Olympiad problems

2003 Gheorghe Vranceanu, 1

Let $ M $ be a set of nonzero real numbers and $ f:M\longrightarrow M $ be a function having the property that the identity function is $ f+f^{-1} . $ [b]1)[/b] Prove that $ m\in M\iff -m\in M. $ [b]2)[/b] Show that $ f $ is odd. [b]3)[/b] Determine the cardinal of $ M. $

2000 Junior Balkan Team Selection Tests - Romania, 3

Tags: algebra , ceiling function , cardinal

Let be a real number $ a. $ For any real number $ p $ and natural number $ k, $ let be the set $$ A_k(p)=\{ px\in\mathbb{Z}\mid k=\lceil x \rceil \} . $$ Find all real numbers $ b $ such that $ \# A_n(a)=\# A_n(b) , $ for any natural number $ n. $ $ \# $ [i]denotes the cardinal.[/i] [i]Eugen Păltânea[/i]

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) . $