MathDB

4

Part of 2024 Assara - South Russian Girl's MO

Problems(2)

Sides are more than diameter of incircle

Source: Assara 2024 J4

10/14/2024
Is there a described nn-gon in which each side is longer than the diameter of the inscribed circle a) at n=4n = 4? b) when n=7n = 7? c) when n=6n = 6? P.A.Kozhevnikov
geometrycombinatorial geometry
XY passes from fixed point; parabolas

Source: Assara 2024 S4

10/14/2024
A parabola pp is drawn on the coordinate plane — the graph of the equation y=−x2y =-x^2, and a point AA is marked that does not lie on the parabola pp. All possible parabolas qq of the form y=x2+ax+by = x^2+ax+b are drawn through point AA, intersecting pp at two points XX and YY . Prove that all possible XYXY lines pass through a fixed point in the plane. P.A.Kozhevnikov
geometryalgebraconicsparabola