EE-MO 2005

The height drawn on the hypotenuse of a right triangle divides the hypotenuse into two sections with a length ratio of $9: 1$ and two triangles of the starting triangle with a difference of areas of $48$ cm $^2$ . Find the original triangle sidelengths.

Let $a, b$ and $c$ be arbitrary integers. Prove that $a^2 + b^2 + c^2$ is divisible by $7$ when $a^4 + b^4 + c^4$ divisible by $7$ .

Rein solved a test on mathematics that consisted of questions on algebra, geometry and logic. After checking the results, it occurred that Rein had answered correctly $50\%$ of questions on algebra, $70\%$ of questions on geometry and $80\%$ of questions on logic. Thereby, Rein had answered correctly altogether $62\%$ of questions on algebra and logic, and altogether $74\%$ of questions on geometry and logic. What was the percentage of correctly answered questions throughout all the test by Rein?