Chapter 119 - 89: Early Departure, Early Return
July second.
Nine o’clock in the morning sharp.
The electric bell on the corridor wall of the university’s academic building rang right on time.
The ringing lasted for more than ten seconds.
When it stopped, the only sound left in the corridor was the incessant chirping of cicadas from the trees outside.
Third floor, Examination Room Four.
At the front of the room.
The proctor picked up a box cutter from the lectern and sliced open the seal.
He pulled out a stack of exam papers from within.
"Pass them back from the front."
The teacher split the papers into several stacks and handed them to the students in the first row of each column.
Chen Zhuo sat in the second-to-last row, by the window.
The person in front of him turned around and handed him the remaining papers.
He took them.
He took one for himself and passed the last one to the student in the final row.
The sound of rustling paper instantly filled the classroom.
On the first page was the first free-response question.
It was a Diophantine equation problem concerning integer solutions.
The question presented a high-degree indeterminate equation and asked for all its positive integer solutions.
Chen Zhuo’s left hand rested flat on the edge of the paper.
His right hand held a pen, its tip landing directly in the answer area.
He wrote a congruence relation on the paper.
Using modular arithmetic, he analyzed the parity of both sides of the equation.
Black ink glided across the white paper, leaving behind even script.
He quickly narrowed the range of possible values for the variables.
Then, with a few simple algebraic transformations, he factored out the common terms.
The structure of the equation was deconstructed.
He listed the final three possible cases and verified them one by one.
He wrote the resulting integer solutions at the very bottom.
Without any pause.
He turned the page.
His eyes fell upon the second question, a polynomial inequality.
The problem had many given conditions, the constraints between the variables intertwined.
Chen Zhuo glanced at the pattern of the coefficients.
In the blank space next to the problem, he constructed an auxiliary function.
Using the Cauchy-Schwarz inequality, he bounded the numerator.
The direction of the inequality sign changed.
The originally complex algebraic expression was stripped of its convoluted shell.
Following the result of the bounding, he wrote down the final step of the proof.
He drew the concluding geometric symbol.
The exam room was quiet, the air conditioner blowing a gentle breeze.
Chen Zhuo’s pace remained steady and smooth.
The movement of his wrist was minimal; only his fingers controlled the path of the pen’s tip.
Time ticked by, second by second.
Sunlight shone through the glass window to his left, cutting a bright patch of light onto his desk.
Forty minutes into the exam.
Chen Zhuo turned to the sixth page of the exam.
It was a combinatorial counting problem.
It asked to calculate the number of ways to place a certain number of pieces on a specially ruled chessboard such that they satisfy a non-attacking condition.
Chen Zhuo finally reached for the blank scratch paper.
He didn’t draw the massive chessboard.
Instead, he wrote down a few simple recurrence symbols on the paper.
He transformed the placement rules for the entire board into a linear recurrence relation.
He listed the initial values for the first three terms.
Then he wrote the characteristic equation.
He solved for the characteristic roots.
Rows upon rows of calculations appeared on the scratch paper.
He substituted the characteristic roots into the template for the general term formula, using the method of undetermined coefficients to find the constants.
He arrived at the final expression.
Then, he transcribed this process, with clear logic, into the answer area on the exam paper.
One hour and ten minutes.
Chen Zhuo’s exam booklet was turned to the last page.
This was the final, most challenging problem of the entire exam.
A pure plane geometry proof.
There was no diagram.
Only a text description.
Given several fixed points on the circumference of a circle, tangents were drawn through these points.
The tangents intersected with other secant lines.
New line segments were drawn connecting the points of intersection.
Finally, it required proving that three of the newly created intersection points were collinear.
Chen Zhuo’s gaze swept over the text twice.
He pushed the scratch paper to the side.
Holding the pen in his right hand, he placed the tip directly onto the blank answer area at the bottom of the page.
He abandoned the traditional path of Euclidean geometry.
He introduced the complex plane onto the paper.
He defined the core circumcircle from the problem as the unit circle on the complex plane.
In this coordinate system.
The fixed geometric points represented by the capital letters A, B, and C in the problem.
Under Chen Zhuo’s pen, they became the lowercase complex numbers a, b, and c.
Because they were all on the unit circle.
Their complex conjugates were simply equal to their reciprocals: 1/a, 1/b, and 1/c.
Chen Zhuo’s pen tip moved across the paper at a steady pace.
Lines of black script arranged themselves on the white paper.
Those tangents and secants hidden in the text.
Were written by him directly as algebraic equations of a complex number z and its conjugate.
The tangent equation.
The secant equation.
The coordinates of the intersection points.
He didn’t need to find their positions on a diagram.
He only needed to solve the two algebraic equations simultaneously.
And find the expression for the intersection point z.
It became a problem of pure algebraic calculation.
All he had to do was follow the rules of algebraic operations and deduce step by step.
The fraction bars were drawn perfectly straight.
The equal signs were aligned vertically.
Chen Zhuo’s handwriting was steady.
When he encountered polynomial multiplication.
He quickly wrote out several parentheses on the scratch paper next to him.
After expanding each term, combining like terms, and canceling identical factors in the numerator and denominator to get a clean, simplified result.
He copied this result into the answer area on the exam.
Not a single circle, not a single line was drawn on the scratch paper.
It was filled with letters, fractions, and conjugate symbols.
The ceiling fan overhead was still turning.
The second hand of the quartz clock above the blackboard ticked away, one space at a time.
Chen Zhuo’s concentration was entirely focused on the tip of his pen.
He was now tackling the final proof of collinearity for the three points.
On the complex plane.
To prove three points are collinear, one only needs to prove that the ratio of the complex numbers formed by these three points is a real number.
And the necessary and sufficient condition for a complex number to be real is that it is equal to its own conjugate.
Chen Zhuo wrote a long fraction on the exam paper.
The numerator and denominator of the fraction contained the complex expressions for all the intersection points he had previously calculated.
There were many variables, and the structure was long.
He began to find the conjugate of this fraction.
It was a tedious, cumbersome calculation process.
Substitution.
Expansion.
Finding a common denominator.
Chen Zhuo’s wrist moved steadily across the paper.
Line after line of equations was laid down.
As the final step of simplifying the fraction was completed.
The complex numerator and denominator canceled each other out.
To the right of the equals sign appeared a form completely identical to the original fraction.
Chen Zhuo placed a period at the end of the equation.
The proof was complete.
He stopped writing.
He lifted his head and glanced at the clock above the blackboard.
Ten-twenty.
There was still an hour and forty minutes until the end of the exam.
The patch of sunlight on his desk had shifted.
The air in the exam room had become a bit stuffy.
Chen Zhuo put down his pen.
He flipped the exam back to the first page.
He scanned the admission ticket number he had filled in from beginning to end.
He confirmed there were no blank spaces left.
He put his eraser and pen back into his pencil case.
He zipped it up.
He pushed back his chair and stood up.
The chair let out a short SCRAPE as it was dragged across the floor.
Chen Zhuo picked up the exam paper and scratch paper from his desk.
He grabbed his admission ticket and ID card from the top-left corner, along with his pencil case.
He walked out from the aisle between the desks.
He walked to the front of the classroom.
He laid the exam paper and scratch paper flat on the lectern.
The teacher sitting behind the lectern looked up.
He glanced at Chen Zhuo, then at the clock on the wall.
The teacher lowered his head, his gaze falling on the exam Chen Zhuo had turned in.
The paper was open to the final, most challenging problem on the last page.
The blank answer area was completely filled.
It was covered in neatly arranged complex equations.
The teacher’s pen-holding hand paused for a moment as his eyes followed the steps of the derivation downward.
He saw the definition of the unit circle.
He saw the expression for the intersection points.
He saw the final proof of the expression being equal to its conjugate.
The teacher looked up at Chen Zhuo.
Chen Zhuo stood before the lectern, his expression calm.
The teacher gathered the papers and placed them under a transparent folder.
He gave Chen Zhuo a slight nod.
Chen Zhuo nodded back.
He turned and walked toward the back door of the classroom.
He pulled the door open and walked out.
The door closed behind him.
The corridor was empty.
Only the branches and leaves of a few tall plane trees outside the window swayed in the gentle breeze.
Chen Zhuo walked down the stairs.
The light in the first-floor lobby was a bit dim.
He pushed open the glass main doors of the academic building.
The ten-thirty morning sun poured down.
The glaring light made him squint slightly.
A wave of heat washed over him. Distorted heat haze shimmered up from the asphalt road.
The air was still very humid, mixed with the smell of leaves.
In front of the academic building was an open rubberized track and field.
A red running track, a green lawn.
Chen Zhuo walked along the tree-lined path at the edge of the field.
He walked to a small tuck shop at the end of the field.
Two vending machines were set against the wall.
He stopped.
He fished a few coins from his pocket.
He inserted them into the coin slot.
The coins rolled down the track, making a few metallic clinking sounds.
He pressed the button.
A whirring sound came from inside the machine.
A can of ice-cold Coke dropped into the collection slot.
Chen Zhuo bent down to retrieve it.
Holding the Coke, he walked over to an old plane tree.
Its trunk was thick, and in its shade sat a wooden bench with peeling paint.
Chen Zhuo sat down on the bench.
He placed his pencil case and admission ticket on the empty space beside him.
He hooked a finger under the pull tab with one hand.
He yanked it upward.
A PSSSSH of cold air and carbonation escaped.
He tilted his head back and took a large gulp of the ice-cold Coke.
The cold liquid rushed into his stomach.
Chen Zhuo lowered the can.
He leaned against the hard wooden backrest of the bench.
He rested his hands on his knees.
He looked at the empty field before him and the glass windows of the distant academic building.
The sound of cicadas circled overhead.
An occasional, warm breeze blew past.
He sat quietly on the bench, waiting for the exam to end.
The angle of the sun slowly shifted, and the shadow on the bench shortened a little.
The rubber track on the field gave off a faint scorched smell in the high heat.
Time passed, minute by minute.
Twelve o’clock sharp.
A crackle of static came from the campus broadcast speakers.
Then, a series of urgent bells signaling the end of the exam rang throughout the school.
"The exam is over. All candidates, please stop writing."
An electronic voice announced over the broadcast.
The academic building fell silent for a few seconds.
The doors to classrooms on every floor were pushed open.
People began to pour down the stairwells.
A few minutes later, the entrance of the academic building was crowded with students.
The team coaches stood outside the cordon line.
Chen Zhuo sat on the bench, watching the crowd.
He finished the last mouthful of Coke in the can.
He tossed the crushed, empty can into a nearby trash bin.
It made a CLANG.
He picked up his pencil case and admission ticket from beside him.
He stood up and walked along the tree-lined path toward the school gate.
In the crowd.
He saw Coach Xu holding up a red folding umbrella and several backpacks.
Chen Zhuo approached the cordon line.
Zhou Kai was already standing next to the coach.
He had his backpack on, and the hair on his forehead was damp with sweat, clinging to his skin.
Lin Yi squeezed out of the crowd, walked over to Chen Zhuo, and leaned against the railing of the cordon line.
They waited a little longer.
Zhang Bai, Mo Xiaoyu, and Li Nanbai slowly emerged with the flow of people.
The faces of all three were a bit pale, showing the exhaustion that comes from prolonged, intense mental effort.
No one said a word.
Coach Xu looked at them.
He closed the red umbrella in his hand.
"The exam’s over, so it’s over. Put your admission tickets away safely."
Li Nanbai let out a long breath.
He reached out and rubbed his stomach.
"What’s for lunch?"
"Something not so sweet."
Chen Zhuo replied casually.
Coach Xu pointed to the street outside the school gate.
"Let’s go. First, we’ll find a place with air conditioning, have a good lunch, and then rest this afternoon. After that, I’ll take you guys for a look around Shanghai."
Chen Zhuo stuffed his pencil case into the backpack Coach Xu had been holding for him.
He slung it over one shoulder.
He strode forward and followed them.
The summer sun was still glaring.
The shadows of the seven of them stretched long on the asphalt road.