Raymond’s Simple AM-GM inequality Trick
Featured problem for today is from Raymond John Diaz, another Math quizzer and school heartthrob. He is a graduate of Pedro Guevara Memorial National High School. Currently, he is taking up BS Applied Mathematics at University of the Philippines – Los Baňos (UP-LB).
He first submitted a problem about geometry but we both agreed to change the problem to a different one since that problem was quite vague. He says he wants to pursue Actuary-the highest paying professionals. Raymond is on his 4th level in algebra in brilliant.
Here are some of his achievements
– 2011, 2012 and 2014 MMC Regional Finals (team) 1st Runner up
-15th and 16th PMO Area Stage qualifier -Southern Tagalog
-Southern Tagalog Invitational Mathematical Challenge a.k.a Mathematch (Team 1st runner up)(individual Top scorer written phase)
-MCL Cup Math Wizards (team champion)
Here is the problem he wants to share about am-gm inequality.
First we need to multiply the two expressions at the left side of the equation
Then we need to group it as sum of their reciprocals and subtract 4 from both sides
Now we let
By the AM-GM inequality for , we have the lowest value of
Square both sides
Multiply both sides by
Therefore the lowest value of (i) + (ii) + (iii) + (iv) + (v) + (vi) equal to