OH NO! Shiningsunnyday has died! What are we going to do?!?

Shiningsunnyday may be gone... but his dream, a USAMO dream... must live on. And I shall carry the ember of his dream, and it shall ignite into a bonfire! For this USAMO dream must live on!

Cough, um anyway.

As some of you may know, Shiningsunnyday has appointed me as a contributor to his blog. So now let's begin with some math shall we?

First, let us introduce one of the most fundamentals of inequalities.

The Trivial Inequality:

for all real numbers .

Now obviously, an inequality problem never screams to us "Hey it's me! Look I'm an !", we have to use some cleverness and manipulations first in order to prove our inequalities.

A final note about inequality problems: When asked to prove that a certain inequality is true, you can think backwards but write your solution forwards. That is, you can go from said inequality and work your way backwards to a true statement, but just be sure that all your steps are reversible and that write your proof beginning with you true statement.

A Classic Problem:

Prove:

Motivation+Solution

Another Problem:

Prove (Source: Intermediate Algebra)

Solution

Hope y'all liked the problems!