Does there exist a set M ? (IMO SL 1987-P9)

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R+ Functional Equation

Mathdreams 4

N 14 minutes ago by Tony_stark0094

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(Andrew Brahms, USA)

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Let $ABC$ be a triangle and $I$ its incenter. The lines $BI$ and $CI$ intersect the circumcircle of $ABC$ again at $M$ and $N$ , respectively. Let $C_1$ and $C_2$ be the circumferences of diameters $NI$ and $MI$ , respectively. The circle $C_1$ intersects $AB$ at $P$ and $Q$ , and the circle $C_2$ intersects $AC$ at $R$ and $S$ . Show that $P$ , $Q$ , $R$ and $S$ are concyclic.

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Does there exist a set M ? (IMO SL 1987-P9)

Amir Hossein 3

N Aug 22, 2010 by mavropnevma

Does there exist a set $M$ in usual Euclidean space such that for every plane $\lambda$ the intersection $M \cap \lambda$ is finite and nonempty ?

Proposed by Hungary.

Remark

I'm not sure I'm posting this in a right Forum.

3 replies

Amir Hossein

Aug 19, 2010

mavropnevma

Aug 22, 2010

Does there exist a set M ? (IMO SL 1987-P9)

G H J

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Amir Hossein

5452 posts

Amir Hossein

#1 h Aug 19, 2010, 10:18 AM • 3 Y

Y by Adventure10 and 2 other users

Does there exist a set $M$ in usual Euclidean space such that for every plane $\lambda$ the intersection $M \cap \lambda$ is finite and nonempty ?

Proposed by Hungary.

Remark

I'm not sure I'm posting this in a right Forum.

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Jorge Miranda

157 posts

Jorge Miranda

#2 h Aug 22, 2010, 1:46 PM • 4 Y

Y by Adventure10 and 3 other users

Solution

Let $M=\{(t^5,t^3,t):t\in\mathbb{R}\}$ , and suppose $\lambda$ has $ax+by+cz+d=0$ as it's cartesian equation (with $a,b,c$ not all zero).

Then $M\cap \lambda$ is nonempty because $at^5+bt^3+ct+d=0$ always has a real solution (because a polynomial of odd degree always has a real root), and it's finite because $at^5+bt^3+ct+d=0$ has a finite number of solutions (because a non-constant polynomial has a finite number of roots).

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pco

23501 posts

pco

#3 h Aug 22, 2010, 1:50 PM • 1 Y

Y by Adventure10

Jorge Miranda wrote:

Solution

Quite quite nice !!!
Great congrats !

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mavropnevma

15142 posts

mavropnevma

#4 h Aug 22, 2010, 3:36 PM • 1 Y

Y by Adventure10

Notice how this is related to the famous moment curve $(t,t^2,t^3)$ , used in solving Crum's problem of infinitely many neighbourly convex polyhedra (Tietze, Besicovitch). But of course we need each coordinate of odd degree ...

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