SEEMOUS 2013

Find all continuous functions $f:[1,8]\to\mathbb R$ , such that $$\int^2_1f(t^3)^2dt+2\int^2_1f(t^3)dt=\frac23\int^8_1f(t)dt-\int^2_1(t^2-1)^2dt.$$

Let $M,N\in M_2(\mathbb C)$ be two nonzero matrices such that $$M^2=N^2=0_2\text{ and }MN+NM=I_2$$ where $0_2$ is the $2\times2$ zero matrix and $I_2$ the $2\times2$ unit matrix. Prove that there is an invertible matrix $A\in M_2(\mathbb C)$ such that $$M=A\begin{pmatrix}0&10&0\end{pmatrix}A^{-1}\text{ and }N=A\begin{pmatrix}0&01&0\end{pmatrix}A^{-1}.$$

Find the maximum value of $$\int^1_0|f'(x)|^2|f(x)|\frac1{\sqrt x}dx$$ over all continuously differentiable functions $f:[0,1]\to\mathbb R$ with $f(0)=0$ and $$\int^1_0|f'(x)|^2dx\le1.$$

Let $A\in M_2(\mathbb Q)$ such that there is $n\in\mathbb N,n\ne0$ , with $A^n=-I_2$ . Prove that either $A^2=-I_2$ or $A^3=-I_2$ .