Photo

The United States has won the 57th International Mathematical Olympiad, the world’s most prestigious problem-solving competition for high school students.

The competition, held July 6-16 in Hong Kong, included teams from over 100 countries. The winning U.S. team score was 214 out of a possible 252, ahead of the Republic of Korea (207) and China (204). Rounding out the top ten were Singapore (196), Taiwan (175), North Korea (168), Russian Federation (165), United Kingdom (165), Hong Kong (161) and Japan (156).

Congratulations to the U.S. team: students Ankan Bhattacharya, Michael Kural, Allen Liu, Junyao Peng, Ashwin Sah, and Yuan Yao, head coach Po-Shen Loh, and deputy coach Razvan Gelca.

This week we’ll be featuring two of the problems from this year’s six-problem test, with our discussion moderated by Po-Shen Loh himself.

Here are the two problems — our challenges for this week.

The first challenge is IMO 2016 Problem 2:

Find all positive integers n for which each cell of an n×n table can be filled with one of the letters I, M and O in such a way that:

• in each row and each column, one third of the entries are I, one third are M and one third are O; and • in any diagonal, if the number of entries on the diagonal is a multiple of three, then one third are I, one third are M, and one third are O.

Our second challenge is a bit more difficult. Here’s an introduction by Dr. Loh:

This year’s IMO featured an unusually large number of non-standard problems which combined multiple areas of mathematics into the same investigation. The most challenging problem turned out to be #3, which was a fusion of algebra, geometry, and number theory. On that question, the USA achieved the highest total score among all countries, ultimately contributing to its overall victory — a historic repeat #1 finish (2015 + 2016), definitively breaking the 21-year drought since the last #1 finish in 1994, and the first consecutive #1 finish in the USA’s record.

Let’s give it a try. Here’s IMO 2016 Problem 3:

Let P = A 1 A 2 … A k be a convex polygon on the plane. The vertices A 1, A 2, …, A k have integral coordinates and lie on a circle. Let S be the area of P. An odd positive integer n is given such that the squares of the side lengths of P are integers divisible by n. Prove that 2S is an integer divisible by n.

That concludes this week’s challenges. If you’d like to try your hand at the remaining four problems in this year’s IMO exam, you’ll find the complete test here. To obtain a perfect score for the test, you must successfully provide the correct answer for each problem, and also prove that each answer is correct.

A Short Conversation with U.S. Team Coach Po-Shen Loh

Photo

I had the good fortune to be in Hong Kong last week for the IMO, and caught up with U.S. coach Po-Shen Loh after the event. Following are excerpts from our conversation:

Gary Antonick: Congratulations on your victory. Were you surprised to win?

Po-Shen Loh: The win was a surprise, and now it means the future will be stronger for the US. Something interesting happened this year: the US training program students were estimating the chance that this year’s US team would win the IMO. Their estimate was 40%. To me, that number is huge. If you did that kind of odds three years ago, you wouldn’t have had that high a percent. There are a lot of very good teams out there. What it means is that last year’s victory has given a big confidence boost.

How you perform on something is not just a function of what you know, but how intensely you engage during the activity. This year, the US team went in with a very contagious We Can Do It attitude. The United States has won before, so winning is possible.

This year was actually a rebuilding year for the USA — four of the team members are not in their final year of school — but they were able to still win.

G.A.: This year’s test seemed to emphasize creativity more than previous years. How would you say the problems were creative?

P-S.L: This year’s problems — particularly #3, which we shared above — fused different subjects together in creative ways. Whenever you have fusion, as in cuisine, then there is creativity involved.

G.A.: How was the U.S. team selected and trained?

P-S.L: Students are selected from an original pool of hundreds of thousands of entrants through a series of selection exams run by the Mathematical Association of America, the official organizers for the USA participation in the IMO. The six student team is then brought together prior to the IMO for a three-and-one-half-week training program at Carnegie Mellon University, which is also organized by the MAA.

Participating in the training camp will be the current year’s six-person team, along with fifty-four additional students who might be on the team in future years. This year we also included ten international students — students on IMO teams from other countries. We paid for their airfare, hotel, food and teaching.

G.A.: Wait — you trained some of the team members from other IMO teams?

P-S.L.: Ten of them, including several gold medalists. Two of the four gold medalists on the Singapore team, for example, trained with the US. One of them commented that one of the techniques he used to solve one of the problems in this year’s IMO was from our training program.

This was such a big success that next year, we’re inviting 30 internationals.

G.A.: Do any other countries do this?

P-S.L.: Not at this scale. It’s counter-intuitive. First, bringing in the international students gives the top US students peers. They always tell you — if you’re the smartest guy in the room, you’re in the wrong room. So we bring in these peers, who are actually at the same level as these top six. Of course that increases the level.

Secondly, when the students come to the IMO, there’s no culture shock. You’ve suddenly landed and you look over there — oh my goodness. There’s the South Koreans. There’s the Chinese. There’s the Singaporeans. These guys must be amazingly good.

I remember when I went to the IMO that was exactly the experience I had.

But if you train together you’ve already been in this international experience for the prior three and a half weeks.

G.A.: Who thought of this idea to train so many members of so many different teams?

P-S.L.: My idea. I think of some pretty strange stuff.

G.A.: What does IMO training look like?

P-S.L.: We have math class from 8:30 in the morning until about 3:00 PM, with time for lunch. And then a seminar at 7:30 PM for an hour.

Every other day the day runs until 6:00 because we have a practice test.

There’s a video about this: An Inside Look at the MAA’s Mathematical Olympiad Summer Program.

At this event that we run — it’s a very different atmosphere. You’re not there to beat anyone. You just work together for 3 1/2 weeks.

How does the US IMO training program differ from other training programs?

The US program did not stress that much mechanics. That was actually a worry that some people had — if you come to the US training program, you might not get enough training.

G.A.: What separates the top performers in the IMO from the rest of those at this event?

P-S.L.: I don’t want to say it’s genetic. A lot of people say you must be born with something very special. Maybe once in a while, you may see something like that. But people are basically the same.

Here’s an example. Suppose I tell you to memorize the following:

First of all, what is gravity?

Now — how about the following? Could you memorize and rewrite the same way? Impossible.

প্রথম সব, মাধ্যাকর্ষণ কি ?

But it’s not impossible. All the people in Bangladesh can do it.

What I’m referring to — you have some concepts in your brain called letters and words. When you look at the English version of this question about gravity, your brain is not memorizing where the squiggles are. Your brain has conceptualized it already, and then compresses the information. You have no problem dealing with it.

So — what’s the difference between a top performer and everyone else? If you look at mathematics — if you’ve built the concept structure, when you reason about the problem, you’re reasoning in large concepts. You’re working with large concepts and putting them together.

It’s not a miracle. It’s just about whether your brain has partitioned the concept map, and whether you can deal with entire concepts as primitive arguments as apposed to working with each little letter at a time.

G.A.: Thank you, Po.

For a gentler introduction to some concepts in this year’s International Mathematical Olympiad, National Coach Po-Shen Loh highlights several through his puzzles this week on Expii here.

Want to stay in shape for next year’s Math Olympiad? Each week, Po-Shen Loh posts a set of five questions on Expii, ramping up in difficulty from accessible to intense. In collaboration with the recent film The Man Who Knew Infinity about the intriguing mathematical genius Srinivasa Ramanujan, who overcame impossible odds to change the future of mathematics, they are scouring the world for undiscovered mathematical ability, in the Spirit of Ramanujan Talent Search.

That’s it for this week. As always, once you’re able to read comments for this post, use Gary Hewitt’s Enhancer to correctly view formulas and graphics. (Click here for an intro.) And send your favorite puzzles to gary.antonick@NYTimes.com.

Solution

Check reader comments on Friday for solutions and a recap by Po-Shen Loh.