Chapter 58 - 46: Witness Dong’s Connections
Han Sibo looked at Liang Xiaodong’s dejected expression and seemed to feel a bit guilty. After some thought, he said, "How about this? I’ll go ask Ge Chi from the class next door. His math is better than mine, so he might already have an idea how to solve it. As soon as we figure this out, I’ll go play badminton with you."
However, at the mention of the name "Ge Chi," the expression on Liang Xiaodong’s face changed in an instant.
He clapped a hand on Han Sibo’s shoulder, stopping him as he was about to get up.
"Hold on!"
Ge Chi was from the class next door. His grades were on par with Han Sibo’s, and in math, he was even a little stronger. He was one of the top math students in their year.
Liang Xiaodong didn’t know Ge Chi very well.
Han Sibo, on the other hand, would sometimes compare answers and discuss problems with Ge Chi in the hallway after exams.
At times like that, Liang Xiaodong could only stand off to the side, bored out of his mind, unable to chime in, feeling like a useless third wheel.
Many teenagers go through a phase with some strange hangups.
When they see their best friend chatting up a storm or having a great time with someone else, they get inexplicably annoyed.
"Don’t bother with him," Liang Xiaodong said stubbornly, puffing out his chest. "It’s just one stupid math problem, right? I can solve it too! Let me see it!"
Hearing this, Han Sibo turned his head and gave him a look that clearly said, ’Are you sure?’
Han Sibo knew his best friend’s math abilities all too well.
On regular exams, he’d consistently score just a little over 100.
Although, on that one pop quiz a month ago, it was like Liang Xiaodong had flipped a switch. He’d turned in his paper first and even gotten the highest score in the class, a real moment of glory for him.
But the glory was short-lived. On this final exam, he was back to his old self, with a score as unremarkable as ever.
Still, seeing Liang Xiaodong’s defiant look, Han Sibo didn’t have the heart to call his bluff. He just silently pushed the worksheet with the problem over to him.
"Here. This is the one."
[Given that the non-negative numbers a, b, c, and d satisfy a + b + c + d = 6, prove that (a - b)(a - c)(a - d)(b - c)(b - d)(c - d) ≤ 27.]
"..."
A subtle, awkward silence suddenly fell.
Liang Xiaodong stared at the problem. Twenty seconds later, he decisively gave up.
He silently picked up his phone, still holding out hope. ’Maybe some guru posted the solution online?’ he thought.
"Don’t bother searching," Han Sibo’s voice drifted over. "If you’re looking online, I already have. There’s nothing. And by the way, I also tried using an AI. Didn’t work."
Liang Xiaodong froze, then sullenly put his phone down.
His eyes darted around for a moment before a thought suddenly struck him. He slapped his thigh.
"Don’t worry! Your bro has connections. I’ll find someone to help!"
"You’re not thinking of asking your grandpa, are you?" Han Sibo guessed. He knew Liang Xiaodong’s grandfather was an academician at the National Academy of Sciences, a titan in the academic world.
"Nope," Liang Xiaodong said, shaking his head.
He might love to fool around, but he was quick-witted and knew how to get his priorities straight.
He’d just heard his dad mention yesterday that his grandfather had gone to the Imperial Capital for important matters. He wouldn’t dare bother the old man with such a trivial thing while he was busy with serious work.
The person he thought of was the seller from the secondhand app who had helped him ace that pop quiz.
"Just you wait," Liang Xiaodong said, giving Han Sibo a wink and a mysterious smile. "You’re about to see just how deep your brother Dong’s connections run."
He expertly opened the secondhand app and navigated to "My Account -> My Transactions -> Purchased Items" to find the seller from last time, silently praying they’d be online.
He sent a message: [Hey, are you there? It’s an emergency!]
To his surprise, a reply popped up in the chat box almost instantly.
[Seller]: [I’m here.]
An instant reply!
Overjoyed, Liang Xiaodong didn’t waste a second. He snapped a photo of the problem and sent it over. Then, he quickly clicked on the seller’s "High School Math Tutoring" listing, left a comment asking them to change the price to 100, and paid without hesitation.
Once all that was done, he slapped his phone down on the desk and shot Han Sibo a smug look, chin held high.
Han Sibo watched him with a skeptical look, his mind filled with doubt.
Less than twenty minutes later, Liang Xiaodong’s phone sounded with a DING.
The seller had sent two images.
Liang Xiaodong opened them and took a look. Holy cow. Two full pages were covered in dense, meticulously neat mathematical symbols and derivations.
Without even looking at the details, he forwarded the images directly to Han Sibo’s WeChat.
"Done."
Han Sibo stared at the two images in his WeChat, his confusion deepening.
’It took so little time? For such a difficult proof? Could this helper Liang Xiaodong found really solve it in an instant?’
’What kind of skill level is that? Not even the best math teacher at school could be this fast!’
Filled with suspicion, he opened the images, picked up a pen, and began to verify the calculations on a piece of scratch paper as he read:
Condition for equality: a = 4cos^2(π/18), b = 4cos^2(5π/18), c = 4cos^2(7π/18), d = 0
Step 1: f(a,b,c,d) ≤ 27 is equivalent to abs(f(a,b,c,d)) ≤ 27.
And since abs(f(a,b,c,d)) is symmetric with respect to a, b, c, and d, we can assume without loss of generality that a ≥ b ≥ c ≥ d.
It is then clear that abs(f(a,b,c,d)) ≤ f(a + d, b, c, 0).
Therefore, the problem is reduced to proving the following for non-negative numbers where a + b + c = 6:
(a - b)(a - c)(b - c)abc ≤ 27.
Step 2: Let f(a, b, c, λ) = (a - b)(a - c)(b - c)abc - λ(a + b + c - 6)...
...
Liang Xiaodong didn’t disturb him, simply sitting beside him and scrolling through his phone, waiting quietly.
The minutes ticked by, one by one.
An hour later, Han Sibo finally let out a long, slow breath and set down his pen.
First, a flash of dawning comprehension lit up his face, which was then immediately replaced by a look of profound, almost drunken amazement.