The integers $1, 2, 3, \ldots, 2016$ are written on a board. You can choose any two numbers on the board and replace them with their average. For example, you can replace $1$ and $2$ with $1.5$, or you can replace $1$ and $3$ with a second copy of $2$. After $2015$ replacements of this kind, the board will have only one number left on it. (a) Prove that there is a sequence of replacements that will make the final number equal to $2$. (b) Prove that there is a sequence of replacements that will make the final number equal to $1000$.

Let $ a_1 \geq a_2 \geq \ldots \geq a_n$ be real numbers such that for all integers $ k > 0,$ \[ a^k_1 \plus{} a^k_2 \plus{} \ldots \plus{} a^k_n \geq 0.\] Let $ p \equal{}\max\{|a_1|, \ldots, |a_n|\}.$ Prove that $ p \equal{} a_1$ and that \[ (x \minus{} a_1) \cdot (x \minus{} a_2) \cdots (x \minus{} a_n) \leq x^n \minus{} a^n_1\] for all $ x > a_1.$

Ana and Banana are playing a game. First Ana picks a word, which is defined to be a nonempty sequence of capital English letters. (The word does not need to be a valid English word.) Then Banana picks a nonnegative integer $k$ and challenges Ana to supply a word with exactly $k$ subsequences which are equal to Ana's word. Ana wins if she is able to supply such a word, otherwise she loses. For example, if Ana picks the word "TST", and Banana chooses $k=4$, then Ana can supply the word "TSTST" which has 4 subsequences which are equal to Ana's word. Which words can Ana pick so that she wins no matter what value of $k$ Banana chooses? (The subsequences of a string of length $n$ are the $2^n$ strings which are formed by deleting some of its characters, possibly all or none, while preserving the order of the remaining characters.) [i]Proposed by Kevin Sun

The sequence of positive integers $a_0, a_1, a_2, . . .$ is defined by $a_0 = 3$ and $$a_{n+1} - a_n = n(a_n - 1)$$ for all $n \ge 0$. Determine all integers $m \ge 2$ for which $gcd (m, a_n) = 1$ for all $n \ge 0$.

Let $n$ be a fixed odd positive integer. For each odd prime $p$, define $$a_p=\frac{1}{p-1}\sum_{k=1}^{\frac{p-1}{2}}\bigg\{\frac{k^{2n}}{p}\bigg\}.$$ Prove that there is a real number $c$ such that $a_p = c$ for infinitely many primes $p$. [i]Note: $\left\{x\right\} = x - \left\lfloor x\right\rfloor$ is the fractional part of $x$.[/i]

Let $a_1, a_2, a_3, . . . , a_n, . . . $ be an infinite sequence of natural numbers in which $a_1$ is not divisible by $5$. Suppose $a_{n+1} = a_n + b_n$ where bn is the last digit of $a_n$, for every $n$. Prove that the sequence $\{a_n\}$ contains infinitely many powers of 2.

Let $(x_n)$ be an infinite sequence of real numbers from interval $(0, 1)$. An infinite sequence $(a_n)$ of positive integers is defined as follows: $a_1 = 1$, and for $i \ge 1$, $a_{i+1}$ is equal to the smallest positive integer $m$, for which $[x_1 + x_2 + \ldots + x_m] = a_i$. Show that for any indexes $i, j$ holds $a_{i+j} \ge a_i + a_j$. [i]Proposed by Nazar Serdyuk[/i]

Let a sequence $(x_n)$ satisfy :$x_1=1$ and $x_{n+1}=x_n+3\sqrt{x_n} + \frac{n}{\sqrt{x_n}}$,$\forall$n$\ge1$ a) Prove lim$\frac{n}{x_n}=0$ b) Find lim$\frac{n^2}{x_n}$

Define the numbers $a_0, a_1, \ldots, a_n$ in the following way: \[ a_0 = \frac{1}{2}, \quad a_{k+1} = a_k + \frac{a^2_k}{n} \quad (n > 1, k = 0,1, \ldots, n-1). \] Prove that \[ 1 - \frac{1}{n} < a_n < 1.\]

A finite sequence $x_1,\dots,x_r$ of positive integers is a [i]palindrome[/i] if $x_i=x_{r+1-i}$ for all integers $1 \le i \le r$. Let $a_1,a_2,\dots$ be an infinite sequence of positive integers. For a positive integer $j \ge 2$, denote by $a[j]$ the finite subsequence $a_1,a_2,\dots,a_{j-1}$. Suppose that there exists a strictly increasing infinite sequence $b_1,b_2,\dots$ of positive integers such that for every positive integer $n$, the subsequence $a[b_n]$ is a palindrome and $b_{n+2} \le b_{n+1}+b_n$. Prove that there exists a positive integer $T$ such that $a_i=a_{i+T}$ for every positive integer $i$.

Let $a_0>0$ be a real number, and let $$a_n=\frac{a_{n-1}}{\sqrt{1+2020\cdot a_{n-1}^2}}, \quad \textrm{for } n=1,2,\ldots ,2020.$$ Show that $a_{2020}<\frac1{2020}$.

Let $a_0,a_1,a_2...$ be a sequence of natural numbers with the following property: $a_n^2$ divides $a_{n-1} a_{n+1}$ for $\forall$ $n\in \mathbb{N}$. Prove that, if for some natural $k\geq 2$ the numbers $a_1$ and $a_k$ are coprime, then $a_1$ divides $a_0$.

Let $\{x_n\}$ be a sequence of natural numbers such that \[(a) 1 = x_1 < x_2 < x_3 < \ldots; \quad (b) x_{2n+1} \leq 2n \quad \forall n.\] Prove that, for every natural number $k$, there exist terms $x_r$ and $x_s$ such that $x_r - x_s = k.$

Let $\{x\}$ denote the fractional part of $x$, $x - [x]$. The sequences $x_1, x_2, x_3, ...$ and $y_1, y_2, y_3, ...$ are such that $\lim \{x_n\} = \lim \{y_n\} = 0$. Is it true that $\lim \{x_n + y_n\} = 0$? $\lim \{x_n - y_n\} = 0$?

The sequence $(a_n)_{n\geq1}$ is defined as: $$a_1=2, a_2=20, a_3=56, a_{n+3}=7a_{n+2}-11a_{n+1}+5a_n-3\cdot2^n.$$ Prove that $a_n$ is positive for every positive integer $n{}$. Find the remainder of the divison of $a_{673}$ to $673$.