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

In summary, the conversation discusses a 2006 Math Olympiad question about assigning areas to sides of a convex polygon and proving that the sum of these assigned areas is at least twice the area of the polygon. Two proofs are provided, with one being a pdf document and the other being a link to another person's proof.
  • #1
lark
163
0
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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  • #2
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
 
Last edited by a moderator:

What is a beautiful geometry proof?

A beautiful geometry proof is a logical and elegant way of demonstrating the truth of a geometric statement. It involves using a series of steps and principles to explain and justify why a certain geometric relationship or theorem is true. A beautiful proof is not only mathematically correct, but also visually appealing and satisfying.

Why is it important to strive for beauty in a geometry proof?

Beauty in a geometry proof signifies clarity, simplicity, and efficiency. A beautiful proof is easy to understand and remember, making it useful for future applications. It also reflects the elegance and orderliness of mathematics, and can inspire further exploration and discovery.

What are some key elements of a beautiful geometry proof?

A beautiful geometry proof typically includes clear diagrams, concise and logical reasoning, and a variety of geometric principles and theorems. It may also involve creative and unexpected approaches, as well as a sense of symmetry, balance, and harmony.

How can one improve their skills in creating beautiful geometry proofs?

Practice, practice, practice! Just like any other skill, the ability to create beautiful geometry proofs takes time and effort to develop. Study and understand the fundamental principles of geometry, and try to approach problems from different angles. Seek feedback and learn from other proofs, and continually challenge yourself with more complex and interesting problems.

Can beauty be subjective in a geometry proof?

While beauty in a geometry proof is often associated with simplicity and elegance, it can also be subjective to some extent. Different mathematicians may have different preferences and opinions on what makes a proof beautiful. However, a truly beautiful proof should be able to stand the test of time and be appreciated by a wide range of audiences.

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