IR-MO 2009

Let $z$ be a complex non-zero number such that $Re(z),Im(z)\in \mathbb{Z}$ . Prove that $z$ is uniquely representable as $a_0+a_1(1+i)+a_2(1+i)^2+\dots+a_n(1+i)^n$ where $n\geq 0$ and $a_j \in \{0,1\}$ and $a_n=1$ .

Time allowed for this problem was 1 hour.

A sphere is inscribed in polyhedral $P$ . The faces of $P$ are coloured with black and white in a way that no two black faces share an edge. Prove that the sum of surface of black faces is less than or equal to the sum of the surface of the white faces.

Time allowed for this problem was 1 hour.

Sone of vertices of the infinite grid $\mathbb{Z}^{2}$ are missing. Let's take the remainder as a graph. Connect two edges of the graph if they are the same in one component and their other components have a difference equal to one. Call every connected component of this graph a branch. Suppose that for every natural $n$ the number of missing vertices in the $(2n+1)\times(2n+1)$ square centered by the origin is less than $\frac{n}{2}$ . Prove that among the branches of the graph, exactly one has an infinite number of vertices.

Time allowed for this problem was 90 minutes.

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