Olympiad problems

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2003 SNSB Admission, 4

Prove that the sets $$ \{ \left( x_1,x_2,x_3,x_4 \right)\in\mathbb{R}^4|x_1^2+x_2^2+x_3^2=x_4^2 \} , $$ $$ \{ \left( x_1,x_2,x_3,x_4 \right)\in\mathbb{R}^4|x_1^2+x_2^2=x_3^2+x_4^2 \}, $$ are not homeomorphic on the Euclidean topology induced on them.

2006 Miklós Schweitzer, 9

Tags: topology , homeomorphism , absolute continuity

Does the circle T = R / Z have a self-homeomorphism $\phi$ that is singular (that is, its derivative is almost everywhere 0), but the mapping $f:T \to T$ , $f(x) = \phi^{-1} (2\phi(x))$ is absolutely continuous?

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