Let $a_n$ be the last digit of the sum of the digits of $20052005...2005$, where the $2005$ block occurs $n$ times. Find $a_1 +a_2 + \dots +a_{2005}$.

Let $ABC$ be an acute-angled triangle and let $AN$, $BM$ and $CP$ the altitudes with respect to the sides $BC$, $CA$ and $AB$, respectively. Let $R$, $S$ be the pojections of $N$ on the sides $AB$, $CA$, respectively, and let $Q$, $W$ be the projections of $N$ on the altitudes $BM$ and $CP$, respectively. (a) Show that $R$, $Q$, $W$, $S$ are collinear. (b) Show that $MP=RS-QW$.

The monetary unit of a certain country is called Reo, and all the coins circulating are integers values of Reos. In a group of three people, each one has 60 Reos in coins (but we don't know what kind of coins each one has). Each of the three people can pay each other any integer value between 1 and 15 Reos, including, perhaps with change. Show that the three persons together can pay exactly (without change) any integer value between 45 and 135 Reos, inclusive.

Let $ABC$ be a isosceles triangle, with $AB=AC$. A line $r$ that pass through the incenter $I$ of $ABC$ touches the sides $AB$ and $AC$ at the points $D$ and $E$, respectively. Let $F$ and $G$ be points on $BC$ such that $BF=CE$ and $CG=BD$. Show that the angle $\angle FIG$ is constant when we vary the line $r$.

We say that a number of 20 digits is special if its impossible to represent it as an product of a number of 10 digits by a number of 11 digits. Find the maximum quantity of consecutive numbers that are specials.

On the cartesian plane we draw circunferences of radii 1/20 centred in each lattice point. Show that any circunference of radii 100 in the cartesian plane intersect at least one of the small circunferences.