Olympiad problems

2021 Thailand TSTST, 2

Let $f:\mathbb{R}^+\to\mathbb{R}^+$ be such that $$f(x+f(y))^2\geq f(x)\left(f(x+f(y))+f(y)\right)$$ for all $x,y\in\mathbb{R}^+$. Show that $f$ is [i]unbounded[/i], i.e. for each $M\in\mathbb{R}^+$, there exists $x\in\mathbb{R}^+$ such that $f(x)>M$.

1998 Moldova Team Selection Test, 10

Tags: bounded , sequence , unbounded , digit , product of digits , recurrence relation , number theory

Let $P(x)$ denote the product of all (decimal) digits of a natural number $x$. For any positive integer $x_1$, define the sequence $(x_n)$ recursively by $x_{n+1} = x_n + P(x_n)$. Prove or disprove that the sequence $(x_n)$ is necessarily bounded.

1981 Austrian-Polish Competition, 6

Tags: recurrence relation , algebra , sequence , unbounded , bounded

The sequences $(x_n), (y_n), (z_n)$ are given by $x_{n+1}=y_n +\frac{1}{x_n}$,$ y_{n+1}=z_n +\frac{1}{y_n}$,$z_{n+1}=x_n +\frac{1}{z_n} $ for $n \ge 0$ where $x_0,y_0, z_0$ are given positive numbers. Prove that these sequences are unbounded.

2020 China Northern MO, P1

Tags: sequence , unbounded , recursion , algebra

The function $f(x)=x^2+ \sin x$ and the sequence of positive numbers $\{ a_n \}$ satisfy $a_1=1$, $f(a_n)=a_{n-1}$, where $n \geq 2$. Prove that there exists a positive integer $n$ such that $a_1+a_2+ \dots + a_n > 2020$.

1982 Austrian-Polish Competition, 4

Tags: sequence , bounded , unbounded , digit , product of digits , recurrence relation , number theory

Let $P(x)$ denote the product of all (decimal) digits of a natural number $x$. For any positive integer $x_1$, define the sequence $(x_n)$ recursively by $x_{n+1} = x_n + P(x_n)$. Prove or disprove that the sequence $(x_n)$ is necessarily bounded.

2021 Azerbaijan EGMO TST, 2

Tags: algebra , sequence , unbounded

Given a non-decreasing unbounded sequence $a_n,$ construct a new sequence $b_n$ as follows $$b_n = \frac{a_2 - a_1}{a_2} + \frac{a_3 - a_2}{a_3} + ... + \frac{a_n - a_{n-1}}{a_n}$$ Prove that $b_n$ is also unbounded.

2024 Olimphíada, 3

Tags: algebra , limit , bounded , unbounded

A sequence of positive real numbers $a_1, a_2, \dots$ is called $\textit{phine}$ if it satisfies $$a_{n+2}=\frac{a_{n+1}+a_{n-1}}{a_n},$$ for all $n\geq2$. Is there a $\textit{phine}$ sequence such that, for every real number $r$, there is some $n$ for which $a_n>r$?