Olympiad problems

1988 Tournament Of Towns, (195) 2

Let $N$ be the orthocentre of triangle $ABC$ (i .e. the point where the altitudes meet). Prove that the circumscribed circles of triangles $ABN, ACN$ and $BCN$ each have equal radius.

2007 Sharygin Geometry Olympiad, 17

What triangles can be cut into three triangles having equal radii of circumcircles?

circumcircle geometry circumradius

2017 Argentina National Math Olympiad Level 2, 5

Let $ABCD$ be a convex quadrilateral with $AB = BD = 8$ and $CD = DA = 6$. Let $P$ be a point on side $AB$ such that $DP$ is bisector of angle $\angle ADB$ and let $Q$ be a point on side $BC$ such that $DQ$ is bisector of angle $\angle CDB$. Calculate the radius of the circumcircle of triangle $DPQ$. Note: The circumcircle of a triangle is the circle that passes through its three vertices.

geometry equal segments angle bisector circumradius

2007 Swedish Mathematical Competition, 3

Let $\alpha$, $\beta$, $\gamma$ be the angles of a triangle. If $a$, $b$, $c$ are the side length of the triangle and $R$ is the circumradius, show that \[ \cot \alpha + \cot \beta +\cot \gamma =\frac{R\left(a^2+b^2+c^2\right)}{abc} \]

trigonometry geometry circumradius