Singapore Invitational Mathematical Challenge

# SIMC Diary #0

### Introduction

This series of diaries, namely 7/5, 22-28/5 will be published.I am one of the representatives of HKUST(Hong kong university of science and technology, ust for short) to participate in the Singapore invitational mathematical challenge (SIMC for short), SIMC is a modelling challenge held two years once, and this year is the 5th SIMC, and because I can take part in SIMC again, this diary can be a note for me.

Last Saturday ~7/5 was the last training session, but I will write about the whole day, as the whole day is actually extreme exciting.

Notice that I will publish the diary of 22-28/5 on 30/6, as I can’t discuss the problems publicly online.

### 7th May,2016

Waking up on half past eight is definitely not a good day to start, but as I am going to HKUST at half past eleven for the pre-trip briefing session. That’s not a big issue. Since before going to ust is neither important nor fascinating, I will just skip this part. I have arrived to ust one hour and 30 minutes earlier, so I first went to the bookshop, there is a planner from July 2016 to December 2016 I want to buy, but I didn’t as my parents are watching me. Secondly, I went to ust library, the library was open access for public, that’s means you could go into the library although you weren’t a library card holder. But now, you have to bring a library card in order to get into library, that’s make me a little bit upset. At least I found a good math magazine ~Crux Mathematicorum, but then I can’t help myself stop imagining sitting in library, doing the contest corner in this magazine, that’s will definitely make a part of my prefect day. At eleven o’clock, I went to the pre trip briefing session, not as fantastic as I thought, because they type my name wrongly!!! I will put the Powerpoint in here.

SIMC Pre-trip Briefing

After eating my lunch and start my last lesson, Dr.Avery Ching talk about SIMC 2014 Past Challenge Question Problem Part A and Part B(question 2), He first talk about Part B problem 2, I recommend you to see and think the problem first in order to understand what I am saying, I won’t rephrase all what Dr.Ching said, instead I will only show you the solution and some motivation and the things Dr.Ching want to talk about.

#### Question 2a

Instead a typical modelling problem, this problem is about a proof, the problem ask us to minimise E(A), a very first sight method is to find the expression of E(A) in terms of $d_1$, $d_2$ ,…, $d_n$, and find the partial derivatives of them, and find all critical points, and find the minimum. You can easily find this method long and tedious, so our alternatives is use some non-calculus method. So we rephrase the problem

Problem 2a

Prove that $E(d_1,d_2,...,d_n) \neq E(\frac{iL}{n},\frac{iL}{n},...,\frac{iL}{n})$, where equality holds only if $d_1=d_2=...=d_n$.

Proof. It is trivial when the equality holds, if the equality don’t hold, then there exist $i,j$ such that $d_i \neq d_j$. And then we have the following lemma to greatly simplify the problem

Lemma 1. if $d_i \neq d_j$ for a certain $i,j$, then there exist $m$ such that $d_m \geq d_{m+1}$.

Proof of Lemma 1. Assume the contrary, then we have $d_1=d_2=...=d_n$, contradiction.

So that we can simplify the problem into only considering three bus stop, we need to show that if the two distance of the two bus stop are the same, then the E(A) will be minimum , then I can derive the problem. We now compare the two situation, first is $d_1=d_2$, second is not. See figure 0.1

Let $x$ be the distance of the last two bus stop in the second case, and without loss of generality, we can let the distance of leftmost and rightmost bus stop be 1 and $x<1$ then we can divide the distance into the 5 situation, see figure 0.2

Then passengers in each interval will only take a specific bus. Then we calculate the probability of passengers from an interval to another interval.Then we find the distance passengers has moved, compare them to get the desired result.

#### Part A and Question 2b

For Part A, we motivate of this example, assuming you are in ust, and have to travel through 1. University of Hong Kong 2. Chinese University of Hong Kong  3. Paris 4. MIT 5. Harvard of course you best route is 1,2–>3–>4,5, it’s because 1,2 and 4,5 are relatively close to each other. In computer science, It is called k-clustering algorithm.

In question 2b, our method is to observe the smaller cases, consist a formula with variable n and E(A). And use calculus? The answer is no. We quote from Dr.Ching:

Calculus is neither the only nor best method to get maximum or minimum.

So we motivate the two example to be the end of this lesson:

1 Given a fixed perimeter, find a two dimensional object with biggest area.

2 Given integer $n$, find integer $k$ such that $\binom nk$ is biggest.

This is the end of the training session.

That’s not mean it’s the end of the day.

#### Night

Back to school, Tonight was the music concert, held by us, secondary 3(in US, it is Grade 9) students, as we would change classmates in next year, and we have been classmates since grade 7, so it is an important concert and students will really cry with each other.

In the first half of the concert, We sang the song~A Whole New World and played Memory(From Cats Musical). I was so nervous that I actually couldn’t play the flute.

In the next half, We sang two Chinese song, we thanked our three class teachers. And we have a class selfie on the concert! People cried, I didn’t, I didn’t have a good memory with them. We at last sang the school song, and the concert ended. The curtain of this day is closed.

Hint of 1: It is a circle you would guess. So consider four non-cyclic points, Can you maximize the area of it with fixed perimeter?