IMC [1]

The sixth blue comet showed up during the following sleep session, casting its glow with impeccable timing, as if someone choreographed its appearance, just before the competition...

I woke up early, with plenty of time until 1 PM. Figured I'd pick up some Latin, you know, just in case those dream entities drop some ancient knowledge on me. Gotta stay prepared.

I have also made a call with Olivia. The conversation began tame, but we drifted away, we got into a discussion about the best way to have sex in a Ferrari.

Olivia was like, "We both gotta dangle our legs off, you know?"

But then I threw in, "What if I lay back, legs out, and you ride me?"

Then she's all like, "But what if I tumble out of the car?" mock protesting and all.

"No worries, we'd go all airborne, finish before we hit the ground. Safety first!"

"I think that's too far..." she laughed.

"Is it, though?"

Yea... I should not have mentioned that.

We talked about how our day-to-day life is going, but I have to admit, I held back a few details... 

As for Railey, we hammered out a deal – she's gonna whip up a fresh design for the app. We're sticking to the OG colors: blue and green, though.

Will still kept the grind at his desk doing more math.

While some people push it to the last tick of the clock, I figured redirecting my brain's energy to something else might be the smoother move.

At 12:20 PM together with Nadya, Isaac, and the other guy on the team... we made our way toward one of the halls prepared for this year's competition.

This wasn't the only hall in the game – with over 300 players repping from every corner of the globe, there wouldn't be squeezing all that talent into one tight spot.

In our case, they assigned us to a hall directly in the Math Department of the college, together with around 100 other participants.

I am talking about an expansive battleground - a whopping 4,000 square feet of open space.

The spacing and arrangement of tables were strategically done by the organizers, 13 rows deep and 8 across.

The spacing was on point, just enough so no one was getting a sneak peek at the competition papers of their rivals.

At the very front stood a stage, lifted slightly above the ground and entering inside I could already see the jury sitting in their respective seats.

You could see an observant on each side of the hall, not for the purpose of checking if anyone cheats though. The answers couldn't be found online, anyway.

No, these keen eyes are there in case someone raises one of those colored papers sitting on each table.

Blue if someone lacked a bottle of water,

Yellow if nature calls,

And lastly, Red for the brave soul who is throwing in the towel because, well, they've got a pot of water hitting a boiling point back at home. You got to have your priorities right.

We lined up at one table flanking the entrance and waited for our turn to draw the number that would tell us where to sit.

In the meantime, the staff was collecting any electronics that we might still have in our possession, and I did indeed have my beloved Motorola with me.

I took it out of my pocket and from the corner of my eye I noticed a message:

'Dear, Maximillian. We have moved your thesis from the editor's office to the proof phase.'

Well... Good. Now I needed to get in the zone, though.

Luck seemed to be on my side as I snagged the number '2' from the draw.

With everyone settling into their designated spots, a hush fell over the hall like a blanket. It was like the entire hall collectively zeroed in on the stage. A stillness settled over the room, so thick you could almost touch it.

In that pivotal moment, an elder professor, with a mop of long white hair, rose from the stage. Sporting a white t-shirt adorned with a Rubik's Cube, he surveyed the hall. In the midst of the silence, he declared,

"Everyone! I'm Professor Simon Donald, and it's a pleasure to have you here at the London Imperial College. This... is an immensely important competition. I'm positive that at least some of you will start your legacy right here."

"But enough of the chit-chat. You will have four hours for the first day of the IMC... The clock starts ticking the moment Professor Terence Tao puts the start time on that blackboard.", he said pointing to the blackboard which hung in the center of the stage.

Professor Terence Tao rose and made his way to the blackboard. He stood in front of it and gazed toward us, making a quick comment, "I came here with my students from the University of California, but I left my calculator at home, so you'll need to solve these yourself today... anyway... Best of luck, and may your proofs be as solid as my coffee in the morning."

Right... No one laughed. Typical mathematician humor.

'13:03' - the professor scribbled the time on the blackboard, and it was time to begin.

And as the chalk settled, you could hear the papers rustling, as everyone dove into the first question:

'Let f : [0; +∞)  -> R be a continuous function such that limit x -> +∞ of f(x) = L exists (it may be finite or infinite).'

'Prove that the limit as n tends to infinity of the definite integral from 0 to 1 of f(nx) equals to L'

For sure, given the limit rolling off to infinity, I could technically run through all the scenarios – finite limits for convergence or infinity for divergence. But let's be real, that'd be a marathon, not a sprint.

Yeah, I'd have to set up an epsilon dance for the convergence party, heading to zero, and then toss in some wild, high-value shindig for the divergence.

Let's keep it a buck fifty. That's at least an hour on the clock. There's got to be a faster route...

Going through the arsenal of theorems, I hit a stop at L'Hopital's rule. It seemed a bit basic, I mean, they wouldn't throw such a straightforward curveball, would they?

Let's say I expand this to a t-parametric integral F(t) = definite integral from 0 to t of f

I could now state that the integral given in the question is equal to F(t)/t for any t>0

And as t tends to infinity and the function itself also tends to infinity...

Got a little hyped right there.

The use of L'Hopital's rule can easily prove that the integral is indeed equal to L.

lim t→∞ = F(t)/t = lim t→∞ F'(t)/1 = lim t→∞ f(t) = L

It follows from the previous statement that the definite integral from 0 to 1 of f(nx) equals L.

End of proof.

Well, that was quick, no doubt. I was curious to see how the other contestants tackled this problem.

Looking around, I saw some people nibbling at their nails and some pinching themselves, trying to make their brains work properly.

Dang, one girl straight up pulled the Red card. She looked European, going through it, visibly stressed out. Tears streaming down her face as she bounced from the hall.

But back to the grind, flipping through the next set of questions. They weren't so fun anymore, for example, the third one had to do with polynomials.

'Prove that there are only finitely many numbers n such that all roots of the given nonconstant polynomial are real'

Straight off the bat, no shortcuts in sight. Rocking the Viete formulas, I focused on the asymptotic behavior of the terms with the highest degrees.

It took me around 30 minutes to conclude.

The rest of the game was heavy on matrices and figures on a plain. Both were descriptive proofs.

For one question, I went all out, dropping a two-pager to prove that a sequence of matrices was packing n + 1 distinct integer eigenvalues, each with its own special multiplicity.

Wrapped it all up, threw in some neat squares at the bottom of each proof, and filled 'em in with my trusty black pen.

I've got to sneak in some fun. They left me with a cool two-hour buffer, might as well enjoy the bonus time.

...

"STOP!", Professor Terence Tao stood by the blackboard and wrote down the time.

'17:03'

The Professors started strolling between the tables, snagging those proof-packed papers. The time has ended.

Reluctantly, the students gave in their papers, clutching onto them like they were giving up hard-earned cash. 

"Maximillian Sullivan, papers please...", Professor Terence Tao was the one to come up to me

"Here",  I replied, with a confident grin.

He shot me a rare look, like spotting a unicorn. He gave my papers a swift check and hit me with a smile.

The students gathered up, and the hall started emptying out.

Day 1, officially ended.