### International Mathematics Competition for University Students

July 22 – 28 2018, Blagoevgrad, Bulgaria

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### Problem 3

Problem 3. Determine all rational numbers $\displaystyle a$ for which the matrix

$\displaystyle \begin{pmatrix} a & -a & -1 & 0 \\ a & -a & 0 & -1 \\ 1 & 0 & a & -a \\ 0 & 1 & a & -a \end{pmatrix}$

is the square of a matrix with all rational entries.

(Proposed by Daniël Kroes, University of California, San Diego)

Hint: Let $\displaystyle A = \begin{pmatrix} a & -a & -1 & 0 \\ a & -a & 0 & -1 \\ 1 & 0 & a & -a \\ 0 & 1 & a & -a \end{pmatrix}$ and suppose that $\displaystyle A=B^2$. What can be the minimal polynomial of $\displaystyle B$?