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"Consider finitely many points in the plane such that, if we choose any three points A,B,C among them, the area of triangle ABC is always less than 1. Show that all of these points lie within the interior or on the boundary of a triangle with area less than 4."

I am unsure if I am just dumb but this makes no sense to me whatsoever. The question clearly states that we can choose ANY 3 points: A,B,C within some triangle (OR on the boundary of) with an area less than 4. So here is what makes no sense to me: lets say the triangle that has an area less than 4 is triangle XYZ. So ALL these points we can choose from "lie within the interior or on the boundary of triangle" XYZ. So if I choose ABC to be XYZ then we have a triangle with area less than 4 but potentially greater than 1. So doesn't the question kind of contradict itself? Because all the points lie within the triangle with area less than 4.

Daniel R
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Pookie
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  • No. The area of the triangle is less than 1 is given. – abc... Jun 18 '18 at 01:05
  • What the problem is claiming is that if you have a fixed, finite set $S$ of points in the plane such that for every $3$ distinct points of $S$, the area of the triangle with those $3$ vertices is less than $1$, then the entire set $S$ can be covered by some triangle with area less than $4$. – quasi Jun 18 '18 at 01:09
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    Reference: this is question B3 on the 2016 Putnam: https://kskedlaya.org/putnam-archive/2016.pdf – Bob Krueger Jun 18 '18 at 02:52

2 Answers2

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I think you're confusing an implication with its converse. The problem asks you to prove an implication concerning a finite set $S$ of points in the plane. The hypothesis in this implication is that, for any three points $A,B,C$ in $S$, the triangle $\triangle ABC$ has area less than $1$. I'll call this hypothesis $H$ for short. The conclusion you're supposed to deduce from this hypothesis is that the whole set $S$ lies within a triangle of area less than $4$. I'll call this conclusion $C$ for short. So you're supposed to prove the implication $H\implies C$.

You've shown (by taking $A,B,C$ very near the corners of a triangle of area $4$) that it's entirely possible for $C$ to be true while $H$ is false. That refutes the implication $C\implies H$. But so what; that's not the implication you were asked to prove.

Andreas Blass
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To rephrase the question:

Given: that any triangle formed from any $3$ points of some finit set $S$ has an area of less than $1$.

Show: that you can pick points $X$, $Y$, $Z$ such that

1) $S$ is inside of the triangle formed by $X$, $Y$, $Z$

2) the triangle formed by $X$, $Y$, $Z$ has area less than 4

Waleed Dahshan
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