Olympiad problems

2003 Romania National Olympiad, 1

[b]a)[/b] Determine the center of the ring of square matrices of a certain dimensions with elements in a given field, and prove that it is isomorphic with the given field. [b]b)[/b] Prove that $$ \left(\mathcal{M}_n\left( \mathbb{R} \right) ,+, \cdot\right)\not\cong \left(\mathcal{M}_n\left( \mathbb{C} \right) ,+,\cdot\right) , $$ for any natural number $ n\ge 2. $ [i]Marian Andronache, Ion Sava[/i]

2018 Brazil Undergrad MO, 7

Tags: isomorphism , superior algebra , graph theory

Unless of isomorphisms, how many simple four-vertex graphs are there?

2018 IMC, 2

Tags: linear algebra , isomorphism

Does there exist a field such that its multiplicative group is isomorphism to its additive group? [i]Proposed by Alexandre Chapovalov, New York University, Abu Dhabi[/i]

2006 Cezar Ivănescu, 2

Tags: group theory , abstract algebra , matrices , isomorphism , group morphism

Prove that the set $ \left\{ \left. \begin{pmatrix} \frac{1-2x^3}{3x^2} & \frac{1+x^3}{3x^2} & \frac{1+x^3}{3x^2} \\ \frac{1+x^3}{3x^2} & \frac{1-2x^3}{3x^2} & \frac{1+x^3}{3x^2} \\ \frac{1+x^3}{3x^2} & \frac{1+x^3}{3x^2} & \frac{1-2x^3}{3x^2}\end{pmatrix}\right| x\in\mathbb{R}^{*} \right\} $ along with the usual multiplication of matrices form a group, determine an isomorphism between this group and the group of multiplicative real numbers.