Introduction

Whether it’s helping with my USAMTS problems or with that pesky Calc homework problem, Wolfram Alpha is pretty great. I mean, I remember plugging in this really complex two-variable system of equations, and it just solved it! Ok, as a mathematician, that could be counted as shortcutting, but I do this since it’s inproductive to just keep bashing at an unrealistic system of equations that you cannot solve. I’ll have a section on the exact problem below, but I think it’s pretty cool. As my appreaciaion for Wolfram increased, back in September I interviewed with Wolfram and became a part of the Wolfram Student Ambassadors program.

I just wanted to make this article pretty short, and I really just wanted to highlight some pretty cool things that you can do with Wolfram notebooks. Recently, on Wolfram Community, I posted this article that summarized my physics research and the cool properties of pendulums you can model with Wolfram. That article got me upgraded to a Featured Contributer, so I thought I’d talk a little about that here.

Mathematica

So Mathematica is Wolfram’s mathematical computation language that many professionals in STEM fields use. Unlike other systems, Mathematica applies intelligent automation in every part of the system, from algorithm selection to to plot layout and and user interface design.

Above, for my article I created a pendulum diagram with Mathematica and the code is pasted above with the pendulum below it. Of course, the code looks pretty intimidating, but actually a lot of Mathematica is just writing what you want in words. I’m exactly painting which line I want where or how high I want a certain circle. But otherwise, the implementation isn’t too advanced so it’s very beginner friendly.

Above is another example of a dense Mathematica block in my article. While it looks pretty intimdiating, for any of you who are familiar with advanced physics, I just rewrote a Euler Lagrange equation above! In essence, I wanted to graph the solutions to a Lagrangian motion equation and a regular graphing tool like Desmos would not be able to compute that so I turned to Mathematica.

If you want to check out my full article, you can find it here!

Wolfram Alpha

As a prize from placing in the USA Mathematical Talent Search (USAMTS) competition, I got a year-long subscription to Wolfram Alpha Pro and Mathematica. It’s pretty nice! At least with Wolfram Alpha, it tackles those pesky equations that I wouldn’t be able to solve by hand. Here’s a problem from Year 33 of USAMTS…

And there we go! Although, the equation that I plugged into above was after subsituting equation 1 into equation 2 it still gave us $\frac{1011}{1010}$ as the answer right away! I’m sure there’s a more elegant way to solve it, but this is a nice way to check our answers.

And currently I have Wolfram Premium which is nice because if I plug in a triple integral it’ll give me the steps to solve the problem so that I can understand it. If you are a math interested student, rather than buying Wolfram Premium, I encourage you to try out USAMTS!

Wolfram Player

I also just wanted to highlight Wolfram Player very quickly before I ended this article. I used Wolfram Player for my research project, and it’s a way to make animations or simulations with Wolfram language since other softwares cost money or take up a bunch of storage. Here’s a screenshot of some cool demonstrations down below. There’s a whole archive of them on Wolfram and they are all incredibly intricate.

Conclusion

Overall, Wolfram is very cool and learning Mathematica, Wolfram Language, or any other Wolfram related languages will be incredibly helpful whether you are a student, researcher, or just anyone who’s interested in solving complex problems! Wolfram has a bunch of practice guides online so following any of those should be good enough to get a good sense of the language. Stay tuned for the next article.

Introduction

I recently entered the John Locke Institute Global Essay Competition which is held by the John Locke Institute sponsored by Oxford and Princeton. It’s an essay competition that encourages you to explore college level academic writing through thought provoking questions that go beyond the bounds of school essays. You can receive many prizes through the competiton, and I myself got a Commendation for the essay I submitted into the History section.

While this deviates a bit from my usual content, I find that it’s actually really similar to writing a research paper, but just in a different subject. You still have to research the topic, cite references, be specific, etc. For anybody looking to enter next year, my essay is down below to see what academic writing for a thought provoking question is like.

Which has a bigger effect on history: the plans of the powerful or their mistakes?

Imagine Jackson Pollock’s works: he was famous for his widely used “drip technique” to create drip artworks (Google Arts and Culture, n.d.). He utilized a can of paint and artfully dripped the paint onto a horizontal canvas. From that, would you argue that his paintings are mistakes or planned? He doesn’t know where the paint exactly goes, so is it a mistake? He would move back and forth across his sprawling canvas, directing the paint onto the areas he felt it needed to be. As he moved back and forth, he would adjust his plan according to how the paint naturally fell. We might say that the powerful don’t know exactly where the paint may fall when it leaves their brush, but each flick of the wrist and each movement along the canvas has a purpose. Mistakes, regardless if made by the powerful or not, are the opportunities that other powerful people take advantage of in order to make plans that change history.

We can interpret the question as such: do the powerful rewrite history or is history written on their mistakes? After defining needed terms, and analyzing various historical contexts in international affairs from the past century, we can logically assess the effects of plans and mistakes based on European philosophy. I will demonstrate this by asserting the intertwined relationship between mistakes and plans overlaid with the events of the World Wars and their effects. I contend that the World Wars and some of their effects left the largest imprints in history, and it’s demonstrated that they are all plans of the powerful. By following this argument, I propose that the plans of the powerful have a bigger effect on history than their mistakes.

To begin, we elucidate the difference between mistakes, effects, and power. A mistake is an action that could be perceived as wrong based on precedents: previous wars or events in this case. A direct consequence of this definition is that a mistake is also considered a matter of perspective. For instance, the Vietnam War was fought during the Cold War in the 1960s and 70s over advances of communism and resulted in more than a million deaths (Spector 1998). Some consider it a mistake from the sheer number of lives lost, but others thought it was a necessary measure in the Cold War and aided the effort to permeate democratic values throughout Asia. It’s clear both mistakes and plans can exert a profound influence on written history, and these effects fall into two categories: indirect and direct. Indirect effects are events propagated from another event, like how decolonization of nations was an effect of the Bolshevik Revolution. Although, in order to avoid any logical fallacies, this paper considers only direct effects or very close indirect effects. By definition, Vladimir Lenin is powerful: he had “control and influence over people and events” (Oxford Dictionary n.d). He created a plan, starting with overthrowing the Russian monarchy to then establish a communist government (Britannica 1998). Generally speaking, employing a plan is a method for achieving some end even if those intentions or effects prove to be disastrous. Lenin’s plan ignited a chain reaction instigating the stage for some of the most influential cultural reforms, such as Third World nationalism and globalization (Britannica 1998). To precisely determine the extent of a “bigger effect”, this paper measures how many are positively or negatively affected by a plan or mistake.

As the French philosopher and Nobel Laureate Albert Camus stated in his book The Plague, “good intentions may do as much harm as malevolence, if they lack understanding” (Stewart 2011). It’s interpreted that mistakes can do as much harm as malevolence, but not necessarily that they are malevolent. Camus’s philosophy brings into question whether a certain plan that lacks understanding can be classified as a mistake. This would make sense, as a plan that failed to understand precedents and proceeded in spite of them is definitely a mistake. For this paper, certain plans are classified as mistakes when appropriate. For clarity, if a plan is also considered a mistake, it’s still a plan. This is important for our argument. To further clarify, if you remove the intention from a mistake, then that makes the mistake an accident, which changes the meaning of it entirely. We choose to explore Camus’s idea further.

WWI (World War 1) began in 1914 with the assassination of Archduke Ferdinand, who was the heir apparent to the Austro-Hungarian throne. While Ferdinand and his wife were riding in a car, a lone gunman named Gavrilo Princip fatally shot the Archduke (Britannica 1998). At the time, the Archduke was heading to Sarajevo to check on the Austro-Hungarian army as the Inspector General of the Army (Shackelford, n.d.). His “good intentions” were simply that he wanted to check on the state of the army for future invasions. The tensions between Serbia and the empire were at an all-time high after the Balkan Wars (Wagnleitner 1999), so the Archduke choosing to ride in an open top car exemplified his lack of “understanding” for the situation he was in. According to Camus, this was a royal mistake. Later, the imperial government declared war on Serbia due to the latter consistently refusing the demands sent to them (Lindsay 2014). In a highly volatile environment, the Emperor Franz Joseph (the powerful) took advantage of Princip killing Ferdinand, as a result of the Archduke’s mistake, and declared war to put an end to the troublesome minorities in the multicultural empire. Thus, WW1 was a plan that took advantage of the Archduke’s death. The effects were quite disastrous, causing around 20 million deaths (WW2 Museum, n.d.). Although, one could argue that since WW1 was planned, then if it “lacked understanding”, would it be a mistake? WW1 being a mistake or not is too subjective since there are valid arguments to both sides. Regardless of if it was a mistake or not, it was still planned. Arguably, the biggest effect of this war was also a plan.

The Treaty of Versailles was signed by Germany and the Allied Nations in 1919, which formally ended WWI, one of the deadliest global conflicts in history (Crooks, n.d.). There were “good intentions” from the start since a treaty is where the parties came to an agreement, according to its etymology at least. It was inherently good from the drafting committee’s perspective. Although, the plan included various guilt clauses, such as the War Clause which forced Germany to take the blame for the war which included loss of territories, reduction in military forces, and general reparations (Crooks, n.d.). This angered the Germans and sowed the seeds for WW2 (World War 2). The committee drafting the treaty clearly lacked “understanding” of Germany’s perspective, which was their grave mistake demonstrating Camus’s point. As with WWI, WW2 was not simply a mistake, but rather a plan that took advantage of the initial mistake, which was the Treaty of Versailles. After the Treaty’s failure, Adolf Hilter used the unfairness of the treaty as leverage to gain political power in Germany. He would become the Chancellor of Germany (the powerful) and consolidate power and implement totalitarian rule in 1933, setting the stage for his plan to start WW2 (USHMM, n.d.). Technically, Hilter was not forced to invade Poland and start WW2, which demonstrates some intention in his actions. Thus, WW2 was planned and took advantage of the Treaty of Versailles. The effects of this bloodbath was essentially the most catastrophic war in history amounting to 35 - 60 million people dying, making it the highest wartime death toll in history (WW2 Museum, n.d). Furthermore, the cataclysmic impact of WW2 still reverberates through the annals of history: the Holocaust, Japanese Internment, and the Nanjing Massacre.

As an indirect consequence of WW2, the South African government wanted to strengthen their control so they heightened the racial segregation laws and institutionalized white minority rule. The architect of the Apartheid, Henrik Verwoerd, was a South African politician and had a large part in shaping the Apartheid policy in the late 1940s (SAHO, n.d.). Still following Camus’s logic, the “good intentions” here were to strengthen the nation, but racial segregation was horribly misunderstood as the best way to salvage the nation and thus Apartheid was a mistake. It is stated in the Universal Declaration of Human Rights that, “all are equal before the law and are entitled without any discrimination to equal protection of the law,” which was implemented by the UN the same year as Apartheid (United Nations, 1948). The South African lawmakers failed in “understanding” this principle as part of their government’s responsibility. Luckily, as with any form of oppression, people do revolt, and so arose the Anti-Apartheid movement. The Free Nelson Mandela campaign was a plan enacted through an international collaboration against Apartheid. Powerful organizations like the International Aid and Defence Fund backed the cause, and leading powers like the United Kingdom would create plans, like holding rallies and festivals to destabilize cultural legitimacy Apartheid. The global collective of cultural leaders were amending a grave mistake through various plans, which ultimately succeeded. Apartheid was eliminated in the early 1990s, and this eliminated the last trace of institutionalized segregation that plagued our world for more than a century. In the modern world, a precedent was set for much needed equality between all races.

It goes to show that a plan’s influence is universal, extending its reach to the social, cultural, and political realms. The mistakes discussed in this paper were a result of a plan, but not pure accidents. For example, arguably the biggest accident in the past century, the Deepwater Horizon Oil Spill in 2010, which afflicted millions of people, was not a mistake of the powerful but rather their workers. Many of those mistakes tend to play out in a similar fashion or their effects are not as widespread as the examples above. Plans and mistakes have a convoluted bond, but it remains true that the powerful use their influence to exploit mistakes and formulate plans off of them. To finish Camus’s thoughts, he further stated, “There can be no true goodness […], without the utmost clear-sightedness” (Stewart, 2011). Most of the powerful above intended to do what was good for their country, and having “clear-sightedness” or a vision is what gets the job done. From WWI to the present time is a cascading chain of plans falling in place one after the other. The paintbrush the powerful wield is a purposeful tool; with intent, one can paint a masterpiece that is called written history.

Final Thoughts

Whew! That was a long essay. Generally, with academic essays, after your hook and introduction, it’s a good idea to insert a paragraph that gives an outline for your essay. It’s almost like the “abstract” of your paper. And it’s great to ground your ideaes and reasoning in some type of philosphy, whether it be Eastern or Western. I inserted bits of Albert Camus’s Philosphy into the essay above, but you can really choose anyone. Overall, this was a great experience to learn how to write at a college level, and I encourage you all to participate!

Introduction

I recently attended the PROMYS program as a second year, and I wanted to encapsulate a bit of the cool math that I learned there! The primary class that I took was Prime and Zeta Functions, where we learned about zeta functions, Dirichlet Functions, Gauss sums, etc. It was all so cool! In my attempt to not forget any of it, I’ll give a beginner’s introduction to Reimann Zeta Functions (the beginning of the course).

Square Free Integers

$\textbf{Question: How many square free integers are there less than 100? 1000? 5000?}$

We define a square free integer as an integer that doesn’t contain any squares in it’s prime factorization. For example, 10 is a square free integer because it’s $2 \cdot 5$. Although, 12 isn’t a square free integer because it’s $2^2 \cdot 3$.

Is there a way we can sift out all the numbers that have squares in their prime factorizations? There is an intuitive way to think about this. There are approximately 3/4 integers that don’t have factors of 4. Similarly, approximately 8/9 integers don’t have factors of 9. 24/25 integers don’t have factors of 25. Notice how we skipped 16 because we already counted the factors of 4. Thus, we are only looking at prime squares in order to avoid overcounting. So we have a probablitic number of square free numbers.

For square free integers less than 100, there are approximately… \(\frac{3}{4} \cdot \frac{8}{9} \cdot \frac{24}{25} \cdot \frac{48}{49} \cdot 100 = 62.69387...\)

This is an approximation for the amount of square free integers. We can write each term as $1 - \frac{1}{p^2}$ where it almost looks like a geometric series. Although, how can we know that $N \cdot \prod_{}^{} 1 - \frac{1}{p^2}$ is actually a number? We can’t. Which is why it’s an approximation. Notice that if we expand $\prod_{}^{} 1 - \frac{1}{p^2}$, we get…

This looks like to be $\sum_{1}^{\infty} \frac{1}{n^2}$ except there are some pesky negative and positive terms. Those just turn out to be evaluted by the mobius function $\mu(n)$. So we get, the above sequence as $\sum_{1}^{\infty} \frac{\mu(n)}{n^2}$.

How can we get rid of the negative signs? We can make it a geometric series. This would require writing… $\frac{1}{1 - \frac{1}{p^2}}$ over all p prime. This is also known as an Euler Product.

where $s = 2$ when we are looking for square free integers. Expanding the Euler product, we get $(\frac{1}{1 - \frac{1}{2^2}})(\frac{1}{1 - \frac{1}{3^2}})(\frac{1}{1 - \frac{1}{5^2}})…$ for s = 2, and p prime where all the terms are in the from $\frac{a}{1-r}$ which is the formula for the sum of an infinite geometric series. That’s equivalent to…

Above is equivalent to $\sum_{1}^{\infty} \frac{1}{n^2}$ where we don’t have the messy positive and negative signs.

Zeta and Convergence

Notice, that if we have $\zeta(2)$, then we must have $\zeta(3)$, $\zeta(4)$, … and so on. An interesting question to ask is whether these functions converge for certain s? We know that $\zeta(1)$ doesn’t converge because this is the harmonic series. If you have taken AP Calculus BC, the following fact will be very familiar.

The above fact can be proved in a variety of ways. Although one way is a favorite of mine because of how intuitive it is. First let’s write out all the terms….

Notice that $\frac{1}{2^s} + \frac{1}{3^s}$ is bounded by $2 \cdot \frac{1}{2^s}$. Similiarly $\frac{1}{4^s} + \frac{1}{5^s} + \frac{1}{6^s} + \frac{1}{7^s}$ is bounded by $4 \cdot \frac{1}{4^s}$ and so on…

In this way we can say that $\sum_{n = 1}^{\infty} \frac{1}{n^s}$ is bounded by $2 \cdot \frac{1}{2^s} + 4 \cdot \frac{1}{4^s} + 8 \cdot \frac{1}{8^s} + … = \frac{1}{2^{s-1}} + \frac{1}{4^{s-1}} + \frac{1}{8^{s-1}} + …$ which is a geometric series and has a finite value because the geometric ratio is $\frac{1}{2^{s-1}}$ is less than 1 because $s > 1$. Thus, $\sum_{n = 1}^{\infty} \frac{1}{n^s}$ is bounded.

Knowing this fact, we can change the question to find what values the $\zeta$ functions converge to. To find what value $\zeta(2)$ converges to, we can look at the $sin(x)$ and find two polynomial expansions of it and equate the coefficents. $sin(x)$ is a good function to use because it’s linear approximation is really clean, $sin(x) \approx x$.

Sin(x) and Taylor Series

It’s a known fact that the Taylor expansion of $sin(x)$ is $\sum_{1}^{\infty} (-1)^k \frac{x^{2k+1}}{(2k+1)!}$ where the coefficent of $x^2$ is $\frac{-1}{6}$. We divide the whole series by x, because $\lim_{x \to \infty} \frac{sinx}{x} = 1.$

We also know that we can write a polynomial in terms of it’s roots, so can we do the same for sin(x)? The roots of sin(x) are $0, \pm \pi, \pm 2\pi$, … In terms of a polynomial this is…

This is a difference of squares, but we are aiming to find the value of the zeta function. So the coefficent of $x^2$ in the above polynomial is just $\frac{-1}{\pi^2} \cdot \zeta(2)$ which means that $\frac{-1}{6} = \frac{-1}{\pi^2} \cdot \zeta(2)$. or $\zeta(2) = \frac{\pi^2}{6}$.

We could apply this same method to find $\zeta(4)$ because we can just square both polynomials for $sin(x)$ and equate the coeffcients. Even for $\zeta(3)$ we can just square the $sin(x)$ expression and multiply an additional $sin(x)$. The only problem with this method is that as you get larger $s$, it becomes harder to use the distributive law on the polynomials.

So what we solved above is called the Basel problem where we find the value where the zeta function for $s = 2$ converges. The Reimann Zeta Function forms the basis of the Reimann Hypothesis! The Hypothesis is about finding the zeros of the zeta function and it’s one of the most fundmental unsolved problems in mathematics (because if proven or disproven it reveals some results about the distribution of primes).

Final Thoughts

The zeta function is really interesting, and I found it quite cool when I was learning about it over the summer. I thought about creating an article on Gauss sums or the Dirichlet function but that seemed to troublesome because most of the cool results from those two are all about the proofs and very minimal computation. The zeta function laters connects with the Dirichlet function to prove that there are an infinite amount of primes of the form $a \mod m$ where (a,m) = 1. Do some research on the Dirichlet function it’s really cool! Be on the lookout for an article about my experiences at the PROMYS program.

Introduction

Christaan Huygens, a highly gifted Dutch physicist and astronomist who is most known for his work on the centrifugal force (yes, that one concept you may have hated in AP Physics), the wave theory of light, and lastly … dun dun dun, the pendulum clock! Whenever you spend the hours away procrasinating, be thankful to Huygens for that clock you always glance over. No really! Clocks before Huygen’s clock were highly inaccurate and times would be as much as an hour off. Now, isn’t that helpful?

Huygen’s Early Life

Huygens was born to a wealthy family on April 14th, 1629, and was immediately put into challenging science and math courses from childhood through private tutors. His father was a famous diplomat and advisor to the House of Orange – yes, the royal family of the Netherlands. Through his father’s connections he was able to meet and talk with famous mathematicians. While Huygens was being homeschooled, among his father’s friends (like Galileo or Marin) Huygens met Rene Descartes who was most famous for his discovery of the connection between geoemtry and algebra. Makes sense why he was so gifted.

Huygen’s Pendulum Clock

Ok, so for some context, the clocks before Huygen’s pendulum clock lost 15 minutes a day, while Huygen’s new pendulums clock saw to only have a loss of 15 seconds a day. How did Huygen’s come up with such an incredible idea? Yes, he was a genius, but he also was inspired from Galileo’s findings. Galileo was experimenting with pendulums and found a property that made them useful timekeepers. It was the property of isochronism or that the period of swing changes based on string length. We know this from the formula, $T = \sqrt{\frac{l}{g}}$ where $l$ = string length and $g$ = 9.8 $\frac{m}{s^2}$. Although there is a restriction, you can’t have the rod length of the pendulum be too long because the amplitude will be too large. This will cause the pendulum to leave the cycloid curve and lose it’s isochronism.

Above is a drawing of what Huygen’s pendulums would look like. Not quite the simple string and metal bob you might’ve been thinking of.

Other Scientific Discoveries

Besides being the father of the pendulum clock, he is also the Father of Titan. Let me explain. Titan is the moon of Saturn, and Huygens had a part in discovering it. Around 5 - 10 years before Huygens found the beauty of the pendulum, he was playing around with telescopes.

In fact, he discovered the law of refraction to derive the focal distance of lenses. This allowed him to release his own “brand” of telescopes. Yes, the Gucci of telescopes maybe. In 1665, with one of his new telescopes he pointed out Saturn’s largest mooon, Titan.

Timeskipping to 1689, Huygens decided to meet the other great English contemporary of his time. Isaac Newton. They debated about the well known wave vs. particle theory of light. Newton was a firm believer in the theory that light traveled in packets and Huygens believed that light traveled in waves. Most people backed up Newton, mostly because of his fame and education. Don’t get me wrong though, Huygens theory was recognized and integrated into the theory of light that we know today. Although, it took a little more then a century for that to happen.

All in all, Huygens legacy still continues with his theory of light and Titan. In fact, NASA just sent a rover on Titan to explore it in 2005! Appropriately, the mission was called Huygens. Although, the legacy didn’t end there …

Huygen’s Synchronzing Pendulums

After he patented his pendulum clock, he got quite sick. This sickness tormented him throughout his old age, but his scientific curiosity endured and didn’t stop there. In 1665 (36 years old), while he was sick with depressive illness, he was laying in his bed playing with his pendulums. After a while of watching them, he noticed that they exhibited the same motion. They they were … synchronized? He experimented a little and found that no matter how the pendulums were set into motion, within thirty minutes, the pendulums were synchronized. They had the exact same motion, period, amplitude, and all. Just that they were swinging in the opposite direction, but their motions were exactly the same.

For context, the pendulums were quite large and the rod length of them was a little larger than a meter. Although, not everyone thought this idea was cool. When he presented his ideas to the Royal Society, they were unimpressed and chalked it up to the inaccuracy of the pendulums and that it was a mere coincidence.

Anyway, this brought up a whole set of questions. Like, how did the pendulums synchronize? Or, what factors affect the rate of synchronization? Some of these are answered, and some aren’t. 300 years later, there’s much research to be done.

Further Research

I was incredibly fascinated by Huygen’s ideas and decided to do some physics research around this. In fact, you can actually find my research write up on my website which is currently under peer review right now. On that page, I’m planning on posting more documents from math research at my local university, so be sure to look out!

Final Thoughts

Overall, Huygens was a pretty cool dude. Although, usually with the greatest scientific contemporaries, I can find some interesting fun facts about their hobbies and such, but I couldn’t find that with Huygens. Maybe he was very invested with his scientific discoveries (and the fact that he died quite early).

Introduction

I think that having a website is great for anyone whether it’s advertising your personal portfolio or creating a blog (like me!) My website is just a culmination of all the things that I enjoy in life from art to math and it’s a little passion project of mine. On top of that, it’s really easy to begin whether you can code or not! Of course, there are templates on Squarespace but there is something personal about coding your own website from scratch.

Learning HTML and CSS

So HTML and CSS by itself is pretty easy, but I used the book “HTML & CSS” by Jon Duckett. This was an incredible book, and it made learning it so easy because it was very visual. Each page had one sentence max and it wasn’t loaded with 50 sentences of content for a simple command. It got to the point and it made HTML & CSS very accessible. I’m convinced that every book about a coding language should be made exactly like this and then more people would actually enjoy coding.

You can learn along the way since most of the commands that you’ll need you can easily find on w3schools or Stackoverflow where I found cool functions like the header changing color when you hover over it.

Overall, HTML & CSS isn’t hard to learn, you just need to practice it (project based learning!).

Hosting the Site

You have to download the code onto your Github so that you can see the Github environment to see when your changes get published. To Git Push your changes you have to select the source control icon on the side and then select Sync Changes. It’ll ask you for a message to put that will appear in your Github main page. Usually I make silly simple messages like “new blog lolol” or “math facts bro” since I don’t have many folllowers on my Github.

You will also have a local host, where you can see changes that you made actively but they aren’t published live time. To activate it, you have to make a command with your sh key in the terminal. In summary, you won’t embarrass yourself by placing an image in a random spot.

Static Site Generator

Ok, so we have the website made directly from scratch or we have the one made from a static site generator. A static site generator just makes the repetitive task of writing HTML and CSS easier. For example, let’s say that you have 6 different pages on your website and you have a nav bar that you want present on all pages. To insert that, you have to copy and paste that code for the header into each page. Ok, so that doesn’t sound that bad. Although, the bad part is when you have to change a detail in the header, and then you have to change that on all the pages. A static site generator prevents all this so that you can just make your changes in one place and it’ll update on all your pages.

The generator that I use is Jekyll, and I chose this because it’s blog aware. Meaning, that it has a blog template type of thing embedded it’s super easy to download. It takes Markdown files and translates them into webpages, and using Markdown is really efficent and easy to learn along the way.

Of course, there are many other generators, like Gatsby.

Customizing

First, to make the website look nice you need a good background image, but even before that – you need a color scheme. I chose yellow and forest green for my scheme, but any colors would really work. I then chose a bold home screen image on Unsplash where you can use the images royalty free. I’m still working on beautifying this website.

Below is the image I ended up choosing:

Of course, anything beyong that it really free terrain. If you want more interactive aspects, you probably have to utilize Javascript. For example, I wanted to create a scrollable gallery on my Art page and had to use a tad bit of Javascipt (shown below).

There are just so many things to do with a website! Have fun with it.

Google Analytics

If you have a blog (like me) or a project on your website where you want people to look at it – you probably want analytics. This allows you to look where your viewers are from, how old they are, or most importantly, how many you have. It’s really easy to activate. All you have to do is connect your website url to the website, and then you will see all your stats. Below is an example of what my page looks like …

For example, the one that I use the most is the “Reports snapshot” feature which tells you how many users you’ve had in that week and what countries they are from although it’s hard to discern whether they are bots or not; I’ve gotten a lot from Cambodia recently. It’s pretty addicting to look at the analytics and just see who is looking (when someone besides my friends and family looks I get pretty excited).

Final Thoughts

There are a lot of improvements needed for this website, but most of them are aesthetic wise. For example, I would like to make my home page a bit more pleasing to look at. I’m also hoping to figure out a comments section or a reaction system and just overall would like the webiste to look a little more clean.

Introduction

While I was doing some AMC and AIME problems, I encountered Pascal’s triangle endless times. It came up a lot in combinatorial problems, and it’s quite cool how theorems like the Hockey Stick Identity come from it. I also always thought mathematicians like Gauss had a interesting childhood, I mean he solved the $\frac{n(n+1))}{2}$ problem in Elementary School. So, was Pascal the same? Let’s find out…

Keep in mind that Pascal was alive during the 1600s, which was a time of the Great Awakening, so there is comparably more information on him than someone like Ptolemy. This time period was also a time where exploration amongst the European powers was in fashion, but since Pascal was mostly in France, they would deal with the Thirty Years War. So similarly, Pascal chose the exploration of mathematics.

Early Life

Blaise Pascal was born in Clemont, France in 1623 and he was a member of the aristocratic class. The Pascals had about $10 - 20 million dollars of wealth in today’s currency.

It’s safe to say that Blaise was a child prodigy. When he was younger, since he was part of the higher classes, he was easily exposed to Math and Science since he didn’t have any chores to do. They had a couple maids for that – sounds nice doesn’t it? His father recognized his talent and took him to the activites of high society, like intellectuals talking over a cup of tea. In fact, Pascal himself participated in these talks as a toddler to his pre teen years and was acknowledged by the famous philospher and mathematician Descartes.

It was at the bright age of 16 years old, that he wrote a treatise about a certain mathematical concept. It wasn’t just any treatise, but it was Pascal’s Theorem. It deals with Projective Geometry, so if you’re into that stuff, I talk about it later in the article.

One thing I find quite funny about his childhood is that when he was one years old, he was gravely sick. His parents chalked it up to the work of a witch – this was a little before the Salem witch trials. There was an old woman who was basically the babysitter for Pascal, so when he wasn’t getting better, his parents blamed the old woman. Shockingly, she confessed that she cast an evil spell on him which made him sick (This obviously isn’t true she must’ve felt very pressured). She then threw two cats out the window (they died, the poor cats!) and rubbed some leafy paste on his stomach. Well, he got better miraculously after that little ritual. But, we don’t know what happened to that lady.

This is foretelling of his ill health in his adult life. Many say that he was cursed, but maybe it’s just karma for killing those two cats.

Later

At the time it was popular for many young artistocrats to follow their father’s footsteps, and follow in their life work. So Pascal’s father was working as a tax supervisor, and the young mathematician would work on his projects and studies while occassionally helping his father. He saw that his father was working with calculators often and saw that it was incredibly tedious. He then began his 3 year project on a new calculator. MIT would’ve surely taken him, but that would be a severe downgrade.

But the real story is that during this time, around the 1630’s, France was fighting in the Thirty Years War (one of the most brutal wars in European history). France kept taking out government bonds, and this made Pascal’s family standing drop from aristocrat to a low middle class. Pascal’s father disagreed with the regional cardinal’s fiscal policies, so he fled France leaving his three children behind. Soon enough, his daughter, Jacequline, started to perform at childrens plays (with the cardinal in attendance) that the cardinal forgave his family. Pascal’s father was then reinstated as the regional tax collector, and had boatloads of work. Hence, Blaise Pascal’s invention.

This is Pascal's Mechanical Calculator, stored in the Conservatoire National de Art et Metiers, Paris.

He designed the project when he was 19 years old, but then he went on to create 50 prototypes till 1945 until he finally presented his findings. In fact, they were so impressive that he was given “royal privilege” by King Louis XIV. Think of it as the modern patent, but more prestigous back then.

He contributed much to the field of physics, besides being a mathematician. He created the all famous …

$\textbf{Pascal’s Law:}$ states that when there is an increase in pressure at any point in a confined fluid, there is an equal increase at every other point in the container. So basically you use the relation, $\frac{F_1}{A_1} = \frac{F_2}{A_2}$

It’s much easier with a picture. If you look down below…

Basically a small force on a small surface area would be able to lift a big car on a big surface area through the property of ratios. Due to his early contribution to the barometer, he had the unit, the Pascal (unit for pressure) named after him. Very cool! He had many more inventions but this article would take an hour to read if I included them all.

In the same year, at thirty years old, he created Pascal’s triangle, a conveinent tabular representation of the binomial coefficents, and one of the most useful tools in probablity.

Major Math Theorems

$\textbf{Pascal’s Theorem:}$

In simple terms, it can be used to prove the collinearity of three intersections among six points in a circle. The image from Brilliant, explains this quite simply.

Pascal’s Theorem just states that for any 6 points that are in the circle, then we have the intersections of AB and DE, AF and CD, and BC and EF are all collinear. It’s quite useful in Olympiad math!

$\textbf{Pascal’s Triangle:}$

This is Pascal's Triangle.

The triangle is created by adding the two numbers above it to get a new number. For example, 1 + 1 = 2, 1 + 2 = 3, and so on. This is a triangle that you have certaintly seen before and maybe have had to memorize a couple of the rows here. Where does it actually apply?

Well, the most simplest explanation is that if you have $(x+1)^3$, that’s $x^3 + 3x^2 + 3x + 1$. The coefficents match up to the 3rd row of the triangle. Similarly, for any $(x+1)^n$, the coefficents will follow the nth row of the triangle.

Although, I said that this would appear in probability as well. If I change the numbers of the triangle to look a little different we get…

This is Pascal's Triangle, but rewrittern with binomial coefficents.

While it may look different, the numbers exactly match up with the coefficent. Look down below for a quick crash course on binomial coefficents…

Now, many theorems come about from Pascal’s triangle. For example, the way that a triangle is perpetuated because you two consecutive numbers to get a new one below. This is $\binom{n}{k} + \binom{n}{k-1} = \binom{n+1}{k}$ which is useful for math competitions! There are a gazillion others that I won’t go into, but visit AOPS or Brilliant if you are interested in looking at those!

Final Moments

This is an interesting little tidbit, but when Pascal was 23 years old, his father slipped on a carpet amd broke his hip (possibly fatal). Somehow, this caused him to have a religous revelation. Apparently, this was enough to convince him to switch from Roman Catholicism to Christianity, and some think that Pascal thought his whole family was cursed. So he thought switching his religion might help and he proceeded to live a ascetic from then on.

Going in a full circle, his illness when he was younger came back in a new form at the ripe age of 36. He first got slightly sick, but refused all the doctors because of his philosphy that “Sickness is the natural state of Christians.”

This is Blaise Pascal's Death Mask at 39 years old: a cast taken over the corpse.

Later when his sister died, Jacqeuline, he was incredibly devastated because she was with him through thick and thin. So this caused him to go a little insane. In 1662, Pascal’s health incredibly worsened and he was mostly bed ridden, and now he kinda realized that he should probably go to a hospital. Although, the hospital thought that he was too mentally insane to be in a hospital. He was moved to Paris during this time, and he was experiencing major convulsions and shocks, but still no one except small time doctors would help him. He died on August 19, 1662 and his parting words were “May God Never Abandon Me.”

An autopsy after his death showed that he suffered from brain lesions, grave problems with his stomach and other organs, tuberculosis, stomach cancer, etc. This time the old woman couldn’t go and help him, but he left Earth as a great scientist.

Introduction

I’ve used LaTeX since probably 7th grade, and it’s helped me tremoundously. Being a math buff, there was no way I couldn’t learn this incredible language that makes mathematical formulas in a document appear seamless. Even to just to change the format of the document, you can edit the preamble of the document which is superior to any Google Doc or Word feature I’ve encountered.

Installation of LateX on MacOS

It’s pretty simple to install LaTeX for MacOS. All you need to do is to download MacTex (https://www.tug.org/mactex/) and let that download.

MacTex does automatically download a TeX Editor, but I personally think VS Code is easier to work with. So in conjunction with downloading MacTex, you can just download VS Code. All you need to do is to download the LaTeX Workshop extenion on VS Code and you can download LaTeX files.

It’s helpful because if you are running a website and you have MathJax running on it, you can have LaTeX on it as well.

For other installations, visit the official webiste: LaTeX-Project

How to format a document

I’ve been involved with math competitions, like USAMTS, which require you to send in worked out solutions and LateX is a great way to format them. Usually if you work in Overleaf they give you the basic commands like $\texttt{\section{}}$ or all the packages at the top. An example of one of my handouts is below…

On the left is the code space, you can see things like a header (\lhead or \rhead) for either a left or right header. There is also some extra stuff like bolding words is $\texttt{\textbf{}}$ and there are font sizes like $\texttt{\Huge{}}$. A useful trick is also putting $\texttt{\vspace{3mm}}$ between paragraphs for that extra space beacuse LaTeX won’t automatically space out your document. On the right is what the a LaTeX document would look like.

Below is the general document setup I use, with a color box embedded as an example. The setup is really simple, it’s just a matter of packages.

Most documents have a bunch of packages at the top, and there are only a couple basic ones that I find really useful. You input a package that you need into the top of your document by enclosing it in $\texttt{\usepackage{}}$.

• TiKZ package: is typically used for graphics, like making colorful boxes. You can see an example of those on my Generating Functions Handout
• Article package: is the most basic class, used for making articles
• Biblatex package: is useful for citations in scientific articles, and you add citations by putting your citation in $\texttt{\addbibresource{}}$
• Geometry package: controls the size of the document, it’s margins, and other page setup functions
• Graphicx pacakge: allows you to include images into your document
• Hyperref pacakge: allows you to embed hyperlinks into your document.

Most of the packages I listed above are general use packages, meaning you will probably need them no matter the type of document. Keep in mind that there are over 4000 packages, so it’s best as you discover the packages as you find commands that require them.

Basic commands

Whenever you type something in LateX, you have to surround it by dollar signs for it to render. You can use double dollar signs on each side for the rendering to be centered. Below is a table of some of the more useful commands…

These are the most basic commands, but I encourage you to look up the more complicated ones on Overleaf (https://www.overleaf.com/learn).

Other Fun Sites

Sometimes you want a smiley face or a heart in your document, but you don’t know how to render one? Well, DeTeXify can help! Basically you draw the symbol you want and it’ll spit out the command, package, and mode. I made all the shapes of the McDonald’s chicken nuggets on this site, (Yes they are supposed to be shapes: a bell, ball, boot, and tie) This is defintely a site to be bookmarked.

For you nerds who end up really liking LaTeX, I would try playing TeXnique. It’s a game where you are given a theorem and you have to type it out in LaTeX in under a minute. A good pastime if I must say. I’ve learned many obscure math theorems from this site (Ex. Coarea Formula) - great for Trivia buffs!

Overleaf also has a lot of fun templates that you can copy and paste into your TeX Editor. I actually ended up using a couple of the Resume ones to contact professors and for scholarships.

Final Thoughts

LaTeX isn’t really too complicated to learn, it just needs a lot of practice to be fluent at! I would always type up my AOPS class math solutions in LaTeX which really enforced my skills. Learning LaTeX is a great skill to add to your resume and serves well in the long run. Have fun learning!

Introduction

Independent research is actually quite common and it’s how most prolific ISEF qualifers start their journey! In general, it’s nice to have a resume of research to learn various skills from computing to engineering. Honestly, I was very overwhelmed with trying to find a topic. Lots of the websites that I checked out either had that staple exploding Volcano or a topic that exceeded the 500 word limit. There was never an in between!

I’ll be referencing my research quite a bit in this article, so you can check out my abstract on my research page.

Finding a Topic

The hardest step is always finding a suitable topic. The data collection is a piece of cake compared to that. I can already see the frustrated faces of you guys at home: many of you are probably screaming, “How to find a good topic”?! Well, if you are a high schooler you will probably want some level of complexity to your research, and I’ll get into that a bit. If in middle school, I would suggest looking at websites such as Science Buddies (https://www.sciencebuddies.org) which is a great website to branch off.

I found mine by searching through Youtube, and I often looked at Veritasium’s Youtube Channel (https://www.youtube.com/@veritasium) for physics research ideas. Another way to get a topic is to simply cold-email a bunch of local professors who would be willing to supervise your project and give you ideas. Usually, this won’t work so it’s best to do a bunch of literature reviews or watch videos as I did.

The best strategy is to take a well-known problem and twist it in a unique way that no one has done. For example, my project dealt with synchronizing pendulums which was a common problem investigated by top schools like Harvard and UCLA. I researched what has been done so far, and nothing regarding the properties of the pendulum on synchronization has been done. So I did just that.

Honestly, just reading it plainly, the topic doesn’t sound super groundbreaking. I mean changing string lengths to see how synchronization in coupled pendulums change? To my suprise at the time, synchronization is used in end-to-end encrypted systems as an encryption key. The lesson is find a topic that you are passionate about, then find useful applications later.

Now that a topic is secured, what’s next?

Collecting the Data

This is the most crucial part of the experiment. I needed to collect the synchronization times for each string length that I set the coupled pendulums to have. For my project I created physical pendulums but I also had a computer model of my coupled pendulums. This was to corroborate the data from my man made pendulums because they may not be super accurate. Luckily, to my favor, the data from both models matched without any changes from my part.

Creating the pendulums was quite hard and I had to recruit the help of my brother to help complete them. Let me just say, there were a lot of sticks and hammers all over my workspace (shown below).

The pendulums that I created are also shown below, though they show hex nuts on them instead of bobs. I found that the hex nuts carried a lot of air resistence and kept twisting in the air; I changed it to a bob later because of the hassle it caused.

Most data collection is quite menial. Whether you are running a computer program or simulations, it’s never fun. But it has to be accurate. Sometimes, my pendulums would be off center on the platform or the platform would roll off the cans so I would have to redo those cases. Most of my data collecting was me sitting in front of my apparatus with my lab notebook: of course, I had a pencil in one hand and my stop watch in the other.

The most important thing is to keep a neat lab notebook because it documents your whole research journey. I made an effort to include my analysis of data in there and not just my quantative numbers. Judges at science fairs also really appreciate a neat notebook.

Writing a Report

Most papers make great use of LaTeX which is a great typesetting language for papers including a lot of math. I would check out my LateX tutorial on my blog page. Even if your project isn’t quite math heavy, LaTeX is still very useful. It just makes the paper look nice and tidy. An example would be my research paper on my webiste. You can also look on arXiv for examples of papers that utilize LaTeX (which most do).

When analyzing your data, I would make sure you have a good amount of statistical analysis. For me, this was adding a regression and error bars onto my graphed data.

When writing my paper I kept messing up on including “we” or “I” in the paper. The whole research paper should be done in passive voice, which was a hard task considering I take AP Lang.

Presenting at Science Fair

Presenting at a science fair is somehow ten times more nervewrecking than any other presentation that I’ve given. Maybe it’s the judges in close proximity. Or maybe it’s their clipboards and eyes quietly judging your project.

I’ve seen many impressive projects that were ISEF worthy, but the judges didn’t buy them. The main reason is that the judges left more confused than coming in. Science is about communicating your thoughts clearly, and the judges are testing that skill.

Many can spit out fancy vocabulary, but it’s no use if no one understands. In fact, the clarity of my project was one of the most postive comments on my rubric. The feedback rubric you get is another great reason to participate in science fairs!

When setting up my project, I also had one of my pendulums there with a video of the pendulums synchronizing. This did a lot of the talking for me and was helpful in being more concise. Below is a photo of my poster that I used for my school, county, and tri state science fairs.

There really is no right way to create your poster since the science is more significant. You could easily go to Staples and print out your poster as one large sheet. That’s a method I saw a lot of people doing, and I’ll admit, it does make your project look extravagant. I didn’t feel like doing that, so I simply made a google slides presentation and printed those slides out.

Final Thoughts

The moral is, you don’t need a phD level project to seem impressive or do well in science fairs. As long as your data and presentation is good, you will go very far. It will also help you learn way more than doing something you don’t understand!

What’s USAMTS?

USAMTS is a free, high school oriented, proof based contest that is funded by NSA. The contest consists of three rounds, each with 5 problems and worth 5 points. The contest is fully remote and is a month long since the problems are hard enough to not get immediately. The contest is a great way to think about problems constantly and sharpen your math skills.

Problem 1

So this puzzle was a bit different from the others because there was no givens in the sense that there was no X’s or Y’s given to us. Although we were given the hearts and stars which was helpful. There are some essential things to note here which would be that the row of stars on the top needed just one X in front of it since each star is already in a row with one other X and Y.
It’s the same case for the rightmost column with stars. The first “given” that I found was that below the star in the second to bottommost row, that needs a Y below it. From there you know that the Y in the second to right column needs to be in the bottom 3 squares.

From there it’s a whole chain of “givens” that provide you the answer.

Problem 2

This problem was pretty straightforward, at first I was about to go with the wrong answer that there is no function because I was simply thinking that only polynomials could only work although there are many functions such as piecewise functions. I chose a function with mod 5 because I though that could cycle in 4 times so luckily that worked. I spent a lot of fruitless time researching cyclic functional equations and basically trying to manipulate $\frac{1-x}{1+x}$ since that was a solution, but it just didn’t output integer solutions. So sometimes just thinking about the problems for a little bit of time can get you the answer.

Problem 4

This was a problem that I spent a lot of time on but I still got it wrong. The answer had something to do with ellipses and noticing that O and H could be the foci of the ellipse. I still don’t know how I misunderstood it, but I thought that the area would be the whole circle because when I made a system of equations and graphed the soutions, it seems that it filled the entire area. I am still unsure of what I misunderstood.

Anyway I created a system of equations with the centroid of the triangle and the equation of a circle with a radius 10. I got 5 equations with 6 variables and I put it into Wolfram Alpha and got many solutions that could work so I’m assuming there was just some disconnect with the problem.

Problem 5

This was quite the tricky problem, and I spent a lot of time on this one too. The strategy that I took was finding configurations for smaller values instead of 2022 outright. For $n = 2$, we notice that a octogon works out. Although, finding $n = 3$ was really hard for some reason. I tried just drawing an octagon and drawing three dots on each side and putting dots on the interior of the octogon. Although I saw that you had to have the points all inside some “enclosed” region. For example, when I drew an octogon the other points that I draw should be in the inside and not on the outside otherwise one of the lines with a slope of -1 or 1 wouldn’t work. Like how a square’s corners wouldn’t work if we wanted to cover 2 points. When I looked at the solution it was one of those where I don’t think I would’ve been able to get it but it didn’t feel that unfamiliar either.

Final Thoughts

This round was significantly harder than the other two to me at least because the proofs felt more involved especially for 3, 4, and 5. While the problems that I got right were probably more minimal compared to the ones that I got in the other rounds, the problems were hard and nice to think about. Overall, USAMTS is a good contest to solidfy your proof skills, because usually if you do AMC or AIME you have enough knowledge to do some of the proofs. The puzzle also gives everyone an easy 5 points since you just need to be persistent.

Good news is that I’m on the leaderboard for rounds 1 and 2 of the contest so far, so let’s hope that my score isn’t totally demolished by this round.

Now that USAMTS is done, I guess it’s time to work on the other upcoming olympiads. Maybe I’ll make a post after the f=ma when discussion period is allowed.

What’s USAMTS?

USAMTS is a free, high school oriented, proof based contest that is funded by NSA. The contest consists of three rounds, each with 5 problems and worth 5 points. The contest is fully remote and is a month long since the problems are hard enough to not get immediately. The contest is a great way to think about problems constantly and sharpen your math skills.

Problem 1

I don’t know if this is just me but I felt like this puzzle was harder than other years. I spent quite a lot of time on it. In fact, for this problem the only clue that I had for a while was that 5, 3, and 1 had to go into the center white squares. To be honest, it was hard to see how to not solve this without coding. Although, I eventually got the answer.

A starting strategy that I had was to list what numbers could go in each box. This strategy didn’t work if the squares were “lone” meaning that there are no given squares around it so you kinda just had to leave those alone. Following the diagram below,

we notice that the square with 4 on it could only be 2, 4, or 6. The square with 9 on it could only be 7 or 9 since there is the 21 below it. I approached the problem like this and made a giant table of the possiblites that could be made with each square. Basically after playing around with it, there was a point I realized that some odd numbers like 7, 9, and 11 pair up with 19, 15, and 23 in some pairing order. So that’s how I based my casework. Once you placed those three pairs all the other numbers usually fell into place to arrive at the solution.

Problem 2

This problem was actually really easy, although I read the problem wrong. Just applying the basic principles of expected value gets you the answer. The only values we have are 0, 1, and 2 since Grogg flips a coin only once a day. I started doing the case where he flipped it infinite times a day.

Let’s call probablity where he fufills the second condition of getting a cookie, T.

It’s a nice algebra trick. Just don’t be like me and actually read the problem. I do think an interesting extension would be if he flipped it infinite times a day which would turn into summing infinite amounts of infinite series; actually, I don’t know if that is possible but cool to think about.

Problem 3

This problem was interesting and honestly pretty straightforward. I first experimented with small values of n. We know that with n = 3 we cannot do anything, since we will always be stuck in a cycle of just having two numbers the same and then an outlier.

Although, notice that with n = 4 we can come up with a winning strategy. So let’s say we have the numbers a, b, c, and d. Then we can create the sequence, $\frac{a+b}{2}, \frac{a+b}{2}, \frac{c+d}{2}, \frac{c+d}{2}$. Then we can average out $\frac{c+d}{2}$ and $\frac{a+b}{2}$ to get some average “v”. Then our sequence becomes v, v, v, and v. This gave me the incorrect assumption that it was actually only the even numbers that worked.

So my proof was an incorrect proof of how the odd numbers could never work out. I used proof by contradiction but I think I messed up when I said that you can split the sequence into two parts and how one of those parts will have an extra number in it. That’s some pretty flawed reasoning now that I look at it since it doesn’t take into account all of the cases. Can you tell I submitted this problem set at 9:58?

Anyway the correct answer was that any composite $n$ works, so techinically I was kinda correct i just had a subset of the answer.

This makes a lot more sense actually. So let’s say we have some composite n, then it has some postive divisor that isn’t 1 or itself. If $d_1 * d_2 = n$, where $d_1, d_2$ are not 1 or n then we have can $d_2$ groups of $d_1$ numbers each. Let’s average the $d_1$ numbers in each groups so we achieve $d_2$ different numbers with $d_1$ of the same number in each.

Now, let’s say that we take a number from each of the $d_2$ groups to get some average. We can repeat this $d_1$ times to get a final average. I don’t know why I didn’t think of this since it’s a simple generalization from even numbers.

Problem 4

This problem was quite fun. Actually I think I got partial credit on this one since I messed up one of the cases. I started off by graphing the cases for small $k$ like 3, 4, or 5. It was pretty easy to see the pattern with even k.

You could just alternate between two rows that had two colors each in it to give $c_k = 4$ for even k. I messed up with the odd k, I guess this case was more complicated though. I also only considered one quadrant of a $k \times k$ coordinate system. So I would only consider the box: 2k x 2k squared centered at (0,0) since from further on it’ll repeat.

Then I tested out the odd ones, which gave two different answers. With $k = 3$, I first got 8 colors since I tried my strategy with even k but then that didn’t quite work out so I added extra colors. Guess I missed a color somewhere since the answer was 9 for k = 3. Pretty sad about that actually because I spent a good 1 hour carefully graphing this thing.

Although for when $k > 3$, luckily I got that case and noticed that there only needed to be 5 colors. With this problem it was more so experimenting and guessing. The proof came naturally afterwards with direct proofs.

Problem 5

I don’t know why but the problem 5’s have been really good this year. This was also a problem that I knew I could solve pretty easily. There is actually enough information given to coordinate bash this problem so that’s the approach that I took and then I just named Point E some point $(e,0)$. Although, the solution given on the USAMTS webiste using Cyclic quads is a lot more cooler.

We can utilize the fact that the line connecting the distance between the circumcenters is perpendicular to the radical axis. Since the radical axis is EF which gives us a really nice way to coord bash. All we need to do is find the circumcenters.

Just find the perpendicular bisectors of each side of the triangle and find where they intersect. I checked my work using a Wolfram Alpha Widget and then you can just repeat this process for each triangle. We recall the fact that if we have a line of slope $m$, then slope of a line perpendicular to that is $\frac{-1}{m}$.

You’ll end up with $y = \frac{s(x−e)}{1+3r−2e}$ as the equation. It’s simple to notice that we don’t want $e$ in our x and y coordinates so we have to get rid of “$e$” somehow. This can be done by noticing the $e$ in the numerator and the $2e$ in the denominator, so $x$ has to be $\frac{1+3r}{2}$ to cancel out that 2e in the bottom.

This means $(\frac{1+3r}{2}, \frac{s}{2})$ is our answer. I liked this problem! I did wish I saw the cyclic quadliateral approach though.

Final Thoughts

Overall, this round was pretty approachable. Good news, I got a 21 on the first round so let’s hope I get around the same for this round. But I made a lot more mistakes so we will see. Round 3 looks tough but there is winter break for a reason I guess.