2017 Guts #35: USAYNO Geometry
Source:
February 21, 2017
USAYNO
Problem Statement
Welcome to the USAYNO, where each question has a yes/no answer. Choose any subset of the following six problems to answer. If you answer problems and get them all correct, you will receive points. If any of them are wrong (or you leave them all blank), you will receive points.Your answer should be a six-character string containing 'Y' (for yes), 'N' (for no), or 'B' (for blank). For instance if you think 1, 2, and 6 are 'yes' and 3 and 4 are 'no', you should answer YYNNBY (and receive points if all five answers are correct, 0 points if any are wrong).(a) Does there exist a finite set of points, not all collinear, such that a line between any two points in the set passes through a third point in the set?(b) Let be a triangle and be a point. The isogonal conjugate of is the intersection of the reflection of line over the -angle bisector, the reflection of line over the -angle bisector, and the reflection of line over the -angle bisector. Clearly the incenter is its own isogonal conjugate. Does there exist another point that is its own isogonal conjugate?(c) Let be a convex figure in a plane, and let be the largest pentagon that can be inscribed in . Is it necessarily true that the area of is at least the area of ?(d) Is it possible to cut an equilateral triangle into pieces, and rearrange the pieces into a square?(e) Let be an acute triangle and be a point in its interior. Let lie on respectively so that bisects , bisects , and bisects . Is it necessarily true that ?(f) Let be the surface area of the -dimensional unit sphere, and let be the surface area of the -dimensional unit sphere. Is ?[color = red]The USAYNO disclaimer is only included in problem 33. I have included it here for convenience.