Source: Serbian selection contest for the IMO 2025
Determine the smallest positive real number such that there exists a sequence of positive real numbers ,, with the property that for every it holds that: Proposed by Pavle Martinović
Hint Needed
Let be a triangle and let and be points on the sides and , respectively , such that is parallel to . Let be any point interior to triangle , and let and be the intersections of with the lines and , respectively. Let be the second intersection points of the circumcircles of triangles and . Prove that the points are collinear .
Clifford's chain of circles, concurrent Simson lines
kosmonauten31140
an hour ago
Source: My own, but most likely already known
Let ,,, be circles having a common point . Denote by the intersection point, other than , of and (). Let be the common point of the 4 circles ,,,.
Let be the Simson line of with respect to . Define ,, cyclically.
Let be the Simson line of with respect to . Define ,, cyclically.
Prove that if ,,, are concurrent, then, ,,, are also concurrent.
In a scalene triangle , is the point of tangency of the incircle with the side . Points and are the intersections of the angle bisectors of and with the circumcircle of , respectively. Let be the antipodal point of in the circumcircle of , and let be the antipodal point of in the circumcircle of .
Prove that triangles and are similar.