Let ${a_n}$ and ${b_n}$ be positive real sequence that satisfy the following condition: (i) $a_1=b_1=1$ (ii) $b_n=a_n b_{n-1}-2$ (iii) $n$ is an integer larger than $1$. Let ${b_n}$ be a bounded sequence. Prove that $\sum_{n=1}^{\infty} \frac{1}{a_1a_2\cdots a_n}$ converges. Find the value of the sum.

A fair six-sided die is rolled five times. The probability that the five die rolls form an increasing sequence where each value is strictly larger than the one that preceded can be written in the form $m/n$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

Let $n$ and $k$ be positive integers. A baby uses $n^2$ blocks to form a $n\times n$ grid, with each of the blocks having a positive integer no greater than $k$ on it. The father passes by and notice that: 1. each row on the grid can be viewed as an arithmetic sequence with the left most number being its leading term, with all of them having distinct common differences; 2. each column on the grid can be viewed as an arithmetic sequence with the top most number being its leading term, with all of them having distinct common differences, Find the smallest possible value of $k$ (as a function of $n$.) Note: The common differences might not be positive. Proposed by Chu-Lan Kao

9. Let [i]k[/i] be a positive integer and let $x_0, x_1, x_2, \cdots$ be an infinite sequence defined by the relationship $$x_0 = 0$$ $$x_1 = 1$$ $$x_{n+1} = kx_n +x_{n-1}$$ For all [i]n[/i] $\ge$ 1 (a) For the special case [i]k[/i] = 1, prove that $x_{n-1}x_{n+1}$ is never a perfect square for [i]n[/i] $\ge$ 2 (b) For the general case of integers [i]k[/i] $\ge$ 1, prove that $x_{n-1}x_{n+1}$ is never a perfect square for [i]n[/i] $\ge$ 2

A bounded sequence $ (x_n)_{n\ge 1}$ of real numbers satisfies $ x_n \plus{} x_{n \plus{} 1} \ge 2x_{n \plus{} 2}$ for all $ n \ge 1$. Prove that this sequence has a finite limit.

For reals $x\ge3$, let $f(x)$ denote the function \[f(x) = \frac {-x + x\sqrt{4x-3} } { 2} .\]Let $a_1, a_2, \ldots$, be the sequence satisfying $a_1 > 3$, $a_{2013} = 2013$, and for $n=1,2,\ldots,2012$, $a_{n+1} = f(a_n)$. Determine the value of \[a_1 + \sum_{i=1}^{2012} \frac{a_{i+1}^3} {a_i^2 + a_ia_{i+1} + a_{i+1}^2} .\] [i]Ray Li.[/i]

Let $n>1$ be an integer. Call a number beautiful if its square leaves an odd remainder upon divison by $n$. Prove that the number of consecutive beautiful numbers is less or equal to $1+\lfloor \sqrt{3n} \rfloor$.

Define a sequence recursively by $x_0=5$ and \[x_{n+1}=\frac{x_n^2+5x_n+4}{x_n+6}\] for all nonnegative integers $n.$ Let $m$ be the least positive integer such that \[x_m\leq 4+\frac{1}{2^{20}}.\] In which of the following intervals does $m$ lie? $\textbf{(A) } [9,26] \qquad\textbf{(B) } [27,80] \qquad\textbf{(C) } [81,242]\qquad\textbf{(D) } [243,728] \qquad\textbf{(E) } [729,\infty]$

Let a sequence be defined as follows: $a_0=1$, and for $n>0$, $a_n$ is $\tfrac{1}{3}a_{n-1}$ and is $\tfrac{1}{9}a_{n-1}$ with probability $\tfrac{1}{2}$. If the expected value of $\textstyle\sum_{n=0}^{\infty}a_n$ can be expressed in simplest form as $\tfrac{p}{q}$, what is $p+q$?

Let be a sequence of natural numbers $ \left( a_n \right)_{n\ge 1} $ such that $ a_n\le n $ for all natural numbers $ n, $ and $$ \sum_{j=1}^{k-1} \cos \frac{\pi a_j}{k} =0, $$ for all natural $ k\ge 2. $ [b]a)[/b] Find $ a_2. $ [b]b)[/b] Determine this sequence.

Anja has to write $2014$ integers on the board such that arithmetic mean of any of the three numbers is among those $2014$ numbers. Show that this is possible only when she writes nothing but $2014$ equal integers.

Let $N>s$ be positive integers. Electricity park has a number of buildings; exactly $N$ of them are power plants, and another one of them is the headquarter. Some pairs of buildings have one-way power cables between them, satisfying: (i) The cables connected to a power plant will only send the power out of the plant. (ii) For each non-headquarter building, there is a unique sequence of cables that can transport the power from that building to the headquarter. A building is [b]$s$-electrifed[/b] if, after removing any one cable from the park, the building can still receive power from at least $s$ different power plants. Find the maximum possible number of $s$-electrifed buildings. [i]Note: There seems to be confusion about whether a power plant is $1$-electrified. For the sake of simplicity let's say that any power plant is not $s$-electrified for any $s\geq 1$.[/i] [i]Proposed by usjl[/i]

In a sequence with the first term is positive integer, the next term is generated by adding the previous term and its largest digit. At most how many consequtive terms of this sequence are odd? $ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ 6 $

Let be two positive real numbers $ a,b, $ and an infinite arithmetic sequence of natural numbers $ \left( x_n \right)_{n\ge 1} . $ Study the convergence of the sequences $$ \left( \frac{1}{x_n}\sum_{i=1}^n\sqrt[x_i]{b} \right)_{n\ge 1}\text{ and } \left( \left(\sum_{i=1}^n \sqrt[x_i]{a}/\sqrt[x_i]{b} \right)^\frac{x_n}{\ln x_n} \right)_{n\ge 1} , $$ and calculate their limits. [i]Dumitru Acu[/i]

A sequence of polynomials $P_m(x, y, z), m = 0, 1, 2, \cdots$, in $x, y$, and $z$ is defined by $P_0(x, y, z) = 1$ and by \[P_m(x, y, z) = (x + z)(y + z)P_{m-1}(x, y, z + 1) - z^2P_{m-1}(x, y, z)\] for $m > 0$. Prove that each $P_m(x, y, z)$ is symmetric, in other words, is unaltered by any permutation of $x, y, z.$