Foreword

I’ve loved since I was a child. People who know me might not be able to figure out how what I’m about to say is related to me. If you read this, you might think I’m a nerd. But I like to say, “You are an intellectual badass.” (I know it’s cheezy! But I’ve always wanted to say this in a Batman voice (the Christian Bale one, of course).

Since I was young, I’ve been in love with math. Yes, you did hear correctly. But I will always be sorry that I let my love down.

‘Yes, I have betrayed you, my love. Maths’

My mother is a math teacher, and since I was a child she has always encouraged me to look into the amazing world of math. I really liked the way math worked. I was always surprised by the charm, the art, and the technique.

But I was always stuck with math from books and its derivatives. I had never heard of the vast world of math until I was in the seventh grade.

The Bangladesh Regional Mathematical Olympiad used to be held at my school. I also took part in the Dhaka regional Olympiad for the first time that year. And since I didn’t know anything, the problems I had to solve beat me up. But I wasn’t upset; I was so happy that it made me want to learn more. So, I trained myself and came back the next year, but…

I got beat up again. This time, I understood. Maybe I wasn’t given the smarts to figure these out. Maybe I’m not as good as others. I started to lose confidence, so I stopped taking part and left the world of math. stopped working out. I wasn’t sure of myself enough.

…………………………………………………………….

One day, as I was getting ready for a test at college, I went over all the math in detail. Before I go any further, I need to say that, until then, I had lost interest in math. So, when I took the math test, I knew all the questions and had practiced them, but I couldn’t figure out how to answer them. How come?

I felt something. That was the worst betrayal of all time. I was blank. I felt a deep sense of emptiness.

I was learning math by heart instead of understanding it. I cried (Yes, I cried a lot.).

But that was when I felt like I could do it again. It was too late, though. It’s too late to start over. Too late to join the Olympics as a mathematician…

The rest of the story could be told later. But the main point of this writing isn’t to tell my story. It’s to tell the readers which problem set has me so dazed that I can’t look away. So let’s get started.

The Legend of Problem 6

No one is surprised by the fact that math is hard, so when you talk about the hardest math problem ever, things start to get a little crazy. Take Question 6, which has a simple name but is so hard that it can make mathematicians cry.

The problem was given at the International Math Olympiad in Australia in 1988. The sixth question was one of the hardest ones so far.

On the last day of the Australian Olympiad in 1988, the organizers decided to throw the kids a huge curveball. This problem has stuck out as one of, if not the hardest problem ever.

Australian-American mathematician Terence Tao, who won the 2006 Fields Medal, the mathematician’s “Nobel Prize,” got a 1 out of 7 when he tried it. This shows how hard it was.

Problem Statement

Let a and b be positive integers such that ab + 1 divides a2 + b2. Show that a2 + b2 / ab + 1 is the square of an integer.

So, I went on to search for the solutions. How the f did they solve it. The Solution to this problem is stated below.

Solution

Let k = a 2+b 2/ ab+1 . Fix k and consider all pairs (a, b) of nonnegative integers (a, b) satisfying the equation a 2 + b 2 / ab + 1 = k, that is, consider S = { (a, b) ∈ N × N | a 2 + b 2/ ab + 1 = k }. We claim that among all such pairs in S, there exists a pair (a, b) such that b = 0 and k = a2 . In order to prove this claim, suppose that k is not a perfect square and suppose that (A, B) ∈ S is the pair which minimizes the sum a + b over all such pairs (if there exist more than one such pair in S, choose an arbitrary one). Without loss of generality, assume that A ≥ B > 0. Consider the equation x2 + B2/xB + 1= k, which is equivalent to x2 − kB • x + B2 − k = 0 as a quadratic equation in x. We know that x1 = A is one root of this equation. By Vieta’s formula, the other root of this equation is x2 = kB − A = B2 – k/ A . The first equation implies that x2 is an integer, the second that x2 != 0, otherwise, k = B2 would be a perfect square, contradicting our assumption. Also, x2 cannot be negative, for otherwise, (x 2) 2 − kBx2 + B 2 − k ≥ (x 2) 2 + k + B 2 − k > 0, a contradiction. Hence x2 ≥ 0 and thus (x2, B) ∈ S. Because A ≥ B, we have x2 = B2 − k /A < A, so x2 + B < A + B, contradicting the minimality of A + B.

I know that most of you don’t get a single thing about it. But it’s so interesting that I couldn’t stop looking at it. Google “Vieta Jumping” to learn more about the legend of the number 6.

“You’re just too good to be true. Can’t take my eyes off you.”Frank Sinatra

Abridged & Slightly Re-Phrased

Published in Amber 6th Edition – Esonance 2019 | Read Here