## Problems of the Miklos Schweitzer Competition 2014

**Problem 1.** Let be a positive integer. Let be a familiy of sets that contains more than half of all subsets of an -element set . Prove that from we can select sets that form a separating family of , i.e., for any two distinct elements of there is a selected set containing exactly one of the two elements.

**Problem 2.** let and let be non-degenerate subintervals of the interval . Prove that

where the summation is over all pairs of indices such that and are not disjoint.

**Problem 3.** We have points in the plane, no three of them collinear. The points are colored with two colors. Prove that from the points we can form empty triangles (they have no colored points in their interiors) with pairwise disjoint interiors, such that all points occuring as vertices of the triangles have the same color.

**Problem 4.** For a positive integer , let be the number of sequences of positive integers such that and for . We make the convention . Let be the unique real number greater than such that . Prove that

- (i) .
- (ii) There exists no number such that .

**Problem 5.** Let be a non-real algebraic integer of degree two, and let be the set of irreducible elements of the ring . Prove that

**Problem 6.** Let be a representation of a finite -group over a field of characteristic . Prove that if the restriction of the linear map to a finite dimensional subspace of is injective, then the subspace spanned by the subspaces () is the direct sum of these subspaces.

**Problem 7.** Lef be a continuous function and let be arbitrary. Suppose that the Minkowski sum of the graph of and the graph of (i.e. the set has Lebesgue measure zero. Does it follow then that the function is of the form , with suitable constants ?

**Problem 8.** Let be a fixed integer. Calculate the distance

where runs over polynomials of degree less than with real coefficients and runs over functions of the form

defined on the closed interval , where and .

**Problem 9.** Let , and let be a convex body, i.e. a compact convex set with nonempty interior. Define the barycenter of the body with respect to the weight function by the usual formula

Prove that the translates of the body have pairwise distinct barycenters with respect to .

**Problem 10.** To each vertex of a given triangulation of the two dimensional sphere, we assign a convex subset of the plane. Assume that the three convex sets corresponding to the three vertices of any two dimensional face of the triangulation have at least one point in common. Show that there exist four vertices such that the corresponding convex sets have at least one point in common.

**Problem 11.** Let be a random variable that is uniformly distributed on the interval , and let

Show that, as , the limit distribution of is the Cauchy distribution with density function .

Source: http://www.bolyai.hu/SCH_angol_2014.pdf

any ideas for problem #8? Seems like a cool problem

Indeed, it is a cool problem, but I don’t have any idea on how to solve it. I haven’t thought much about the problems, since it’s a busy period for me.

Problem 1 seems like a nice problem, but the bound is strong as well, I suppose.

Any ideas?

In any case, is it necessary that the first problem is ought to be easier than other problems?

What are the steps to being participate of Miklos Schweitzer Competition?, Any advice, I’ll grateful.

I tried to participate to the competition, and in one year I even submitted some solutions. Unfortunately it is not really possible to participate if you’re not from Hungary. At least that was the policy then. I don’t know if it has changed. You should consult their webpage http://www.bolyai.hu/schweitzer.htm and eventually ask the organizers themselves if foreign participation is possible.