Olympiad problems

2022 Thailand TSTST, 1

Let $ABCD$ be a parallelogram with $AC=BC.$ A point $P$ is chosen on the extension of ray $AB$ past $B.$ The circumcircle of $ACD$ meets the segment $PD$ again at $Q.$ The circumcircle of triangle $APQ$ meets the segment $PC$ at $R.$ Prove that lines $CD,AQ,BR$ are concurrent.

2024 Brazil Team Selection Test, 4

Tags: geometry , AZE IMO TST , IMO Shortlist , TST

Let $ABCD$ be a cyclic quadrilateral with $\angle BAD < \angle ADC$. Let $M$ be the midpoint of the arc $CD$ not containing $A$. Suppose there is a point $P$ inside $ABCD$ such that $\angle ADB = \angle CPD$ and $\angle ADP = \angle PCB$. Prove that lines $AD, PM$, and $BC$ are concurrent.

2024 Azerbaijan IMO TST, 1

Tags: number theory , IMO Shortlist , AZE IMO TST , TST

Determine all ordered pairs $(a,p)$ of positive integers, with $p$ prime, such that $p^a+a^4$ is a perfect square. [i]Proposed by Tahjib Hossain Khan, Bangladesh[/i]

2022 Taiwan TST Round 1, G

Tags: geometry , IMO Shortlist , AZE IMO TST , Hi

Let $ABCD$ be a parallelogram with $AC=BC.$ A point $P$ is chosen on the extension of ray $AB$ past $B.$ The circumcircle of $ACD$ meets the segment $PD$ again at $Q.$ The circumcircle of triangle $APQ$ meets the segment $PC$ at $R.$ Prove that lines $CD,AQ,BR$ are concurrent.