## Miklos Schweitzer 2012 Problems

**1.** Is there any real number for which there exist two functions such that

but the function which associates to the -th decimal digit of is not recursive?

**2.** Call a subset of the cyclic group *rich* if for all there exists such that are all in . For what is there a constant such that for each odd positive integer , every *rich* subset has at least elements?

**3.** Prove that if a -chromatic graph’s edges are coloured in two colors in any way, there is a subtree with vertices and edges of the same color.

**4.** Let be a convex shape in the dimensional space, having unit volume. Let be a Lebesgue measurable set with measure at least , where . Prove that dilating from its centroid by the ratio of , the shape obtained contains the centroid of .

**5.** Let be four dimensional linear subspaces in such that the intersection of any two contains only the zero vector. Prove that there exists a linear four dimensional subspace in such that all four vector spaces are two dimensional.

**6.** Let be matrices with complex elements such that and , where denotes the commutator of the matrices. Prove that is the identity matrix. **[Solution]
**

**7.** Let be a simple curve, lying inside a circle of radius , rectifiable and of length . Prove that if , then there exists a circle of radius which intersects in at least distinct points.

**8.** For any function consider the function for which for any .

a) Is it true that if is Lebesgue measurable then is also Lebesgue measurable?

b) Is it true that if is Borel measurable then is also Borel measurable?

**9.** Let be the complex unit disk , and a real number. Suppose that is a holomorphic function such that and . Prove that

**10.** Let be a knot in the -dimensional space (that is a differentiable injection of a circle into ), and be the diagram of the knot (that is the projection of it to a plane so that in exception of the transversal double points it is also an injection of a circle). Let us color the complement of in black and color the diagram in a chessboard pattern, black and white so that zones which share an arc in common are coloured differently. Define the black graph of the diagram, a graph which has vertices in the black areas and every two vertices which correspond to black areas which have a common point have an edge connecting them.

a) Determine all knots having a diagram such that has at most spanning trees. (Two knots are not considered distinct if they can be moved into each other with a one parameter set of the injection of the circle)

b) Prove that for any knot and any diagram , has an odd number of spanning trees.

**11.** Let be independent random variables with the same distribution, and let . For what real numbers is the following statement true:

Miklos Schweitzer competition 2012 (many thanks to Eles Andras for the translation of the problems from Hungarian to English)

Great! Thank you.

You’re welcome. 🙂