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 [i]fatal [/i] 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. [color=#f00]PS. A help in translation of the 2nd paragraph is welcome[/color]. [hide=Original wording]Diremos 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 $m$.[/hide]

Let $n$ be a positive integer. Prove that in a regular $6n$-gon, we can draw $3n$ diagonals with pairwise distinct ends and partition the drawn diagonals into $n$ triplets so that: [list] [*] the diagonals in each triplet intersect in one interior point of the polygon and [*] all these $n$ intersection points are distinct. [/list]

$A, B, C$ are adjacent vertices of a regular $2n$-gon and $D$ is the vertex opposite to $B$ (so that $BD$ passes through the center of the $2n$-gon). $X$ is a point on the side $AB$ and $Y$ is a point on the side $BC$ so that $XDY = \frac{\pi}{2n}$. Show that $DY$ bisects $\angle XYC$.

Find all natural numbers $n$ for which there exists a convex polyhedron satisfying the following conditions: (i) Each face is a regular polygon. (ii) Among the faces, there are polygons with at most two different numbers of edges. (iii) There are two faces with common edge that are both $n$-gons.

a) Prove that for $n = 2,3,4,...$ holds: $$\sin a + \sin 2a + ...+ \sin (n-1)a=\frac{\cos a \left(\frac{a}{2}\right) - \cos \left(n-\frac{1}{2}\right) a}{2 \sin \left(\frac{a}{2}\right)}$$ b) A point on the circumference of a wheel, which, remaining in a vertical plane, rolls along a horizontal path, describes, at one revolution of the wheel, a curve having a length equal to four times the diameter of the wheel. Prove this by first considering tilting a regular $n$-gon. [hide=original wording for part b]Een punt van de omtrek van een wiel dat, in een verticaal vlak blijvend, rolt over een horizontaal gedachte weg, beschrijft bij één omwenteling van het wiel een kromme die een lengte heeft die gelijk is aan viermaal de middellijn van het wiel. Bewijs dit door eerst een rondkantelende regelmatige n-hoek te beschouwen.[/hide]

Two regular polygons, a $7$-gon and a $17$-gon are given. For each of them two circles are drawn, an inscribed circle and a circumscribed circle. It happened that rings containing the polygons have equal areas. Prove that sides of the polygons are equal. (3)