2024: Year in Review

Sometimes, we forget how far we’ve come.

This blog is now a little more than a year old! We’ve done well to keep up our weekly publishing streak, with 52 posts this year (and 17 more in the tail of 2023). Let’s celebrate the turn of the year with a consolidation of our writings. 

Algebra

Choo Ray (8) broke down Vieta Jumping, showing us that beyond identifying the key idea, it was also important to double down on the analysis, and draw connections between features of the problem and of the solution. Wee Kean (14) gave a concise introduction to linear algebra, and explained a geometric (Combi-Nullstellensatz-flavoured) application via dimension counting of a space of polynomials. 

Dylan (39) showcased ISL 2023 A7, exposing the geometry hidden behind a discrete averaging of the square root function. Gabriel (46) explained his approach towards a piecewise inequality problem: investigating small cases motivated an induction proof, with terms grouped by size.

Drew (35) demonstrated the visualisation of Cauchy-type functional equations with “banned regions” of the graphs. Pengchong (40) introduced the idea of “substituting zero” into a functional equation with domain the positive reals, formalised with the notion of limits in analysis. 

Combinatorics

Wee Kean (-11) investigated a circular chip-firing problem, highlighting the idea of labelling and tracking chips. David (-10) showed terminating and uniqueness properties of general chip-firing processes by studying the directed graph of configurations. 

Dylan (-16) used small cases to identify the inductive nature of a construction combi problem. Xu Chen (-7) tackled ISL 2018 C5, a construction optimisation problem, by using small cases to guide and motivate bounds and construction strategies.

Yan Sheng (38) deep-dived into a problem about bookshelf sorting, repeatedly optimising monovariants to obtain better quantitative bounds. Drew (45) elucidated ISL 2023 C5 and USA TST 2017 P4 by explaining how a global strategy (tracking auxiliary historical data) sidesteps the need for a clever local strategy. Yu Peng (-1) showcased dynamical programming as a useful perspective for some algorithmic combi problems, by searching for/building a structure incrementally.

David (4) showed us how indicator functions underlie the Inclusion-Exclusion principle. 

Yu Peng (-12) taught us about flows and the Max-Flow-Min-Cut Theorem, providing a fresh perspective to some results in graph theory. Choo Ray (18) explained the value of understanding different solutions to the same problem, illustrating it with a graph theory problem involving cycle avoidance.

David (7) taught us about coupling: pairing or correlating outcomes/states, with enumerative, algorithmic, and probabilistic applications. Wee Kean (29) introduced us to impartial game theory with Nim, explaining how to deduce the win states.

Aloysius (22) explained how logic puzzles teach us about parity, double counting, and notions of topology. Etienne (9) showcased the Ham Sandwich Theorem (Borsuk-Ulam: generalisation of the intermediate value principle) as a way of thinking and arguing topologically, even in the discrete setting. 

Glen (11) shared several problems about rectangle tilings, how to use the structure of junction-pairs, and an alternative approach with resistor networks. Lucas (17) showcased colouring strategies and invariants in grid combi. Jeck (23) introduced a combinatorial generalisation of line arrangements for which Sylvester-Gallai, and several other results in discrete planar geometry, still hold. 

Geometry

Ker Yang (24) demonstrated the ubiquity of Riem’s Theorem (a configuration of two circles and parallel lines). Glen (51) showed us how to see inversions and Möbius transformations as symmetries of the plane acting on the diagram, and how to understand conjugates of these actions. 

Glen (43) defended the Euclidean geometry bashers by illustrating the surgical power of coordinate geometry by introducing overarching principles and tools, and outlining the bash strategies in various examples. 

Dylan (48) tackled SMO Junior 2024 P1, a geometric optimisation problem, ultimately explaining how to visualise the optimisation as water in a cup. 

Number Theory

Jit (1) taught us about solutions to Pell’s equation using the multiplicative norm on $\mathbb Z[\sqrt d]$, and its cubic generalisation. Dylan (2) explored recurrence and modular properties of the Fibonacci numbers. 

Choo Ray (-5) uncovered the rabbit hole beneath an innocent Diophantine involving tetrahedral numbers. Jit (10) showed us a proof of Lagrange’s 4-Squares Theorem. 

Wee Kean (0) introduced some natural quantities in analytic number theory (such as the prime-counting function), and described algorithms to compute them, along with their computational complexities. 

Contests

Sheldon (-14) shared his most memorable ISLs, giving insight into what makes a problem beautiful. David (25) Marie-Kondoed several CTST problems, stripping away the noise to extract the aesthetic core. 

Glen (-15, 27) documented his raw solving process of SMO Open 2023/24, giving an inside look to his approach towards problem solving. 

Dylan (28) described the "IMO syllabus" based on past problems, and shared his take on the IMO journey. David (30, 31) documented his livesolve experience of IMO 2024 with Sheldon, Yan Sheng, and Aloysius. Glen (32), Andrew (33), and Etienne (34) shared their insights and opinions on each of three IMO 2024 problems, in their respective capacities (coordinator, PSC, observer). 

David (52) shared his approaches to solving some Putnam 2024 problems.

Problem Creation/Pedagogy

Choo Ray (37) shared his thought process towards creating an original problem, inspired by certain controlled sequences of the recursion $n\mapsto n + \varphi(n)$. 

David (41, 42, 44) revealed the story behind his SMO 2024 P4 problem proposal, detailing its roots in machine learning theory/convex geometry, and introducing us to the Rademacher complexity and other ideas in learning theory.

Shi Cheng (36) reflected on his olympiad journey, sharing what worked and what didn’t. Dylan (-9) shared about several broad themes of mathematical intuition. Gabriel (26) taught us to draw connections between different problems/solutions, distil the shared structures, and pay attention to resemblances detected by our subconscious. 

Xu Chen (6) illustrated the importance of visualising objects, such as functional equations and game strategies. Yu Peng (13) reminded us that if your solution/strategy falls short, not all is lost: by identifying the deficiencies and optimising your approach, you might just succeed! Lucas (-2) described the power of wishful thinking and squeezing constraints (e.g. showing an optimal configuration must be symmetric) to narrow one’s search space for constructions. 

Yan Sheng (50) taught us how to dice onions while keeping piece sizes roughly equal (though knife safety remains a worry!). 

Beyond Olympiads

Yan Sheng (12) revealed his all-time favourite theorems, giving us a glimpse of the rich structures and ideas in various fields of mathematics.

Etienne (-4) investigated rectangular tilings of squares, naturally leading towards algebraic numbers and continued fractions. David (-13) shared about bounding the degree of polynomials with small coefficients satisfying certain divisor relations. Jit (21) explained why most algebraic plane curves have few solutions with both coordinates being roots of unity. 

Zhao Yu (3) and Dylan (16) shared about elliptic curves: hidden additive structure among solutions to a cubic Diophantine equation, with surprising consequences. Jit (-8) introduced the p-adics and the dynamical behaviour of iterated polynomial maps.

David (15) explained a prime factorisation algorithm via lattices and an apt notion of “reduced” basis, introducing ideas in cryptography.

Yan Sheng (5) explained how some abstract logic theory proves its own inconsistency, illustrating the difference between “implicit” and “explicit” existence. Wee Kean (49) introduced Turing machines as an abstraction of computability and algorithms, building to Hilbert’s 10th problem.

Glen (-3) used properties of hyperbolic space (i.e. spacetime) and its symmetries to prove non-standard theorems in Euclidean geometry. 

Yan Sheng (19) explained how applying Vieta’s Theorem to a polynomial with infinite factors, establishes the $\pi^2/6$ identity. David (20) utilised known properties of common probability distributions to compute probabilities and averages without calculus. 

David (47) introduced some analysis and topology, explained the Baire Category Theorem (sparse closed subsets of the real numbers are “small”), and described some peculiar consequences. Yan Sheng (-6) described various approximation schemes of $\sqrt 2$, with increasingly better rates of convergence.

Cosmic Gratitude

To Aloysius, Andrew, Choo Ray, David, Drew, Dylan, Etienne, Gabriel, Glen, Jeck, Jit, Ker Yang, Lucas, Pengchong, Sheldon, Shi Cheng, Wee Kean, Xu Chen, Yan Sheng, Yu Peng, and Zhao Yu, thank you for supporting this dream. We appreciate you taking time out of your usual work/study/military commitments to stay connected, and contribute to this shared creation.

And thanks to our loyal readers too; hope the blog has been, and will continue to be, a source of inspiration and mathematical ecstasy. 

Till next year,

Simon