MathDB

Let

\Delta

be the set of all triangles inscribed in a given circle, with angles whose measures are integer numbers of degrees different than

45^\circ,90^\circ

and

135^\circ

. For each triangle

T\in\Delta

f(T)

denotes the triangle with vertices at the second intersection points of the altitudes of

T

with the circle.
(a) Prove that there exists a natural number

n

such that for every triangle

T\in\Delta

, among the triangles

T,f(T),\ldots,f^n(T)

(where

f^0(T)=T

and

f^k(T)=f(f^{k-1}(T))

) at least two are equal. (b) Find the smallest

n

with the property from (a).

Problem Statement