The grading phase of the German Federal Math Competition (Bundeswettbewerb Mathematik) has ended, and I am happy to have received a first price, i.e. to have solved all the problems correctly. In case you are interested, you may find my solutions in German here.
Problem 1. Find all quadrupels (a, b, c, d) of positive real numbers that satisfy the following two equations:
ab + cd=8, abcd = 8 + a + b + c + d.
Problem 2. On a table lie 2022 matches and a regular dice that has the number on top. Now Max and Moritz play the following game:
Alternately, they take away matches according to the following rule, where Max begins: The player to make a move rolls the dice over one of its edges and then takes a way as many matches as the top number shows. The player that cannot make legal move after some number of moves loses.
For which can Moritz force Max to lose?
Problem 3. In an acute triangle ABC with AC<BC, lines m_a and m_b are the perpendicular bisectors of sides BC and
AC, respectively. Further, let M_c be the midpoint of side AB. The Median CM_c intersects m_a in point S_a and m_b in point S_b; the lines AS_b und BS_a intersect in point K.
Prove: <ACM_c = <KCB.
Problem 4. Some points in the plane are either colored red or blue. The distance between two points of the opposite color is at most 1. Prove that there exists a circle with diameter sqrt(2) such that no two points outside of this circle have same color. It is enough to prove this claim for a finite number of colored points.
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