By the end of January, we see that both stars expected to qualify for the USAMO have patiently waited for the announcement of the team result in the intercom during the AP Physics course:
"Good morning, everyone! We successfully defended our state title at the Vans Math Contest, formerly known as the Square Root of the Answer! The VMC contestants are summoned to the mathematics department office with their shoe sizes, and they will receive their prizes in one or two business days" the principal announces on the intercom.
"Shoe sizes? What are yours?" Imélie asks.
"Ten for me" Geneviève answers.
"Eight" Krista answers.
"But why are they asking for the contestants' shoe sizes?" Imélie continues, without having officially entered the VMC. "If you won pairs of shoes..."
"I remember this sequence last year. I would love something other than Skate Clouds this year; I still wear the Skate Clouds that Cory's dad gave me when we won last year"
"That's the prize of winning at the VMC's state phase?" asks Krista, surprised of what she won at this tournament.
Marcia wears 9, Cory, 11 and Vontae, 12. Then their shoe sizes form an arithmetic sequence. Gen, you have more charisma than all other female mathletes that can win it all at the VMC, or that could qualify for the EGMO or the IMO! Trent's words resonate in her head, with a falsetto voice. More charisma? Mathletic contests aren't won with charisma, and certainly not the EGMO or the IMO! she thinks, while, upon arriving at the mathematics department office, Trent shows a catalog of Vans shoes on a tablet to both girls. If Krista very rapidly chooses Vans Skate Clouds, because she wants something other than these strong safety cleats that Ted gave her, Gen, however, wanted something other than that.
In the meantime, in the neighboring parish, Tara and her math teammates are pleasantly surprised to see the neighboring parish qualify, and she wonders what the rest of the year will be made of, beyond intensive practices of multi-variable calculus.
"We qualified for the Vans Math Contest final, and we are asked our shoe sizes? They dared wait until this point to ask us that?" asks one of the boys on the team.
"Enjoy these free shoes. We'll only have one pair of shoes so choose wisely" Tara advises her teammates. "Don't be barefooted cobblers..."
"Tara, how do you know we win free shoes?" Trevor asks Tara.
"You didn't forget that I date a footballer at another school at the final? There all their mathletic successes put people achieving them under the spotlight! And he talked about how his teammates dated female mathletes..."
"Venomous Agendas fans seemed to treat the first VMC round as a formality, and, if they didn't qualify for the final, it would be a massive disappointment for them" Annette then tells the group.
"Our state will have three teams at the final. They publicly announced the finalists in a list sorted by state and then in alphabetical order. Benjamin-Franklin will also be at the final" another neighboring parish guy shows them the list of finalists.
Some VMC finalists are also names that people in both parishes recognize because they were in the Math Madness as well: University High, Village School, Whitman, Georgetown Prep, TJHSST, to name only some of them. Several mathletics powerhouses. Clearly a high level of competition await the Venomous Agendas and the neighboring parish. When Zack sees Tara again the next day...
"Oh Tara, when I think about it, I can definitely say that with you, it's like dating one of the mathletes my teammates date"
"Even though I'm qualified for the grand final of the Vans Math Contest, look at this new pair of shoes I won!" Tara points at her shoes.
"Vans Skate Clouds! All three wear these at our school; strangely this line of shoes seems to be fashionable among the girls at the VMC's national final... Otherwise it fits you well. Not like Krista with her football cleats!"
"I suppose you know your shoes..."
Marcia is with Randy, who has a confession to make about the VMC's role in his successes in the single-variable calculus course.
"Regardless of the warnings issued by your teammates, dating you forces me to make an effort in class. And taking part in the VMC for extra credit, too. As of today, I have an A in calculus BC" Randy then tells Marcia about Calculus BC.
"When I started out as a tutor of the football team, you thought that I was simply distributing answers to homework and the cats got your teammates' tongues faster than they actually did. If you told me a year ago that you would get an A in calculus BC, I would never have believed you. But now that you're there, ask me for help in multi-variable calculus if you need to, and you can take it easy on the final in May" Marcia confides to him about her past as a tutor.
"You push me hard, granted, not as hard as Gen and Krista mentally push each other, but you're very direct. At least when I feel forced to give my tongue to the cats"
"We won't be able to see each other as often as before because this month represents the high season for me as an ACT semi-private tutor. Nevertheless, starting next week, if you know you will need multi-variable calculus in college, you can continue going to the practices of the team. You grew a lot academically, and I would be lying if I said I left your teammates indifferent in class"
"On that count, however, Ted was very lucky to be able to date Krista at the right time, but I might be wondering why Zack was willing to date a girl in the neighboring parish, when Imélie was probably within his league"
"Enough talk of this, I just want another kiss" Marcia announces before kissing Randy. "For the rest of the year, we will train for the VMC final without Gen and Krista: they can get by with only one week of review"
At the same time, both girls prepared popcorn for four people while they are getting ready to watch X+Y with their respective boyfriends at Krista's home. And with hot chocolate cups ready to infuse if they're thirsty.
"Before the film starts, I must say that you both made a lot of mathematical progress. When I first met you, you said you were lost in the multi-variable calculus material; now you're doing it" Gen speaks while her eyes are on Cory.
"It would never have been possible without you two" Ted adds.
"Ted, I accepted to go to a football game for you, I found that a little mind-numbing. You will all have a better idea of what we will go through in the coming months after having watched X+Y" Krista tells him back.
"We already know; on Boxing Day I saw you calculate curvatures, torsions, Frenet frames pedal curves, evolutes, the whole kit, without complication" Krista's dad tells both boys.
"Dad? You want to watch a film with us? Our coach recommended that to us"
"It's a real warning for anyone dreaming of the EGMO or the IMO. I think personally that the parish newspaper exaggerates but still..." Gen issues her warning to Krista's dad.
From a purely mathematical standpoint, it's a warning. But most people will watch X+Y not for the mathematics, but as a drama, she thinks, while both girls are perplexed by their respective boyfriends' reactions to the film.
But why is she making me watch X+Y as Éliane made Curtis watch Mean Girls when he dated her? Ted wonders, when they aren't even halfway through the film. Cory, however, can't help himself but wonder how Gen, and Krista as well, differ from Zhang or the other girl at the training camp, or even from other female competitors at the IMO, real or fictional.
"Now, a practice question: find the solutions to this system for integer a, b and c: 2^a+3^b+1=6^c" Krista asks the people in the room.
"What's this?" Ted asks. "I don't remember being asked this on the Math Madness! And it's definitely not a VMC question"
"You know the AIME has two constraints on the answers; we talked about this when the manifest of the qualifiers was released! It looks like a question from a proof contest instead!" Cory points out.
"First step: subtract one from the left-hand side. We then have 2^a+3^b=5(6^(c-1)+...+1)" Gen starts solving the problem.
"But this also means that the left-hand side of this new equation must be a multiple of five" Krista adds.
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"The big question is: how to ensure that we have a multiple of five? One must wonder how to exploit the cycles of the unit digit on the left-hand side so that their sum can be either five or fifteen" Gen continues.
For 2^a, we get 2, 4, 8, 6 in a cycle. The same thing for 3^b gives 3, 9, 7, 1, cyclically also. Of all possible combinations of sums of these quantities, the only ones giving a multiple of 5 are 2+3, 7+8, 1+4 or 9+6, therefore we must have the same parity on a and b.
"But if we take the original equation, the sum must be a multiple of six, which excludes any solutions with fifteen, sixteen cannot be a multiple of six" Krista points out.
"At the same time, for c equal or greater than three, the solution must be a multiple of eight because you can rewrite six to the c as two to the c times three to the c"
"But why are you looking for a multiple of eight?" Cory asks, perplexed.
"You're way too fast to be true!" Ted exclaims.
"Suppose there is a solution for c equal or greater than three. The left-hand side must also be a multiple of eight as well. But, under modulo eight, a power of three can only take the values one or three, and a power of two is a multiple of eight for a equal or greater than three" Gen answers Cory's question.
"It follows the only way to get a multiple of eight on the left-hand side requires b equal to two, which is in contradiction with the previous step since having three modulo eight on the second term requires an odd power of three. Conclusion, c can only take the values one and two" Krista draws her conclusion.
And the three combinations are then (1,1,1), (5,1,2) and (3,3,2).
"Even if we were aiming for a proof contest, I think a lesson on proof methods in greater depth than what was given to us in the geometry course will be useful. And will be useful to everyone attending the math team practices" Gen suggests the people in the room.
"How will that serve us at the VMC final?" Cory asks.
"If, at the VMC final, there was a question of the kind « Show that, if a condition X holds, we get property Y », you will not be lost. Long story short, in such a case, the solution must be clearly written. Last year, the SRA final was an amalgamation of past year multi-variable calculus finals question of prestigious universities" Geneviève explains regarding Cory's question, but Randy could benefit as much as Krista, just in a different context.
The five team members, especially both stars, are in a hurry to complete their homework in their other courses with all speed on Saturday morning so they can study for the AIME to the extent possible. And not just by solving problems from other playoff games, or past problems on the AIME. I must not stress out over the 11 points goal, which I know will make me qualify for the USAMO no matter what's happening, but at 133.5, in a bad year, an 8 would suffice to qualify. That said I must leave absolutely nothing to chance, and Krista either. This year, it took 94.5 and 90 points to go to the AIME depending on whether we took the A or B version of the AMC12 respectively, she thinks, while, next Tuesday, she knows, deep down, there will be at most three that could qualify for the USAMO. Marcia must then accomplish a miracle to qualify since Marcia needs 2 more points on the AIME, and 11 points on the AIME are all but trivial to score.
Speaking of the AIME itself, which forced all participants to miss an entire afternoon of classes, it becomes clear who took part in the Math Madness playoffs and who didn't. The problems are, yes, on the same level, as playoff questions, but whose who didn't participate must go slower, like Randy.
Those who participated fly through 3-4 questions, solving these initial questions as rapidly as in the playoffs, but get stuck in the second third for some of them. Including, unfortunately, Cory or Vontae. Marcia holds on, and resists the temptation to panic, to which Randy struggles against. It goes without saying that Gen and Krista are in a much better state of mind than even Marcia! They are driven by a burning desire to break their personal records while several contestants has for sole experience of the competition a handful of questions belonging to the first half that are asked in another competition, and only one question from the second half. The question that caused the Venomous Agendas' loss.
I am the math geek of the team, even if I no longer go to the off-season practices, unlike Ted... aargh! What's happening to me? What are these problems? Did I attend all these training sessions of the math team for nothing? Randy thinks while he is about to give his tongue to the cats after not even an hour. Last year's male mathletes proved that success in calculus didn't guarantee that one can succeed in these competitions! Is that how my mathletic career will end, as ephemeral as it is? he keeps thinking while he hyperventilates and trembles at the same time as he starts feeling powerless over some of the remaining problems. This is worse than us vs Iowa last year; am I paying the price for my mathletic inexperience?
At the same time as his worst football memories resurface, Randy seems to struggle to even hand in his copy because he gets dizzy at every step he makes. He suddenly remembers being unable to counter Iowa's wide receiver and, on multiple occasions, being powerless in plays leading to touchdowns in an atrocious loss of 77-0 in the second game of the 2022 season.
Meanwhile, in the calculus BC lecture that students in the competition missed, the substitute wonders what happened with Cory, knowing that the regular teacher is proctoring the AIME with ten students and soon fewer. And the substitute knew nothing about the calculation of definite integrals...
"The reason why I teach the calculus BC course this afternoon is because the regular teacher proctors the AIME, and it must be said, our very own Venomous Agendas are participating. It's also the reason why Cory is absent, and Randy as well. Since I cannot help you with the material, I suggest reading the pages in your textbook on the material of the day, partial fraction integration as well as trigonometric substitution" the substitute teacher announces in front of the class.
"Since you know nothing about the material, I must go up front. For fractions sometimes it can be useful to separate a big fractions in several sub-fractions whose integrals are easier to calculate. And, since the integral of a sum is the sum of the individual integrals..." Imélie explains while the substitute wonders how she decompose fractions in its partial fractions so quickly.
"Imélie, if you are so good, then why aren't you on the math team?" the substitute proceeds to ask her.
"I was part of it two years ago and I was traumatized by the AMC-twelve. It was unacceptable to me at the time, but since the time I realized even our stars aren't perfect..."
"This bad experience happened before this stuck-up girl became stuck-up!" Another student claims by referring to Geneviève. The mathletic season was very short back then.
"No, Gen might be a girl that assumes herself intellectually, but she is all but stuck-up! She is an academic inspiration source for all of us, especially in classes where we need mathematics!" Imélie exclaims. "With her future Venomous Agendas classes will know that girls can perform in math and sciences!"
"You start to sound like Tara at the dinner..." another student talks behind her back.
"Tara only talked about stereotype threat, as how one doesn't perform as well when the stereotype is invoked" Imélie retorts.
Oh... for each girl like Gen there are other girls that just gave up faced with the difficulties of mathletic contests! Just how much can Imélie reveal about women's mathletics! She might be a girl on the math and science part of the infamous « priority sorting list » the administration uses to ensure that the students most likely to finish atop a course are not all in the same section, she appears to lack resilience, the substitute thinks, believing Imélie can still save the pedagogical deal.
After some explanations about trigonometric substitution for integral calculations from Imélie, the substitute comments on the previous math team coach.
"Last time I replaced a teacher for a math lecture, I replaced the previous math team coach before he left to teach at Permian. But that was years and years ago"
"Ah! Everything is clear now! Never Permian would have played us in the Math Madness regular season, if it wasn't personal vengeance of a teacher or an administrator! From everything I heard about Permian, I think it's first come, first serve to coach academic teams there" another student adds his remarks.
Randy was, yes, the first to abandon the contest after less than an hour, so he only missed one period of class. But other participants, such as the infamous « second line representative » of the team in the Math Madness, do not delay doing the same. Especially not when the bell rings and the second line representative admits defeat. But the participants holding on know it's the regular bell and there's still time remaining.
For most people in the mathletic world, the image they have of elite female mathletes is they come from elite institutions like, you know, TJHSST... decidedly we are by far the weakest school to even be in the EGMO discussion, at least if we talk about schools on an overall level, Trent thinks while the parish newspaper speculates disproportionately about the school's two stars about the EGMO and the IMO.
It started with a line or two in an article that covered the end of their Math Madness run and, in the latest edition, the parish newspaper plays the comparison game with past participants of the state at the USAMO, as well as with girls likely to qualify for the EGMO or the IMO, irrespective to their residency or their home institution.
But after halftime there are only the five members of the great team for the VMC final remaining; the other five all gave their tongues to the cats. Halftime marked by the starting bell of the 2nd course they are missing. More than ever both stars work harder, but the other three start showing signs of weakness.
An attrition race is already underway, with Vontae giving up after spending nearly two hours, followed by Cory, who gives out almost three minutes later, and none of the two harbor any illusion about the next step. The last hour happens in a room that's a shadow of its former self, Trent alone with the three girls that made the pride of the parish at the Math Madness, the AMC12 and the VMC in chronological order. All three girls that made so many in the parish fight hard. Even Marcia suddenly starts believing she has a chance to go to the next stage, but the last problem of the contest bring her back to the harsh reality.
How did I solve a #15 problem on the AIME that quickly in the quarter-final game against Whitman, lost by an irreducible fraction of a second, and this problem #15 seems to take an eternity? And once again there is no guarantee that I can solve it. There is no shame in abandoning before the end on a contest like this one, Marcia thinks, while this painful memory from the playoffs resurfaces. And Whitman has lost by an irreducible fraction of a second too, but against University High, in the semi-final, and Adlai Stevenson High, champions of the Midwest, won it all. It may very well have been us in the same position! she thinks before snapping out of it, while the other two girls appear to be very focused and determined to solve this ultimate problem #15.
But even Marcia's return to reality seems to be short-lived, because the more she tries to solve this problem #15, the more she sweats. And she has a heat stroke on top of that: not good for her! Ouch, ouch and re-ouch. She simply picks three digits randomly for the question before her overheating brain starts to fry. Poor girl! She loses consciousness shortly afterward and Trent is forced to pick the copy from her player's hands.
And with very little time left, the other two girls, visibly tired, finished solving the tournament's ultimate problem. With a sense of accomplishment, they turn to their teammate that lost consciousness. Still reeling in, Marcia hears her teammates closing in on her:
"Where am I?" Marcia asks.
"You're still in the room where we took the AIME. You must have lost consciousness and panic" Geneviève answers her while Krista realizes that Marcia has a fever.
"My brain is fried... I need rest and cold drinks" she adds in a hoarse voice.