MathDB

4

Part of 2023 Durer Math Competition (First Round)

Problems(2)

5 points conyclic

Source: (2022 -) 2023 XVI Dürer Math Competition Regional E4

5/25/2024
Let kk be a circle with diameter ABAB and centre OO. Let C be an arbitrary point on the circle different from AA and BB. Let DD be the point for which OO, BB, DD and CC (in this order) are the four vertices of a parallelogram. Let EE be the intersection of the line BDBD and the circle kk, and let FF be the orthocenter of the triangle OACOAC. Prove that the points O,D,E,C,FO, D, E, C, F lie on a circle.
geometryConcyclic
ant travels inside a circular disc

Source: (2022 -) 2023 XVI Dürer Math Competition Regional E+4

5/25/2024
We are given an angle 0o<ϕ≤180o0^o < \phi \le 180^o and a circular disc. An ant begins its journey from an interior point of the disc, travelling in a straight line in a certain direction. When it reaches the edge of the disc, it does the following: it turns clockwise by the angle ϕ\phi , and if its new direction does not point towards the interior of the disc, it turns by the angle ϕ\phi again, and repeats this until it faces the interior. Then it continues its journey in this new direction and turns as before every time when it reaches the edge. For what values of ϕ\phi is it true that for any starting point and initial direction the ant eventually returns to its starting position?
combinatoricsgeometrycombinatorial geometry