By Alex Zhai
Gargoyle staff reporter
Posted Friday, June 2, 2006, The OG, opinions

[Note: Sophomore Alex Zhai finished second in the country in the 2006 USA Mathematical Olympiad, scoring 41 out of a possible 42 points. He leaves Sunday to participate in the Mathematical Association of America's four-week Mathematical Olympiad Summer Program at the University of Nebraska in Lincoln. If Alex is selected to be one of the six members of the U.S. team, he will travel to Ljubljana, Slovenia, to compete in the International Mathematical Olympiad from July 10 to 18.]

In a small, dusty, and dimly lit room with shelves filled with old math materials, I stared down at the following problem:

“For integral m, let p(m) be the greatest prime divisor of m. By convention, we set p(±1) = 1 and p(0) = ∞. Find all polynomials f with integer coefficients such that the sequence {p(f(n2)) - 2n}n ≥ 0 is bounded above.” [See *note at the bottom of this article for a more detailed explanation.]

It was April 18, the first day of the 35th annual United States of America Mathematical Olympiad (USAMO), a two-day test spanning a total of nine hours. You might expect a test like this to have more than a hundred problems, but this test had only six.

To many people, math contests test how many types of problems you have seen before. Preparation consists of first remembering how to do all of the different types of problems and then practicing to do them quickly.

But while experience matters in the USAMO, none of this year's 431 competitors had seen any of the problems before. The USAMO does not generally test the contestants' abilities to match problems to particular theorems or formulas. Instead, the participants must explore each problem for an extended period of time, employing creativity, experience, solid reasoning, and a dash of luck to find a solution, which usually involves several unobvious steps.

Having attempted previous years' problems, I knew I had a good chance of placing in the top 30. Most contestants solve only one or two out of the six problems, so I had hoped to build an early lead by solving at least two problems on the first day.

However, problem two had been uncharacteristically easy, so if I would have any lead going into day two, it most likely rested on the third and hardest question of the first day. That was the question I was reading on that April afternoon in a back room of the Math House.

The problem looked intimidating. For a moment, I grew distracted and thought about the several hundred other students around the country who could be looking at the same problem, some of them with ambitions of scoring near the top, others just content with having qualified for the USAMO.

Even though the test is ultimately a competition, the pursuit of solving its problems transcends a numerical score. As takers of the USAMO, we had taken up a struggle to learn — and, moreover, to find — what we did not yet know, following the innate human desire to broaden and deepen understanding. I wondered what everyone else was thinking, but I quickly put these thoughts away as I settled down to attempt the problem.

After a little thought, one can see that the function f(x) = k works, where k can be any number other than zero. But the problems of the USAMO require proof: Not only did I have to find all polynomials f(x) which satisfied the conditions of problem three, but I also had to show that no other polynomials worked.

I tried to find some more polynomials that worked, but 30 minutes later, it didn't seem like I was making any progress. For another half hour, I came up with a few ideas, but none that looked at all promising.

When we are learning math, a particular problem is usually an illustration of some technique we were taught. We are supposed to be able to figure it out, so we try all the techniques we know until we find one that works. Most of the time, not being able to solve a problem is an indication that we just need to review the material more carefully.

But an Olympiad problem is more of an adventure than an assignment. It requires you to explore its properties, searching for interesting patterns and slowly piecing them together. The solution often falls naturally after looking around enough.

There was no rush — nearly two hours remained — but clearly I had not found a good angle to look at question number three, so I laid my head down for a short rest to clear my mind, and that's when I made my first step toward a solution.

I realized that I would have to deal with the hardest part of the problem first: the primes. Instead of looking for properties of f(n) itself, I needed to examine the primes that divided f(n). Eventually, I was able to prove that it is possible to choose some number x so that f(x) is divisible by an arbitrarily large prime number (in more mathematical terms, {p(f(n))} is not bounded above).

It didn't take long after that to prove that f(x) is divisible by (4x - k2), where k is some odd number. Suddenly, I knew I had made much progress — it would be easy to bound {p(f(n2)) - 2n} if I could break f(n2) down into factors. As it turned out, I did not have time to work out the last detail for a complete solution, but I thought I would receive significant partial credit for coming close. (Here is the official solution.)

Have you ever thought about something for a long time, and then, in a sudden burst of insight, you think you're starting to “get it”? The third question on the 2006 USAMO was like that to me. I had not nailed the problem on the test, but walking out of the Math House that day, I felt like I understood how it worked at a much deeper level.

Believe it or not, Olympiad math is not something I take too seriously, even though I enjoy it. In 10 years, I doubt I will directly need a good portion of the math I've learned. However, the experience of trying problems such as those on the USAMO is nonetheless valuable. It teaches you to try to comprehend things that seem at first incomprehensible. It teaches how to think, and when it comes down to it, thinking is what makes life interesting.

*Mathematical terms can be looked up at Wolfram Research's mathworld. The sequence {p(f(n2)) - 2n}n ≥ 0 just represents the infinite list of numbers p(f(0)) - 0, p(f(1)) - 2, p(f(4)) - 4, p(f(9)) - 6, p(f(16)) - 8, and so on. If the sequence is bounded above, that means there is some fixed number that is greater than all the numbers in the sequence. The problem asks us to find what polynomials f will make the given sequence bounded above. One such polynomial is f(x) = 1, because p(f(n2)) - 2n = p(1) - 2n = 1 - 2n is always less than 2, no matter what non-negative number n is. The full solution is given at the American Mathematics Competitions Web site.