## Putnam 2017 A3 – Solution

**Problem A3.** Denote . Then is continuous and . We can see that

Now note that on we have so . Furthermore, on we have so multiplying with we get . Therefore

To prove that goes to we can still work with . Note that the negative part cannot get too big:

As for the positive part, taking we have

Next, note that on

We would need that the last term be larger than . This is equivalent to . Since is continuous on , it is bounded above, so some delta small enough exists in order for this to work.

Grouping all of the above we get that

Since we get that goes to .

## 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.