DE-BUNDESWETTBEWERB 2021

A cube with side length $10$ is divided into two cuboids with integral side lengths by a straight cut. Afterwards, one of these two cuboids is divided into two cuboids with integral side lengths by another straight cut. What is the smallest possible volume of the largest of the three cuboids?

The fraction $\frac{3}{10}$ can be written as a sum of two reciprocals in exactly two ways: $$\frac{3}{10}=\frac{1}{5}+\frac{1}{10}=\frac{1}{4}+\frac{1}{20}$$ a) In how many ways can $\frac{3}{2021}$ be written as a sum of two reciprocals? b) Is there a positive integer $n$ not divisible by $3$ with the property that $\frac{3}{n}$ can be written as a sum of two reciprocals in exactly $2021$ ways?

Consider a triangle $ABC$ with $\angle ACB=120^\circ$ . Let $A', B', C'$ be the points of intersection of the angular bisector through $A$ , $B$ and $C$ with the opposite side, respectively. Determine $\angle A'C'B'$ .