HMMT 二月 2017 · 冲刺赛 · 第 35 题

HMMT February 2017 — Guts Round — Problem 35

[ 20 ] (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 3 F . Is it necessarily true that the area of P is at least the area of F ? 4 (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 re- spectively so that P D bisects ∠ BP C , P E bisects ∠ CP A , and P F bisects ∠ AP B . Is it necessarily true that AP + BP + CP ≥ 2( P D + P E + P F )? (f) Let P be the surface area of the 2018-dimensional unit sphere, and let P be the surface 2018 2017 area of the 2017-dimensional unit sphere. Is P > P ? 2018 2017

解析

[ 20 ] (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 3 F . Is it necessarily true that the area of P is at least the area of F ? 4 (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 re- spectively so that P D bisects ∠ BP C , P E bisects ∠ CP A , and P F bisects ∠ AP B . Is it necessarily true that AP + BP + CP ≥ 2( P D + P E + P F )? (f) Let P be the surface area of the 2018-dimensional unit sphere, and let P be the surface 2018 2017 area of the 2017-dimensional unit sphere. Is P > P ? 2018 2017 Proposed by: Alexander Katz Answer: NYYYYN