HMMT 2017 U35

(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 $A B C$ be a triangle and $P$ be a point. The isogonal conjugate of $P$ is the intersection of the reflection of line $A P$ over the $A$-angle bisector, the reflection of line $B P$ over the $B$-angle bisector, and the reflection of line $C P$ over the $C$-angle bisector. Clearly the incenter is its own isogonal conjugate. Does there exist another point that is its own isogonal conjugate? (c) Let $F$ be a convex figure in a plane, and let $P$ be the largest pentagon that can be inscribed in $F$. Is it necessarily true that the area of $P$ is at least $\frac{3}{4}$ the area of $F$ ? (d) Is it possible to cut an equilateral triangle into 2017 pieces, and rearrange the pieces into a square? (e) Let $A B C$ be an acute triangle and $P$ be a point in its interior. Let $D, E, F$ lie on $B C, C A, A B$ respectively so that $P D$ bisects $\angle B P C, P E$ bisects $\angle C P A$, and $P F$ bisects $\angle A P B$. Is it necessarily true that $A P+B P+C P \geq 2(P D+P E+P F)$ ? (f) Let $P_{2018}$ be the surface area of the 2018-dimensional unit sphere, and let $P_{2017}$ be the surface area of the 2017-dimensional unit sphere. Is $P_{2018}>P_{2017}$ ?