"What? Someone solved the problem on the little blackboard in the hallway?"
"Huh? Really? Let's go take a look!"
In every senior-year classroom at USTC High School in Fei City, similar conversations took place.
For a moment, all the seniors who had been diligently reviewing for the college entrance exam just over three months away stopped studying at the same time. They rushed out of their classrooms and gathered at a corner of the academic building's hallway.
A little blackboard stood in that corner, with a math problem written on it.
For a finite nonempty set H composed of positive integers, let gcd(H) denote the greatest common divisor of all elements in H. Prove that for every finite nonempty subset A of positive integers, ∑N∈A(-2)^|N|-1gcd(H)>0
The little blackboard and the problem on it had appeared in that corner that morning. Throughout the morning, nearly every senior had come over to look and try to solve it.
Yet before evening self-study began, not a single person had completed this "bounty."
The problem was extremely difficult, fully reaching the level of a Math Olympiad problem. Without specialized Math Olympiad training, it was basically impossible to solve.
And even with Math Olympiad training, it was still impossible without sufficient skill—students who had participated in Math Olympiads and even won awards in last year's Chinese High School Olympiad Mathematics Competition had admitted that they could not solve it either.
But at this very moment, the little blackboard that had originally contained only the problem now displayed a neat proof.
And it was remarkably brief—only six lines.
"Brilliant! So that's how it can be solved!"
"What's this Bézout's Identity used here? How come I've never even heard of it?"
"Huh? What's so brilliant about it? Bro, explain it to me. I can't understand any of it!"
The students gathered around discussed it animatedly. Those who understood wore looks of amazement, while those who did not were utterly baffled.
People were different. It was perfectly normal for some to understand while others did not.
But at this moment, whether they understood it or not, all the students present shared one thing in common: before seeing this answer, none of them had been able to solve the problem. That made an even sharper contrast with the student who had solved it.
"That's incredible. They solved it just like that. Does that mean they have a chance to take part in USTC's independent admissions?"
"Yeah. After all, Professor Liu from USTC came up with this problem. He said that as long as one of us could solve it, he'd be willing to give that student a chance to take part in USTC's independent admissions, with the possibility of being admitted at a reduced score during the exam."
"That's way too sweet."
The seniors present all showed envious expressions.
As students of USTC High School, they were naturally very familiar with the University of Science and Technology of China. As one of the country's top universities and a member of the C9 League, USTC held immense appeal for students whose grades were not stable enough to guarantee admission into top universities.
Getting into USTC through independent admissions would obviously be easier. Not to mention that solving this problem might also give them a chance to be admitted with a reduced score during USTC's independent admissions exam.
More than ninety percent of the students present were surely tempted.
But there was only one question now—who exactly was the student who had solved the problem?
They could see only students watching. No one stepped forward to claim they had solved it.
As for pretending?
Sorry, there was a surveillance camera right overhead. They were not idiots.
"The teachers are here. Everyone, make way."
Just then, several teachers arrived from behind the gathered students. The students quickly parted, allowing the teachers to make their way inside.
The students all knew the teacher leading them. He was their Grade Director, Chen Liang. Wearing a broad smile, he said to a middle-aged man beside him, "Professor Liu, don't forget what you said yesterday. Give the student who solved that problem an opportunity."
Clearly, the middle-aged man beside him was Liu Bin, the professor from USTC.
Liu Bin said with a chuckle, "Director Chen, I definitely wouldn't lie to you about this. Given the relationship between our two schools, resources will naturally be tilted toward your side. As long as that problem was truly solved, I certainly won't go back on my word."
"That's good, that's good."
Chen Liang laughed heartily and nodded.
For a Grade Director like him, sending one more student from his school to a university like USTC meant one more achievement credited to him.
By then, they had finally passed through the crowd of students and arrived in front of the little blackboard.
Liu Bin stepped forward and began examining it.
He had written this problem. Besides being a mathematics professor at USTC, he had another identity: a member of the Chinese Math Olympiad Exam Question Committee.
Every year, the Chinese Mathematics Olympiad included problems written by him, and he had also participated in the question-setting work for the international IMO.
So he knew very well just how difficult the problem he had written was.
Otherwise, how could so many talented students at USTC High School have been stumped by it for nearly an entire day?
Still, what pleased him was that the problem had ultimately been solved.
And the solution was...
Exceptionally elegant!
The standard solution he had written himself took ten lines to prove it, while the little blackboard before him used only six.
Moreover, his own proof had not used Bézout's Identity.
To others, the most ingenious part of this proof lay in its use of Bézout's Identity. But as a mathematics professor and veteran problem setter, Liu Bin could see another brilliant aspect: Bézout's Identity had been inserted at exactly the right point, making it possible to complete the proof in only six lines.
Simple, clear, and... a dazzling display of skill!
"Very good. Very impressive."
Liu Bin nodded in satisfaction, then turned toward the students and asked, "Which student solved this problem? You should know the reward for solving it. Come explain your line of thought to me now, and the reward will be yours."
The students immediately stirred again, all turning their heads and casting envious looks around them as they searched for someone raising a hand.
Yet even after quite a while, no one stepped forward.
The atmosphere became awkward for a moment.
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