DK-GEORGMOHR 2005

This figure is cut out from a sheet of paper. Folding the sides upwards along the dashed lines, one gets a (non-equilateral) pyramid with a square base. Calculate the area of the base.

Determine, for any positive real number $a$ , the number of solutions $(x,y)$ to the system of equations $$\begin{cases} |x|+|y|= 1 x^2 + y^2 = a \end{cases}$$ where $x$ and $y$ are real numbers.

The point $P$ lies inside $\vartriangle ABC$ so that $\vartriangle BPC$ is isosceles, and angle $P$ is a right angle. Furthermore both $\vartriangle BAN$ and $\vartriangle CAM$ are isosceles with a right angle at $A$ , and both are outside $\vartriangle ABC$ . Show that $\vartriangle MNP$ is isosceles and right-angled.

Fourteen students each write an integer number on the board. When they later meet their math teacher Homer Grog, they tell him that no matter what number they erased on the board, then the remaining numbers could be divided into three groups at once sum. They also tell him that the numbers on the board were integer numbers. Is it now possible for Homer Grog to determine what numbers the students wrote on the board?