What is the elegant proof of this 2006 Math Olympiad question?

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SUMMARY

The discussion centers on a 2006 Math Olympiad problem regarding convex polygons, specifically proving that the sum of the maximum triangle areas assigned to each side of a polygon P is at least twice the area of P. Two elegant proofs are provided in the linked PDF documents, with the first proof being collaboratively completed by a forum member. The links to these proofs are crucial for understanding the methodologies used in the proof.

PREREQUISITES
  • Understanding of convex polygons and their properties
  • Familiarity with geometric proofs and area calculations
  • Basic knowledge of mathematical Olympiad problems
  • Ability to read and interpret mathematical PDFs
NEXT STEPS
  • Review the 2006 Math Olympiad problem statement and its requirements
  • Study the provided proofs in the linked PDFs for detailed methodologies
  • Explore additional geometric proofs related to convex polygons
  • Investigate other Math Olympiad problems for similar proof techniques
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Mathematics enthusiasts, competitive math students, educators, and anyone interested in advanced geometric proofs and Olympiad-level problem-solving strategies.

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Assign to each side b of a convex polygon P the maximum area of a triangle that has b as a side and is contained in P. Show that the sum of the areas assigned to the sides of P is at least twice the area of P.

This was a 2006 Math olympiad question. A beautiful proof is at http://camoo.freeshell.org/IMO_2006_6.pdf" . Two proofs there, actually.

The person who helped me complete the first proof made a pdf of his contribution, it's at http://artofproblemsolving.com/Forum/download/file.php?id=27098"

Laura
 
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I apparently can't edit that posting any more. But the link to the pdf with the other person's proof has changed, to http://www.artofproblemsolving.com/Forum/download/file.php?id=27177"

Laura
 
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