013\}$ satisfy $f(j) < f(i) + j - i$ for all integers $i,j$ such that $1 \leq i < j \leq 2013$ ?
Let $n$ be positive integer. Define a sequence $\{a_k\}$ by
\[a_1=\frac{1}{n(n+1)},\ a_{k+1}=-\frac{1}{k+n+1}+\frac{n}{k}\sum_{i=1}^k a_i\ \ (k=1,\ 2,\ 3,\ \cdots).\]
(1) Find $a_2$ and $a_3$.
(2) Find the general term $a_k$.
(3) Let $b_n=\sum_{k=1}^n \sqrt{a_k}$. Prove that $\lim_{n\to\infty} b_n=\ln 2$.
50 points
On a circle there are $2n+1$ points, dividing it into equal arcs ($n\ge 2$). Two players take turns to erase one point. If after one player's turn, it turned out that all the triangles formed by the remaining points on the circle were obtuse, then the player wins and the game ends.
Who has a winning strategy: the starting player or his opponent?
There are $n$ boxes which is numbere from $1$ to $n$. The box with number $1$ is open, and the others are closed. There are $m$ identical balls ($m\geq n$). One of the balls is put into the open box, then we open the box with number $2$. Now, we put another ball to one of two open boxes, then we open the box with number $3$. Go on until the last box will be open. After that the remaining balls will be randomly put into the boxes. In how many ways this arrangement can be done?
Let $ U$ be a real normed space such that, for an finite-dimensional, real normed space $ X,U$ contains a subspace isometrically isomorphic to $ X$. Prove that every (not necessarily closed) subspace $ V$ of $ U$ of finite codimension has the same property. (We call $ V$ of finite codimension if there exists a finite-dimensional subspace $ N$ of $ U$ such that $ V\plus{}N\equal{}U$.)
[i]A. Bosznay[/i]
We are given one red and $k>1$ blue cells, and a pack of $2n$ cards, enumerated by the numbers from $1$ to $2n$. Initially, the pack is situated on the red cell and arranged in an arbitrary order. In each move, we are allowed to take the top card from one of the cells and place it either onto the top of another cell on which the number on the top card is greater by $1$, or onto an empty cell. Given $k$, what is the maximal $n$ for which it is always possible to move all the cards onto a blue cell?