Category Archives: Math

Simon Marais Mathematics Competition 2019

This year’s contest had some interesting problems. I’ll present some example solutions to the less technically challenging of the lot. Problem A1: Let . Define a recursive sequence by letting be the sum of and the product of its distinct … Continue reading

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Complex Numbers are Very Real

A brief while ago I had a discussion with some people who claimed that complex numbers may be very hard to wrap your head around. This is arguably quite true, as intuition can be different for everyone. In this post, … Continue reading

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Cheap Nullstellensatz

I was recently asked to prove a non-exhaustive set of cases for the weak Nullstellensatz, and the proof was so deceptively simple I decided to share it. The main theorem goes along the lines of this. Nullstellensatz: Let be an … Continue reading

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ISL 2017 A1

Problem A1: Let and be positive integers such that     If , prove that the polynomial     has no positive roots. Solution A1: We appeal to a similar strategy that is used in IMO 2012-2. Write     … Continue reading

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IMO 2018 Day 2

Day 2 had a bunch of really nice problems with more advanced techniques required to solve them. Problem 6 is also hard and I will be writing it up at a later time. Problem 4: A site is any point … Continue reading

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IMO 2018 Day 1

I have solved problems 1 and 2. I have not yet attempted problem 3 but I believe that it is beyond me, especially within a time frame of four and a half hours. Problem 1: Let be the circumcircle of … Continue reading

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Revenge of the Finite Fields

A few days ago, I posted some proofs and a construction of finite fields. It turns out there are a lot more nice constructions and proofs and problems related to finite fields, so I’ll be sharing some of them here. … Continue reading

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Finite Fields for Mortals

I’ve been studying a bunch of cryptography lately, and I thought it would be nice to present finite fields in some relatively easy to understand way that doesn’t require too much prior knowledge. All that is really needed here is … Continue reading

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UIUCTF2017 – OldTV

Not many people solved this despite the author and I believing that this should have been an easy challenge. We were given the following program: Initially, you should try to verify the signature and compute , and realize that it … Continue reading

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Pullbacks of Differential Forms

Let be a smooth map between manifolds and let be a smooth -form on . We have the natural push forward/total differential given by , where is a curve satisfying and , but this also gives a natural way to … Continue reading

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