Olympiad problems

1974 Czech and Slovak Olympiad III A, 4

Tags: algebra , polynomial , function , bounded , set

Let $\mathcal M$ be the set of all polynomial functions $f$ of degree at most 3 such that \[\forall x\in[-1,1]:\ |f(x)|\le 1.\] Denote $a$ the (possibly zero) coefficient of $f$ at $x^3.$ Show that there is a positive number $k$ such that \[\forall f\in\mathcal M:\ |a|\le k\] and find the least $k$ with this property.

2015 Romania Team Selection Tests, 2

Tags: bounded , recurrence , algebra , sequence

Let $(a_n)_{n \geq 0}$ and $(b_n)_{n \geq 0}$ be sequences of real numbers such that $ a_0>\frac{1}{2}$ , $a_{n+1} \geq a_n$ and $b_{n+1}=a_n(b_n+b_{n+2})$ for all non-negative integers $n$ . Show that the sequence $(b_n)_{n \geq 0}$ is bounded .

1992 Putnam, B5

Tags: determinant , bounded

Let $D_n$ denote the value of the $(n -1) \times (n - 1)$ determinant $$ \begin{pmatrix} 3 & 1 &1 & \ldots & 1\\ 1 & 4 &1 & \ldots & 1\\ 1 & 1 & 5 & \ldots & 1\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 1 & 1 & 1 & \ldots & n+1 \end{pmatrix}.$$ Is the set $\left\{ \frac{D_n }{n!} \, | \, n \geq 2\right\}$ bounded?

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]

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*}