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Found problems: 3

2022 IMC, 7
Let $A_1, \ldots, A_k$ be $n\times n$ idempotent complex matrices such that $A_iA_j = -A_iA_j$ for all $1 \leq i < j \leq k$. Prove that at least one of the matrices has rank not exceeding $\frac{n}{k}$.
idempotence rank linear algebra

2015 Romania National Olympiad, 1
Let be a ring that has the property that all its elements are the product of two idempotent elements of it. Show that: [b]a)[/b] $ 1 $ is the only unit of this ring. [b]b)[/b] this ring is Boolean.
idempotence ring theory superior algebra

2017 District Olympiad, 4
Let $ A $ be a ring that is not a division ring, and such that any non-unit of it is idempotent. Show that: [b]a)[/b] $ \left( U(A) +A\setminus\left( U(A)\cup \{ 0\} \right) \right)\cap U(A) =\emptyset . $ [b]b)[/b] Every element of $ A $ is idempotent.
idempotence ring theory superior algebra