Course Notes

I live-TeX notes to math courses that I take, online or in-person, official or self-taught. These will appear here.

• (Spring 2021) Arizona Winter School in Number Theory, notes
• (Fall 2020) Representation Theory (ICTP et al), notes
• (Fall 2020) Groups, Rings, and Fields (Math 250A @ UC Berkeley), notes

Problem Writing

I have written for many math contests over the years. Most prominently, I am the youngest Problem Czar for the Harvard-MIT Math Tournament. A (currently out-of-date) compendium of problems I've written can be found here.

Here are some of my favorites, roughly in order of ascending difficulty.

• (HMNT 2021 I2) Suppose $ a $ and $ b $ are positive integers for which $ 8 a^a b^b = 27 a^b b^a $. Find $ a^2 + b^2 $.
• (HMMT 2023 Algebra #5) Suppose $ E $, $ I $, $ L $, $ V $ are (not necessarily distinct) nonzero digits in base ten for which
  • the four-digit number $ \underline{E} \ \underline{V} \ \underline{I} \ \underline{L} $ is divisible by $ 73 $, and
  • the four-digit number $ \underline{V} \ \underline{I} \ \underline{L} \ \underline{E} $ is divisible by $ 74 $.
  Compute the four-digit number $ \underline{L} \ \underline{I} \ \underline{V} \ \underline{E} $.
• (NEMO 2019 I14) Find all primes $ p \geq 5 $ such that $ p $ divides $ (p-3)^{p-3} - (p-4)^{p-4} $.
• (Mock ARML I8) Regular hexagon $ ABCDEF $ has side length 1. Equilateral triangle $ XYZ $ is drawn such that $ A $ is on $ \overline{XY} $, $ C $ is on $ \overline{YZ} $, and $ E $ is on $ \overline{ZX} $. If $ 70\% $ of the interior of $ ABCDEF $ is also withn the interior of $ XYZ $, find $ XY $.
• (HMNT 2021 I10) Find all real numbers $ (x,y,z) $ for which \[ x + xy + xyz = 1, \quad y + yz + xyz = 2, \quad z + xz + xyz = 4. \]
• (OMO Spring 2020/25) Let $ \mathcal{S} $ denote the set of positive integer sequences (with at least two terms) whose terms sum to $ 2019 $. For a sequence of positive integers $ a_1, a_2, \dots, a_k $, its value is defined to be \[ V(a_1, a_2, \dots, a_k) = \frac{a_1^{a_2} a_2^{a_3} \cdots a_{k-1}^{a_k}}{a_1! a_2! \cdots a_k!}. \] Find the sum of the values over all sequences in $ \mathcal{S} $.
• (ELMO 2021/5) Let $ n $ and $ k $ be positive integers. Two infinite sequences $ \{s_i\}_{i\geq 1} $ and $ \{t_i\}_{i\geq 1} $ are equivalent if, for all positive integers $ i $ and $ j $, $ s_i = s_j $ if and only if $ t_i = t_j $. A sequence $ \{r_i\}_{i\geq 1} $ has equi-period $ k $ if $ r_1, r_2, \ldots $ and $ r_{k+1}, r_{k+2}, \ldots $ are equivalent.
  Suppose $ M $ infinite sequences with equi-period $ k $ whose terms are in the set $ \{1, \ldots, n\} $ can be chosen such that no two chosen sequences are equivalent to each other. Determine the largest possible value of $M$ in terms of $ n $ and $ k $.
• (HMMT 2023 Algebra #10) Let $ \zeta = e^{2\pi i/99} $ and $ \omega = e^{2\pi i/101} $. The polynomial \[ x^{9999} + a_{9998}x^{9998} + \dots + a_1x + a_0 \] has roots $ \zeta^m + \omega^n $ for all pairs of integers $ (m,n) $ with $ 0 \leq m < 99 $ and $ 0 \leq n < 101 $. Compute $ a_{9799} + a_{9800} + \dots + a_{9998} $.
• (USA TST 2023/3) Consider pairs $ (f,g) $ of functions from the set of nonnegative integers to itself such that
  • $ f(0) \geq f(1) \geq f(2) \geq \dots \geq f(300) \geq 0 $;
  • $ f(0)+f(1)+f(2)+\dots+f(300) \leq 300 $;
  • for any 20 nonnegative integers $ n_1, n_2, \dots, n_{20} $, not necessarily distinct, we have \[ g(n_1+n_2+\dots+n_{20}) \leq f(n_1)+f(n_2)+\dots+f(n_{20}). \]
  Determine the maximum possible value of $ g(0)+g(1)+\dots+g(6000) $ over all such pairs of functions.