Olympiad problems

2004 Alexandru Myller, 3

Prove that the number of nilpotent elements of a commutative ring with an order greater than $ 8 $ and congruent to $ 3 $ modulo $ 6 $ is at most a third of the order of the ring.

2010 Laurențiu Panaitopol, Tulcea, 2

Tags: freshman s dream , frobenius , nilpotence

Let be two $ n\times n $ complex matrices $ A,B $ satisfying the equations $ (A+B)^2=A^2+B^2 $ and $ (A+B)^4=A^4+B^4. $ Show that $ (AB)^2=0. $

2015 District Olympiad, 4

Tags: superior algebra , ring theory , nilpotence , abstract algebra

Let $ m $ be a non-negative ineger, $ n\ge 2 $ be a natural number, $ A $ be a ring which has exactly $ n $ elements, and an element $ a $ of $ A $ such that $ 1-a^k $ is invertible, for all $ k\in\{ m+1,m+2,...,m+n-1\} . $ Prove that $ a $ is nilpotent.

2011 District Olympiad, 4

Tags: abstract algebra , ring theory , nilpotence , cardinal , superior algebra , group theory

Let be a ring $ A. $ Denote with $ N(A) $ the subset of all nilpotent elements of $ A, $ with $ Z(A) $ the center of $ A, $ and with $ U(A) $ the units of $ A. $ Prove: [b]a)[/b] $ Z(A)=A\implies N(A)+U(A)=U(A) . $ [b]b)[/b] $ \text{card} (A)\in\mathbb{N}\wedge a+U(A)\subset U(A)\implies a\in N(A) . $

2011 Laurențiu Duican, 2

Tags: nilpotence , linear algebra

Let be a field $ \mathbb{F} $ and two nonzero nilpotent matrices $ M,N\in\mathcal{M}_2\left( \mathbb{F} \right) $ that commute. Show that: [b]a)[/b] $ MN=0 $ [b]b)[/b] there exists a nonzero element $ f\in\mathbb{F} $ such that $ M=fN $ [i]Dorel Miheț[/i]