OMJ 2019

Let $a$, $b$ be the positive integers greater than $1$. Prove that if $$\frac{a}{b}, \frac{a-1}{b-1}$$ differ by 1, then both are integers.

Let $ABCD$ be the isosceles trapezium with bases $AB$ and $CD$, such that $AC = BC$. The point $M$ is the midpoint of side $AD$. Prove that $\sphericalangle ACM = \sphericalangle CBD$.

The point $D$ lies on the side $AB$ of the triangle $ABC$. Assume that there exists such a point $E$ on the side $CD$, that $$\sphericalangle EAD = \sphericalangle AED \text{and} \sphericalangle ECB = \sphericalangle CEB.$$ Show that $AC + BC > AB + CE$.