Let $c$ be a positive real number. Prove that $c$ can be expressed in infinitely many ways as a sum of infinitely many distinct terms selected from the sequence $\left( \frac{1}{10n} \right)_{n\in \mathbb{N}}$

We attempt to cover the plane with an infinite sequence of rectangles, overlapping allowed. (a) Is the task always possible if the area of the $n$th rectangle is $n^2$ for each $n$? (b) Is the task always possible if each rectangle is a square, and for any number $N$, there exist squares with total area greater than $N$?

A sequence $\left(a_n\right)$ with $a_1 = 1$ satisfies the following recursion: In the decimal expansion of $a_n$ (without trailing zeros) let $k$ be the smallest digest then $a_{n+1} = a_n + 2^k.$ How many digits does $a_{9 \cdot 10^{2010}}$ have in the decimal expansion?

Is $ \sqrt{2} $ the limit of a sequence of numbers of the form $ \sqrt[3]{n} - \sqrt[3]{m} $, where $ n, m = 0, 1, 2, \cdots $.

We consider two sequences of real numbers $x_{1} \geq x_{2} \geq \ldots \geq x_{n}$ and $\ y_{1} \geq y_{2} \geq \ldots \geq y_{n}.$ Let $z_{1}, z_{2}, .\ldots, z_{n}$ be a permutation of the numbers $y_{1}, y_{2}, \ldots, y_{n}.$ Prove that $\sum \limits_{i=1}^{n} ( x_{i} -\ y_{i} )^{2} \leq \sum \limits_{i=1}^{n}$ $( x_{i} - z_{i})^{2}.$

Given three numbers $x, y, z$ denote the absolute values of the differences of each pair by $x_1,y_1, z_1$. From $x_1, y_1, z_1$ form in the same fashion the numbers $x_2, y_2, z_2$, etc. It is known that $x_n = x,y_n = y, z_n = z$ for some $n$. Find $y$ and $z$ if $x = 1$.

[u]Round 1 [/u] [b]p1. [/b]Twelve people, some are knights and some are knaves, are sitting around a table. Knaves always lie and knights always tell the truth. At some point they start up a conversation. The first person says, “There are no knights around this table.” The second says, “There is at most one knight at this table.” The third – “There are at most two knights at the table.” And so on until the 12th says, “There are at most eleven knights at the table.” How many knights are at the table? Justify your answer. [b]p2.[/b] Show that in the sequence $10017$, $100117$, $1001117$, $...$ all numbers are divisible by $53$. [b]p3.[/b] Harry and Draco have three wands: a bamboo wand, a willow wand, and a cherry wand, all of the same length. They must perform a spell wherein they take turns picking a wand and breaking it into three parts – first Harry, then Draco, then Harry again. But in order for the spell to work, Harry has to make sure it is possible to form three triangles out of the pieces of the wands, where each triangle has a piece from each wand. How should he break the wands to ensure the success of the spell? [b]p4.[/b] A $2\times 2\times 2$ cube has $4$ equal squares on each face. The squares that share a side are called neighbors (thus, each square has $4$ neighbors – see picture). Is it possible to write an integer in each square in such a way that the sum of each number with its $4$ neighbors is equal to $13$? If yes, show how. If no, explain why not. [img]https://cdn.artofproblemsolving.com/attachments/8/4/0f7457f40be40398dee806d125ba26780f9d3a.png[/img] [b]p5.[/b] Two girls are playing a game. The first player writes the letters $A$ or $B$ in a row, left to right, adding one letter on her turn. The second player switches any two letters after each move by the first player (the letters do not have to be adjacent), or does nothing, which also counts as a move. The game is over when each player has made $2011$ moves. Can the second player plan her moves so that the resulting letters form a palindrome? (A palindrome is a sequence that reads the same forward and backwards, e.g. $AABABAA$.) [u]Round 2 [/u] [b]p6.[/b] A red square is placed on a table. $2010$ white squares, each the same size as the red square, are then placed on the table in such a way that the red square is fully covered and the sides of every white square are parallel to the sides of the red square. Is it always possible to remove one of the white squares so the red square remains completely covered? [b]p7.[/b] A computer starts with a given positive integer to which it randomly adds either $54$ or $77$ every second and prints the resulting sum after each addition. For example, if the computer is given the number $1$, then a possible output could be: $1$, $55$, $109$, $186$, $…$ Show that after finitely many seconds the computer will print a number whose last two digits are the same. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The $49$ numbers $2,3,4,...,49,50$ are written on the blackboard . An allowed operation consists of choosing two different numbers $a$ and $b$ of the blackboard such that $a$ is a multiple of $b$ and delete exactly one of the two. María performs a sequence of permitted operations until she observes that it is no longer possible to perform any more. Determine the minimum number of numbers that can remain on the board at that moment.

Let $\{a_n\}^{\infty}_0$ and $\{b_n\}^{\infty}_0$ be two sequences determined by the recursion formulas \[a_{n+1} = a_n + b_n,\] \[ b_{n+1} = 3a_n + b_n, n= 0, 1, 2, \cdots,\] and the initial values $a_0 = b_0 = 1$. Prove that there exists a uniquely determined constant $c$ such that $n|ca_n-b_n| < 2$ for all nonnegative integers $n$.

Let $\Lambda= \{1, 2, \ldots, 2v-1,2v\}$ and $P=\{\alpha_1, \alpha_2, \ldots, \alpha_{2v-1}, \alpha_{2v}\}$ be a permutation of the elements of $\Lambda$. (a) Prove that $$\sum_{i=1}^v \alpha_{2i-1}\alpha_{2i} \leq \sum_{i=1}^v (2i-1)2i.$$ (b) Determine the largest positive integer $m$ such that we can partition the $m\times m$ square into $7$ rectangles for which every pair of them has no common interior points and their lengths and widths form the following sequence: $$1,2,3,4,5,6,7,8,9,10,11,12,13,14.$$

The side lengths of a triangle area $t$ form an arithmetic progression with difference $d$. Find the sides and angles of the triangle. Specifically, solve this problem for $d=1$ and $t=6$.

The [i]fibboican[/i] sequence $a_1,\ a_2,\ \dots$, is defined by $a_1 = a_2 = 1$, and for integers $k \geq 3$, [list] [*] $a_k = a_{k-1} + a_{k-2}$ if $k$ is odd [*] $\frac {1}{a_k} = \frac {1}{a_{k-1}} + \frac {1}{a_{k-2}}$ if $k$ is even. [/list] Prove that, for each integer $m\ge 1$, the numerator of $a_m$ (when written in simplest form) is a power of $2$. [i]Eric Shen (CAN)[/i]

Let $\{a_n\}_{n\geq 1}$ be a sequence of real numbers which satisfies the following relation: \[a_{n+1}=10^n a_n^2\] (a) Prove that if $a_1$ is small enough, then $\displaystyle\lim_{n\to\infty} a_n =0$. (b) Find all possible values of $a_1\in \mathbb{R}$, $a_1\geq 0$, such that $\displaystyle\lim_{n\to\infty} a_n =0$.

Consider all binary sequences (sequences consisting of 0’s and 1’s). In such a sequence the following four types of operation are allowed: (a) $010 \rightarrow 1$, (b) $1 \rightarrow 010$, (c) $110 \rightarrow 0$, and (d) $0 \rightarrow 110$. Determine if it is possible to obtain the sequence $100...0$ (with $2003$ zeroes) from the sequence $0...01$ (with $2003$ zeroes).

a) We call [i]admissible sequence[/i] a sequence of 4 even digits in which no digits appears more than two times. Find the number of admissible sequences. b) For each integer $ n\geq 2$ we denote $ d_n$ the number of possibilities of completing with even digits an array with $ n$ rows and 4 columns, such that (1) any row is an admissible sequence; (2) the sequence 2, 0, 0, 8 appears exactly ones in the array. Find the values of $ n$ for which the number $ \frac {d_{n\plus{}1}}{d_n}$ is an integer.

Given [i]Fibonacci[/i] sequence $(F_n),$ and a positive integer $m$, denote $k(m)$ by the smallest positive integer satisfying $F_{n+k(m)}\equiv F_n(\bmod m),$ for all natural numbers $n$, $s$ is a positive integer. Prove that: a) ${F_{{{3.2}^{s - 1}}}} \equiv 0(\bmod {2^s})$ and ${F_{{{3.2}^{s - 1}} + 1}} \equiv 1(\bmod {2^s}).$ b) $k({2^s}) = {3.2^{s - 1}}.$