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## SEEMOUS 2013 (unofficial sources)

Here are some of the problems of SEEMOUS 2013.

1. Let ${f:[1,8] \rightarrow \Bbb{R}}$ be a continuous mapping, such that

$\displaystyle \int_1^2 f^2(t^3)dt+2\int_1^2f(t^3)dt=\frac{2}{3}\int_1^8 f(t)dt-\int_1^2 (t^2-1)^2 dt.$

Find the form of the map ${f}$.

2. Let ${M,N \in \mathcal{M}_2(\Bbb{C})}$ be nonzero matrices such that ${M^2=N^2=0}$ and ${MN+NM=I_2}$. Prove that there is an invertible matrix ${A \in \mathcal{M}_2(\Bbb{C})}$ such that ${M=A\begin{pmatrix} 0&1\&0\end{pmatrix}A^{-1}}$ and ${N=A\begin{pmatrix} 0&0 \\ 1&0\end{pmatrix}A^{-1}}$.

3. Let ${A \in \mathcal{M}_2(\Bbb{Q})}$ such that there is ${n \in \Bbb{N},\ n\neq 0}$, with ${A^n=-I_2}$. Prove that either ${A^2=-I_2}$ or ${A^3=-I_2}$.