Simple Problem, Tedious Solution.

Photo by Joanna Kosinska on Unsplash

The problem:

Putnam and Beyond, #94

Looks simple, or maybe I shouldn’t say that. Of course, the goal is to minimize both values simultaneously, but I can’t imagine how they could be minimized below 0.

Okay, they obviously go below 0. First, we want to have both a and b being negative; then, we want their squares to have lesser magnitudes than their original values. Preferably, since we want to minimize both, it’s likely going to be a=b.

That occurs when a=b=-0.25, considering the lowest points of quadratic functions. But what if it isn’t? In particular, we haven’t rigorously proved that our intuition is optimal. Somehow we need to show that contradiction occurs otherwise.

Assume that

and

Then

which ultimately fails. We’re looking for contradiction, so, using squares to find minimum bounds?

Hmm, we’ve reached an awkward situation again. Perhaps we can use the fact that

I’ve made an idiotic move. We could’ve simply used

This achieves the desired argument.

After looking at the official solution, it is obviously much more elegant than my solution. One simple inequality is constructed, and easily resolves all difficulties:

Official Solution

I’m guessing that one could think of this solution (with ease) through the following chain-of-thought:

1. Intuitively, the minimum is found, and is confirmed by simple test cases
2. One considers a constructive argument, i.e., manipulating the expression based on the intuitive minimal value to form a seemingly nonnegative answer (and hope it’s indeed nonnegative)
3. We think of ways to prove this intuition by completing the square: use the properties of the maximum value operator, we attempt to find a lower bound for the entire expression
4. Conveniently, we’ll represent both variables in the lower bound; since twice the maximum is larger than the maximum plus the minimum (also considering that the maximum and minimum values have a nice symmetry), we multiply the entire expression by 2
5. Completing the square works, and we prove the final expression is nonnegative, implying that the intuitive minimum bound is indeed correct.

Or, skilled problem-solvers may immediately notice the overarching structure occurring in the expression, intuitively complete the square, done.

Now, I understand that the chain-of-thought described above may be difficult to understand; it’s a basic outline, and a more thorough explanation will be too long. I’ll add systematic explanations at the end of each series.

Finally, let’s see what went on in my solution.

1. First steps were easy; the lower bound was immediately identified
2. I directly proceeded to contradiction. This is because of two reasons: First of all, I did not think too much about the overarching structure of the expression, and proof by contradiction usually works for these “simple” problems. In addition, squaring expressions usually generates “additional terms”; I gave up intuitively thinking that squaring will generate unwanted terms of lower degree, forgetting how an integer term can be added to the expression.
3. Again, during the contradictory proof, I wasted effort attempting a variety of algebraic manipulations without careful thought. Squaring explicitly came to mind, but was not possible since the original expression could indeed obtain a negative value without an added term.
4. At the end, I accidentally noticed how my initial claim using the quadratic function would still work if maximum and minimum values were added, hence obtaining the solution.

And, it’s really that simple (definitely not)!