Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.) [i]Proposed by Hong Kong[/i]

Let $ p $ and $ q $ be given primes and the sequence $ \{ p_n \}_{n = 1}^{\infty} $ defined recursively as follows: $ p_1 = p $, $ p_2 = q $, and $ p_{n+2} $ is the largest prime divisor of the number $( p_n + p_{n + 1} + 2016) $ for all $ n \geq 1 $. Prove that this sequence is bounded. That is, there exists a positive real number $ M $ such that $ p_n < M $ for all positive integers $ n $.

Define the sequence $a_1, a_2, \ldots$ and $b_1, b_2, \ldots$ as $a_0=1,a_1=4,a_2=49$ and for $n \geq 0$ $$ \begin{cases} a_{n+1}=7a_n+6b_n-3, \\ b_{n+1}=8a_n+7b_n-4. \end{cases} $$ Prove that for any non-negative integer $n,$ $a_n$ is a perfect square.

We define a sequence $ \left(a_{1},a_{2},a_{3},\ldots \right)$ by \[ a_{n} \equal{} \frac {1}{n}\left(\left\lfloor\frac {n}{1}\right\rfloor \plus{} \left\lfloor\frac {n}{2}\right\rfloor \plus{} \cdots \plus{} \left\lfloor\frac {n}{n}\right\rfloor\right), \] where $\lfloor x\rfloor$ denotes the integer part of $x$. [b]a)[/b] Prove that $a_{n+1}>a_n$ infinitely often. [b]b)[/b] Prove that $a_{n+1}<a_n$ infinitely often. [i]Proposed by Johan Meyer, South Africa[/i]

We’re playing a game with a sequence of $2008$ non-negative integers. A move consists of picking a integer $b$ from that sequence, of which the neighbours $a$ and $c$ are positive. We then replace $a, b$ and $c$ by $a - 1, b + 7$ and $c - 1$ respectively. It is not allowed to pick the first or the last integer in the sequence, since they only have one neighbour. If there is no integer left such that both of its neighbours are positive, then there is no move left, and the game ends. Prove that the game always ends, regardless of the sequence of integers we begin with, and regardless of the moves we make.

For any odd prime $p$ and any integer $n,$ let $d_p (n) \in \{ 0,1, \dots, p-1 \}$ denote the remainder when $n$ is divided by $p.$ We say that $(a_0, a_1, a_2, \dots)$ is a [i]p-sequence[/i], if $a_0$ is a positive integer coprime to $p,$ and $a_{n+1} =a_n + d_p (a_n)$ for $n \geqslant 0.$ (a) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $(a_0, a_1, a_2, \dots)$ and $(b_0, b_1, b_2, \dots)$ such that $a_n >b_n$ for infinitely many $n,$ and $b_n > a_n$ for infinitely many $n?$ (b) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $(a_0, a_1, a_2, \dots)$ and $(b_0, b_1, b_2, \dots)$ such that $a_0 <b_0,$ but $a_n >b_n$ for all $n \geqslant 1?$ [I]United Kingdom[/i]

Let $n$ be an odd positive integer, and let $x_1,x_2,\cdots ,x_n$ be non-negative real numbers. Show that \[ \min_{i=1,\ldots,n} (x_i^2+x_{i+1}^2) \leq \max_{j=1,\ldots,n} (2x_jx_{j+1}) \]where $x_{n+1}=x_1$.

Let $\alpha\in(0,1]$ be a given real number and let a real sequence $\{a_n\}^\infty_{n=1}$ satisfy the inequality $$a_{n+1}\le\alpha a_n+(1-\alpha)a_{n-1}\qquad\text{for }n=2,3,\ldots$$Prove that if $\{a_n\}$ is bounded, then it must be convergent.

A sequence of integers $\{f(n)\}$ for $n=0,1,2,\ldots$ is defined as follows: $f(0)=0$ and for $n>0$, $$\begin{matrix}f(n)=&f(n-1)+3,&\text{if }n=0\text{ or }1\pmod6,\\&f(n-1)+1,&\text{if }n=2\text{ or }5\pmod6,\\&f(n-1)+2,&\text{if }n=3\text{ or }4\pmod6.\end{matrix}$$Derive an explicit formula for $f(n)$ when $n\equiv0\pmod6$, showing all necessary details in your derivation.

Let $N$ be a positive integer and $A = a_1, a_2, ... , a_N$ be a sequence of real numbers. Define the sequence $f(A)$ to be $$f(A) = \left( \frac{a_1 + a_2}{2},\frac{a_2 + a_3}{2}, ...,\frac{a_{N-1} + a_N}{2},\frac{a_N + a_1}{2}\right)$$ and for $k$ a positive integer define $f^k (A)$ to be$ f$ applied to $A$ consecutively $k$ times (i.e. $f(f(... f(A)))$) Find all sequences $A = (a_1, a_2,..., a_N)$ of integers such that $f^k (A)$ contains only integers for all $k$.

Let a sequence $(a_i)_{i=10}^{\infty}$ be defined as follows: [list=a] [*] $a_{10}$ is some positive integer, which can of course be written in base 10. [*] For $i \geq 10$ if $a_i > 0$, let $b_i$ be the positive integer whose base-$(i + 1)$ representation is the same as $a_i$'s base-$i$ representation. Then let $a_{i + 1} = b_i - 1$. If $a_i = 0$, $a_{i + 1} = 0$. [/list] For example, if $a_{10} = 11$, then $b_{10} = 11_{11} (= 12_{10})$; $a_{11} = 11_{11} - 1 = 10_{11} (= 11_{10})$; $b_{11} = 10_{12} (= 12_{10})$; $a_{12} = 11$. Does there exist $a_{10}$ such that $a_i$ is strictly positive for all $i \geq 10$?

Let $\sum_{n=1}^\infty a_n$ be a divergent series with positive nonincreasing terms. Prove that the series $$\sum_{n=1}^\infty\frac{a_n}{1+na_n}$$diverges.

What is the $7$th term of the sequence $\{-1, 4,-2, 3,-3, 2,...\}$? (A) $ -1$ (B) $ -2$ (C) $-3$ (D) $-4$ (E) None of the above

A sequence $(x_{i})$ is said to have a [i]Cesaro limit[/i] exactly if $\lim_{n\to\infty} \frac{x_{1}+\ldots+x_{n}}{n}$ exists. Find all real-valued functions $f$ on the closed interval $[0, 1]$ such that $(f(x_i))$ has a Cesaro limit if and only if $(x_i)$ has a Cesaro limit.

Consider the sequence of natural numbers $a_n$ defined as $a_0=4$ and $a_{n+1}=\frac{a_n(a_n-1)}{2}$ for each $n\geq 0$. Define a new sequence $b_n$ as follows: $b_n=0$ if $a_n$ is even, and $b_n=1$ if $a_n$ is odd. Prove that for each natural $m$, the sequence \[b_m, b_{m+1}, b_{m+2},b_{m+3}, \dots\] is not periodic.

Let $a_1,a_2,a_3,\ldots$ be an infinite sequence of positive integers such that $a_{n+2m}$ divides $a_{n}+a_{n+m}$ for all positive integers $n$ and $m.$ Prove that this sequence is eventually periodic, i.e. there exist positive integers $N$ and $d$ such that $a_n=a_{n+d}$ for all $n>N.$

Does there exist a sequence $(a_n)_{n\in\mathbb N}$ of distinct positive integers such that: (i) $a_n<1999n$ for all $n$; (ii) none of the $a_n$ contains three decimal digits $1$?

A sequence $\{an\}$ of positive integers is defined by \[a_n=\left[ n +\sqrt n + \frac 12 \right] , \qquad \forall n \in \mathbb N\] Determine the positive integers that occur in the sequence.

Let $a_1 <a_2 < \ldots$ be an increasing sequence of positive integers. Let the series $$\sum_{i=1}^{\infty} \frac{1}{a_i }$$ be convergent. For any real number $x$, let $k(x)$ be the number of the $a_i$ which do not exceed $x$. Show that $\lim_{x\to \infty} \frac{k(x)}{x}=0.$

Show that the sequence $$ \sqrt{7} , \sqrt{7-\sqrt{7}}, \sqrt{7-\sqrt{7-\sqrt{7}}}, \ldots$$ converges and evaluate the limit.

Let $n$ be a given positive integer. (a) Do there exist $2n+1$ consecutive positive integers $a_0,a_1,\ldots,a_{2n}$ in the ascending order such that $a_0+a_1+\ldots+a_n=a_{n+1}+\ldots+a_{2n}$? (b) Do there exist consecutive positive integers $a_0,a+1,\ldots,a_{2n}$ in ascending order such that $a_0^2+a_1^2+\ldots+a_n^2=a_{n+1}^2+\ldots+a_{2n}^2$? (c) Do there exist consecutive positive integers $a_0,a_1,\ldots,a_{2n}$ in ascending order such that $a_0^3+a_1^3+\ldots+a_n^3=a_{n+1}^3+\ldots+a_{2n}^3$? [hide=Official Hint]You may study the function $f(x)=(x-n)^3+\ldots+x^3-(x+1)^3-\ldots-(x+n)^3$ and prove that the equation $f(x)=0$ has a unique solution $x_n$ with $3n(n+1)<x_n<3n(n+1)+1$. You may use the identity $1^3+2^3+\ldots+n^3=\frac{n^2(n+1)^2}2$.[/hide]

Determine all sequences $(x_1,x_2,\ldots,x_{2011})$ of positive integers, such that for every positive integer $n$ there exists an integer $a$ with \[\sum^{2011}_{j=1} j x^n_j = a^{n+1} + 1\] [i]Proposed by Warut Suksompong, Thailand[/i]

We define the sequence $x_1=n,y_1=1,x_{i+1}=[\frac{x_i+y_i}{2}],y_{i+1}=[\frac{n}{x_{i+1}} ]$. Prove that $min\{ x_1, x_2, ..., x_n\}=[\sqrt{n}]$ .

We define a sequence $ \left(a_{1},a_{2},a_{3},\ldots \right)$ by \[ a_{n} \equal{} \frac {1}{n}\left(\left\lfloor\frac {n}{1}\right\rfloor \plus{} \left\lfloor\frac {n}{2}\right\rfloor \plus{} \cdots \plus{} \left\lfloor\frac {n}{n}\right\rfloor\right), \] where $\lfloor x\rfloor$ denotes the integer part of $x$. [b]a)[/b] Prove that $a_{n+1}>a_n$ infinitely often. [b]b)[/b] Prove that $a_{n+1}<a_n$ infinitely often. [i]Proposed by Johan Meyer, South Africa[/i]

A sequence of real numbers $ a_{0},\ a_{1},\ a_{2},\dots$ is defined by the formula \[ a_{i \plus{} 1} \equal{} \left\lfloor a_{i}\right\rfloor\cdot \left\langle a_{i}\right\rangle\qquad\text{for}\quad i\geq 0; \]here $a_0$ is an arbitrary real number, $\lfloor a_i\rfloor$ denotes the greatest integer not exceeding $a_i$, and $\left\langle a_i\right\rangle=a_i-\lfloor a_i\rfloor$. Prove that $a_i=a_{i+2}$ for $i$ sufficiently large. [i]Proposed by Harmel Nestra, Estionia[/i]