Given a positive integer $N$. Prove that: there are infinitely elements of the [i]Fibonacci[/i] sequence that are divisible by $N$.

Given a sequence $a_1,a_2,\ldots $ of non-negative real numbers satisfying the conditions: 1. $a_n + a_{2n} \geq 3n$; 2. $a_{n+1}+n \leq 2\sqrt{a_n \left(n+1\right)}$ for all $n\in\mathbb N$ (where $\mathbb N=\left\{1,2,3,...\right\}$). (1) Prove that the inequality $a_n \geq n$ holds for every $n \in \mathbb N$. (2) Give an example of such a sequence.

Let $\mathbb Q_{\geq 0}$ be the non-negative rational numbers, $f: \mathbb Q_{\geq 0} \to \mathbb Q_{\geq 0}$ such that $f(z+1) = f(z)+1$, $f(1/z) = f(z)$ for $z\neq 0$, and $f(0) = 0.$ Define a sequence $P_n$ of non-negative integers recursively via $$P_0 = 0,\quad P_1 = 1,\quad P_n = 2 P_{n-1}+P_{n-2}$$ for every $n \geq 2$. Find $f\left(\frac{P_{20}}{P_{24}}\right).$ [i]Proposed by Robert Trosten[/i]

Let $\mathbb{N}$ be the set of all natural numbers. Let $f:\mathbb{N} \to \mathbb{N}$ be a bijective function. Show that there exists three numbers $a$, $b$, $c$ in arithmatic progression such that $f(a)<f(b)<f(c)$

Show that any arithmetic progression where the first term and the common difference are non-zero natural numbers contains an infinite number of composite terms. *A number is composite if it is not prime.

Let $d$ be a positive integer. The seqeunce $a_1, a_2, a_3,...$ of positive integers is defined by $a_1 = 1$ and $a_{n + 1} = n\left \lfloor \frac{a_n}{n} \right \rfloor+ d$ for $n = 1,2,3, ...$ . Prove that there exists a positive integer $N$ so that the terms $a_N,a_{N + 1}, a_{N + 2},...$ form an arithmetic progression. Note: If $x$ is a real number, $\left \lfloor x \right \rfloor $ denotes the largest integer that is less than or equal to $x$.

Let $P_0,P_1,P_2,\ldots$ be a sequence of convex polygons such that, for each $k\ge0$, the vertices of $P_{k+1}$ are the midpoints of all sides of $P_k$. Prove that there exists a unique point lying inside all these polygons.

Evan writes a random positive integer on a board: the integer $k$ has probability $2^{-k}$ of being written. He keeps writing integers in this way repeatedly until he writes an integer that he had written before. He then takes all the integers he has written besides his last, sorts them in the order he first drew them, and also sorts them in increasing order, forming two sequences. For example, if he wrote $5,8,2,3,6,10,2$ in that order then his two sequences would be $5,8,2,3,6,10$ and $2,3,5,6,8,10.$ Find the probability that for all $k \in \{ 1,4,34 \},$ that $k$ was written, and $k$ appears in the same position in both sequences.

Let $x_0 = 5$ and $x_{n+1} = x_n + \frac{1}{x_n} \ (n = 0, 1, 2, \ldots )$. Prove that \[45 < x_{1000} < 45. 1.\]

Define in the plane the sequence of vectors $v_1, v_2, \ldots$ with initial values $v_1 = (1, 0)$, $v_2 = (-1/\sqrt{2}, 1/\sqrt{2})$ and satisfying the relationship $$v_n=\frac{v_{n-1}+v_{n-2}}{\lVert v_{n-1}+v_{n-2}\rVert},$$ for $n \geq 3$. Show that the sequence is convergent and determine its limit. [b]Note:[/b] The expression $\lVert v \rVert$ denotes the length of the vector $v$.

Let $N$ be the number of integer sequences $a_1, a_2, \dots, a_{2^{16}-1}$ satisfying \[0 \le a_{2k + 1} \le a_k \le a_{2k + 2} \le 1\] for all $1 \le k \le 2^{15}-1$. Find the number of positive integer divisors of $N$. [i]Proposed by Ankan Bhattacharya[/i]

Given a sequence of prime numbers $p_1, p_2,\cdots$ , with the following property: $p_{n+2}$ is the largest prime divisor of $p_n+p_{n+1}+2018$ Show that the set $\{p_i\}_{i\in \mathbb{N}}$ is finite.

Given natural $n$. We shall call "universal" such a sequence of natural number $a_1, a_2, ... , a_k, k\ge n$, if we can obtain every transposition of the first $n$ natural numbers (i.e such a sequence of $n$ numbers, that every one is encountered only once) by deleting some its members. (Examples: ($1,2,3,1,2,1,3$) is universal for $n=3$, and ($1,2,3,2,1,3,1$) -- not, because you can't obtain ($3,1,2$) from it.) The goal is to estimate the length of the shortest universal sequence for given $n$. a) Give an example of the universal sequence of $n2$ members. b) Give an example of the universal sequence of $(n^2 - n + 1)$ members. c) Prove that every universal sequence contains not less than $n(n + 1)/2$ members d) Prove that the shortest universal sequence for $n=4$ contains 12 members e) Find as short universal sequence, as you can. The Organising Committee knows the method for $(n^2 - 2n +4) $ members.

An infinite sequence of positive integers $a_1, a_2, \dots$ is called $good$ if (1) $a_1$ is a perfect square, and (2) for any integer $n \ge 2$, $a_n$ is the smallest positive integer such that $$na_1 + (n-1)a_2 + \dots + 2a_{n-1} + a_n$$ is a perfect square. Prove that for any good sequence $a_1, a_2, \dots$, there exists a positive integer $k$ such that $a_n=a_k$ for all integers $n \ge k$. [size=75](reposting because the other thread didn't get moved)[/size]

In the following, a [i]word[/i] will mean a finite sequence of letters "$a$" and "$b$". The [i]length[/i] of a word will mean the number of the letters of the word. For instance, $abaab$ is a word of length $5$. There exists exactly one word of length $0$, namely the empty word. A word $w$ of length $\ell$ consisting of the letters $x_1$, $x_2$, ..., $x_{\ell}$ in this order is called a [i]palindrome[/i] if and only if $x_j=x_{\ell+1-j}$ holds for every $j$ such that $1\leq j\leq\ell$. For instance, $baaab$ is a palindrome; so is the empty word. For two words $w_1$ and $w_2$, let $w_1w_2$ denote the word formed by writing the word $w_2$ directly after the word $w_1$. For instance, if $w_1=baa$ and $w_2=bb$, then $w_1w_2=baabb$. Let $r$, $s$, $t$ be nonnegative integers satisfying $r + s = t + 2$. Prove that there exist palindromes $A$, $B$, $C$ with lengths $r$, $s$, $t$, respectively, such that $AB=Cab$, if and only if the integers $r + 2$ and $s - 2$ are coprime.