Olympiad problems

2013 Middle European Mathematical Olympiad, 3

Let $ABC$ be an isosceles triangle with $AC=BC$. Let $N$ be a point inside the triangle such that $2 \angle ANB = 180 ^\circ + \angle ACB $. Let $ D $ be the intersection of the line $BN$ and the line parallel to $AN$ that passes through $C$. Let $P$ be the intersection of the angle bisectors of the angles $CAN$ and $ABN$. Show that the lines $DP$ and $AN$ are perpendicular.

2023 India EGMO TST, P6

Tags: geometry , Isosceles Triangle , anglechasing

Let $ABC$ be an isosceles triangle with $AB = AC$. Suppose $P,Q,R$ are points on segments $AC, AB, BC$ respectively such that $AP = QB$, $\angle PBC = 90^\circ - \angle BAC$ and $RP = RQ$. Let $O_1, O_2$ be the circumcenters of $\triangle APQ$ and $\triangle CRP$. Prove that $BR = O_1O_2$. [i]Proposed by Atul Shatavart Nadig[/i]

India EGMO 2023 TST, 6

Tags: geometry , Isosceles Triangle , anglechasing

Let $ABC$ be an isosceles triangle with $AB = AC$. Suppose $P,Q,R$ are points on segments $AC, AB, BC$ respectively such that $AP = QB$, $\angle PBC = 90^\circ - \angle BAC$ and $RP = RQ$. Let $O_1, O_2$ be the circumcenters of $\triangle APQ$ and $\triangle CRP$. Prove that $BR = O_1O_2$. [i]Proposed by Atul Shatavart Nadig[/i]