Math Olympiad Algebra Challenge | You should know this trick !!

Math Olympiad Algebra Problem

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[jeanmarcbonici]

There is a purely algebraic solution! Let s1=a+b+c and s2=a*b+b*c+a*c. You must first add the three equations and you get the equation 2 s1^2-3 s2=690. Secondly, you must take the difference two by two of the three equations, square and add member to member the three equations obtained and you obtain the second equation s1^4-6 s1^2 s2=133506. By eliminating s1 between these two equations you first get s2^2==23232 then s2=-88 Sqrt[3] or s2=88 Sqrt[3].

[mdzeauddin]

It is impossible to get an interior point O in a triangle ABC with sides 11, 13 and 20, such that all the three angles at O will measure 120 degrees!
It is possible, however, when triangle ABC is an equilateral triangle!

[piman]

This was an unfair challenge. The problem was clearly reverse-engineered by drawing three lines of differing lengths a, b, and c radiating out from a point, O, and then forming the triangle ABC with side lengths of 11, 13, and 20 units, with the angles at O all being 120 degrees. While this is definitely possible, it is not intuitive. Nevertheless, it made for a very interesting problem and solution.

[robertibatullin]

The solution is interesting but looks sophisticated and counter-intuitive. I guess there should be a simpler and purely algebraic one.

[WahranRai]

Who told you that a, b, c are sides of triangle ?

[DB-lgsq]

شكرا لكم
ارجو ان توضحوا من البداية طبيعة الاعداد a, b, c

[nlmnrrs]

Need to prove: O exists in the interior of triangle ABC.