CONO-SUR 1989

Two isosceles triangles with sidelengths $x,x,a$ and $x,x,b$ ( $a \neq b$ ) have equal areas. Find $x$ .

Find the sum$$1+11+111+\cdots+\underbrace{111\ldots111}_{n\text{ digits}}.$$

A number $p$ is $perfect$ if the sum of its divisors, except $p$ is $p$ . Let $f$ be a function such that:

$f(n)=0$ , if n is perfect $f(n)=0$ , if the last digit of n is 4 $f(a.b)=f(a)+f(b)$

Find $f(1998)$

Let $n$ be square with 4 digits, such that all its digits are less than 6. If we add 1 to each digit the resulting number is another square. Find $n$

Let $ABCD$ be a square with diagonals $AC$ and $BD$ , and $P$ a point in one of the sides of the square. Show that the sum of the distances from P to the diagonals is constant.

Show that reducing the dimensions of a cuboid we can't get another cuboid with half the volume and half the surface.