Johnny once saw plums hanging, like eggs so big and numbered according to the first natural numbers. He is the first to pick the plum with number $2$. After that, Jantje picks the plum each time with the smallest number $n$ that satisfies the following two conditions: $\bullet$ $n$ is greater than all numbers on the already picked plums, $\bullet$ $n$ is not the product of two equal or different numbers on already picked plums. We call the numbers on the picked plums plum numbers. Is $100 000$ a plum number? Justify your answer.

Catherine lowers five matching wooden discs over bars placed on the vertices of a regular pentagon. Then she leaves five smaller congruent checkers these rods drop. Then she stretches a ribbon around the large discs and a second ribbon around the small discs. The first ribbon has a length of $56$ centimeters and the second one of $50$ centimeters. Catherine looks at her construction from above and sees an area demarcated by the two ribbons. What is the area of that area?

There are $19$ balls in a box, numbered $1$ through $19$. When we go out get that box without looking five different balls, which number has the largest probability of being the difference between the highest and lowest number drawn? Justify you reply .

(a) Prove that for every $x \in R$ holds that $$-1 \le \frac{x}{x^2 + x + 1} \le \frac 13$$ (b) Determine all functions $f : R \to R$ for which for every $x \in R$ holds that $$f \left( \frac{x}{x^2 + x + 1} \right) = \frac{x^2}{x^4 + x^2 + 1}$$