First Thesis

"This is high school level stuff," the short guy in glasses voiced.

"It indeed is, that's why it's a warm-up, Will," Professor Yang agreed.

"Well, anyone with a bit of logical thinking could solve this. Let me try..." Will stood up and made his way to the whiteboard.

"No matter how logical people are, if they don't have enough brain cells, they are still stupid," Nadya commented.

I grinned; Nadya had a knack for making depressing comments.

"If we represent the cards as numbers... Let a card face-down be a 1 and a card face-up be a 0, we might be met with a situation like this..." Will started to write on the whiteboard, creating a binary sequence '1001110110011011...'

"We can see that there are only two possible scenarios for a move: '11' and '10'," he explained as he continued writing.

"In each case after a move, we are left with a decreasing amount of 1's: '10' and '00'," Will continued, pointing out patterns.

"Assuming that the whole series of cards is a binary number instead, a decreasing number of ones, in this manner particularly, signifies a decreasing value and a binary number tending to the value of 0, as it cannot be negative," Will finished before turning towards Professor Yang.

"It cannot be negative indeed," Professor Yang stated, before looking at Professor Milik. "Professor, it's your turn; my questions are too simple."

"Okay, I will give you guys a question from one of the previous IMCs," Professor Milik looked at us meaningfully. "And don't get ahead of yourself; this is a question that I didn't include in the papers that I sent you," he smirked.

'Let C be a nonempty closed bounded subset of the real line and F: C -> C be a nondecreasing continuous function. Show that there exists a point p in C such that f(p) = p.'

"A Calculus question. What do you think?" he finished writing and turned towards us.

"Well... a closed set, so the complement would be a union of open intervals around this set," Isaac said while moving his finger in a circle, trying to add some food for thought.

"Ai, that doesn't really add anything. Maybe we could prove this by contradiction..."

"That could be a good idea, Nadya," Professor Milik approved.

"Not only is it a good idea, but it is also the correct idea. I know how to prove this," I stated confidently, and the entire room looked at me with surprised expressions. Only Professor Milik kept his composure.

"Well, tell us then if you're so sure about that..." I could sense Will's skepticism.

"We can prove that the contrary is not possible for a non-decreasing function," I said confidently as I walked towards the whiteboard. Standing before it, I paused for a moment, thinking about the problem.

Finally, I opened my eyes and lifted the marker.

'If we suppose that f(x) is not equal to x for all x in C, and let [a, b] be the smallest closed interval that contains C. Then, by our hypothesis, f(a) > a and f(b) < b.'

As I wrote up to this point, Professor Yang had a look of surprise on her face.

'Let p = sup{ x in C: f(x) > x}... For all x > p, where x is in C, we have f(x) < x.'

"Damn... it can't be non-decreasing..." Isaac seemed to come to the correct conclusion too.

'Therefore, f(f(p)) < f(p),' I finally wrote.

"Ai! Proving by contradiction that there exists such a point p in C that f(p) = p."

"Exactly!" I exclaimed, pointing the marker at Nadya. The room was filled with a mixture of surprise and admiration.

"Good job, Max. I thought it would take you guys at least an hour to solve this, but I guess we have an outstanding student on the team this year," Professor Milik praised.

"I could probably come to this after a couple more minutes..." Will said pretentiously, attempting to downplay the achievement.

"Indeed you could, Will," Professor Yang said towards Will with a slightly patronizing tone, "Before we move on to solve more questions, I will explain to you guys what the visit to London will look like. We're taking off in the morning of the 19th of September."

"The IMC is hosted by Imperial College London, and as usual, the day before the competition, students from many countries will have joint lectures."

"This year, 44 students from universities all over the USA, count yourself in, will join together with the Australian students. Be prepared to listen to a lecture given by Professor Kevin Buzard. You might also get to ask some questions to Professor Terence Tao. He is gonna be joining as a guest"

" The competition will take place on the 21st, and on the 22nd, there will be some free time to move around London or converse with fellow mathematicians. Do what you want. But on the morning of the 23rd, we're coming back to the USA"

"Alright, cool, but is this a team thing?" I spoke up.

"Ai, Max. You didn't even check the rules?" Nadya's got this startled look.

"Yeah... maybe skipped that part."

"Ehh... It's a solo thing, and honestly, that's for the better," Will mumbled under his breath.

"I don't know, man. Seeing how Max just breezed through that problem, I'm kind of glad he's rolling with us. We will not be solving problems as a team, Max, but the total points of the team average in the end, you feel me," Isaac drops this with a chill vibe, totally opposite to how he barged into class.

After Professor Yang provided us with insights into accommodation in London, we delved into tackling several additional questions for the next three hours.

While I managed to swiftly devise solutions for most of them within the initial minutes, I intentionally reserved some for the others. Isaac and Nadya shone, marking them as the standouts.

But, it was evident that Will was great at Algebra, and contributed to related questions.

Heading back to the dorm, it hit me: if I'm diving into the math scene, might as well go big. Time to lay down the groundwork for my rep by crafting a solid thesis.

Normally, that's a move you'd make as a third-year in the Math Department, and here I am, not even a math major. But you know what? Ideas are brewing in my head already. Gotta make some waves.

Back when I was cruising through these questions about Inverse Functions, I stumbled upon this crazy link between them and the Fourier series, you know, the stuff I first got into when I dipped my toes into electronics.

It hit me like, "What if we could flip signals in communication systems by breaking down these inversion functions with the Fourier Partial Series?"

If I pull this off, it ain't just about polishing up signal quality and dialing down the noise; that's just the tip of the iceberg.

We're talking about unleashing a mathematical playbook that could straight-up turbocharge the performance of our gadgets.

It's like giving these devices a whole new set of tools to flex with and trust me, that's huge in the electronics game.

But before that, a necessary pit stop was in order.

I whipped out my trusty Motorola and opened up my company's account.

Dreamland Net:

Balance: $10,107

A grin stretched from ear to ear on my face. The hard work was paying off big time, the business was on the up and up. 

I zapped a chunky 5000$ of that success into my personal bank account with an express transaction and set my course for the nearest computer shop.

I went all out at the computer shop, treating myself to the cream of the tech crop: an RTX 3080, a slick Ryzen processor, a beastly 64GB of DDR5 RAM, and a whopping 2TB of SSD memory.

When I got back to my room, Rick was sitting there programming, Turns out, this dude had a solid three years under his belt at Microsoft, but he was craving the joys of a student's life.

Who knows maybe I could find a use for this guy...

I mean... what is up with me... He might prove helpful.

So I got the PC all setup, booted it up, and my mind snapped back to the thesis. Brainstorming and sifting through electrical analyses and mathematical frameworks already in existence.

The next week, I attended the meetings with the team and the profs, hashing out ideas, and solving previous year's questions.

As soon as that's done, I always run straight back to my room, putting my whole heart into creating something out of nothing.

Non-stop in the zone.

The end of the week came and I had it.

When I looked at it, it wasn't perfect, it needed editing, but..., but there it was.

'Optimizing Electronic Signal Processing Using Analytical Exploration of Inversion Functions through Fourier Partial Series'

My sweat poured into every word and equation.