Olympiad problems

1981 Austrian-Polish Competition, 6

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.

1975 Czech and Slovak Olympiad III A, 2

Tags: algebra , system of equations , floor function , inequalities , bounded , square , function

Show that the system of equations \begin{align*} \lfloor x\rfloor^2+\lfloor y\rfloor &=0, \\ 3x+y &=2, \end{align*} has infinitely many solutions and all these solutions satisfy bounds \begin{align*} 0<\ &x <4, \\ -9\le\ &y\le 1. \end{align*}

2004 Germany Team Selection Test, 1

Tags: calculus , inequalities , bounded , algebra , sequence

Consider pairs of the sequences of positive real numbers \[a_1\geq a_2\geq a_3\geq\cdots,\qquad b_1\geq b_2\geq b_3\geq\cdots\] and the sums \[A_n = a_1 + \cdots + a_n,\quad B_n = b_1 + \cdots + b_n;\qquad n = 1,2,\ldots.\] For any pair define $c_n = \min\{a_i,b_i\}$ and $C_n = c_1 + \cdots + c_n$, $n=1,2,\ldots$. (1) Does there exist a pair $(a_i)_{i\geq 1}$, $(b_i)_{i\geq 1}$ such that the sequences $(A_n)_{n\geq 1}$ and $(B_n)_{n\geq 1}$ are unbounded while the sequence $(C_n)_{n\geq 1}$ is bounded? (2) Does the answer to question (1) change by assuming additionally that $b_i = 1/i$, $i=1,2,\ldots$? Justify your answer.

2002 IMO Shortlist, 2

Tags: inequalities , algebra , sequence , bounded

Let $a_1,a_2,\ldots$ be an infinite sequence of real numbers, for which there exists a real number $c$ with $0\leq a_i\leq c$ for all $i$, such that \[\left\lvert a_i-a_j \right\rvert\geq \frac{1}{i+j} \quad \text{for all }i,\ j \text{ with } i \neq j. \] Prove that $c\geq1$.

2005 Germany Team Selection Test, 1

Tags: algebra , sequence , bounded

Let $a_0$, $a_1$, $a_2$, ... be an infinite sequence of real numbers satisfying the equation $a_n=\left|a_{n+1}-a_{n+2}\right|$ for all $n\geq 0$, where $a_0$ and $a_1$ are two different positive reals. Can this sequence $a_0$, $a_1$, $a_2$, ... be bounded? [i]Proposed by Mihai Bălună, Romania[/i]