AT-OEMO-BEG 2020

Let $a$ be a real number and $b$ a real number with $b\neq-1$ and $b\neq0.$ Find all pairs $(a, b)$ such that $$\frac{(1 + a)^2 }{1 + b}\leq 1 + \frac{a^2}{b}.$$ For which pairs (a, b) does equality apply? (Walther Janous)

How many positive five-digit integers are there that have the product of their five digits equal to $900$ ?

(Karl Czakler)

Given is an isosceles trapezoid $ABCD$ with $AB \parallel CD$ and $AB> CD$ . The projection from $D$ on $AB$ is $E$ . The midpoint of the diagonal $BD$ is $M$ . Prove that $EM$ is parallel to $AC$ .

(Karl Czakler)

Find all positive integers $a$ for which the equation $7an -3n! = 2020$ has a positive integer solution $n$ .

(Richard Henner)