We hopped into the car and she pointed to the gearshift lever, a sleek black knob nestled in the center console. "Here's your gear shifter. This is a nice Audi, so it should be smooth and precise."
Lisa then guided me to the pedals. "These are your three pedals. On the right, you've got the gas pedal. In the middle, the brake pedal. And on the left, the clutch pedal."
I nodded, taking in the details. "Got it. So, clutch in, shift, and let it out smoothly, right?"
"Exactly," Lisa confirmed, impressed by my quick grasp of the basics. "You're off to a good start, Max.
With Lisa's expert guidance, I started up the car, clutch pedal down, shifted into first gear, and began to push the gas pedal as gently as possible. It felt like second nature to me, as if I had been doing it for years.
Within just a few minutes, I was confidently driving through the training center, smoothly changing gears and executing flawless turns.
My driving skills seemed to have matured in no time, and I was cruising through the training course with ease, impressing both myself and Lisa.
Lisa couldn't help but express her astonishment. "This is wild, Max, but you seem confident enough to hit the road. Let's give it a whirl."
With that, I took off, cautiously easing onto the suburban streets of Chicago. Every mistake I made, no matter how minor, Lisa was right there to provide immediate guidance and corrections.
I never repeated any of those errors, and it felt like I had a natural knack for driving. It was as if I'd been born behind the wheel. Just an hour later I was cruising through like a seasoned driver.
"I've only seen a single person like you before, but he was a pro-Go Kart driver. You learn so fast it's unbelievable. I don't think you really need those extra six hours on the books, but would you mind coming back? This is nice and comfortable, and I want to see just how good you can get. haha"
As I made my way back to the education center, I didn't even need Lisa's assistance for directions. I had it all memorized, every turn and landmark.
In the following week, I stepped up my driving game and even got the hang of drift parking in some seriously tight spots. Lisa was having fun driving in the passenger seat. In one week I got even better at driving than my instructor.
While I was busy leveling up my driving skills, Olivia and I decided to explore the Illinois Mountains, and we even conquered the summit of Chicago Peak. We were really getting some cardio. Building up our shape.
But, as luck would have it, Olivia had to bail on me in Chicago the following week. She hightailed it back to New York to wrap up the break with her family.
The PT-GAI Weather app was on fire, racking up downloads day after day. It started with a hundred, then five hundred, and before I knew it, we hit the two-thousand mark.
Two weeks in, we were sitting pretty at eight thousand downloads, and the in-app purchases were starting to take off.
In those first two weeks, I'd already pulled in over $200. I knew this was just the start, and eventually, companies might want to work with me. To make deals and partnerships official, I realized I needed to start a company.
It was going to be a small one-person operation, so I wasted no time setting up the structure and officially registered the business under the name "Dreamland Net." I figured keeping the name broad and versatile would be a smart move, especially if I ever decided to start more ventures down the line.
With "Dreamland Net" officially in the game, I went ahead and opened a separate bank account for the business. The next step was to file for a patent under the company's name.
Everything was locked and loaded. I connected the business bank account to handle in-app transactions and shifted my focus to an email I had received from Professor Milik.
"Hi, Max. The IMC Olympiad is primarily designed for students in their first, second, third, or fourth year of university. This year, we're sending four students, including one team leader, and you'll meet them later. The Olympiad primarily features proof-based questions in the areas of Algebra, Calculus, and Number Theory. I'm attaching three books and 400 pages of the previous year's material for you to study. Go through as much as you can."
"P.S. The Book of Proof is my personal preference, there are other books out there too. :)"
Other than the Book of Proof Professor Milik also sent me a Proof based Calculus book as well as 'Calculus by Spivak' that seemingly has some crazy hard problems in it.
I wanted to skip through reading those books and go straight into problems from previous years. And so I took a random algebra problem under consideration and started thinking about a solution. The question went like this:
Suppose that a, b, c are real numbers in the interval [−1, 1] such that:
'1 + 2abc ≥ a^2 + b^2 + c^2'
Prove that:
'1 + 2(abc)^n ≥ a^2n + b^2n + c^2n'
for all positive integers n.
I thought about it for some time, going through lemmas that have their ground in asymmetric and symmetric matrixes, but couldn't think of anything that would fit into this. After a while, I moved to vector theorems, inner products, and dot products. Finally, it clicked.
"Cauchy-Schwarz inequality!"
It is known that the absolute value of the inner product is always less or equal to the product of the vector norms, in other words, by Cauchy-Schwarz I know that:
'(a^(n-1) +a^(n−2)bc + . . . + b^(n-1)c)^2 ≤ (|a|^(n-1) + |a|^(n-2)|bc| + ... + |bc|^(n-1))^2 ≤
(1 + |bc| + . . . + |bc|^(n−1))^2 ≤ (1 + |b|^2 + . . . + |b|^2(n−1))(1 + |c|^2 + . . . + |c|^2(n−1)'
I can rewrite the given constraint as:
'(a − bc)^2 ≤ (1 − b^2)(1 − c^2)'
And I could already see the answer, multiplying the Cauchy-Schwarz inequality by this constraint and simplifying I obtained:
'(a^n − b^(n)*c^(n))^2 ≤ (1 − b^n)(1 − c^n)'
And then further transformed it into:
'1 + 2(abc)^n ≥ a^2n + b^2n + c^2n'
That's a wrap on the proof. Following the tradition, I left a small black rectangle at the bottom of the page, just like the mathematicians do.
After successfully proving it on my own, I couldn't resist the temptation to check out the proof provided at the back of the PDF. To my surprise, their approach was entirely different. They used the concept of symmetric matrices, showing that it's semidefinite and determinant, which ultimately proved that the inequality is true.
This got me even more pumped about math. There are multiple paths to prove something, and as long as the calculations remain flawless, all these methods are valid.
I kept at it, tackling more of those Olympiad questions. Most of them were a piece of cake for me, and I could prove 'em without breaking a sweat. But there were a few that turned into real brain-busters, and a couple had me scratching my head for days.
In between all that learning grind, I aced the driving exam out of the whazoooo, with no mistakes to be found. The examiner happened to be a young, attractive blonde. She didn't even need to write down a single word; instead, she just kicked back and enjoyed the ride.
For the next two weeks, I shifted gears and got deep into those books Professor Milik hooked me up with. Those tomes became my secret stash of theorems and ideas, beefing up my math toolkit so I could whip out some slick and airtight proofs.
I also decided to send a message to Rick Rosby, the guy that at this point I've met on two different occasions, and I still didn't know what to think about him.
'Yo, Aliens guy, you renting an apartment in Massachusetts or snagging a dorm room?'
'Hey, I'm gonna be staying at a dorm. Name's Rick, by the way.'
'Yeah? You wanna room together? I'm bunking at the dorm too.'
'Hmm, sounds ok. But you gotta quit with the aliens guy thing.'
'Sure thing. In two weeks, I'm hitting up MIT. We should link up and sort out the paperwork.'
'Right on, see you there then.'
'Kk, Aliens guy.'
'...'
Over the last two weeks, I knocked out those books Professor Milik hooked me up with and proved some more shit from power series to sequences, prime numbers, and proving integral values. Every solved problem felt like a slam dunk.
Six weeks had flown by in the blink of an eye, and there I was, aboard a flight headed to Massachusetts. As I checked my company bank account, a smile appeared on my face as I saw the balance approaching the 5-digit mark.
But just when I thought things couldn't get any more surprising, out of the corner of my eye, I spotted someone I never would have expected to encounter here.