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Omni-MATH / (a) Does there exist a finite set of points, not all collinear, such that a line…

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(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 ABCABC be a triangle and PP be a point. The isogonal conjugate of PP is the intersection of the reflection of line APAP over the AA-angle bisector, the reflection of line BPBP over the BB-angle bisector, and the reflection of line CPCP over the CC-angle bisector. Clearly the incenter is its own isogonal conjugate. Does there exist another point that is its own isogonal conjugate? (c) Let FF be a convex figure in a plane, and let PP be the largest pentagon that can be inscribed in FF. Is it necessarily true that the area of PP is at least 34\frac{3}{4} the area of FF? (d) Is it possible to cut an equilateral triangle into 2017 pieces, and rearrange the pieces into a square? (e) Let ABCABC be an acute triangle and PP be a point in its interior. Let D,E,FD, E, F lie on BC,CA,ABBC, CA, AB respectively so that PDPD bisects ∠BPC,PE\angle BPC, PE bisects ∠CPA\angle CPA, and PFPF bisects ∠APB\angle APB. Is it necessarily true that AP+BP+CP≥2(PD+PE+PF)AP+BP+CP \geq 2(PD+PE+PF)? (f) Let P2018P_{2018} be the surface area of the 2018-dimensional unit sphere, and let P2017P_{2017} be the surface area of the 2017-dimensional unit sphere. Is P2018>P2017P_{2018}>P_{2017}?
Plain-text mathematical notation (without MathML)
(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 (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 ∠BPC,PE bisects ∠CPA, and PF bisects ∠APB. Is it necessarily true that AP+BP+CP≥2(PD+PE+PF)? (f) Let P₂₀₁₈ be the surface area of the 2018-dimensional unit sphere, and let P₂₀₁₇ be the surface area of the 2017-dimensional unit sphere. Is P₂₀₁₈>P₂₀₁₇?
Original LaTeX notation
(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 $P_{2018}>P_{2017}$?

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Source and history

task_id
f5277071-be9c-51ed-aa20-cca98b075818
task_key
test--f5277071-be9c-51ed-aa20-cca98b075818
task_revision_id
1
config / split
/ test

Official source

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