Define sequence $(a_n):a_1=1,a_2=2,a_{n+2}=\begin{cases} 5a_{n+1}-3a_n,\text{if }a_n\cdot a_{n+1}\text{ is even}\\ a_{n+1}-a_n,\text{if }a_n\cdot a_{n+1}\text{ is odd} \end{cases}$ Prove that for all $n\in\mathbb{Z}_+$, $a_n\neq0$.

Let $(a_n)_{n=0}^{\infty}$ be a sequence of real numbers defined as follows: [list] [*] $a_0 = 3$, $a_1 = 2$, and $a_2 = 12$; and [*] $2a_{n + 3} - a_{n + 2} - 8a_{n + 1} + 4a_n = 0$ for $n \geq 0$. [/list] Show that $a_n$ is always a strictly positive integer.

Let $n > 1$ be a positive integer. A 2-dimensional grid, infinite in all directions, is given. Each 1 by 1 square in a given $n$ by $n$ square has a counter on it. A [i]move[/i] consists of taking $n$ adjacent counters in a row or column and sliding them each by one space along that row or column. A [i]returning sequence[/i] is a finite sequence of moves such that all counters again fill the original $n$ by $n$ square at the end of the sequence. [list] [*] Assume that all counters are distinguishable except two, which are indistinguishable from each other. Prove that any distinguishable arrangement of counters in the $n$ by $n$ square can be reached by a returning sequence. [*] Assume all counters are distinguishable. Prove that there is no returning sequence that switches two counters and returns the rest to their original positions.[/list] [i]Mitchell Lee and Benjamin Gunby.[/i]

Let $m$ be a positive integer and $\{a_n\}_{n\geq 0}$ be a sequence given by $a_0 = a \in \mathbb N$, and \[ a_{n+1} = \begin{cases} \displaystyle \frac{a_n}2 & \textrm { if } a_n \equiv 0 \pmod 2, \\ a_n + m & \textrm{ otherwise. } \end{cases} \] Find all values of $a$ such that the sequence is periodical (starting from the beginning).

For each integer $a_0 > 1$, define the sequence $a_0, a_1, a_2, \ldots$ for $n \geq 0$ as $$a_{n+1} = \begin{cases} \sqrt{a_n} & \text{if } \sqrt{a_n} \text{ is an integer,} \\ a_n + 3 & \text{otherwise.} \end{cases} $$ Determine all values of $a_0$ such that there exists a number $A$ such that $a_n = A$ for infinitely many values of $n$. [i]Proposed by Stephan Wagner, South Africa[/i]

Let $C$ be a fixed circle, $u > 0$ be a fixed real and let $v_0 , v_1 , v_2 , \ldots$ be a sequence of positive real numbers. Two ants $A$ and $B$ walk around the perimeter of $C$ in opposite directions, starting from the same starting point. Ant $A$ has a constant speed $u$, while ant $B$ has an initial speed $v_0$. For each positive integer $n$, when the two ants collide for the $n$−th time, they change the directions in which they walk around the perimeter of $C$, with ant $A$ remaining at speed $u$ and ant $B$ stops walking at speed $v_{n-1}$ to walk at speed $v_n$. (a) If the sequence $\{v_n\}$ is strictly increasing, with $\lim_{n\rightarrow \infty} v_n = +\infty$, prove that there is exactly one point in $C$ that ant $A$ will pass "infinitely" many times. (b) Prove that there is a sequence $\{v_n\}$ with $\lim_{n\rightarrow\infty} v_n = +\infty$, such that ant $A$ will pass "infinitely" many times through all points on the circle $C$.

A sequence $\{a_i\}$ is defined by $a_1 = c$ for some $c > 0$ and $a_{n+1} = a_n + \frac{n}{a_n}$. Prove that $\frac{a_n}{n}$ converges and find its limit.

The Fibonacci sequence is defined by $ F_1 \equal{} F_2 \equal{} 1$ and $ F_{n\plus{}1} \equal{} F_n \plus{}F_{n\minus{}1}$ for $ n > 1$. Let $ f(x) \equal{} 1985x^2 \plus{} 1956x \plus{} 1960$. Prove that there exist infinitely many natural numbers $ n$ for which $ f(F_n)$ is divisible by $ 1989$. Does there exist $ n$ for which $ f(F_n) \plus{} 2$ is divisible by $ 1989$?

a) Prove that there exists a sequence of digits $\{c_n\}_{n\geq 1}$ such that or each $n\geq 1$ no matter how we interlace $k_n$ digits, $1\leq k_n\leq 9$, between $c_n$ and $c_{n+1}$, the infinite sequence thus obtained does not represent the fractional part of a rational number. b) Prove that for $1\leq k_n\leq 10$ there is no such sequence $\{c_n\}_{n\geq 1}$. [i]Dan Schwartz[/i]

Let $(a_n)$ be defined by: $$ a_1 = 2, \qquad a_{n+1} = a_n^3 - a_n + 1 $$ Consider positive integers $n,p$, where $p$ is an odd prime. Prove that if $p | a_n$, then $p > n$.

The sequence $\{a_n\}$ is defined as follows: $a_1$ is a positive rational number, $a_n= \frac{p_n}{q_n}$, ($n= 1,2,…$) is a positive integer, where $p_n$ and $q_n$ are positive integers that are relatively prime, then $a_{n+1} = \frac{p_n^2+2015}{p_nq_n}$ Is there a$_1>2015$, making the sequence $\{a_n\}$ a bounded sequence? Justify your conclusion.

The infinite sequence of 2's and 3's \[\begin{array}{l}2,3,3,2,3,3,3,2,3,3,3,2,3,3,2,3,3, \\ 3,2,3,3,3,2,3,3,3,2,3,3,2,3,3,3,2,\cdots \end{array}\] has the property that, if one forms a second sequence that records the number of 3's between successive 2's, the result is identical to the given sequence. Show that there exists a real number $r$ such that, for any $n$, the $n$th term of the sequence is 2 if and only if $n = 1+\lfloor rm \rfloor$ for some nonnegative integer $m$.

The sequence $\{a_1,a_2,\ldots\}$ is recursively defined by $a_1 = 1$, $a_2 = 1$, $a_3 = 2$, and \[ a_{n+3} = \frac 1{a_n}\cdot (a_{n+1}a_{n+2}+7), \ \forall \ n > 0. \] Prove that all elements of the sequence are integers.

You are given $n\ge 4$ positive real numbers. It turned out that all $\frac{n(n-1)}{2}$ of their pairwise products form an arithmetic progression in some order. Show that all given numbers are equal. [i](Proposed by Anton Trygub)[/i]

[b]p1.[/b] Let $n$ be the number so that $1 - 2 + 3 - 4 + ... - (n - 1) + n = 2012$. What is $4^{2012}$ (mod $n$)? [b]p2. [/b]Consider three unit squares placed side by side. Label the top left vertex $P$ and the bottom four vertices $A,B,C,D$ respectively. Find $\angle PBA + \angle PCA + \angle PDA$. [b]p3.[/b] Given $f(x) = \frac{3}{x-1}$ , then express $\frac{9(x^2-2x+1)}{x^2-8x+16}$ entirely in terms of $f(x)$. In other words, $x$ should not be in your answer, only $f(x)$. [b]p4.[/b] Right triangle with right angle $B$ and integer side lengths has $BD$ as the altitude. $E$ and $F$ are the incenters of triangles $ADB$ and $BDC$ respectively. Line $EF$ is extended and intersects $BC$ at $G$, and $AB$ at $H$. If $AB = 15$ and $BC = 8$, find the area of triangle $BGH$. [b]p5.[/b] Let $a_1, a_2, ..., a_n$ be a sequence of real numbers. Call a $k$-inversion $(0 < k\le n)$ of a sequence to be indices $i_1, i_2, .. , i_k$ such that $i_1 < i_2 < .. < i_k$ but $a_{i1} > a_{i2} > ...> a_{ik}$ . Calculate the expected number of $6$-inversions in a random permutation of the set $\{1, 2, ... , 10\}$. [b]p6.[/b] Chell is given a strip of squares labeled $1, .. , 6$ all placed side to side. For each $k \in {1, ..., 6}$, she then chooses one square at random in $\{1, ..., k\}$ and places a Weighted Storage Cube there. After she has placed all $6$ cubes, she computes her score as follows: For each square, she takes the number of cubes in the pile and then takes the square (i.e. if there were 3 cubes in a square, her score for that square would be $9$). Her overall score is the sum of the scores of each square. What is the expected value of her score? PS. You had better use hide for answers.