OPM 2005

In line for a SuperRockPop concert were 2005 people. With the aim of offering $3$ tickets for the "backstage", the first person in line was asked to shout "Super", the second "Rock", the third "Pop", the fourth "Super", the fifth "Rock", ` the sixth "Pop" and so on. Anyone who said "Rock" or "Pop" was eliminated. This process was repeated, always starting from the first person in the new line, until only $3$ people remained. What positions were these people in at the beginning?

Consider the triangles $[ABC]$ and $[EDC]$ , right at $A$ and $D$ , respectively. Show that, if $E$ is the midpoint of $[AC]$ , then $AB <BD$ .

On a board with $a$ rows and $b$ columns, each square has a switch and an unlit light bulb. By pressing the switch of a house, the lamp in that house changes state, along with the lamps in the same row and those in the same column (those that are on go out and the that are off light up). For what values of $a$ and $b$ is it possible to have just one lamp on, by pressing a series of switches?

A natural number $n$ is said to be abundant if the sum of its divisors is greater than $2n$ . For example, $18$ is abundant because the sum of its divisors, $1 + 2 + 3 + 6 + 9 + 18$ , is greater than $36$ . Write every even number greater than $46$ as a sum of two numbers abundant.

Considers a quadrilateral $[ABCD]$ that has an inscribed circle and a circumscribed circle. The sides $[AD]$ and $[BC]$ are tangent to the circle inscribed at points $E$ and $F$ , respectively. Prove that $AE \cdot F C = BF \cdot ED$ .

Prove that there is a unique function $f: N\to N$ , that verifies $$f(a + b)f(a - b) = f(a^2)$$ , for any $a, b\in N$ such that $a > b$ .