Let $n$ be a positive integer. We start with $n$ piles of pebbles, each initially containing a single pebble. One can perform moves of the following form: choose two piles, take an equal number of pebbles from each pile and form a new pile out of these pebbles. Find (in terms of $n$) the smallest number of nonempty piles that one can obtain by performing a finite sequence of moves of this form.

The limit of the sum of an infinite number of terms in a geometric progression is $ \frac {a}{1 \minus{} r}$ where $ a$ denotes the first term and $ \minus{} 1 < r < 1$ denotes the common ratio. The limit of the sum of their squares is: $ \textbf{(A)}\ \frac {a^2}{(1 \minus{} r)^2} \qquad\textbf{(B)}\ \frac {a^2}{1 \plus{} r^2} \qquad\textbf{(C)}\ \frac {a^2}{1 \minus{} r^2} \qquad\textbf{(D)}\ \frac {4a^2}{1 \plus{} r^2} \qquad\textbf{(E)}\ \text{none of these}$

Consider an infinite chessboard whose rows and columns are indexed by positive integers. At most one coin can be put on any cell of the chessboard. Let be given two arbitrary sequences ($a_n$) and ($b_n$) of positive integers ($n \in N$). Assuming that infinitely many coins are available, prove that they can be arranged on the chessboard so that there are $a_n$ coins in the $n$-th row and $b_n$ coins in the $n$-th column for all $n$.

Let $n \ge 3$ be an odd integer. Each cell is a $n \times n$ board painted in yellow or blue. Let's call the sequence of cells $S_1, S_2,...,S_m$ [i]path [/i] if they are all the same color and the cells $S_i$ and $S_j$ have one in common an edge if and only if $|i - j| = 1$. Suppose that all yellow cells form a path and all the blue cells form a path. Prove that one of the two paths begins or ends at the center of the board.

Let $\alpha$ be the unique real root of the polynomial $x^3-2x^2+x-1$. It is known that $1<\alpha<2$. We define the sequence of polynomials $\left\{{p_n(x)}\right\}_{n\ge0}$ by taking $p_0(x)=x$ and setting \begin{align*} p_{n+1}(x)=(p_n(x))^2-\alpha \end{align*} How many distinct real roots does $p_{10}(x)$ have?

Consider sequences $a_0$,$a_1$,$a_2$,$\cdots$ of non-negative integers defined by selecting any $a_0$,$a_1$,$a_2$ (not all 0) and for each $n$ $\geq$ 3 letting $a_n$ = |$a_n-1$ - $a_n-3$| 1-In the particular case that $a_0$ = 1,$a_1$ = 3 and $a_2$ = 2, calculate the beginning of the sequence, listing $a_0$,$a_1$,$\cdots$,$a_{19}$,$a_{20}$. 2-Prove that for each sequence, there is a constant $c$ such that $a_i$ $\leq$ $c$ for all $i$ $\geq$ 0. Note that the constant $c$ my depend on the numbers $a_0$,$a_1$ and $a_2$ 3-Prove that, for each choice of $a_0$,$a_1$ and $a_2$, the resulting sequence is eventually periodic. 4-Prove that, the minimum length p of the period described in (3) is the same for all permitted starting values $a_0$,$a_1$,$a_2$ of the sequence

We call [i]word [/i] a sequence of letters $\overline {l_1l_2...l_n}, n\ge 1$ . A [i]word [/i] $\overline {l_1l_2...l_n}, n\ge 1$ is called [i]palindrome [/i] if $l_k=l_{n-k+1}$ , for any $k, 1 \le k \le n$. Consider a [i]word [/i] $X=\overline {l_1l_2...l_{2014}}$ in which $ l_k\in\{A,B\}$ , for any $k, 1\le k \le 2014$. Prove that there are at least $806$ [i]palindrome [/i] [i]words [/i] to ''stick" together to get word $X$.

In the plane a point $O$ is and a sequence of points $P_1, P_2, P_3, \ldots$ are given. The distances $OP_1, OP_2, OP_3, \ldots$ are $r_1, r_2, r_3, \ldots$ Let $\alpha$ satisfies $0 < \alpha < 1.$ Suppose that for every $n$ the distance from the point $P_n$ to any other point of the sequence is $\geq r^{\alpha}_n.$ Determine the exponent $\beta$, as large as possible such that for some $C$ independent of $n$ \[r_n \geq Cn^{\beta}, n = 1,2, \ldots\]

Let $A,B$ be two points in the plane with integer coordinates $A=(x_1,y_1)$ and $B=(x_2,y_2)$. (Thus $x_i,y_i\in\mathbb Z$, for $i=1,2$.) A path $\pi:A\to B$ is a sequence of [b]down[/b] and [b]right[/b] steps, where each step has an integer length, and the initial step starts from $A$, the last step ending at $B$. In the figure below, we indicated a path from $A_1=(4,9)$ to $B1=(10,3)$. The distance $d(A,B)$ between $A$ and $B$ is the number of such paths. For example, the distance between $A=(0,2)$ and $B=(2,0)$ equals $6$. Consider now two pairs of points in the plane $A_i=(x_i,y_i)$ and $B_i=(u_i,z_i)$ for $i=1,2$, with integer coordinates, and in the configuration shown in the picture (but with arbitrary coordinates): $x_2<x_1$ and $y_1>y_2$, which means that $A_1$ is North-East of $A_2$; $u_2<u_1$ and $z_1>z_2$, which means that $B_1$ is North-East of $B_2$. Each of the points $A_i$ is North-West of the points $B_j$, for $1\le i$, $j\le2$. In terms of inequalities, this means that $x_i<\min\{u_1,u_2\}$ and $y_i>\max\{z_1,z_2\}$ for $i=1,2$. [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvYi9hL2I4ODlmNDAyYmU5OWUyMzVmZmEzMTY1MGY3YjI3YjFlMmMxMTI2LnBuZw==&rn=VlRSTUMgMjAxNC5wbmc=[/img] (a) Find the distance between two points $A$ and $B$ as before, as a function of the coordinates of $A$ and $B$. Assume that $A$ is North-West of $B$. (b) Consider the $2\times2$ matrix $M=\begin{pmatrix}d(A_1,B_1)&d(A_1,B_2)\\d(A_2,B_1)&d(A_2,B_2)\end{pmatrix}$. Prove that for any configuration of points $A_1,A_2,B_1,B_2$ as described before, $\det M>0$.

We define the following operation which will be applied to a row of bars being situated side-by-side on positions $1, 2, \ldots ,N$. Each bar situated at an odd numbered position is left as is, while each bar at an even numbered position is replaced by two bars. After that, all bars will be put side-by- side in such a way that all bars form a new row and are situated on positions $1, \ldots,M$. From an initial number $a_0 > 0$ of bars there originates a sequence $(a_n)_{n \geq 0}$, where an is the number of bars after having applied the operation $n$ times. [b](a)[/b] Prove that for no $n > 0$ can we have $a_n = 1997$. [b](b)[/b] Determine all natural numbers that can only occur as $a_0$ or $a_1$.

The sequence $\{y_{n}\}_{n \ge 1}$ is defined by \[y_{1}=y_{2}=1,\;\; y_{n+2}= (4k-5)y_{n+1}-y_{n}+4-2k.\] Determine all integers $k$ such that each term of this sequence is a perfect square.

Find all polynomials with integer coefficients $P$ such that for all positive integers $n$, the sequence $$0, P(0), P(P(0)), \cdots$$ is eventually constant modulo $n$. [i]Proposed by Ivan Chan Kai Chin[/i]

The kingdom of Anisotropy consists of $n$ cities. For every two cities there exists exactly one direct one-way road between them. We say that a [i]path from $X$ to $Y$[/i] is a sequence of roads such that one can move from $X$ to $Y$ along this sequence without returning to an already visited city. A collection of paths is called [i]diverse[/i] if no road belongs to two or more paths in the collection. Let $A$ and $B$ be two distinct cities in Anisotropy. Let $N_{AB}$ denote the maximal number of paths in a diverse collection of paths from $A$ to $B$. Similarly, let $N_{BA}$ denote the maximal number of paths in a diverse collection of paths from $B$ to $A$. Prove that the equality $N_{AB} = N_{BA}$ holds if and only if the number of roads going out from $A$ is the same as the number of roads going out from $B$. [i]Proposed by Warut Suksompong, Thailand[/i]

Let $p \equiv 2 \pmod 3$ be a prime, $k$ a positive integer and $P(x) = 3x^{\frac{2p-1}{3}}+3x^{\frac{p+1}{3}}+x+1$. For any integer $n$, let $R(n)$ denote the remainder when $n$ is divided by $p$ and let $S = \{0,1,\cdots,p-1\}$. At each step, you can either (a) replaced every element $i$ of $S$ with $R(P(i))$ or (b) replaced every element $i$ of $S$ with $R(i^k)$. Determine all $k$ such that there exists a finite sequence of steps that reduces $S$ to $\{0\}$. [i]Proposed by fattypiggy123[/i]

Let $\mathbb N$ denote the set of positive integers. Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that: (i) The greatest common divisor of the sequence $f(1), f(2), \dots$ is $1$. (ii) For all sufficiently large integers $n$, we have $f(n) \neq 1$ and \[ f(a)^n \mid f(a+b)^{a^{n-1}} - f(b)^{a^{n-1}} \] for all positive integers $a$ and $b$. [i]Proposed by Yang Liu[/i]