DK-GEORGMOHR 2019

Which positive integers satisfy that the sum of the number's last three digits added to the number itself yields $2029$ ?

Two distinct numbers a and b satisfy that the two equations $x^{2019} + ax + 2b = 0$ and $x^{2019}+ bx + 2a = 0$ have a common solution. Determine all possible values of $a + b$ .

Seven positive integers are written on a piece of paper. No matter which five numbers one chooses, each of the remaining two numbers divides the sum of the five chosen numbers. How many distinct numbers can there be among the seven?

Georg writes a positive integer $a$ on a blackboard. As long as there is a number on the blackboard, he does the following each day: $\bullet$ If the last digit in the number on the blackboard is less than or equal to $5$ , he erases that last digit. (If there is only this digit, the blackboard thus becomes empty.) $\bullet$ Otherwise he erases the entire number and writes $9$ times the number. Can Georg choose $a$ in such a way that the blackboard never becomes empty?

In the figure below the triangles $BCD, CAE$ and $ABF$ are equilateral, and the triangle $ABC$ is right-angled with $\angle A = 90^o$ . Prove that $|AD| = |EF|$ .