## Problems of SEEMOUS 2020

**Problem 1.** Consider such that

where is the *adjugate* matrix of , i.e., the matrix whose elements are where is the determinant obtained from , eliminating the line and the column .

Find the maximum number of matrices verifying the two equations above such that any two of them are not similar.

**Problem 2.** Let be a real number. Calculate:

**(a)** .

**(b)** .

**Problem 3.** Let be a positive integer, and such that and

Find the rank of .

**Problem 4.** Consider , and let be a -periodic and differentiable function which satisfies on and

**(a)** Prove that for every the equation has a unique solution in the interval denoted .

**(b)** Let and . Prove that and study the convergence of the series and .

*Source: https://seemous2020.auth.gr/*

## Putnam 2019 – Problem B2

## Putnam 2019 – Problem B5

**B4.** Let be the th Fibonacci number, defined by and for all . Let be the polynomial of degree such that for all . Find integers and such that .

## Putnam 2019 – Problem B4

**B4.** Let be the set of functions that are twice continuously differentiable for and that satisfy the following two equations (where scripts denote partial derivatives):

For each , let

Determine and show that it is independent of the choice of .

## Putnam 2019 A2

**A2.** In the triangle , let be the centroid, and let be the center of the inscribed circle. Let and be the angles at the vertices and , respectively. Suppose that the segment is parallel to and that . Find .

## Putnam 2019 B3

**B3.** Let be a real orthogonal matrix and let be a unit column vector (that is ). Let , where is the identity matrix. Show that if is not an eigenvalue of , then is an eigenvalue of .