DK-GEORGMOHR 2017

A system of equations $$\begin{cases} x^2 \, ? \, z^2 = -8 y^2 \, ? \, z^2 = 7 \end{cases}$$ is written on a piece of paper, but unfortunately two of the symbols are a little blurred. However, it is known that the system has at least one solution, and that each of the two question marks stands for either $+$ or $-$ . What are the two symbols?

Georg has a board displaying the numbers from $1$ to $50$ . Georg may strike out a number if it can be formed by starting with the number $2$ and doing one or more calculations where he either multiplies by $10$ or subtracts $3$ . Which of the board's numbers may Georg strike out? Example: Georg may strike out $26$ because it may, for example, be formed by starting with $2$ , multiplying by $10$ , subtracting $3$ three times, multiplying by $10$ and subtracting $3$ twenty-eight times.

The figure shows an arc $\ell$ on the unit circle and two regions $A$ and $B$ . Prove that the area of $A$ plus the area of $B$ equals the length of $\ell$ .

Let $A, B, C$ and $D$ denote the digits in a four-digit number $n = ABCD$ . Determine the least $n$ greater than $2017$ satisfying that there exists an integer $x$ such that $$x =\sqrt{A +\sqrt{B +\sqrt{C +\sqrt{D + x}}}}.$$

In a chess tournament, each pair of players play one game. A lost game yields 0 points, a won game yields 1 point and a tied game yields $\frac12$ point. After the tournament, it turns out that in each group of three players, at least one got $1 \frac12$ points in the games against the two others. What is the largest number of players that may have participated?