Jack won his round easily, without any complications. After another round, the final finally arrived.
The two finalists were Dan and Jack; the recognition for the highest level of mathematical knowledge already belonged to Preparatory School Nada, but the two of them would compete to decide who would be the winner.
Both stepped onto the stage at the same time.
—At last we will know who is the best.—said Jack.
Dan stood at his lectern facing Jack.
—Yes, just don't complain when I beat you.
—Beat me? Do you really think you can beat me? Tell me, how many times have I already defeated you in competitions like this?
Dan rolled his eyes.
—Ah… Well, that is in the past, so it no longer matters.
(By the way, the organizers replaced the tables with lecterns for the final so that everything could be seen more clearly.)
—Are both participants ready?—asked the teacher in charge of this round.
—Yes.
—Yes.
—Very well, let's begin!
Noemí watched from below beside the teacher; she wanted to see which of the two was better, and since she could not choose between them, she decided to support both.
—First problem: Prove that
p(x) = x³ + 3x² − 4x − 12 is divisible by (x + 2).
Dan pressed the button first, making a small sound.
—Very well, what is your answer, Dan?
Dan showed his sheet of notes.
—Quotient: x² + x − 6
—CORRECT. One point for Dan.
Dan looked at Jack with a mocking smile for having taken the lead; Jack frowned when he saw that smile.
—Second problem: Solve 2sin²x + 3cosx − 3 = 0 in the interval [0, 2π).
Jack pressed the button faster than Dan.
—x = 0, π/3, 5π/3.
—CORRECT. One point for Jack.
He returned the same smile to Dan, which made Dan's eye twitch slightly.
—Third problem: Equation of the tangent line to the circle x² + y² − 6x + 4y − 12 = 0 that passes through the point (8, 0).
Dan pressed his button first.
—Slopes m = 3/4 and m = −21/4
—Correct. One point.
—Fourth problem: Calculate the limit:
lim (x→0) [√(1 + x) − 1 − (x/2)] / x²
Jack pressed his button.
—−1/8
—Correct. One point.
—Fifth problem: An urn contains 5 red balls, 4 blue balls, and 3 green balls. Find the probability of drawing exactly 2 red balls and 1 green ball when taking 3 balls out without replacement.
Dan was faster.
—3/22
—Correct. One point.
—Sixth problem: Find the sum of the first 20 terms of the series ∑ (3n + 1) / (n(n + 2)).
This time it was Jack.
—Approximately 7.85
—Very good. One point.
—Seventh problem: Solve the inequality (x² − 5x + 6) / (x − 4) ≤ 0
—x ∈ [2, 3]
—Correct. One point for Jack.
—Eighth problem: Given z = 1 + i√3, calculate z⁵.
—16 − 16i√3
—That is right. One point for Dan.
—Ninth problem: Given vectors a = (2, −3, 1) and b = (1, 4, −2). Find the unit vector perpendicular to both.
—(2, 5, 11) / (5√6)
—Excellent. One more point for Dan.
—Tenth question: Find the critical points and classify them for the function f(x,y) = x² + 4xy + 3y² − 6x − 12y + 5.
—Relative minimum at the point (3, 0)
—One point for Jack.
Noemí was impressed by their level of knowledge.
—They are amazing, they are so evenly matched.
After eight more questions, the score stood as follows: 9 points in favor of Jack and 9 points in favor of Dan.
—Very well, this will be the final question to decide who will be the winner.
Dan and Jack stared at each other; Dan's hand squeezed gently around the pen he was holding. Jack inhaled and exhaled to focus his mind.
—Final question!... Calculate the limit:
lim (x→0) [ (sin(x) − x + (x³/6)) / x⁵ ]
Both began writing on their sheets of paper, their hands moving quickly and their brains working at full capacity.
And the first one to press the button was...
—What is your answer... Dan?
—1/120
Jack stopped writing and looked at the teacher.
—That answer is...
Correct.
—Dan has won this competition.
Dan clenched his fists and smiled broadly, closing his eyes.
—Yes!
Noemí gave a small smile and began to applaud, happy for Dan and also for Jack for having made it all the way to the final.
The other students from the different participating schools began to applaud Dan's victory.
—Congratulations... Dan—said Jack, trying to hide his frustration.
Dan received his winner's trophy.
—At last I have beaten you!—he continued smiling happily.
—Yes, at last. Well, with your victory it stands at five wins for me and one for you.
Dan rolled his eyes.
—Don't remind me, let me enjoy this at least once.—He shrugged his shoulders.
Jack received his medal for second place.
—But I assure you that in the next competition we face each other, I will beat you easily.
—Yes, yes, of course.
Jack walked down from the stage, and Dan murmured to himself.
—When he says it like that, it sounds frightening...
Dan also stepped down from the stage, and Noemí approached him.
—Many congratulations on your victory, Dan.
Dan blushed slightly and looked away.
—T-thank you very much, Noemí.
Jack walked over to them.
—Well, you are turning red.—he said to Dan.
Noemí turned toward him.
—Ah, congratulations to you too for reaching the final.
—Thank you. What do you say if on the way back I tell you how I studied for this competition?
Dan frowned and smiled, trying to hide his annoyance.
—How about we sit in the same seats we had on the way here instead?
The teacher approached the three of them.
—As you wish. But we should get going already; it is getting late.
—Whatever you prefer—said Jack.
On the trip back, Dan and Noemí sat together again. Noemí struggled to keep her eyes open; she felt very tired.
Dan noticed this.
—If you are tired, you should go to sleep.
—Yes... You are right.—Noemí closed her eyes and began to drift off.
Dan stayed still, letting her rest. Noemí soon fell fast asleep; unable to stay upright, she began to lean to one side.
Dan grew concerned; nervously, he slowly raised his hand to support her by her right cheek and gently guided her to rest her head against his shoulder.
Noemí did not wake up, remaining calm and deeply asleep. She snuggled softly against his shoulder.
Dan tried to tremble as little as possible so as not to disturb her, though he was a bundle of nerves inside.
The journey continued peacefully.