2005 Canadian MO Problems/Problem 4: Difference between revisions
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==Problem== | ==Problem== | ||
Let <math>ABC</math> be a triangle with circumradius <math>R</math>, perimeter <math>P</math> and area <math>K</math>. Determine the maximum value of <math>KP/R^3</math>. | Let <math>ABC</math> be a triangle with circumradius <math>R</math>, perimeter <math>P</math> and area <math>K</math>. Determine the maximum value of <math>KP/R^3</math>. | ||
==Solution== | ==Solution Outline== | ||
Use the formula <math>K=\dfrac{abc}{4R}</math> to get <math>KP/R^3=\dfrac{abc(a+b+c)}{4R^4}</math>. Then use the extended sine law to get something in terms of sines, and use AM-GM and Jensen's to finish. (Jensen's is used for <math>\sin A+\sin B+\sin C \le \dfrac{3\sqrt3}{2}</math>. | |||
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Latest revision as of 07:13, 1 November 2022
Problem
Let 
be a triangle with circumradius 
, perimeter 
and area 
. Determine the maximum value of 
.
Solution Outline
Use the formula 
to get 
. Then use the extended sine law to get something in terms of sines, and use AM-GM and Jensen's to finish. (Jensen's is used for 
.
