A combi salad from mathematical olympiad treasures

For discussing Olympiad Level Combinatorics problems
Thephysimatician
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A combi salad from mathematical olympiad treasures

Unread post by Thephysimatician » Tue Dec 22, 2020 10:08 pm

Let n and k be two natural numbers and let S be a set of n points such that
(a) no three points of S are collinear.
(b) for any point P of S there are at least k points of S which are equidistant from P.
Prove that k<1/2+(2n)^(1/2)

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Mehrab4226
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Location:Dhaka, Bangladesh

Re: A combi salad from mathematical olympiad treasures

Unread post by Mehrab4226 » Tue Dec 22, 2020 11:38 pm

Question LaTeXed:
Let $n$ and $k$ be two natural numbers and let $S$ be a set of n points such that
(a) no three points of $S$ are collinear.
(b) for any point $P$ of $S$ there are at least $k$ points of $S$ which are equidistant from $P$.
Prove that $k< \frac{1}{2}+(2n)^{\frac{1}{2}}$
Better:Prove that $k< \frac{1}{2}+\sqrt{2n}$
The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
-Henri Poincaré

tanmoy
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Location:Rangpur,Bangladesh

Re: A combi salad from mathematical olympiad treasures

Unread post by tanmoy » Wed Dec 23, 2020 12:13 am

"Questions we can't answer are far better than answers we can't question"

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