2012 JBMO Problems/Problem 1
Let be positive real numbers such that . Prove that When does equality hold?
The LHS rearranges to . Since we have that . Therefore, the LHS rearranges again to .
Now, distribute the on the RHS into the parenthesis and multiply the LHS and RHS by 2 to get Let and similarly for and .
The inequality now simplifies to Note that because , , and are positive real numbers less than , , , and are always positive real numbers. Rearranging terms shows that this further simplifies to which equals By the trivial inequality we know that this is always true. Finally, we have equality when and Solving the equations yields that equality holds when
Solution by Someonenumber011 :)