Category Archives: Math

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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A Bad Attempt At Connecting Differential Forms And Multivariable Calculus

So after taking MVC, we’ve all been through those tedious proofs that and . Here, we give a unified way to view these identities. We start by giving the notion of a tangent space of a point in , which … Continue reading

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Free Objects and Universal Properties, Because Most General is Too Wordy

So we often see these constructions come up. For a set , we define the free group as the group of words under string concatenation of . Similarly, the free monoid over is just words given by under concatenation. We … Continue reading

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