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Journal Publications
Journal Publications
Published in Parabola (December 2023; Volume 59, Issue 3) - the journal of the Mathematics and Statistics Department of the University of New South Wales.
Published in Parabola (December 2023; Volume 59, Issue 3) - the journal of the Mathematics and Statistics Department of the University of New South Wales.
Appeared in 'Most Popular Articles' - January 2024; February 2024, Parabola UNSW
The topic of my paper was inspired by a problem I encountered at the Bangladesh National Mathematical Olympiad a couple of years ago. The problem asked for the value of T(P) for a particular P. I could only solve it partially, building up to the casework along with the initial equation and inequalities. On a more sustained effort back home, I managed to derive the general algorithm to find T(P) for any P, which was the motivation behind working out the much simpler general formula, which I later found was first presented in 1985 by one of my favorite Number Theory authors - George E. Andrews - via the method of partitions.
Md Faiyaz Siddiquee Taaroop
Published in Parabola (August 2025; Volume 61, Issue 2) - the journal of the Mathematics and Statistics Department of the University of New South Wales.
Published in Parabola (August 2025; Volume 61, Issue 2) - the journal of the Mathematics and Statistics Department of the University of New South Wales.
The geometric proof employs the classic construction of the three squares on the three sides, but (at least to me) much more intuitive than Euclid's subsequent construction. I think that the equivalence between squares ACHI and ACKL (see the paper!) is quite nice.
The mechanical proof occurred to me as a pleasant accident during my initial days of Olympiad physics preparation. Essentially, I approached the same problem in two different ways, once simply using Pythagoras and once by using a more mechanical argument; both led to the same result, so, walking backwards, I developed a mechanical proof of the theorem that is still very close to my heart.
Md Faiyaz Siddiquee Taaroop
School Conference Presentation
School Conference Presentation
Exploring Depressed Cubics, General Cubics, Tartaglia's Method & Cardano's Method
Exploring Depressed Cubics, General Cubics, Tartaglia's Method & Cardano's Method
Faiyaz_Cubic_Equation.pdfPrepared in LaTeX Beamer for my 9th grade Student-Led Conference in Mathematics. The presentation involves the introduction to Tartaglia's Method (generalized by the well-known Cardano's Method) for solving cubic equations missing the x^2 term (known as depressed cubics).
Md Faiyaz Siddiquee Taaroop
Solution to the General Cubic Equation.pdfIn this note, the elaborated and generalized algebraic solutions are given along with the explanation of Cardano's method.
Md Faiyaz Siddiquee Taaroop
Problem Ideas
Problem Ideas
Pathogen Antibody Problem.pdfPathogen-Antibody InteractionÂ
Pathogen-Antibody InteractionÂ
A nice biology-oriented Lokta-Volterra style problem; quite a pretty way of demonstrating the power of ansatz in solving differential equations.
Md Faiyaz Siddiquee Taaroop
Expository Notes
Expository Notes
Gaussian Distribution and all that.pdfGaussian Distribution and All That
Gaussian Distribution and All That
Probably the longest expository math note I have written to date! Covers the Herschel-Maxwell derivation of the Gaussian/Normal distribution, insights into the Gaussian Integral, a tiny bit of statistics, and how everything works together to give us the final form of this ubiquitous distribution. This note was hugely motivated by 3blue1brown's video series.
Md Faiyaz Siddiquee Taaroop
arithmetic_geometric_pytriples.pdfAn Interesting Observation on Arithmetic and Geometric Sequences forming Pythagorean Triples
An Interesting Observation on Arithmetic and Geometric Sequences forming Pythagorean Triples
The title is quite self-explanatory. This is a result I accidentally came across while punching my dearest Casio fx-991ex calculator (now sadly discontinued) in Math class, allowing me to easily find evenly-spaced Pythagorean triples. The second part (geometric progression) is quite interesting as well.
Md Faiyaz Siddiquee Taaroop
factorial.pdfClosed-form expression for the number of digits in n factorial
Closed-form expression for the number of digits in n factorial
The problem seemed interesting, so I gave it a go (plus I wanted to hone my LaTeX skills). The result also seems derivable from Stirling's approximation.
Md Faiyaz Siddiquee Taaroop
calc-of-var-parabola.pdfUsing the Calculus of Variations to Find Turning Points of Parabolas
Using the Calculus of Variations to Find Turning Points of Parabolas
Can obviously be generalized!
This is an insanely beautiful idea from the Calculus of Variations with nice applications in the understanding of the Least Action Principle in classical mechanics and the idea of the Lagrangian. Essentially, small perturbations/changes from a turning point (minima in the context of least action) will only have a SECOND ORDER change in the corresponding function. This means that expanding the change in the function using the Taylor Series in FIRST ORDER and setting it equal to zero gets the job done!
Md Faiyaz Siddiquee Taaroop
JRC_Camp_Problem.pdfA Gentle Introduction to Problem-Solving using Induction
A Gentle Introduction to Problem-Solving using Induction
A nice problem with a nice solution :)
Md Faiyaz Siddiquee Taaroop
3-sided fair coin.pdfHow thick is a 3-sided fair coin?
How thick is a 3-sided fair coin?
Inspired by one of Matt Parker's (from Standup Maths) excellent videos!
My result, although obtained from a slightly different approach, seems to agree with the upper bound mentioned in the video.
Md Faiyaz Siddiquee Taaroop
Derivative and Integral of Inverse Functions.pdfDerivatives and Integrals of Inverse Functions
Derivatives and Integrals of Inverse Functions
An interesting geometrical analysis meant to show how certain integrals and derivatives can be approached in a visually intuitive way. Apart from the general formulae and derivations. the specific cases of the trigonometric functions are mentioned since I found them quite intriguing.
Md Faiyaz Siddiquee Taaroop
Quadratic.pdfA New Perspective on Quadratic Equations
A New Perspective on Quadratic Equations
Happy New Year 2025!
Md Faiyaz Siddiquee Taaroop
Spherical Trigonometry.pdfSpherical Trigonometry
Spherical Trigonometry
Should have written this a long time ago! I went through a large portion of my olympiad astronomy career not knowing the proofs of these widely used theorems from spherical trigonometry. This is a mathematical apology to the part of my mind which keeps on asking "But why?".
Md Faiyaz Siddiquee Taaroop
Buffon's Needle Problem.pdfBuffon's Needle Problem
Buffon's Needle Problem
Back into the past, this was among the extremely intriguing problems that urged me to learn calculus. On a philosophical note, I like to think that pi comes up because in the random process of throwing needles, one essentially utilities all the possible angles in a circle that the needle could take. I think that the setup of the problem leverages that circular aspect. The note also features "my way" of thinking about the integral of inverse functions (for more context, see this note: https://sites.google.com/view/taaroop/mathematics#h.48ypad7akexn).
Md Faiyaz Siddiquee Taaroop
Envelopes and Cardioids.pdfEnvelopes and Cardioids
Envelopes and Cardioids
Inspired by the winning entry by Paralogical (https://youtu.be/fJWnA4j0_ho?si=pJ8_O3Em39bYCWnG) at the first Summer of Math Exposition. This is what makes math beautiful, in my opinion (minus the ugly trigonometry at the end; I just had to do those to make the final answer pretty). The idea in the envelope condition involving nudging the parameter of the original family of curve and how that relates to partial derivatives and helps us find the point of tangency on the envelope is just brilliant in my opinion. Hope you enjoy!
You might also want to check out this note: https://sites.google.com/view/taaroop/physics#h.7ywgjl8bn1gb
Md Faiyaz Siddiquee Taaroop
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