CONO-SUR 1995

Find a number with $3$ digits, knowing that the sum of its digits is $9$ , their product is $24$ and also the number read from right to left is $\frac{27}{38}$ of the original.

There are ten points marked on a circumference, numbered from $1$ to $10$ and join all points with segments. I color the segments, with red someones and others with blue. Without changing the colors of the segments, renumber all the points from the $1$ to the $10$ . Will be possible to color the segments and to renumber the points so that those numbers that were jointed with red are jointed now with blue and the numbers that were jointed with blue they are jointed now with red?

Let $ABCD$ be a rectangle with: $AB=a$ , $BC=b$ . Inside the rectangle we have to exteriorly tangents circles such that one is tangent to the sides $AB$ and $AD$ ,the other is tangent to the sides $CB$ and $CD$ .

1. Find the distance between the centers of the circles(using $a$ and $b$ ).
2. When the radiums of both circles change the tangency point between both of them changes, and describes a locus. Find that locus.

Attachments:

The semicircle with centre $O$ and the diameter $AC$ is divided in two arcs $AB$ and $BC$ with ratio $1: 3$ . $M$ is the midpoint of the radium $OC$ . Let $T$ be the point of arc $BC$ such that the area of the cuadrylateral $OBTM$ is maximum. Find such area in fuction of the radium.

Let $n$ be a natural number and $f(n) = 2n - 1995 \lfloor \frac{n}{1000} \rfloor$ ( $\lfloor$ $\rfloor$ denotes the floor function).

1. Show that if for some integer $r$ : $f(f(f...f(n)...))=1995$ (where the function $f$ is applied $r$ times), then $n$ is multiple of $1995$ .

2. Show that if $n$ is multiple of 1995, then there exists r such that: $f(f(f...f(n)...))=1995$ (where the function $f$ is applied $r$ times). Determine $r$ if $n=1995.500=997500$