FI-MO 2004

The equations $x^2 +2ax+b^2 = 0$ and $x^2 +2bx+c^2 = 0$ both have two different real roots. Determine the number of real roots of the equation $x^2 + 2cx + a^2 = 0.$

$a, b$ and $c$ are positive integers and $$\frac{a\sqrt{3} + b}{b\sqrt{3} + c}$$ is a rational number. Show that $$\frac{a^2 + b^2 + c^2}{a + b + c}$$ is an integer.

Two circles with radii $r$ and $R$ are externally tangent. Determine the length of the segment cut from the common tangent of the circles by the other common tangents.

The numbers $2005! + 2, 2005! + 3, ... , 2005! + 2005$ form a sequence of $2004$ consequtive integers, none of which is a prime number. Does there exist a sequence of $2004$ consequtive integers containing exactly $12$ prime numbers?

Finland is going to change the monetary system again and replace the Euro by the Finnish Mark. The Mark is divided into $100$ pennies. There shall be coins of three denominations only, and the number of coins a person has to carry in order to be able to pay for any purchase less than one mark should be minimal. Determine the coin denominations.