International Mathematics Competition for University Students

July 27 - Aug 2 2015, Blagoevgrad, Bulgaria

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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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    Day 1 questions
    Day 1 solutions
    Day 2 questions
    Day 2 solutions

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Problem 1

1. For any integer $n\ge 2$ and two $n\times n$ matrices with real entries $A,\; B$ that satisfy the equation $$A^{-1}+B^{-1}=(A+B)^{-1}\;$$ prove that $\det (A)=\det(B)$.

Does the same conclusion follow for matrices with complex entries?

Proposed by Zbigniew Skoczylas, Wroclaw University of Technology

Hint: Multiply by $(A+B)$. Notice that $(AB^{-1})^{-1}=BA^{-1}$.

  Solution