This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 3

2022 IMC, 7
Tags: linear algebra , idempotence , rank
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 $\frac{n}{k}$.

2017 District Olympiad, 4
Tags: superior algebra , idempotence , ring theory
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.

2015 Romania National Olympiad, 1
Tags: superior algebra , idempotence , ring theory
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.