It was a beautiful day in the Winter of 1994. I had slept in because it was Saturday, but my older brother, Nikos, was not so lucky. He had just come back from a regional math competition named after the Greek philosopher Θαλής (Thales of Miletus) and he did not look happy. I asked him how he did, but he did not answer; he just reached into his backpack and took out the paper that contained the problems from the competition. Looking back, what I did next defines much of how I approach life since that day: I took the paper and started solving the problems one after the other until I was done. It took me three days – the competition lasted three hours.
The next year, I was old enough to compete in the first round of the Greek Math Olympiad and I convinced a bunch of my friends from school (remember the cool nerds from Geniuses wanted?) to wake up early on a Saturday and go test our mettle against the rest of the Greek population still in school (though university students were too old to compete). You know how people talk about the 1% these days… To make it into the next round you had to be in the 1% of something that none of us knew how to define back then. After all, we were all equally scared of the old guys with the formal suits and the strict faces holding the exam papers. How was I, or any one of my friends from school, different from all the other kids taking the exam throughout the country that day? Did we have a special upbringing? Did our parents answer math questions for fun? At the moment I received the paper with the 4 problems on it, I immediately hoped I had a special upbringing. That my dad was not a high-school dropout and my mom had studied math in college, instead of political science. Damn. These problems looked hard. But I knew what I was supposed to do. So I started solving them, one after the other. Unfortunately, the time ran out after 3 hours and I had only solved 2 and 1/2 problems by then. But after I got home, I said hi to my mom, ate a sandwich and went into my room. I stayed there until I had finished the other 1 and 1/2 problems. Emerging victorious from my room that night, I made another ham-and-cheese sandwich and added extra ketchup to reward myself.
A month later, I learned that I had made it into the 1%. And four months later, into the 0.01% that went on to the final round named after the famous Greek nudist, Αρχιμήδης (Archimedes of Syracuse). My dad, mom, brothers, classmates and, especially, teachers, were incredulous. But that was nothing compared to how I felt. They thought I was a lazy bum, but I knew it for certain. How did I get to reach the finals and make it into the top 20 of all Greek high-schoolers? I was not even in high-school yet! Who knows… All I know is that I would stay up past 3 a.m. on Tuesday nights to solve problems from Crux Mathematicorum with Mathematical Mayhem, when everyone else was fast asleep. And I wouldn’t stay up that late even if my favorite cartoon (Thundercats) was on TV at that time (which is a bit ironic, since my parents thought I was watching TV all night and so considered it appropriate to wake me up using buckets of water).
I didn’t solve these math problems because I was planning to be a mathematician. I disliked math class in school as much as (probably more than) my classmates. But this was different. These problems did not appear at the end of a chapter full of dry equations and ready-made formulas. The problems on Crux Mathematicorum were dancing in the middle of pages surrounded by other problems vying for attention. Any progress I made was solely due to stubbornness. But then, I started unlocking my superpowers one-by-one. And I started seeing patterns, not only mathematical patterns, but patterns in my own thoughts and my own temper when confronting impossible problems. Which came in really handy later in life when I was asked to solve a much harder problem (An intellectual tornado).
So now that you are all primed, here is a problem to keep you occupied for a few hours, or days…
Problem 1: If and , for all non-negative integers , find .