## SEEMOUS 2016 Problem 4 – Solution

## SEEMOUS 2016 – Problems

**Problem 1.** Let be a continuous and decreasing real valued function defined on . Prove that

When do we have equality?

**Problem 2.** a) Prove that for every matrix there exists a matrix such that .

b) Prove that there exists a matrix such that for all .

**Problem 3.** Let be idempotent matrices () in . Prove that

where and is the set of matrices with real entries.

**Problem 4.** Let be an integer and set

Prove that

a)

b) .

Some hints follow.

## 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

## SEEMOUS 2013 + Solutions

Here are some of the problems of SEEMOUS 2013. **Update: the 4th problem has arrived; it is number 3 below.**

**1.** Let be a continuous mapping, such that

Find the form of the map .

*Solution:* Change the variable from to in the RHS integral and DO NOT calculate the last integral in the RHS. Get all the terms in the left and find that it is in fact the integral of a square equal to zero.

**2.** Let be nonzero matrices such that and . Prove that there is an invertible matrix such that and .

*Solution:* One solution can be given using the fact that can be written in that form, but for different matrices .

Another way to do it is to consider applications . We get at once and from these we deduce that and . First note that is not the zero application. Then there exists , i.e. there exists such that . We have . Consider .

Then are not collinear, . Consider now the basis formed by and take to be the change of base matrix from the canonical base to .

**3.** Find the maximum possible value of

over all continuously differentiable functions with and .

**4.** Let such that there is , with . Prove that either or .

*Solution:* Consider the minimal polynomial of , which has degree at most . The eigenvalues of satisfy . We have two cases: either the eigenvalues are real and therefore they are both equal to either they are complex and conjugate of modulus one. In both cases the determinant of is equal to . Therefore, by Cayley Hamiltoh theorem satisfies an equation of the type .

By hypothesis the minimal polynomial divides . If has degree one then and so .

If not, then the minimal polynomial is and we must have

It can be proved that is in fact an integer. (using the fact that the product of two primitive polynomials is primitive)

Since it follows that . The rest is just casework.

## SEEMOUS 2012 Problem 3

a) Prove that if is an even positive integer and is a real symmetric matrix such that then .

b) Does this assertion from a) also hold for odd positive integers ?

## SEEMOUS 2012 Problem 2

Let . Consider the right triangles as in the figure. (More precisely, for every the hypotenuse of is a leg of with right angle , and the vertices and lie on the opposite sides of the straight line ; also for every .)

It is possible for the set of points to be unbounded, but the series to be convergent? Here denotes the measure of .

(A subset of the plane is bounded if, for example is contained in a disk.)

## SEEMOUS 2012 Problem 1

Let be the matrix, where is the remainder of the division of by for . Find the greatest for which .