Olympiad problems

1985 Tournament Of Towns, (094) 2

The radius $OM$ of a circle rotates uniformly at a rate of $360/n$ degrees per second , where $n$ is a positive integer . The initial radius is $OM_0$. After $1$ second the radius is $OM_1$ , after two more seconds (i.e. after three seconds altogether) the radius is $OM_2$ , after $3$ more seconds (after $6$ seconds altogether) the radius is $OM_3$, ..., after $n - 1$ more seconds its position is $OM_{n-1}$. For which values of $n$ do the points $M_0, M_1 , ..., M_{n-1}$ divide the circle into $n$ equal arcs? (a) Is it true that the powers of $2$ are such values? (b) Does there exist such a value which is not a power of $2$? (V. V. Proizvolov , Moscow)

radius geometry circles rotation arc combinatorial geometry speed number theory

2021 JBMO Shortlist, G2

Let $P$ be an interior point of the isosceles triangle $ABC$ with $\hat{A} = 90^{\circ}$. If $$\widehat{PAB} + \widehat{PBC} + \widehat{PCA} = 90^{\circ},$$ prove that $AP \perp BC$. Proposed by [i]Mehmet Akif Yıldız, Turkey[/i]

shortlist geometry arc

2005 Sharygin Geometry Olympiad, 9.3

Given a circle and points $A, B$ on it. Draw the set of midpoints of the segments, one of the ends of which lies on one of the arcs $AB$, and the other on the second.

midpoint geometry arc locus