FI-MO 2012

A secant line splits a circle into two segments. Inside those segments, one draws two squares such that both squares has two corners on a secant line and two on the circumference. The ratio of the square's side lengths is $5:9$ . Compute the ratio of the secant line versus circle radius.

Let $x\ne 1,y\ne 1$ and $x\ne y.$ Show that if $$\frac{yz-x^2}{1-x}=\frac{zx-y^2}{1-y},$$ then $$\frac{yz-x^2}{1-x}=\frac{zx-y^2}{1-y}=x+y+z.$$

Prove that for all integers $k\geq 2,$ the number $k^{k-1}-1$ is divisible by $(k-1)^2.$

Let $k,n\in\mathbb{N},0<k<n.$ Prove that $$\sum_{j=1}^k\binom{n}{j}=\binom{n}{1}+ \binom{n}{2}+\ldots + \binom{n}{k}\leq n^k.$$

The [i]Collatz's function[i] is a mapping $f:\mathbb{Z}_+\to\mathbb{Z}_+$ satisfying $$f(x)=\begin{cases} 3x+1,& \text{as }x\text{ is odd} x/2, & \text{as }x\text{ is even.} \end{cases}$$ In addition, let us define the notation $f^1=f$ and inductively $f^{k+1}=f\circ f^k,$ or to say in another words, $f^k(x)=\underbrace{f(\ldots (f}_{k\text{ times}}(x)\ldots ).$ Prove that there is an $x\in\mathbb{Z}_+$ satisfying $$f^{40}(x)> 2012x.$$