AT-OEMO-REG 2020

Let $a$ be a positive integer. Determine all $a$ such that the equation $$\biggl( 1+\frac{1}{x} \biggr) \cdot \biggl( 1+\frac{1}{x+1} \biggr) \cdots \biggl( 1+\frac{1}{x+a} \biggr)=a-x$$ has at least one integer solution for $x$ . For every such $a$ state the respective solutions.

(Richard Henner)

The set $M$ consists of all $7$ -digit positive integer numbers that contain (in decimal notation) each of the digits $1, 3, 4, 6, 7, 8$ and $9$ exactly once. (a) Find the smallest positive difference $d$ of two numbers from $M$ . (b) How many pairs $(x, y)$ with $x$ and $y$ from M are there for which $x - y = d$ ?

(Gerhard Kirchner)

Let a triangle $ABC$ be given with $AB <AC$ . Let the inscribed center of the triangle be $I$ . The perpendicular bisector of side $BC$ intersects the angle bisector of $BAC$ at point $S$ and the angle bisector of $CBA$ at point $T$ . Prove that the points $C, I, S$ and $T$ lie on a circle.

(Karl Czakler)

Find all quadruples $(p, q, r, n)$ of prime numbers $p, q, r$ and positive integer numbers $n$ , such that $$p^2 = q^2 + r^n$$ (Walther Janous)