An ARML Experience

Personal Anecdote time: When I was a senior in high school I was on the Chicago Area ARML B team. Back then they only had an  A and B team. That means, among the elite Chicago area problems solvers, I was a pretty good, but by no means the best. Tryouts were based strictly on accumulated points and there was no untimed testing or other accommodations. If you didn't make the cut, you were out, like any other real sport. Fast forward many years. I was a math teacher at an elite high school and one of the coaches for the school's math team. Coaches brought their students to ARML tryouts, and then, for amusement, worked on the problems the kids were doing themselves. I think we all wanted to see if we still had it. Also, alas, Homo Sapiens has the unfortunate tendency of never missing an opportunity to establish a pecking order it seems.  Anyway, I don't remember the exact problem, but a problem very similar to the following was flashed on the overhead. 

"What is the minimum value of (2x^2)/y + (3y)/x^2 in the first quadrant of the Cartesian plane? 

My colleague, and ostensibly a friendly acquaintance, was sitting next to me. I answered the question within 5 seconds without writing anything except the answer on a sheet of paper. The answer was marked correct immediately. It was practically a reflex response on my part, and, to quote a great action movie, "they die hard".  If you have been trained properly the skills are still in there.  Extra credit if you know the movie I am referring to. Hint: it is not "Die Hard".

My "colleague" then loudly said, effectively to the entire room of fellow teachers,

"Colleague": "What!? YOU got it?" 

The "YOU" really was verbally bolded. Nerds are often not adept at phoniness in real time, to our credit I might add, and I appreciate honesty.  I think he imagined his academic pedigree would "win the day", but, as I've discussed elsewhere, math problems don't  care if you went to a fancy school.

Actually, these were not "power problems" so they were meant to be done quickly and this was no great feat. The dialogue continues below.

Me: "Uh, yeah, and please don't sound so surprised, I was on the ARML team in high school."

Yes, I said this with the characteristic Erickson dead pan sarcasm.  Since he was still scribbling furiously, I said, "Are you trying to do this using gradients?" Gradients were the obvious "sucker play".

"Colleague": "Uh, yeah." 

Me: "Oh, dear."

I just made up the "Oh dear"  because it's funny but that wasn't part of my standard shtick back then. Normally, I would never disturb someone by talking to them while they are trying to do a problem, but he had initiated the dialog.

My colleague had not been on the Chicago area ARML team as he was not from the Chicago area. I instantly made a further deduction: my colleague had never even been a serious math team kid, as evidenced by the fact that only a complete rookie would try and use standard mathematics when an obvious mathematical ninjitsu tactic was available. My colleague was actually aware of the fact that I was on the ARML team in high school which makes this even more odd unless you understand the larger context.

Time was up. The organizer asked how I did it and I told him. He smiled. None of this was a surprise to him I'm sure. He was a teacher at Evanston Twp., had coached the ARML team when I was a student, so, of course, so I grew up on his problems.

The unfortunate fact is that I have many such anecdotes. Why did I share this with you? To give you some more advice of course. If you pursue mathematics it is very likely that you will run into people who drastically over estimate their own capabilities and drastically under estimate yours.

These perceptions will likely be exacerbated by all sorts of factors having nothing to do with actual merit or talent: race/ethnicity, gender, social class, accents/speech patterns, degrees, academic pedigrees to name a few such factors.  Protests and lawsuits aside, this is not likely to change anytime soon. I was told most of this as a child by my mother explicitly so none of this was a shock to me and I think it made me more resilient. Sure, I get angry that the universe isn't a fair place, but honestly, who told you life was fair? Just watch a nature program, and see a little bunny rabbit minding his own business get ripped to bloody shreds by a fox if you need evidence that fairness has nothing to do with it (Darwinian struggle seems to be the subtext of these programs).

My advice, if you really love your subject, is to just keep studying it. The people over estimating themselves tend to get rather complacent and lazy, confusing titles with talent, credentials with creativity, and they stop studying intensely.  They will occasionally say stupid things to you and you will have to learn to calmly, but politely and perhaps a bit indirectly, point out to them that they are being stupid and that maybe they should read the book "Inequalities" by Hardy, Littlewood, and Polya. It kind of becomes a fun game when you get good at it but you do need to know your stuff if you're going to play it.

Incidentally, as an exercise, you should try and solve the ARML tryout type of problem quickly by hand or in your head. 

Mathematica makes such problems trivial of course, but I grant you it is a bit like bringing a howitzer to a knife fight. Still, if you need to solve math problems, but, after all of this, you're saying to yourself, "Nobel was right: I hate you, I hate mathematicians, and I hate math dammit!" then maybe you should just get good at Mathematica or some similar programming language. We can find the answer easily as follows:

In[12]:= Minimize[{(2*x^2)/y + (3*y)/x^2, x > 0 && y > 0}, {x, y}]

Out[12]= {2 Sqrt[6], {x -> 1, y -> Sqrt[2/3]}}

So the minimum is 2 Sqrt[6].

My colleague was trying to solve it like this

In[14]:= Grad[(2*x^2)/y + (3*y)/x^2, {x, y}]

Out[14]= {(4 x)/y - (6 y)/x^3, 3/x^2 - (2 x^2)/y^2}

In[17]:= Solve[(4 x)/y - (6 y)/x^3 == 0 && 3/x^2 - (2 x^2)/y^2 == 0, {x, y}] // Quiet

Out[17]= {{y -> -Sqrt[(2/3)] x^2}, {y -> Sqrt[2/3] x^2}}

Which tells you that the minimum values potentially occur along a parabola containing the point 

(1, sqrt(2/3)) and then you can plug in and get the answer.

Answer: Of course, the quick way to do this is to simply apply the arithmetic-geometric mean inequality. Really lovely! You don't need to know where the minimum occurs to know what the minimum is and you can see that the expression was designed so that when you multiply the two terms of the sum everything cancels and you get 6.

Incidentally, I initially hated inequalities but came to love them and recognized when they were applicable. Correct that, I was always trying to apply them! To paraphrase the old "hammer/nail" trope:  the world looked like an optimization problem to me and I just needed to hit it with the right inequality! 

Note: Undoubtedly, there will be those who doubt the veracity of the above anecdote or some other anecdote (I have lots of them).  If by some chance you are one of the principles involved and you, to quote Woody Allen, "resemble these remarks," feel free to come forward and identify yourself to contradict the above narrative and we can sit down for a polygraph test together.

Incidentally, the action movie I referred to above is Ronin.