Volume 2: Chapter 4 – The World Stage
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[Scene: Class 1A – Morning Announcement]
(The classroom is buzzing. Students whisper. Stare at Kenta. He ignores them.)
Takahashi Kenji: (leaning over) "Kenta. Everyone's staring."
Kenta: (not looking up) "They always stare."
Takahashi Kenji: "Not like THIS. This is different."
(The door opens. Professor Hayashi enters. Behind him, the school principal. And a man in a suit Kenta doesn't recognize.)
Professor Hayashi: "Morik. Come with us."
Kenta: (standing) "Again?"
Principal: "This is important, Morik. National importance."
(The class gasps. Kenta follows them out.)
Takahashi Kenji: (whispering) "National importance?! What does that mean?!"
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[Scene: Principal's Office]
(Kenta sits across from the man in suit. Professor Hayashi and the principal stand behind.)
Man in Suit: "Kenta Morik. I'm from the Ministry of Education."
Kenta: (calm) "Okay."
Man in Suit: (impressed by his calm) "You've been selected to represent Japan in the International Mathematical Olympiad."
Kenta: (blinking) "The what?"
Professor Hayashi: (excited) "It's the biggest competition in the world, Kenta! The best young minds from every country! Solving problems that would make PhD students cry!"
Principal: "Japan hasn't won in twelve years. Twelve years, Morik."
Man in Suit: "We've reviewed your record. Your performance at the inter-school competition. Professor Hayashi's recommendation. We believe you're our best chance."
Kenta: (thinking) "World stage. Every country. Pressure."
Kenta: (aloud) "When is it?"
Man in Suit: "Three weeks. In Singapore. You'll be part of a six-person team."
Kenta: "Why me? I'm only a first-year."
Professor Hayashi: (proudly) "Because you're better than any upperclassman we have. Better than any student I've ever taught."
Man in Suit: "We need your answer, Morik."
Kenta: (pause) (thinking) "Father... what would you say?"
Kenta: (aloud) "I'll do it."
(Professor Hayashi grabs him in a hug. The principal cheers. The ministry man smiles.)
Professor Hayashi: (crying already) "THAT'S MY BOY!"
Kenta: (stiff) "Sensei... you're hugging me."
Professor Hayashi: (not letting go) "I DON'T CARE!"
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[Scene: School Gate - After School]
(Kenta walks out. Sui and Yui are waiting. They already know.)
Sui: (screaming) "KENTA-KUN!"
Yui: (also screaming) "THE OLYMPIAD!"
(They crash into him. Hug from both sides.)
Sui: (crying) "You're going to REPRESENT JAPAN!"
Yui: (also crying) "THE WHOLE COUNTRY!"
Kenta: (ears red) "It's just another test."
Sui: "IT'S NOT JUST ANOTHER TEST! IT'S THE WORLD!"
Yui: "And you're going to WIN!"
Kenta: (quietly) "I'll try."
Sui: (grabbing his face) "You don't TRY. You DO. You're Kenta Morik."
Yui: (grabbing his other cheek) "And we believe in you."
Kenta: (looking at them both) "I know."
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[Scene: Kenta's Room - Training Montage]
(For three weeks, Kenta studies. Problems sent by the ministry. Past Olympiad questions. Theories he's never seen before.)
(Sui and Yui are there. Every day. They don't study with him. They just... exist. Bring food. Water. Quiet support.)
Sui: (placing tea beside him) "Break."
Kenta: (not looking up) "In a minute."
Sui: (grabbing his pen) "NOW."
Kenta: (looking up) "You're bossy."
Sui: (grinning) "That's why you love me."
Yui: (bringing snacks) "Both of us!"
Kenta: (almost smiling) "I do love you. Both."
Sui & Yui: (in unison, melting) "KENTA-KUN!"
(He turns back to his problems. But he's smiling.)
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[Scene: Singapore - Competition Day]
(Huge auditorium. Students from fifty countries. Flags everywhere. Cameras. Tension.)
Kenta: (thinking) "This is bigger than I imagined."
Coach: (beside him) "Nervous?"
Kenta: "No."
Coach: (laughing) "You really are something."
(The problems are distributed. Six problems. Four and a half hours for each of two days. Three problems per day. The hardest math problems on Earth.)
Kenta: (reading Day 1, Problem 1) "Interesting."
(He begins.)
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[DAY 1 - PROBLEM 1]
[International Mathematical Olympiad 2024 - Day 1 Problem 1]
Determine all real numbers x for which there exists a real number y such that:
x² + y² ≤ 1 and (x + y)² ≥ 4xy
Kenta: (analyzing) "This is an inequality problem with geometric interpretation."
(He writes rapidly:)
Kenta's Solution:
Step 1: Expand (x + y)² ≥ 4xy
x² + 2xy + y² ≥ 4xy
x² - 2xy + y² ≥ 0
(x - y)² ≥ 0
Step 2: This is always true for all real x, y. So the second condition gives no restriction.
Step 3: The only condition is x² + y² ≤ 1. This means (x, y) lies inside the unit circle.
Step 4: For any real x, does there exist a y such that x² + y² ≤ 1?
Step 5: For a given x, we need y² ≤ 1 - x². This requires 1 - x² ≥ 0, so |x| ≤ 1.
Step 6: Therefore, the real numbers x are those with -1 ≤ x ≤ 1.
Final Answer: x ∈ [-1, 1]
Time: 12 minutes.
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[DAY 1 - PROBLEM 2]
[International Mathematical Olympiad 2024 - Day 1 Problem 2]
Let ABC be an acute triangle with AB > AC. Let P be a point on side BC such that ∠BAP = ∠PAC. Let Q be the point on line AP such that QB = QC. Prove that Q lies on the circumcircle of triangle ABC.
Kenta: (drawing the triangle) "Angle bisector theorem. Isosceles triangle properties. Cyclic quadrilateral."
(He fills pages with geometric proofs. Uses angle chasing. Power of a point. Finally:)
Kenta's Conclusion: "Therefore, ∠BQC = ∠BAC, so Q lies on the circumcircle of ABC."
Time: 48 minutes.
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[DAY 1 - PROBLEM 3]
[International Mathematical Olympiad 2024 - Day 1 Problem 3]
Find all functions f: ℝ → ℝ such that for all real numbers x, y:
f(xf(y) + f(x)) = f(xy) + f(x)
Kenta: (thinking) "Functional equation. Need to find all possible functions."
(He tests constant functions. Linear functions. Uses substitutions: let y = 0, let x = 1, etc.)
Kenta's Solution:
Step 1: Let P(x, y) denote the given equation.
Step 2: From P(1, y): f(f(y) + f(1)) = f(y) + f(1). This suggests f(y) + f(1) is constant.
Step 3: After multiple substitutions and cases, two families of solutions emerge:
· f(x) = 0 for all x
· f(x) = x for all x
· f(x) = c for some constant c? Testing shows only these two work.
Final Answer: f(x) = 0 for all x, or f(x) = x for all x.
Time: 72 minutes.
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[End of Day 1 - Kenta solved all three perfectly]
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[DAY 2 - PROBLEM 4]
[International Mathematical Olympiad 2024 - Day 2 Problem 4]
Let n be a positive integer. A sequence of n positive integers is called "good" if it satisfies:
a₁ = 1, a_n = n, and for each k = 1, 2, ..., n-1, a_{k+1} is either a_k + 1 or a_k + 2.
Find the number of good sequences for each n.
Kenta: (reading) "Combinatorics. Recurrence relation."
(He works methodically:)
Kenta's Solution:
Step 1: Let f(n) be the number of good sequences for given n.
Step 2: For any valid sequence, a_k ≤ k + (k-1)? Actually, since we start at 1 and can increase by 1 or 2, the maximum value at position k is 2k-1.
Step 3: The condition a_n = n means we need exactly the right number of +2 steps.
Step 4: Let t be the number of times we add 2. Then total increase = (n-1) + t = n, so t = 1.
Step 5: Exactly one +2 step among n-1 steps. So the number of sequences = number of ways to choose which step is +2.
Final Answer: n-1 sequences.
Time: 25 minutes.
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[DAY 2 - PROBLEM 5]
[International Mathematical Olympiad 2024 - Day 2 Problem 5]
Let a, b, c be positive real numbers such that a + b + c = 1. Prove that:
a/(b² + c²) + b/(c² + a²) + c/(a² + b²) ≥ 9/2
Kenta: (analyzing) "Inequality. Likely uses Cauchy-Schwarz or Nesbitt-type transformations."
(He tries multiple approaches. Finally finds the key:)
Kenta's Proof:
Step 1: By Cauchy-Schwarz: (b² + c²)(1² + 1²) ≥ (b + c)², so b² + c² ≥ (b + c)²/2.
Step 2: Therefore a/(b² + c²) ≤ 2a/(b + c)².
Step 3: Similarly for other terms.
Step 4: Summing: LHS ≤ 2[a/(b+c)² + b/(c+a)² + c/(a+b)²].
Step 5: But by known inequality, a/(b+c)² + ... ≥ 9/4(a+b+c)? Wait, need lower bound not upper.
(He realizes his approach is wrong. Starts over.)
Step 1 (new): Use substitution: Let x = b + c, y = c + a, z = a + b. Then a = (y + z - x)/2, etc.
Step 2: Also x + y + z = 2(a + b + c) = 2.
Step 3: The inequality transforms to something symmetric.
Step 4: Apply Cauchy-Schwarz in Engel form: Σ a/(b² + c²) ≥ (a + b + c)² / Σ a(b² + c²).
Step 5: Then use known inequalities to bound denominator.
(After two hours of careful manipulation, he arrives at the result.)
Final: Inequality holds with equality when a = b = c = 1/3.
Time: 118 minutes.
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[DAY 2 - PROBLEM 6 - THE LEGENDARY FINAL PROBLEM]
[International Mathematical Olympiad 2024 - Day 2 Problem 6]
Let n ≥ 2 be an integer. A set of n integers is called "balanced" if for any two distinct elements x and y, the number (x + y) is divisible by neither x nor y. Determine all n for which there exists a balanced set of n integers.
Kenta: (reading) "This is... deep. Number theory. Divisibility. Construction and impossibility proofs."
(He stares at the problem for ten minutes. Not moving. Just thinking.)
Proctor: (whispering) "Is he okay?"
Coach: "He's thinking. Let him work."
Kenta: (internally) "Need to find condition on n. Try small cases first. n=2? Choose 1 and 2: 1+2=3, not divisible by 1 or 2? 3 divisible by 1 but 3/1=3, so divisible by x? The problem says 'divisible by neither x nor y'. So 3 divisible by 1 means it IS divisible by x. So 1 and 2 fail."
(He tests more small cases. Patterns emerge.)
Kenta's Solution:
Step 1: Try n=2. Need a,b such that a+b not divisible by a and not divisible by b. This means a does not divide b and b does not divide a? Actually a+b ≡ b mod a, so a divides a+b iff a divides b. So condition becomes: a does not divide b and b does not divide a. This is possible: take 2 and 3. 2+3=5, not divisible by 2 or 3. So n=2 works.
Step 2: n=3? Try to find three numbers. This becomes harder. After testing, possible set: {2,3,5}? 2+3=5 divisible by 5? Actually 5 divisible by 5 means divisible by z, not x or y. Wait careful: For x=2, y=3, sum=5. 5 not divisible by 2 or 3, good. For x=2, y=5, sum=7, not divisible by 2 or 5, good. For x=3, y=5, sum=8, not divisible by 3 or 5, good. So {2,3,5} works! So n=3 works.
Step 3: n=4? Try to extend pattern. Perhaps numbers that are pairwise coprime? Choose primes: {2,3,5,7}. Check: 2+3=5 divisible by 5? Yes! 5/5=1, so sum divisible by z. The condition says "divisible by neither x nor y" but says nothing about z. So 2+3=5, 5 is divisible by 5, but 5 is an element of the set. The problem doesn't forbid sum being divisible by another element, only by x and y themselves. So this might be allowed.
(He re-reads the problem carefully.)
Step 4: The condition only restricts divisibility by x and y. Not by other elements. So sets of distinct primes should work for any n? Check n=4 with {2,3,5,7}: All sums are odd+odd=even? Actually 2 is even, others odd. Need to check all pairs: 2+3=5 (not divisible by 2 or 3), 2+5=7 (not divisible by 2 or 5), 2+7=9 (not divisible by 2 or 7), 3+5=8 (not divisible by 3 or 5), 3+7=10 (not divisible by 3 or 7), 5+7=12 (not divisible by 5 or 7). All good! So n=4 works.
Step 5: It seems all n might work? But wait, need to consider parity issues. If we take all odd numbers, sums are even. But even could be divisible by odd? No, even divisible by odd only if odd divides even. For distinct odd primes, sums are even, but odd primes don't divide even numbers (except 2, but 2 not in set). So seems fine.
Step 6: But there's a catch. For n=5, choose five distinct odd primes. Works. So all n should work? But the problem asks "determine all n for which there exists". If all n work, answer is "all positive integers n ≥ 2".
Step 7: But is that too trivial? There must be a restriction. Check n=2 with {1,2} failed because 1 divides everything. So need numbers >1. So choose numbers >1.
Step 8: For any n, choose n distinct prime numbers greater than 2. For any two primes p and q, p+q is even. An even number cannot be divisible by an odd prime p or q (since p and q are odd and greater than 2). Therefore the condition holds.
Step 9: Therefore, for every n ≥ 2, a balanced set exists. Simply take the first n odd primes.
Final Answer: All integers n ≥ 2.
Time: 142 minutes.
(He writes the final line. Sets down his pencil.)
Kenta: (thinking) "Done."
(He raises his hand.)
Proctor: (rushing over) "Yes?"
Kenta: "I'm finished."
Proctor: (staring at his paper) "All six?"
Kenta: "All six."
(The proctor's face goes pale. Then red. Then he runs to the judges' table.)
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[Scene: During the Competition - Other Contestants]
(Other contestants sweat. Frown. Erase. Cry. Some have given up on Problem 6 entirely.)
Contestant from USA: (head in hands) "I can't even start."
Contestant from China: (muttering) "This is impossible."
Contestant from Russia: (staring blankly) "I've never seen anything like this."
(Kenta sits calmly. Reviewing his work. Making small corrections.)
Coach: (to another coach) "Your student?"
Other Coach: "No, that's the Japanese boy. He finished two hours ago."
Coach: "Two hours?!"
Other Coach: "And his solutions are... perfect. The judges are in shock."
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[Scene: Awards Ceremony]
(The auditorium is packed. Tension high. The head judge approaches the podium.)
Head Judge: "Ladies and gentlemen. This year's International Mathematical Olympiad has been historic."
(Crowd murmurs.)
Head Judge: "For the first time in twelve years, the final problem has been solved. Completely. Elegantly. Perfectly."
(Gasps.)
Head Judge: "The solver achieved a perfect score. 42 out of 42. Only the seventh perfect score in the history of this competition."
(The Japanese team grips each other. Coach is praying.)
Head Judge: "The gold medal winner, representing Japan... KENTA MORIK!"
(The auditorium ERUPTS. Kenta stands. Walks to the podium. Calm. Composed.)
Coach: (screaming) "THAT'S OUR BOY!"
Team Members: (hugging each other) "HE DID IT! HE ACTUALLY DID IT!"
(Kenta receives the gold medal. The trophy. A standing ovation from fifty countries.)
Head Judge: (shaking his hand) "Young man, that was the most brilliant performance I've seen in forty years. Your solution to Problem 6 was... elegant. Beautiful. The committee is still discussing it."
Kenta: (quietly) "Thank you."
(He looks at the medal. Thinks of Sui. Yui. Professor Hayashi. His father.)
Kenta: (thinking) "I did it. For all of you."
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[Scene: Video Call - That Night]
(Kenta's phone buzzes. Sui and Yui's faces appear. Both crying.)
Sui: "KENTA-KUN!"
Yui: "YOU WON! YOU ACTUALLY WON!"
Sui: (screaming) "PERFECT SCORE!"
Yui: (also screaming) "THE WHOLE WORLD!"
Kenta: (smiling) "I told you I would."
Sui: (sobbing) "We're so proud! The whole school is going CRAZY!"
Yui: (also sobbing) "Professor Hayashi is crying in his office! He's been crying for HOURS!"
Kenta: (almost laughing) "That sounds like him."
Sui: "When are you coming home?!"
Kenta: "Two days."
Yui: "WE'LL BE AT THE AIRPORT! WITH SIGNS! AND CONFETTI!"
Kenta: "You don't have to—"
Sui: "WE WANT TO!"
Yui: "THE WHOLE SCHOOL WANTS TO!"
Kenta: (ears red) "Okay."
Sui: (kissing the camera) "WE LOVE YOU!"
Yui: (also kissing the camera) "OUR WORLD CHAMPION!"
Kenta: (smiling) "I love you both too. See you soon."
(He hangs up. Looks at the medal in his hand.)
Kenta: (thinking) "Father... I did it. For you."
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[Scene: Airport - Return]
(Kenta walks out of customs. The terminal is PACKED. Students. Teachers. Reporters. Cameras.)
And there, at the front, Sui and Yui. Holding a massive banner:
"WELCOME HOME, WORLD CHAMPION! WE LOVE YOU!"
Sui: (screaming) "KENTA-KUN!"
Yui: (also screaming) "THERE HE IS!"
(They run. Crash into him. Hug.)
Sui: (crying) "YOU DID IT!"
Yui: (also crying) "OUR KENTA-KUN!"
Kenta: (holding them) "I'm back."
Professor Hayashi: (pushing through the crowd, tears streaming) "KENTA! KENTA MORIK!"
(He grabs Kenta in a bear hug. Sobbing openly.)
Professor Hayashi: (choking) "Perfect score! PERFECT SCORE! No one has done that in twelve years! And you're MY student! MY STUDENT!"
Kenta: (stiff but warm) "Sensei... you're crying again."
Professor Hayashi: "I'VE BEEN CRYING FOR TWO DAYS! I DON'T CARE!"
(The crowd laughs. Cheers. Cameras flash.)
Reporter: "Kenta! How does it feel to be a national hero?!"
Kenta: (thinking) "National hero?"
Kenta: (aloud) "I just solved problems."
Sui: (grabbing the mic) "HE'S BEING MODEST! HE'S THE BEST!"
Yui: (grabbing the mic too) "THE SMARTEST! THE MOST WONDERFUL!"
Kenta: (ears red) "Can we go home now?"
Professor Hayashi: (laughing through tears) "Yes. Let's go home. Champion."
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[Scene: School Assembly - Next Day]
(The entire school is in the gymnasium. Kenta stands on stage. Gold medal around his neck. Trophy beside him.)
Principal: (at the podium) "Today, we honor Kenta Morik. The first Japanese student in twelve years to achieve a perfect score at the International Mathematical Olympiad. The first in our school's history."
(Thunderous applause. Standing ovation.)
Principal: "Kenta, would you like to say a few words?"
(Kenta approaches the podium. Looks at the crowd. Finds Sui and Yui in the front row. Crying. Smiling. Waving.)
Kenta: (quietly) "I didn't do this alone."
(The crowd quiets.)
Kenta: "My father taught me to love the stars. Professor Hayashi gave me impossible problems. And two people... two very loud, very bright people... believed in me when I didn't believe in myself."
(He looks at Sui and Yui. They're sobbing.)
Kenta: "This medal is for all of you. For Japan. For my school. For my family. For them."
(He holds up the medal. The crowd erupts.)
Sui: (yelling) "THAT'S OUR BOYFRIEND!"
Yui: (also yelling) "THE WORLD CHAMPION!"
Takahashi Kenji: (bawling) "I'M SO PROUD TO BE HIS DESK NEIGHBOR!"
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[Scene: Kenta's Room - Night]
(Four items on his desk now: Father's astrolabe. Competition trophy. Professor's medal. And now, the Olympiad gold medal.)
Kenta: (looking at them) "Four."
(He picks up his notebook. Flips to the pages where he solved Problem 6. Reads his solution again.)
Kenta: (thinking) "All integers n ≥ 2. So simple. So beautiful."
(His phone buzzes. Two messages.)
Sui: "Kenta-kun... today was perfect. You're perfect. We love you. Sleep well, our world champion. "
Yui: "My prince~ you made history. But to us, you've always been a legend. Congratulations, champion. Goodnight!"
Kenta: (smiling) (typing back to both): "Goodnight. Thank you for everything. For being there. For believing. I love you both. More than any medal."
(Immediate replies:)
Sui: "HE SAID MORE THAN ANY MEDAL!"
Yui: "WE'RE MORE IMPORTANT THAN GOLD! PROGRESS MAXIMUM! "
Sui: "SWEET DREAMS OUR CHAMPION!"
Yui: "DREAM OF US! (AND YOUR NEXT VICTORY!) "
Kenta: (putting down the phone) (looking at the four items on his desk)
Kenta: (thinking) "Father. Professor. Sui. Yui. All of them. All believing."
(He closes his eyes, smiling.)
Kenta: (thinking) "I'll never stop. For them."
(He falls asleep, at peace.)
End of Chapter 4
Ready for Chapter 5?