Home > Uncategorized > SEEMOUS 2013 (unofficial sources)

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}.

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