Categories
Competition Math

Middle Five

Below are sets of videos discussing the “Middle Five” (or problems 6-10) of the AIME starting from 2002. These videos serve to help you get a better understanding of not only the solution, but also the thinking process and motivation behind it. Solving these problems is crucial to taking that next step to qualifying for the USA(J)MO.

Categories
Competition Math

Symmetric Sums

This handout covers all of the material you need to know to solve most AMC and AIME problems that involve either Vieta’s Formulas or Newton Sums. These proofs and descriptions are also followed up by worked example problems as well as exercises left to the reader.

Last Updated: March 9th, 2021

Categories
Competition Math

2002 AIME I

Below are videos discussing problems 6 thru 10 of the 2002 AIME I. They provide in-depth solutions and what I would think when solving the questions. Keep in mind that I am still experimenting with making videos, so please bear with me. Enjoy!

Problem 6

 

Problem 7

Problem 8

Problem 9

Problem 10

You can find more videos on our youtube channel.

Categories
Pure Math

The integers

Construction of the Integers

You may have heard of the integers before. Namely, it’s the set of numbers

    \[\dots, -2, -1, 0, 1, 2,\dots\]

stretching on infinitely. But what fundamental properties does this amazing set satisfy? How much can we derive based on these axioms? Are the theorems we see and take for granted really trivial?

The integers, commonly denoted as \Z, are governed by two operations: addition (+) and multiplication (*). Specifically, we have additive and multiplicative closure, which means

    \[(a+b),\, (a*b)\in\Z\ \forall\, a,b\in \Z.\]

This means that if we take two integers a and b, their sum a+b and their product a*b are integers as well. We also quickly see that the integers form a ring, as for all a,b,c\in \Z we have the following properties:

    \begin{align*} \textbf{Commutativity:}\ & a+b = b+a \\ & a*b = b*a \\ \textbf{Associativity:}\ & (a+b)+c = a+(b+c) \\ & (a*b)*c = a*(b*c) \\ \textbf{Distributivity:}\ & a*(b+c) = (a*b)+(a*c) \\ \textbf{Zero:}\ & \text{There exists an element 0 such that}\ a+0=a \\ \textbf{One:}\ & \text{There exists an element 1 such that}\ a*1=a \\ \textbf{Negativity:}\ & \text{There exists an element}\ a^\prime\ \text{such that}\ a+a^\prime = 0 \end{align*}

With the integers defined, we now look at one of its special subsets: the positive integers. How do we define positivity? We cannot simply say an integer is positive when it’s greater than 0, as we have not defined the notion of order in the integers, and thus cannot say when integers are greater or less than each other.

Exercise: Is there a way to define “greater than” or “less than” with our above axioms?

Let S be a subset of the integers that is closed under addition and multiplication. Remember that this means if we have a,b\in S, then a+b and a*b must also be in S. But this is not enough to construct the positive integers; the integers is a subset of the integers satisfying additive and multiplicative closure. Thus, we stipulate further that the identity 0 cannot be in S, and that for any integer a, only one of a and -a lie in S (here, -a refers to the additive inverse guaranteed by Negativity).

Exercise: Does the positive integers satisfy these two properties?
Exercise: How do we know that 1, -2 are not both in S?

S can either be the positive integers or the negative integers. To simplify things, we call the positive integers \Z^+ and the negative integers \Z^-. Now we can define the symbols > and <. Let a,b,p be integers where a+p=b. We say a<b if p\in \Z^+, and a>b if p\in \Z^-.

Categories
Pure Math

Knot Theory

Written by Kevin Xu

If you’ve ever tied your shoes or made a bracelet, you probably have heard of knots. You start out with a line and then fold it over and under itself until it forms a shape for you to use. However, this is not quite correct in the mathematical world. In mathematics, a knot is an embedding of a circle in the 3D space. You may be confused about this definition, but the gist of it is that a knot is a connected strand that goes over and under itself. Here is the simplest kind of knot:

Figure 1: The unknot

This knot is called the trivial knot or the unknot. Here is the unknot, but in a slightly different form:

Figure 2: Also the knot

A little harder to see, right? We consider this to be the unknot because it can be twisted and turned in 3D space to get the unknot. These twists and turns form equivalence classes for knots. When twisting and turning, there are three moves called Reidemeister moves that we can make:

Figure 3: The Reidemeister moves

It turns out that all possible transformations of twisting and turning a knot can be expressed in these three moves. But how does this have to do with proving two knots are equivalent? Notice that if we had some characteristic of a knot that was constant under these three moves, then every knot in a knot’s equivalent class shares the same characteristic, called an invariant.

 

We can easily find some simple knot invariants. For a knot, we define an intersection to be a place where the knot goes over or under itself. One invariant would be the minimum number of intersections in all knots equivalent to it. Of course this characteristic would stay constant for all knots in its equivalence class; they all point to the same knot!

 

There are many complex and invariants in knot theory, for example tricolorability, or even knot polynomials such as the Jones and Alexander polynomials. Knot theory is a fascinating subject that bears a lot of similarity to braid theory, and thus has applications in both physics and cryptography.

Categories
Pure Math

Calculus

What is Calculus?

Calculus is the study of “continuous change”, dealing with tangent lines to a curve, or function, and the area under a curve. It deals with change at a specific point, with the instantaneous rate of change at a point being known as the derivative, and the area under a curve (between the function and the x-axis) being known as the integral. It is one of the first courses of undergraduate higher-level mathematics, and it is split into two different categories, integral and differential calculus.

Differential and Integral Calculus

Differential calculus is the typical first semester, or half, of an introductory calculus course. This deals with the derivative and introduces the concept of limits, which approximate the value as a function approaches a certain point or infinity. This naturally ties in with the concept of finding a tangent line at a point. This concept of the derivative has applications in many other areas and fields of mathematics.

Integrals are the second half of calculus, the complement of the derivative. Integrals represent the area under a curve, which also has many useful applications in fields like engineering, optimization in computer science, and machine learning.

Applications

Calculus is known as the gateway to higher mathematics, and it has applications in a wide variety of fields. Calculus is used in every branch of the sciences, including computer sciencestatistics, and engineering. One particularly interesting application is in data science or machine learning. The fundamental concept of optimization, or how a machine learning model learns, is through calculus. It uses calculus to minimize the errors made by the computer until an almost-optimal model is reached. It also has many applications in engineering as well as physics, particularly anything that involves motion, speed, and acceleration, which can be represented in terms of functions and their derivatives.

Prerequisites/What you need to know

To understand calculus, you will need to go through the standard high school math curriculum. This is typically the sequence Algebra I -> Algebra 2 -> Trignometry -> Precalculus -> Single-variable Calculus -> Multivariable Calculus

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