Question 58
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 $ABC$ be a triangle and $P$ be a point. The isogonal conjugate of $P$ is the intersection of the reflection of line $AP$ over the $A$-angle bisector, the reflection of line $BP$ over the $B$-angle bisector, and the reflection of line $CP$ 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 $ABC$ be an acute triangle and $P$ be a point in its interior. Let $D, E, F$ lie on $BC, CA, AB$ respectively so that $PD$ bisects $\angle BPC$, $PE$ bisects $\angle CPA$, and $PF$ bisects $\angle APB$. Is it necessarily true that $AP + BP + CP \geq 2(PD + PE + PF)$?
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 $P_{2018} \gt P_{2017}$?
b. Let $ABC$ be a triangle and $P$ be a point. The isogonal conjugate of $P$ is the intersection of the reflection of line $AP$ over the $A$-angle bisector, the reflection of line $BP$ over the $B$-angle bisector, and the reflection of line $CP$ 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 $ABC$ be an acute triangle and $P$ be a point in its interior. Let $D, E, F$ lie on $BC, CA, AB$ respectively so that $PD$ bisects $\angle BPC$, $PE$ bisects $\angle CPA$, and $PF$ bisects $\angle APB$. Is it necessarily true that $AP + BP + CP \geq 2(PD + PE + PF)$?
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 $P_{2018} \gt P_{2017}$?