International Mathematics Competition for University Students

July 22 – 28 2018, Blagoevgrad, Bulgaria

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Ivan is Watching You
Ivan's Office

Day 1
    Problem 1
    Problem 2
    Problem 3
    Problem 4
    Problem 5

Day 2
    Problem 6
    Problem 7
    Problem 8
    Problem 9
    Problem 10

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    Leaders

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    Day 1 questions
    Day 1 solutions
    Day 2 questions
    Day 2 solutions
    Closing Ceremony
        Presentation

Official IMC site

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 Daniƫl Kroes, University of California, San Diego)

  Hint    Solution