2006 Iran MO (3rd Round)

Part of Iran MO (3rd Round)

Subcontests

(8)

1

Kearting [Dividing plane to convex sets]

We have finite number of distinct shapes in plane. A "convex Kearting" of these shapes is covering plane with convex sets, that each set consists exactly one of the shapes, and sets intersect at most in border. http://aycu30.webshots.com/image/4109/2003791140004582959_th.jpg In which case Convex kearting is possible? 1) Finite distinct points 2) Finite distinct segments 3) Finite distinct circles

1

Embedding in Q^n

We mean a traingle in

\mathbb Q^{n}

, 3 points that are not collinear in

\mathbb Q^{n}

a) Suppose that

ABC

\mathbb Q^{n}

. Prove that there is a triangle

A'B'C'

\mathbb Q^{5}

\angle B'A'C'=\angle BAC

. b) Find a natural

m

that for each traingle that can be embedded in

\mathbb Q^{n}

it can be embedded in

\mathbb Q^{m}

. c) Find a triangle that can be embedded in

\mathbb Q^{n}

and no triangle similar to it can be embedded in

\mathbb Q^{3}

. d) Find a natural

m'

that for each traingle that can be embedded in

\mathbb Q^{n}

then there is a triangle similar to it, that can be embedded in

\mathbb Q^{m}

. You must prove the problem for

m=9

m'=6

to get complete mark. (Better results leads to additional mark.)

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