Home > Algebra, Analysis, Olympiad, Probability, Problem Solving > Problems of the Miklos Schweitzer Competition 2014

## Problems of the Miklos Schweitzer Competition 2014

Problem 1. Let ${n}$ be a positive integer. Let ${\mathcal{F}}$ be a familiy of sets that contains more than half of all subsets of an ${n}$-element set ${X}$. Prove that from ${\mathcal{F}}$ we can select ${\lceil \log_2 n\rceil+1 }$ sets that form a separating family of ${X}$, i.e., for any two distinct elements of ${X}$ there is a selected set containing exactly one of the two elements.

Problem 2. let ${k \geq 1}$ and let ${I_1,...,I_k}$ be non-degenerate subintervals of the interval ${[0,1]}$. Prove that

$\displaystyle \sum \frac{1}{|I_i \cup I_j|} \geq k^2,$

where the summation is over all pairs of indices ${(i,j)}$ such that ${I_i}$ and ${I_j}$ are not disjoint.

Problem 3. We have ${4n+5}$ points in the plane, no three of them collinear. The points are colored with two colors. Prove that from the points we can form ${n}$ empty triangles (they have no colored points in their interiors) with pairwise disjoint interiors, such that all points occuring as vertices of the ${n}$ triangles have the same color.

Problem 4. For a positive integer ${n}$, let ${f(n)}$ be the number of sequences ${a_1,...,a_k}$ of positive integers such that ${a_i \geq 2}$ and ${a_1...a_k = n}$ for ${k \geq 1}$. We make the convention ${f(1)=1}$. Let ${\alpha}$ be the unique real number greater than ${1}$ such that ${\sum_{n=1}^\infty n^{-\alpha}=2}$. Prove that

• (i) ${ \sum_{ k = 1}^n f(k)= O(n^\alpha)}$.
• (ii) There exists no number ${\beta<\alpha}$ such that ${f(n)=O(n^\beta)}$.

Problem 5. Let ${\alpha}$ be a non-real algebraic integer of degree two, and let ${P}$ be the set of irreducible elements of the ring ${\Bbb{Z}[\alpha]}$. Prove that

$\displaystyle \sum_{ p \in P} \frac{1}{|p|^2} = \infty.$

Problem 6. Let ${\rho : G \rightarrow GL(V)}$ be a representation of a finite ${p}$-group ${G}$ over a field of characteristic ${p}$. Prove that if the restriction of the linear map ${\sum_{ g \in G} \rho(g)}$ to a finite dimensional subspace ${W}$ of ${V}$ is injective, then the subspace spanned by the subspaces ${\rho(g)W}$ (${g \in G}$) is the direct sum of these subspaces.

Problem 7. Lef ${f: \Bbb{R} \rightarrow \Bbb{R}}$ be a continuous function and let ${g: \Bbb{R} \rightarrow \Bbb{R}}$ be arbitrary. Suppose that the Minkowski sum of the graph of ${f}$ and the graph of ${g}$ (i.e. the set ${\{(x+y,f(x)+g(y) : x,y \in \Bbb{R}\}}$ has Lebesgue measure zero. Does it follow then that the function ${f}$ is of the form ${f(x)=ax+b}$, with suitable constants ${a,b \in \Bbb{R}}$?

Problem 8. Let ${n \geq 1}$ be a fixed integer. Calculate the distance

$\displaystyle \inf_{p,f} \max_{0 \leq x \leq 1} |f(x)-p(x)|,$

where ${p}$ runs over polynomials of degree less than ${n}$ with real coefficients and ${f}$ runs over functions of the form

$\displaystyle f(x) = \sum_{ k = n}^\infty c_kx^k$

defined on the closed interval ${[0,1]}$, where ${c_k\geq 0}$ and ${\sum_{k=n}^\infty c_k =1}$.

Problem 9. Let ${\rho : \Bbb{R}^n \rightarrow \Bbb{R},\ \rho(x)=e^{-\|x\|^2}}$, and let ${K \subset \Bbb{R}^n}$ be a convex body, i.e. a compact convex set with nonempty interior. Define the barycenter ${s_K}$ of the body ${K}$ with respect to the weight function ${\rho}$ by the usual formula

$\displaystyle s_K = \frac{\int_K \rho(x) x dx}{\int_K \rho(x)dx}.$

Prove that the translates of the body ${K}$ have pairwise distinct barycenters with respect to ${\rho}$.

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 ${U}$ be a random variable that is uniformly distributed on the interval ${[0,1]}$, and let

$\displaystyle S_n = 2\sum_{k=1}^n \sin(2kU\pi).$

Show that, as ${n \rightarrow \infty}$, the limit distribution of ${S_n}$ is the Cauchy distribution with density function ${f(x) =\frac{1}{\pi(1+x^2)}}$.

1. December 7, 2014 at 5:16 pm

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

• December 7, 2014 at 5:40 pm

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.

2. February 24, 2015 at 6:18 pm

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?

3. September 3, 2015 at 8:50 am

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

• October 3, 2015 at 12:40 am

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.