# 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) 1^{st} 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.

**Problem:**

Let

Show that

**Solution:**

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

(i)

(ii)

(iii)

(iv)

(v)

(vi)

By the AM-GM inequality for , we have the lowest value of

Proof:

Square both sides

Multiply both sides by

Therefore the lowest value of (i) + (ii) + (iii) + (iv) + (v) + (vi) equal to