NL-NWO 1992

Four dice are thrown. What is the probability that the product of the number equals $36?$

In the fraction below and its decimal notation (with period of length $4$ ) every letter represents a digit, and different letters denote different digits. The numerator and denominator are coprime. Determine the value of the fraction:

$\frac{ADA}{KOK}=0.SNELSNELSNELSNEL...$

$Note.$ Ada Kok is a famous dutch swimmer, and "snel" is Dutch for "fast".

Consider the configuration of six squares as shown on the picture. Prove that the sum of the area of the three outer squares ( $I,II$ and $III$ ) equals three times the sum of the areas of the three inner squares ( $IV,V$ and $VI$ ).

Attachments:

For every positive integer $n$ , we define $n?$ as $1?=1$ and $n?=\frac{n}{(n-1)?}$ for $n \ge 2$ .

Prove that $\sqrt{1992}<1992?<\frac{4}{3} \sqrt{1992}.$

We consider regular $n$ -gons with a fixed circumference $4$ . Let $r_n$ and $a_n$ respectively be the distances from the center of such an $n$ -gon to a vertex and to an edge. $(a)$ Determine $a_4,r_4,a_8,r_8$ . $(b)$ Give an appropriate interpretation for $a_2$ and $r_2$ $(c)$ Prove that $a_{2n}=\frac{1}{2} (a_n+r_n)$ and $r_{2n}=\sqrt{a_2n r_n}.$ $(d)$ Define $u_0=0, u_1=1$ and $u_n=\frac{1}{2}(u_{n-2}+u_{n-1})$ for $n$ even or $u_n=\sqrt{u_{n-2} u_{n-1}}$ for $n$ odd. Determine $\lim_{n\to\infty}u_n$ .