One week after the AIME, people no longer have this competition in mind except maybe for Geneviève and Krista, for whom there is still hope to get to the next stage. Most people that attend the team practices at this stage only care about the VMC final and consequently are there for multi-variable calculus. Like Imélie, and other people wanting to exploit the generosity of extra credit in calculus BC to reinforce their position, knowing that now extra credit comes from multi-variable calculus.

"Even in the single-variable course we're so focused on the computational aspect that we forget about the artistic aspect of the discipline. Today we'll show the other facet of mathematics that will enlighten you if you plan to declare a major containing mathematics courses in college, proof methods" Trent begins the sesson.

"Geneviève talked about how the VMC final will contain problems of the kind « Show that, if condition X holds, then we have Y »" Cory comments.

"To solve that kind of problems, what matters is the clarity of the argumentation. You have a bit of discretion to determine what the audience is supposed to know or not, a bit like in literature when you're asked to write according to the target audience. A proof is an argument to convince the audience that a mathematical statement is true (or false)" Gen continues talking.

"Gen, that's enough! During the dinner you insinuated that my mathletic playstyle didn't lend itself to a proof competition!" Marcia shouts at her teammate, visibly mad at this description.

"Very well, Marcia, here's your chance to prove otherwise. Let there be N face-down cards, and a move tuns a face-down card face-up as well as the card to its right. Show that this game ends in a finite number of moves" Gen issues a challenge to Marcia.

Marcia works on the board, while the others, who expected to see multi-variable calculus, wonder why Gen chose a problem unrelated to this material for Marcia to prove the contrary of what Gen and Krista think, initially in secret, and openly since the dinner, of her playstyle. After a few moments, Marcia obtains the following on the board:

The deck of cards can be seen as a binary number containing N digits, with 1 being a face-down card and 0 a face-up card. There are two cases to consider: the case 10 and the case 11.

In the case 10 the two digits become 01, while in the cade 11 we obtain 00. Yet, in a binary number, changing a 1 for a 0 in a higher position amounts to reduce the value of a number, which makes the resulting sequence of binary numbers strictly diminishing.

But the sequence of numbers' lower bound is 0, which represents the whole deck face up; yet a strictly diminishing sequence of integers with a positive first term K will have a finite number X of terms before the X+1-th term becomes negative, with K > X.

"Now I must admit that you're much more advanced on proofs than I thought. I was wrong to believe that your mathletic style didn't lend itself well to proof contests. I owe you apologies for what I said at the dinner"

"Thank you, Gen"

Maybe... maybe my father was right! There has to be a reason why Marcia is capable of such bursts of speed, it seems like she has a better mental « gearbox » than the other two. Not that my girlfriend is bad at math, she largely proved she had talent to my eyes, Cory thinks while looking at the solution Marcia gave to the problem, that he recognizes for having watched X+Y with Gen and Krista.

"For everyone else, the most common error is to write a proof in reverse. Starting from the conclusion to then get to the hypotheses might be good to reflect on how the proof works, but not for the proof proper. Another big mistake is to lack clarity under the form of undeclared variables or unclear statements" Trent explains to them.

"Another common error I see my friends make is make leaps of logic with no adequate justification" Imélie comments. "Included in the leaps of logic is to check for particular values without any special significance and then extrapolate"

"Imélie, you claim that your friends make such and such leaps of logic, while you know very well that the courses you and your friends take are very computational!" Vontae accuses her. "You're delirious, Imélie, if you think you must treat a computational exercise as you would a proof problem!"

"Oh no, Vontae, I am not delirious; in fact, I believe the vision people have of math courses is very computational, and same goes of the various mathletics contests. The Math Madness has been a clear success since, even though people followed you because of their inability to solve the problems, the burst of speed was much more entertaining than watching someone write a long proof! Gen issued a warning to the public about that" Imélie is clearly on the defensive.

"My warning was incomplete, but I didn't believe it was good to talk about what was missing because I don't believe the public would understand why warnings should be issued for that. When you're not a mathlete, there are things you can't understand about mathletic competitions..." Gen sighs.

Other problems which, by chance, didn't appear on the Math Madness, on the AMC12, the AIME or the VMC are problems that require to count solutions in the absence of a more efficient method to find the answer, or amount to intensive calculations. During this time, since Gen and Krista know, deep down, they can calculate partial derivatives, multiple integrals of all kinds without complication – after all, they flew through a differential geometry textbook in a week – they can focus on another problem drawn from a past USAMO, while using what Trent had to say about proof-writing methods. That everyone takes notes, so be it.

Like A implies B is equivalent not to B implies A (or the reciprocal) but to non-B implies non-A (the contrapositive). And contradiction proofs (such as in the problem solved immediately after the viewing of X+Y) by exhaustion (when there are cases to consider) or by induction.

"Imélie, there's something wrong. You said that, while I was busy at the AIME, you taught the course and the substitute let you cover partial fraction and trigonometric substitution integration?" asks Trent, surprised to hear about how she could prove everything regarding trigonometric substitution so easily.

"You really did that?" Vontae asks, a little surprised because the usual experience he has of a lecture with a substitute is instead a session where people do homework, either for the course in question or for another course. "You're nice, Imélie..."

"Tell us what traumatized you on the AMC-twelve that you entered two years ago..." asks a student in her section.

"What traumatized me was that I had the impression of having failed a math test for the first time in my life. I was used, until middle school, to have perfect grades, or nearly so, in mathematics. But last year I saw Éliane and Gen advance without perfect preliminary results..." Imélie tells them.

"The big disadvantage of mathletics is that, to do well in it, you need a lot of background knowledge. It's in part for this reason that no one here was remarkable at their mathletic beginnings, but also why Marcia suddenly became such an important contributor while she's a late mathletic bloomer" Vontae explains.

Shortly after this confession from Imélie, they start seeing Jacobians, how to calculate some and how the change in variables happen. But at the end of the training session, Cory makes a proposal to the rest of the team:

A case of content theft: this narrative is not rightfully on Amazon; if you spot it, report the violation.

"One may be wondering if we could host a Duel of the Parishes but with questions about multi-variable calculus only" Cory suggests.

"Ideally it should happen after the USAMO since there's a very real possibility that we can potentially qualify up to three people in it" Krista comments, "but it's more likely that it would be two, or even one"

"The USAMO, you are always talking about this when we're talking about mathletics!" Marcia complains, while she makes another comment. "The week before that is the State Quiz Bowl Championship and we're qualified for this"

"Bravo, Marcia! Show them that we are more than just a mathletics school!" Vontae exclaims, thinking instead about the legal use of the term proof, rather than its mathematical usage.

Until proven otherwise, the first thing that comes to mind of everyone in this state when they hear the words Venomous Agendas is mathletics. Except that Vontae must pay attention to the difference between the legal concept of proof, which is less rigorous in my opinion than the mathematical concept we just covered. Especially since there's a hierarchy of burdens of proof in the legal world, beyond reasonable doubt being the standard in criminal matters, while clear and convincing proof and the preponderance of proof are the next lower two echelons of legal proof, Marcia thinks, while quiz bowl flies under the public's radar.

"Unfortunately, this means I must miss a math team practice per week; we'll finally have regular practices for quiz bowl and not simply a practice or two before contests" Marcia announces to them.

"Marcia, last year you already seemed to have a good foundation for multi-variable calculus, I think I can give you one day off per week at your discretion" Trent announces, before turning to Imélie to ask her a favor. "Do you want to serve as a reserve for the VMC final, if the VMC rules allow for it?"

"Do you want to be my valentine, Imélie? Rest assured that I will not consider your possible reservist status for the VMC for tomorrow" Vontae asks.

"Yes, to both statements"

All these stories about formal proofs in mathematics gives everyone something to think about. Imélie, a mathlete? We have seen it all; she appeared way too perfectionistic to do well as a mathlete, but I am forced to acknowledge that she is worth training as a reservist for the VMC final, assuming the VMC rules allow us to make substitutions between the state phase and the national final, Vontae thinks while wondering what to offer Imélie on Valentine's Day.

However, the following day, in the AP English course, the project is announced for the rest of the winter, in front of a class that waits to see what's happening for the assignment:

"For the rest of the winter, here's the long-term assignment, Journal of a Place. The instructions are fairly simple: visit a meaningful place to you on eight separate days and then write journal entries modeled after chapters in Thoreau's Walden" the teacher announces them.

"Assignments that actually meant something to us included writing out essays for college applications, CVs, cover letters, and tests also asked us to write literary analyses, but this? This is the most ridiculous assignment we got this year!" Randy complains.

"Yeah, this assignment is not going to cut it for us. We spent all year to date doing assignments we know will matter to us outside the course itself on some level, so why, all of a sudden, you shift gears on what assignments we are given?" Geneviève lodges another complaint.

"The real substance of what we are learning here has been in our previous assignments; outside of these assignments you seem to mostly teach us to a test that has severe issues!" Florence adds to this cacophony of complaints lodged regarding this assignment.

This AP English course is very bad, I think it's too late to change teachers, At least not without causing a scheduling conflict, Florence thinks, while complaints increase in intensity about this homework. Even though Florence has a denture about the AP English Literature and Composition Exam, she feels that any attempt to debate the homework's rationale cannot be about the pedagogical problems of this exam. Exam which is more or less a glorified version of the ACT's reading + writing tests to her eyes.

"There are too many complaints! If you want to debate the merits of these complaints, or have an alternative assignment proposal, so be it, I will debate them with you and take as long as necessary to do so. However, please be advised that, even if I could be successfully convinced that your alternative assignment plan is a good plan, I will need the approval of all sections for any changes to this assignment to take effect!" The teacher warns before the debate starts.

"If you guys want to debate this, please by mindful of the following: your arguments must have some support, and also be mindful of not attacking the opposing debater instead of its arguments" Florence warns.

"Before we propose alternatives, we would like to know the following: are the other sections materially different in what they want out of the course vs this section?" another student asks Florence, thinking the problem's core is the need to get approval from all sections.

"From what I have seen, no. But I am not fully convinced of such"

The students know there's a big disconnect between what they want out of this course vs what they effectively get. We need to suggest an alternative to this poorly designed assignment and that's likely to be accepted by all sections at any cost! Randy thinks, while wondering what could be accepted by all sections. Clearly to everyone's eyes there's a big difference between what's done in class vs in homework.

In all likelihood, Imélie is better than I was at the same date last year, and yet I was good enough for a 90% in the unofficial final, Marcia thinks while realizing Imélie proved almost nothing as a mathlete, only in class. There is no other math team member that took or take the calculus BC course concurrently that isn't already entered into the VMC. But why wasn't I offered this last year? How is Imélie different from my last year's self? The team of last year needed a reservist more than this year's, but would that have pushed me to play better in multi-variable calculus if I effectively was?

Alternatives to this homework that they consider deleterious, students in AP English suggest a baker's dozen. As united as they may be in their opposition to this assignment, this is where their unity ends. Meanwhile Geneviève stays behind and just continues solving the problem of the previous day, which is not always obvious when she hears her classmates scream at each other for solutions to this assignment dilemma. An analysis of one of the readings, business correspondence, and they are tired of hearing about potential solutions. But these are the most popular two choices, and in a choice that appears divisive, but the teacher only gives the opportunity to debate it to this section. Fighting over homework content? It's a very bad teacher but we must endure him for another two and a half months; at least he isn't a tough grader, Randy thinks.

"In the end I believe that it would be better to put the alternative assignment choices up for voting across all sections, in which case the aggregate winner will be the alternative assignment so chosen" the teacher announces, convinced by the students that the homework was problematic.

"We cannot assume that all sections will share the same disgust towards this assignment that we harbor. That being said, this section does not support the current assignment at all" Florence explains herself, even though she secretly knows that the other sections aren't so different from hers.

And, obviously, we have Imélie meeting Vontae during recess. She opens a heart of chocolate so that the two of them can eat it together.

"Since you're new on the team, and officially you're a reserve player, I hope you realize that you have an immense chance to be able to play with your inspiration sources. You might have let a bad experience discourage you once, but here's your chance to prove the contrary of what you lived as a mathlete in the past. And on the national stage, no less! Victory at the VMC will be your ultimate proof to the contrary" Vontae tells his valentine.

"Hearing about your teammates talking about it, the VMC seems to be a contest that looks more like what I'm used to; Gen claimed the final was like taking a multi-variable calculus final at, I don't know, Harvard... in which case I would be no more worried about my success in college than she is, so long as I do well in the final"

"Imélie... the only reason why we needed a reservist for the VMC is because the coach makes an allowance for the case where one between Gen and Krista competes at the EGMO. It would be too tight to compete, make them catch up a week in class and then participate to the VMC final to the coach's eyes"

Ouch. By now even the coach seems to believe in it; yet, until the day the AIME took place, he found the parish media were exceedingly sensationalistic towards them and sold what he called mirages. However, the parish media are now considered as the premier reference for mathletic news in Louisiana... I'm forced to acknowledge we should now treat their chances as nonzero until proven otherwise; even Marcia, Annette or Tara, or these girls from elsewhere in the state are not even in the discussion for this contest! Vontae thinks, while possible « proofs to the contrary » in this context may include a poor score on the AIME or, more likely, on the USAMO.