Difference between revisions of "2001 JBMO Problems/Problem 4"
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Let <math>N</math> be a convex polygon with 1415 vertices and perimeter 2001. Prove that we can find 3 vertices of N which form a triangle of area smaller than 1. | Let <math>N</math> be a convex polygon with 1415 vertices and perimeter 2001. Prove that we can find 3 vertices of N which form a triangle of area smaller than 1. | ||
− | == | + | ==Solution== |
The largest side has length at least <math>\frac{2001}{1415}</math>. Therefore, the sum of the other <math>1414</math> sides is <math>\frac{2001\cdot1414}{1415}</math>. Divide these sides into <math>707</math> pairs of adjacent sides, and there exist one pair of sides <math>a,b</math> such that <math>a+b \le \frac{2001\cdot1414}{1415\cdot707}=\frac{2001\cdot2}{1415}</math>. Blah blah blah... we know how to prove <math>\frac{2001\cdot2}{1415} < 2\sqrt2</math> except why would a problem want you to do that.... no idea. | The largest side has length at least <math>\frac{2001}{1415}</math>. Therefore, the sum of the other <math>1414</math> sides is <math>\frac{2001\cdot1414}{1415}</math>. Divide these sides into <math>707</math> pairs of adjacent sides, and there exist one pair of sides <math>a,b</math> such that <math>a+b \le \frac{2001\cdot1414}{1415\cdot707}=\frac{2001\cdot2}{1415}</math>. Blah blah blah... we know how to prove <math>\frac{2001\cdot2}{1415} < 2\sqrt2</math> except why would a problem want you to do that.... no idea. | ||
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~ [https://artofproblemsolving.com/community/user/61542 AwesomeToad] | ~ [https://artofproblemsolving.com/community/user/61542 AwesomeToad] | ||
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Revision as of 23:55, 8 January 2023
Contents
[hide]Problem
Let be a convex polygon with 1415 vertices and perimeter 2001. Prove that we can find 3 vertices of N which form a triangle of area smaller than 1.
Solution
The largest side has length at least . Therefore, the sum of the other sides is . Divide these sides into pairs of adjacent sides, and there exist one pair of sides such that . Blah blah blah... we know how to prove except why would a problem want you to do that.... no idea.
Well, then and the area of the triangle with sides and the angle between them has area as desired.
2001 JBMO (Problems • Resources) | ||
Preceded by Problem 2 |
Followed by Last Question | |
1 • 2 • 3 • 4 | ||
All JBMO Problems and Solutions |