ÖMO 2020

Let $h$ be a semicircle with diameter $AB$ . Let $P$ be an arbitrary point inside the diameter $AB$ . The perpendicular through $P$ on $AB$ intersects $h$ at point $C$ . The line $PC$ divides the semicircular area into two parts. A circle will be inscribed in each of them that touches $AB, PC$ and $h$ . The points of contact of the two circles with $AB$ are denoted by $D$ and $E$ , where $D$ lies between $A$ and $P$ . Prove that the size of the angle $DCE$ does not depend on the choice of $P$ .

(Walther Janous)

The players Alfred and Bertrand put together a polynomial $x^n + a_{n-1}x^{n- 1} +... + a_0$ with the given degree $n \ge 2$ . To do this, they alternately choose the value in $n$ moves one coefficient each, whereby all coefficients must be integers and $a_0 \ne 0$ must apply. Alfred's starts first . Alfred wins if the polynomial has an integer zero at the end. (a) For which $n$ can Alfred force victory if the coefficients $a_j$ are from the right to the left, i.e. for $j = 0, 1,. . . , n - 1$ , be determined? (b) For which $n$ can Alfred force victory if the coefficients $a_j$ are from the left to the right, i.e. for $j = n -1, n - 2,. . . , 0$ , be determined?

(Theresia Eisenkölbl, Clemens Heuberger)