Let $a_1,a_2,\dotsc$ be a sequence of real numbers from the interval $[0,1]$. Prove that there is a sequence $1\leqslant n_1<n_2<\dotsc$ of positive integers such that $$A=\lim_{\substack{i,j\to \infty \\ i\neq j}} a_{n_i+n_j}$$exists, i.e., for every real number $\epsilon >0$, there is a constant $N_{\epsilon}$ that $|a_{n_i+n_j}-A|<\epsilon$ is satisfied for any pair of distinct indices $i,j>N_{\epsilon}$.

The sequence $a_1, a_2, ...$ of natural numbers satisfies $GCD(a_i, a_j)=GCD(i, j)$ for all $i \neq j$. Prove that $a_i=i$ for all $i$.

Every number in the sequence $1, 2, ... , 2021$ is either white or black. At one step Alice can choose three numbers of the sequence and change the color of each of them (white to black and black to white) if one of those three numbers is the arithmetic mean of the other two. Alice wants to perform several steps so that at the end all the numbers in the sequence are black. For which initial colorings of numbers can Alice achieve this?

Given three numbers $x, y, z$, and set $x_1 = |x - y|, y_1 = | y -z|, z_1 = |z- x|$. From $x_1, y_1, z_1$, form in the same fashion the numbers $x_2, y_2, z_2$, and so on. It is known that $x_n = x, y_n = y, z_n = z$ for some $n$. Find all possible values of $(x, y, z)$.

Let the sequence $\{a_i\}^\infty_{i=0}$ be defined by $a_0 =\frac12$ and $a_n = 1 + (a_{n-1} - 1)^2$. Find the product $$\prod_{i=0}^\infty a_i=a_0a_1a_2\ldots$$

Call an odd prime $p$ [i]adjective[/i] if there exists an infinite sequence $a_0,a_1,a_2,\ldots$ of positive integers such that \[a_0\equiv1+\frac{1}{a_1}\equiv1+\frac{1}{1+\frac{1}{a_2}}\equiv1+\frac{1}{1+\frac{1}{1+\frac{1}{a_3}}}\equiv\ldots\pmod p.\] What is the sum of the first three odd primes that are [i]not[/i] adjective? Note: For two common fractions $\frac{a}{b}$ and $\frac{c}{d}$, we say that $\frac{a}{b}\equiv\frac{c}{d}\pmod p$ if $p$ divides $ad-bc$ and $p$ does not divide $bd$. [i]2019 CCA Math Bonanza Individual Round #14[/i]

Define the function $ g(\cdot): \mathbb{Z} \to \{0,1\}$ such that $ g(n) \equal{} 0$ if $ n < 0$, and $ g(n) \equal{} 1$ otherwise. Define the function $ f(\cdot): \mathbb{Z} \to \mathbb{Z}$ such that $ f(n) \equal{} n \minus{} 1024g(n \minus{} 1024)$ for all $ n \in \mathbb{Z}$. Define also the sequence of integers $ \{a_i\}_{i \in \mathbb{N}}$ such that $ a_0 \equal{} 1$ e $ a_{n \plus{} 1} \equal{} 2f(a_n) \plus{} \ell$, where $ \ell \equal{} 0$ if $ \displaystyle \prod_{i \equal{} 0}^n{\left(2f(a_n) \plus{} 1 \minus{} a_i\right)} \equal{} 0$, and $ \ell \equal{} 1$ otherwise. How many distinct elements are in the set $ S: \equal{} \{a_0,a_1,\ldots,a_{2009}\}$? [i](Paolo Leonetti)[/i]

A restricted rook (RR) is a fictional chess piece that can move horizontally or vertically (like a rook), except that each move is restricted to a neighboring square (cell). If RR can only (with at most one exception) move up and to the right, how many possible distinct paths are there to move RR from the bottom left square to the top right square of a standard 8-by-8 chess board? Note that RR may visit some squares more than once. A path is the sequence of squares visited by RR on its way.

Let $a_n$ be the number of n-digit integers formed by $1, 2$ and $3$ which do not contain any consecutive $1$’s. Prove that $a_n$ is equal to $$\left( \frac12 + \frac{1}{\sqrt3}\right)(\sqrt{3} + 1)^n$$ rounded off to the nearest integer.

Let each sequence of capital Bulgarian letters be a word (Note that the Bulgarian alphabet consists of 30 letters). We say that a word is [i]calm[/i], if there is no sequence of the letter $P$ in it (two letters $P$ next to each other). Find a clear expression for $f(n)$ (closed formula), which represents the number of calm words with length $n$. Example: [i]РНТ[/i] and [i]ТТРВ[/i] are calm, but [i]ЖГРР[/i] and [i]ЖРРРП[/i] aren’t.

[b]p1.[/b] Let $a(1), a(2), ..., a(n),...$ be an increasing sequence of positive integers satisfying $a(a(n)) = 3n$ for every positive integer $n$. Compute $a(2019)$. [b]p2.[/b] Consider the function $f(12x - 7) = 18x^3 - 5x + 1$. Then, $f(x)$ can be expressed as $f(x) = ax^3 + bx^2 + cx + d$, for some real numbers $a, b, c$ and $d$. Find the value of $(a + c)(b + d)$. [b]p3.[/b] Let $a, b$ be real numbers such that $\sqrt{5 + 2\sqrt6} = \sqrt{a} +\sqrt{b}$. Find the largest value of the quantity $$X = \dfrac{1}{a +\dfrac{1}{b+ \dfrac{1}{a+...}}}$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a_{1}, \ldots, a_{n}$ be an infinite sequence of strictly positive integers, so that $a_{k} < a_{k+1}$ for any $k.$ Prove that there exists an infinity of terms $ a_{m},$ which can be written like $a_m = x \cdot a_p + y \cdot a_q$ with $x,y$ strictly positive integers and $p \neq q.$

Let $a_1,a_2,a_3,\ldots$ and $b_1,b_2,b_3,\ldots$ be positive integers such that $a_{n+2} = a_n + a_{n+1}$ and $b_{n+2} = b_n + b_{n+1}$ for all $n \ge 1$. Assume that $a_n$ divides $b_n$ for infinitely many values of $n$. Prove that there exists an integer $c$ such that $b_n = c a_n$ for all $n \ge 1$.

Define the sequence of positive integers $\{a_n\}$ as follows. Let $a_1=1$, $a_2=3$, and for each $n>2$, let $a_n$ be the result of expressing $a_{n-1}$ in base $n-1$, then reading the resulting numeral in base $n$, then adding $2$ (in base $n$). For example, $a_2=3_{10}=11_2$, so $a_3=11_3+2_3=6_{10}$. Express $a_{2013}$ in base $10$.

Find all continuous functions $f,g:\mathbb{R}\rightarrow\mathbb{R}$ such that for any sequences $(a_n)_{n\geq 1}$ and $(b_n)_{n\geq 1}$ such that the sequence $(a_n+b_n)_{n\geq 1}$ is convergent, the sequence $(f(a_n)+g(b_n))_{n\geq 1}$ is also convergent.

Consider the following sequence $$(a_n)_{n=1}^{\infty}=(1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,1,\dots)$$ Find all pairs $(\alpha, \beta)$ of positive real numbers such that $\lim_{n\to \infty}\frac{\displaystyle\sum_{k=1}^n a_k}{n^{\alpha}}=\beta$. (Proposed by Tomas Barta, Charles University, Prague)