DK-GEORGMOHR 2003

In a right-angled triangle, the sum $a + b$ of the sides enclosing the right angle equals $24$ while the length of the altitude $h_c$ on the hypotenuse $c$ is $7$ . Determine the length of the hypotenuse.

Within the real numbers, solve the equation $$x^5 + \lfloor x \rfloor = 20$$ where $\lfloor x \rfloor$ denotes the largest whole number less than or equal to $x$ .

Determine the integers $n$ where $$|2n^2+9n+4|$$ is a prime number.

Georg and his mother love pizza. They buy a pizza shaped as an equilateral triangle. Georg demands to be allowed to divide the pizza by a straight cut and then make the first choice. The mother accepts this reluctantly, but she wants to choose a point of the pizza through which the cut must pass. Determine the largest fraction of the pizza which the mother is certain to get by this procedure.

For which natural numbers $n\ge 2$ can the numbers from $1$ to $16$ be lined up in a square scheme so that the four row sums and the four column sums are all mutually different and divisible by $n$ ?