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Proofs: Combinatorial!

Andrei marked \(4n\) points on the plane. He then connected with segments every pair of points which were \(1\) cm apart. He found that amid every \(n+1\) points there were at least two points connected by segments. Prove that Andrei drew at least \(7n\) segments.

Note by Andrei Golovanov
2 years, 2 months ago

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What exactly does this statement mean:

"He found that amid every \(n+1\) points there were exactly two points connected by segments" Raghav Vaidyanathan · 2 years, 2 months ago

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@Raghav Vaidyanathan I think the word exactly should be replaced by atleast. Sudeep Salgia · 2 years, 2 months ago

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@Sudeep Salgia Yes, that is correct. It should say "at least". Thanks for pointing it out. @Raghav Vaidyanathan @Sudeep Salgia Andrei Golovanov · 2 years, 2 months ago

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Comment deleted May 31, 2015

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@Andrei Golovanov But you just said that it is not possible with n=1. Vishnu C · 2 years, 1 month ago

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@Vishnu C The point is that with \(n=1 \), the conditions of the question cannot hold, and hence any given conclusion is true.

It's like "If \(a, b \) are 2 positive reals such that \( a^2 + b^2 = 0 \), then \( a+b = 0\)" is a true statement. Because under all cases where the condition holds (which is no case), the result is true. Note that even though \( a+b = 0 \) can never be a true statement under the restricted condition that (a,b) are 2 positive reals. However, by adding additional conditions, we can make it a statement about nothing, which is trivially true. Calvin Lin Staff · 2 years, 1 month ago

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@Calvin Lin That is, more or less, what the question asks. If this condition holds, he must have drawn \(7n\) segments. Prove why. Andrei Golovanov · 2 years, 1 month ago

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@Calvin Lin So you're basically saying that the condition will hold when pigs fly. Vishnu C · 2 years, 1 month ago

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@Vishnu C That's an amusing way to think. But wait! Swine flu! So pigs flew! So the condition is true! :) Vishnu C · 2 years, 1 month ago

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