## SEEMOUS 2014

**Problem 1.** Let be a nonzero natural number and be a function such that . Let be distinct real numbers. If

prove that is not continuous.

**Problem 2.** Consider the sequence given by

Prove that the sequence is convergent and find its limit.

**Problem 3.** Let and such that , where and is the conjugate matrix of .

(a) Show that .

(b) Show that if then .

**Problem 4.** a) Prove that .

b) Find the limit

## Asymptotic characterization in terms of sequence limits

## Miklos Schweitzer 2013 Problem 7

**Problem 7.** Suppose that is an additive function (that is for all ) for which is bounded of some nonempty subinterval of . Prove that is continuous.

## Continuity in Geometry

Here are a few interesting geometry problems which use continuity problems in their solutions.

**Pb 1.** Consider three parallel lines in the plane . Prove that there exist points such that the triangle is equilateral.

**Pb 2.** Consider a triangle . Prove that is equilateral if and only if for every point in the plane we can construct a triangle with sides .

## Agreg 2012 Analysis Part 4

**A Fixed Point Theorem**

This parts wishes to extend the next result (which can be used without proof) to infinite dimension.

**Theorem.** (Browuer) Consider a finite dimensional normed vector space. Consider a convex, closed, bounded non-void set. If is a continuous application, then has a fixed point in .

1. In endowed with we consider the following application:

Prove that is continuous with values in the unit sphere of , but does not admit any fixed points.

2. Consider a normed vector space, a closed, bounded non-void subset of and a compact application (not necessarily linear). (a compact application maps bounded sets into relatively compact sets)

i) Let . We can cover (which is compact) by a finite number of open balls of radius : with for every . For we define

Prove that is continuous and that there exists such that for we have .

ii) We introduce the application defined by

Prove that for every we have .

## Arcs on a circle

Suppose that we have a finite set of arcs on a circle, with the property that every two of them intersect. Prove that there exists a diameter which intersects all arcs.

## Traian Lalescu student contest 2011 Problem 3

For a continuous function such that prove that if is uniformly continuous, then it is bounded. Prove also that the converse of the previous statement is not true.

Traian Lalescu student contest 2011