problem solving: A Tricky Inequality
Sometimes, all you need to resolve the most difficult problems is to simply calm down, step back, and change your perspectives.
Introductory puzzle: what inequality does this graph represent?
Today’s problem is tricky, but not difficult. Don’t overthink! Also, do note that the Cauchy-Schwarz inequality is involved:
Now, time for the problem.
A Full Solution:
The equality is homogeneous, meaning that all of the terms have equal degree (after expanding). We have product of sums on the right and a square of products (3abc) on the left, inspiring the use of C-S inequality.
The problem is, C-S requires a product of sum of squares; while we can “construct” squares on the right hand side, the current arrangement leads to highly mixed terms which differs from the organized product on the left side.
Now comes the main part. We shift the order of terms so that they’re less “mixed” when we put them inside the C-S inequality.
This gives us:
which allows for a direct application of C-S:
Done!
My Initial Attempt:
I failed to shift my perspectives. The results were, well, catastrophic.
I immediately saw squares on the left and cubes on the right. Two thoughts came to mind:
- “Producing” another term; basically, synthetically adding/multiplying something on both sides, simultaneously.
- Finding and making use of existing patterns
I chose the first option, which was sensible, but wasn’t carried out in a sensible way. Inspired by an example problem, I tried squaring:
I should’ve known that squaring generates more unnecessary terms, getting in the way of what we want to achieve. After realizing my mistake, I attempted to brute-force my way through the original inequality:
Unfortunately, it implied I would have to prove this:
Oh! The exponent of each term inside the square root is even. At this point, the answer was immediate:
I found it to be quite surprising and enjoyable. Hopefully, you also think this is a nice problem!
And that’s it! It’s that simple.
