Olympiad problems

2011 JBMO Shortlist, 4

In a group of $n$ people, each one had a different ball. They performed a sequence of swaps, in each swap, two people swapped the ball they had at that moment. Each pair of people performed at least one swap. In the end each person had the ball he/she had at the start. Find the least possible number of swaps, if: a) $n = 5$, b) $n = 6$.

2009 Oral Moscow Geometry Olympiad, 1

The figure shows a parallelogram and the point $P$ of intersection of its diagonals is marked. Draw a straight line through $P$ so that it breaks the parallelogram into two parts, from which you can fold a rhombus. [img]https://1.bp.blogspot.com/-Df2tIBthcmI/X2ZwIx3R4vI/AAAAAAAAMhQ/8Zkxfq30H8MSCdc66tm33n6jt-QKfGMowCLcBGAsYHQ/s0/2009%2Boral%2Bmoscow%2Bj1.png[/img]

construct geometry rhombus parallelogram

2020 Dürer Math Competition (First Round), P4

Suppose that you are given the foot of the altitude from vertex $A$ of a scalene triangle $ABC$, the midpoint of the arc with endpoints $B$ and $C$, not containing $A$ of the circumscribed circle of $ABC$, and also a third point $P$. Construct the triangle from these three points if $P$ is the a) orthocenter b) centroid c) incenter of the triangle.

construct geometry angle chasing construction homothety