Melancholia Ep. 1: CALCULUS Club Placement Test

Problem Statement

Origin Korean drama Melancholia, Episode 1. Depending on your region, you might be able to watch the episode for free on Rakuten Viki. Mild spoilers ahead.

Promotional poster (from Wikimedia)

Melancholia is set in Asung High School, a seemingly prestigious school that is actually full of corruptions. Ji Yoon-soo (right), who is super passionate about math, is hired to teach competition-level math to raise the school's prestige. There she meets Baek Seung-yoo (left), a student who performs poorly in math classes and likes taking photographs. In reality, Seung-yoo is a math prodigy who went to MIT but came back to Korea as a "failed genius". Yoon-soo notices Seung-yoo's interest in math from his photographs, and after interacting with him, starts to open up his heart as well as his dark past. But their relationship also leads to scandals, and drama ensues.

In Episode 1, Yoon-soo gives the students the following question as a placement test for her CALCULUS Club:

The placement test

Three non-zero vectors $\vec{a}, \vec{b}, \vec{c}$ in the 3-dimensional (Euclidean) space satisfy

$$|x\vec{a} + y\vec{b} + z\vec{c}| \geq |x\vec{a}| + |y\vec{b}|$$

for all real numbers $x, y, z$. Prove that $\vec{a}\perp\vec{b}$, $\vec{b}\perp\vec{c}$, and $\vec{c}\perp\vec{a}$.

Can you solve it? And what dark secrets lie behind this seemingly typical question?

Solution

In the show, the problem was a trick question set up by Yoon-soo, and an anonymous student (who is unsurprisingly revealed as Seung-yoo) correctly points out that the premise is false: there are no non-zero vectors $\vec{a}, \vec{b}, \vec{c}$ satifying the given condition, and as such the statement is treated as always true. ($P\to Q$ is true if $P$ is false. It's like, suppose Jack says "If I win this soccer match I will run 3 laps naked," and later Jack lost the match. Whether Jack chooses to still run out of his own volition or not, we can't say that Jack lied since he didn't say what he would do if he loses.)

To see why, substitute $z = 0$ and square both sides. For all real numbers $x$ and $y$, we must have

$$\begin{aligned} |x\vec{a} + y\vec{b}|^2 &\geq (|x\vec{a}| + |y\vec{b}|)^2 \\ x^2|\vec{a}|^2 + y^2|\vec{b}|^2 + 2xy\vec{a}\T\vec{b} &\geq x^2|\vec{a}|^2 + y^2|\vec{b}|^2 + 2|xy||\vec{a}||\vec{b}| \\ xy|\vec{a}||\vec{b}|\cos \theta &\geq |xy||\vec{a}||\vec{b}| \end{aligned}$$

where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$. Since the vectors are non-zero, we get $xy\cos\theta \geq |xy|$, which is false for at least one of $(x, y) = (2, 2)$ and $(x, y) = (2, -2)$ depending on the sign of $\cos\theta$.

The Story Behind the Problem

Kim Myung-ho's Misery

Science article

I applaud the showrunners for choosing this problem for the corruption-themed show, as there was actually a huge drama behind this problem. The problem appeared as Problem 7 (15/100 points) of Math II in the 1995 entrance exam for Sungkyunkwan University. Kim Myung-ho, who earned a PhD from UMich and was a fresh assistant professor at Sungkyunkwan at the time, participated in the exam grading session and recognized the issue with this problem. While the problem's statement is technically true, it would be hard to come up with a consistent grading rubric, and Kim recommended that no weight should be given for the problem. He promptly reported the issue to the president of the university.

Alas, this pissed off other people in the department, who subsequently threatened Kim with disciplinary actions such as giving him a stern warning, suspending him from teaching for 3 months (without salary of course), and denying his tenure. The last effectively terminated Kim's position at the university.

In 1996, Kim went to court. The Korea Institute for Advanced Study and the Korean Mathematical Society were asked whether the problem has an issue or not, but both of them refused to answer, allegedly due to their ties with Sungkyunkwan University. The district court dismissed the case and Kim appealed to the high court. At this point the story spread abroad, and many prominent mathematicians such as Serge Lang and Michael Atiyah expressed support for Kim. The high court eventually dismissed the case as well.

Kim Myung-ho's Revenge

The story did not end there. In 2003, the laws regarding wrongful refusal of re-appointment were amended (to no longer give the schools free rein to dismiss people), and wrongfully terminated professors were allowed to file lawsuits. Kim did so in 2005 for a few times but got dismissed, with the court citing that Kim did not have the qualifications to be an educator.

Kim became super angry at the court the their posse. He staged protests outside the courthouse and submitted complaint letters to the court. Finally in 2007, he carried a cross-bow to the apartment of judge Park Hong-woo. A fight occurred and an arrow was fired. Kim said that the arrow did not hit anyone, but the judge claimed he was shot. This "cross-bow case" was brought to the Supreme Court. The presented pieces of evidence were questionable (the blood marks were not near the hole in the shirt; the arrows were not broken or blunt; DNA tests were denied by the court; no picture of the wound; etc.). Despite this, Kim was eventually sentenced to 4 years in prison.

Promotional poster (from Amazon)

Kim was released in 2011, after which he continued to be an activist fighting against the Korean judiciary system. That year, the movie Unbowed inspired by the whole incident came out. It brought the case back to the public eye, who now began to sympathize with Kim, and the movie became a box office success.

References

Exported: 2021-11-28T23:50:45.655301