FI-MO 2013

The coefficients $a,b,c$ of a polynomial $f:\mathbb{R}\to\mathbb{R}, f(x)=x^3+ax^2+bx+c$ are mutually distinct integers and different from zero. Furthermore, $f(a)=a^3$ and $f(b)=b^3.$ Determine $a,b$ and $c$ .

In a particular European city, there are only $7$ day tickets and $30$ day tickets to the public transport. The former costs $7.03$ euro and the latter costs $30$ euro. Aina the Algebraist decides to buy at once those tickets that she can travel by the public transport the whole three year (2014-2016, 1096 days) visiting in the city. What is the cheapest solution?

The points $A,B,$ and $C$ lies on the circumference of the unit circle. Furthermore, it is known that $AB$ is a diameter of the circle and $$\frac{|AC|}{|CB|}=\frac{3}{4}.$$ The bisector of $ABC$ intersects the circumference at the point $D$ . Determine the length of the $AD$ .

A subset $E$ of the set $\{1,2,3,\ldots,50\}$ is said to be special if it does not contain any pair of the form $\{x,3x\}.$ A special set $E$ is superspecial if it contains as many elements as possible. How many element there are in a superspecial set and how many superspecial sets there are?

Find all integer triples $(m,p,q)$ satisfying $$2^mp^2+1=q^5$$ where $m>0$ and both $p$ and $q$ are prime numbers.