DK-GEORGMOHR 1999

In a coordinate system, a circle with radius $7$ and center is on the y-axis placed inside the parabola with equation $y = x^2$ , so that it just touches the parabola in two points. Determine the coordinate set for the center of the circle.

A fisherman has caught a number of fish. The three heaviest together make up $35\%$ of the total weight of the catch. He sells them. After that, the three lightest make up together $5/13$ of the weight of the rest. How many fish did he catch?

A function $f$ satisfies $$f(x)+xf(1-x)=x$$ for all real numbers $x$ . Determine the number $f (2)$ . Find $f$ .

Nanna and Sofie move in the same direction along two parallel paths, which are $200$ meters apart. Nanna's speed is $3$ meters per second, Sofie's only $1$ meter per second. A tall, cylindrical building with a diameter of $100$ meters is located in the middle between the two paths. Since the building first once the line of sight breaks between the girls, their distance between them is $200$ metres. How long will it be before the two girls see each other again?

Is there a number whose digits are only $1$ 's and which is divided by $1999$ ?