I love mathematics, especially geometry and number theory. Observing this, my parents have encouraged my interest by identifying avenues through which I can explore math. Before I recount my long history with math, you can check out a couple of expository papers that I recently wrote for the Euler Circle.

I can trace my interest back to fourth grade when I was introduced to interesting math problems by my parents through books from the Art of Problem Solving. The problems were more like patterns to be found and puzzles to be solved, which I found very appealing especially in geometry and number theory. Over the subsequent summer, I attended a Pre-MathCounts course at Star League which covered a wide variety of topics.

To delve deeper into rigorous problem solving, I participated in the Stanford Math Circle for 9 months in the evenings during fifth grade, where the teachers, Dr. Vadim Matov and Dr. Elena Pavlovskaia, posed interesting problems in set theory for geometry, number theory, and algebra. Ideas that I found intriguing were the inclusion-exclusion principle and the pigeonhole principle because, despite their simplicity, they had many applications especially in counting and number theory problems. I was also introduced to proof techniques, such as proof by contradiction. In order to gain understanding of combinatorics in addition to deepening my knowledge in the other areas, I attended an AMC 8 summer camp at Star League after fifth grade.

Intrigued further by number theory problems, I took an online semester course on Number Theory by Justin Stevens at Star League after class over evenings in sixth grade. Among the interesting ideas I learned in this class are generalized rules for divisibility of integers by primes, modular arithmetic and congruences, and Diophantine equations. Through this course, I also learned how to prove by induction. As l learned more, I became more interested in the challenges posed by problems from contests. During the summer after sixth grade, I attended a course for AMC 10 preparation at AlphaStar Academy, which covered advanced concepts in algebra, number theory, geometry and combinatorics.

My eighth grade middle school course on geometry taught me to write proofs rigorously in a step-by-step manner. Over the following summer, I learned trigonometry and calculus from the AoPS books.

During ninth grade, I took Precalculus Honors course at my high school while concurrently learning Multivariable Calculus by Prof. Denis Auroux through MIT OpenCourseWare online video lectures.

During the summer after ninth grade (Summer 2021) I attended two classes at the Euler Circle, each spanning 5 weeks, conducted by Dr. Simon Rubinstein-Salzedo (widely known in AoPS forums for Simon’s favorite factoring trick). The first class was on Combinatorics, and the second one was on Combinatorial Game Theory.

Euler Circle was both an interesting and challenging experience for me. Many of the math topics were well outside the school curriculum that were completely new to me. In the Combinatorics class, I enjoyed learning novel and fascinating ideas such as rook polynomials, exponential generating functions, and special numbers such as the Stirling numbers and Eulerian numbers. In the Combinatorial Game Theory class, the topics I found most intriguing were surreal numbers and impartial games.

I enjoyed the opportunities at the Euler Circle to engage in abstract thinking. For example, I had to dissociate myself from the regular number system and grasp definitions of new number systems, such as the surreal numbers, which has its own arithmetic operations. I also liked the fact that the classes were proof-oriented and not just about finding numerical answers. I was given two problem sets per week on the weekly topics covered. The problem sets contained challenging questions for which I had to write detailed proofs.

Over the course of each Euler Circle class, I had to research a specific topic and write an expository paper. For the Combinatorics class, I chose Catalan objects to investigate and wrote an expository paper about that. For the Combinatorial Game Theory class, I researched and wrote a paper on Classical Impartial Games. Writing these papers offered me the opportunity to dive deep into new areas of mathematics, which was immensely satisfying.

Another aspect of Euler Circle that I liked was that I had to write my proofs using LaTeX. I enjoyed learning the power of LaTeX to precisely format text, equations, tables and figures while creating articles.

Concurrently with high school during my sophomore year, I enjoyed solving interesting problems through the WOOT class (Worldwide Online Olympiad Training from the Art of Problem Solving). New topics that I came across are quadratic residues & primitive roots in modular arithmetic and functional equations.

During the summer after my sophomore year, I had the great opportunity to participate in the Ross Mathematics Program at the Ohio Dominican University. The program was led by Prof. Jim Fowler, Prof. Daniel Shapiro, and Prof. Paul Pollock. It was a one-of-a-kind experience. I thoroughly enjoyed living independently for 6 weeks with a group of students whose interests closely matched mine. It was a truly immersive experience learning proof-based number theory and solving challenging problem sets well into each night. I was introduced to a variety of new topics, and the counselors were awesome, especially Justin Wu. I really wished that the summer would never end.

At Ross, I learned and explored number theory from the perspective of abstract algebra — for example, Fermat’s theorem, Wilson’s theorem, the Unique Factorization Theorem, and their astounding generalizations with rings, groups, and finite fields. Some notable concepts I learned of there are quadratic residues & Quadratic Reciprocity (QR), the various methods with which to prove QR (for example, from a geometric perspective that involves Pick’s theorem), and Dirichlet series — a variation of generating functions that has slightly different properties under multiplication. Also, I was able to participate in fun group projects such as the “Celebration”, in which we proved the Unique Factorization Theorem from ring axioms, order axioms, and the Well-Ordering Principle. Another cool idea I was introduced to was the Lean programming language, which is used for formalizing proofs. 

After my junior year, I was accepted to Canada/USA Mathcamp, which was held at Champlain College in Burlington, Vermont. It was was an entirely different but equally stimulating experience. Every day, we were free to choose among 4 classes every week to attend, organized by several mentors. From daily lectures and problem set solving with friends, I sampled topics like infinite numbers, infinite Ramsey theory, computational structures called “gadgets”, Erdos’s probabilistic method, graph colorings, game theory, and geometric topology. The most interesting concept I remember is the probabilistic method, which is used to give a non-constructive proof of existence of specific objects (like large sets of “almost orthogonal” vectors in n-dimensional space) using probability theory.

Three weeks into the camp, I joined a group project to investigate the evasiveness of graph properties. Graph properties are boolean functions whose inputs are graphs and whose outputs are True or False (like, “Does the graph have a path that visits all vertices once?”), and they are called “evasive” if an algorithm requires knowledge of almost all edges to determine True or False. We read papers that used group theoretic arguments and topology to prove the evasiveness of nontrivial monotonic graph properties.

All the classes I took and the project were very interesting and illuminating … so much more than I had known before, and each of these experiences were only the tip of the iceberg.

I also had the pleasure of participating in the Math Wrangle, a debate-style math competition where we solved a series of problems, presented solutions publicly, cross-examined the opposing team’s solutions, and defended our solutions against the opposing team’s cross examination.

Outside of academic life, Mathcamp staff and friends planned many non-academic events/field trips, like problem solving relays, rock-climbing, and Burlington town travels, which were all definitely worth it.

Math Competitions

Since fifth grade, I have been an avid participant in the AMC competitions. I made the Achievement Roll in AMC 8 in my fifth grade, and the Honor Roll in the next two years. I also made the Honor Roll in the 2020 AMC 10 and 2021 AMC 10, and have been invited to participate in the 2020 AIME, 2020 AOIME, 2021 AIME, 2023 AIME, and 2024 AIME.

Beyond learning mathematics, I also enjoy making Platonic solids, such as dodecahedrons and icosahedrons, with paper. I initially used straightedge and compass to draw the 2-dimensional layout of these solids which I then folded to form such solids. For more precision, I automated the drawing process by writing Python programs. I have also written Python programs to understand how square waves and triangular waves can be decomposed to sinusoidal waves of different frequencies and phase shifts.