Let $ f (n) $ be a function that fulfills the following properties: $\bullet$ For each natural $ n $, $ f (n) $ is an integer greater than or equal to $ 0 $. $\bullet$ $f (n) = 2010 $, if $ n $ ends in $ 7 $. For example, $ f (137) = 2010 $. $\bullet$ If $ a $ is a divisor of $ b $, then: $ f \left(\frac {b} {a} \right) = | f (b) -f (a) | $. Find $ \displaystyle f (2009 ^ {2009 ^ {2009}}) $ and justify your answer.

In an acute angle $ \vartriangle ABC $, let $ AD, BE, CF $ be their altitudes (with $ D, E, F $ lying on $ BC, CA, AB $, respectively). Let's call $ O, H $ the circumcenter and orthocenter of $ \vartriangle ABC $, respectively. Let $ P = CF \cap AO $. Suppose the following two conditions are true: $\bullet$ $ FP = EH $ $\bullet$ There is a circle that passes through points $ A, O, H, C $ Prove that the $ \vartriangle ABC $ is equilateral.

Let $P$ be a regular polygon with $ 4k + 1 $ sides (where $ k $ is a natural) whose vertices are $ A_1, A_2, ..., A_ {4k + 1} $ (in that order ). Each vertex $ A_j $ of $P$ is assigned a natural of the set $ \{1,2, ..., 4k + 1 \} $ such that no two vertices are assigned the same number. On $P$ the following operation is performed: Let $ B_j $ be the midpoint of the side $ A_jA_ {j + 1} $ for $ j = 1,2, ..., 4k + 1 $ (where is consider $ A_ {4k + 2} = A_1 $). If $ a $, $ b $ are the numbers assigned to $ A_ {j} $ and $ A_ {j + 1} $, respectively, the midpoint $ B_j $ is written the number $ 7a-3b $. By doing this with each of the $ 4k + 1 $ sides, the $ 4k + 1 $ vertices initially arranged are erased. We will say that a natural $ m $ is fatal if for all natural $ k $ , no matter how the vertices of $P$ are initially arranged, it is impossible to obtain $ 4k + 1 $ equal numbers through a finite amount of operations from $ m $. a) Determine if the $ 2010 $ is fatal or not. Justify. b) Prove that there are infinite fatal numbers. PS. A help in translation of the 2nd paragraph is welcome. Original wordingDiremos que un natural $m$ es fatal si no importa cómo se disponen inicialmente los vértices de ${P}$, es imposible obtener mediante una cantidad finita de operaciones $4k+1$ números iguales a $m$.

Let $ a_1 <a_2 <... <a_n $ consecutive positive integers (with $ n> 2 $). A grasshopper jumps on the real line, starting at point $ 0 $ and jumping $ n $ to the right with lengths $ a_1 $, $ a_2 $, ..., $ a_n $, in some order (each length occupies exactly once), ending your tour at the $ 2010 $ point. Find all the possible values $ n $ of jumps that the grasshopper could have made.

The vigilantes are a group of five superheroes, such that each one has one and only one of the following powers: hypnosis, super speed, shadow manipulation, immortality and super strength (each has a different power). On an adventure to the island of Philippines, they meet the sorcerer Vicencio, an old wise man who offers them the following ritual to help them: The ritual consists of a superhero $A$ acquiring the gift (s) of $B$ without $B$ acquiring the gift (s) of $A$. Determine the fewest number of rituals to be performed by the sorcerer Vicencio so that each superhero controls each of the five gifts. Clarification: At the end of the ritual, a superhero $A$ will have his gifts and those of a superhero $B$, but $B$ does not acquire those of $A$, but it does keep its own.

Let $ \vartriangle ABC $ be a triangle with $ \angle ACB = 60º $. Let $ E $ be a point inside $ \overline {AC} $ such that $ CE <BC $. Let $ D $ over $ \overline {BC} $ such that $$ \frac {AE} {BD} = \frac {BC} {CE} -1 .$$Let us call $ P $ the intersection of $ \overline {AD} $ with $ \overline {BE} $ and $ Q $ the other point of intersection of the circumcircles of the triangles $ AEP $ and $ BDP $. Prove that $QE \parallel BC $.

$ 15 $ teams participate in a soccer league. Each team plays each of the remaining teams exactly once. If a team beats another team in a match they receive $ 3 $ points, while the loser receives $ 1 $ point. In the event of a tie, both teams receive $ 2 $ points. When all possible league matches are held, the following can be observed: $\bullet$ No two teams have finished with the same amount of points. $\bullet$ Each team finished the league with at least $ 21 $ points. Let $W$ be the team that finished the league with the highest score. Determine how many points $W$ scored and show that there were at least four ties in the league.