## SEEMOUS 2017 Problems

**Problem 1.** Let . Suppose

satisfies

Show that is invertible.

**Problem 2.** Let .

- a) Prove that there exists such that for every the matrix equation
has a unique solution .

- b) Prove that if and is diagonalizable then

**Problem 3.** Let be a continuous function. Prove that

**Problem 4.** a) Let be an integer. Compute .

b) Let be a fixed integer and let be the sequence defined by

Prove that the sequence converges and find its limit.

**Source: **link

**Hints:** 1. This must be a joke 🙂 We know that if for a norm matrix we have then is invertible. The sum of squares is a norm, so implies that is invertible.

2. The equation is . We know that there are at most values for when is not invertible (related to the eigenvalues of ). We have two cases. If one of the eigenvalues is zero, then pick smaller than the distance to the closest eigenvalue to . If is not an eigenvalue, then choose again smaller than the closest eigenvalue to . For note that and since and we get that . Since is diagonalizable we conclude that

Thus, multiplying by and letting go to zero we obtain for zero eigenvalues and zero for non-zero ones. Thus, the limit counts the number of zero eigenvalues, which is precisely (since is diagonalizable).

3. Classic SEEMOUS stuff. Prove that this works when , then extend to polynomials, and then, by density, to continuous functions. Proving this for monomial might get involved, but initial computations show that we have the same terms when performing the integrations by parts on the two sides…

4. coming…

See:

http://math.ubbcluj.ro/~anisiu/IMC/2017seemous.pdf

for a file with solutions.