Some amusing things that I wrote a long time ago, mostly in high school.
Sample 11th Grade KSEA NMC #20
A brief problem with a telescoping product.
How many times must you go around a circle one radian at a time to end up at the same point?
A brief application of modular arithmetic.
An Exercise in Symmetric Sums
A problem requiring the manipulation of the symmetric sums of the roots of a polynomial.
What is the sum of the reciprocals of all integers divisible by primes that are only in a given set?
This problem is solved using infinite geometric series. Incidentally, it comes in handy for the 2018 AMC 12A Problem #19.
A Problem on a Parametrically Defined Parabola
This problem was first posed on the Khan Academy fora from an unknown textbook. I use a fundamental property of parabolas to simplify the final step of problem.
A Problem Applying the Number-Theoretic Properties of Primitive Roots of Unity
It is quite stunning that the \(n^{\textrm{th}}\) primitive roots of unity are given by \(\exp{\frac{2\pi k}{n}}\) for positive integers \(k<n\) such that \(\gcd{\left(k,n\right)}=1\). Here, we exploit this fact to solve a problem.
Parity of a Derivative
For nonconstant differentiable functions with parity, \(f\) and \(f'\) have opposite parity. Why is this so?
Canada 2001 #1
A quick solution to the first problem on the 2001 Canadian Olympiad.
Two Problems Using Cyclic Quadrilateral
These two problems are solved using the properties of cyclic quadrilaterals.
2016 AIME I #6
Happy Birthday Will :)
Ellipse to Line Distance via Non-Isometric Axis Scaling
A novel approach to the problem of finding the shortest distance between an ellipse and a line by reducing the problem to a simpler one. This paper will also discuss an algorithm that can be implemented to find this distance.
Ellipse to Line Distance, Part 2
This continues the previous paper by delving into the more theoretical aspects of the technique. We prove that the technique is valid. This paper also happens to be my HL Mathematics Internal Assessment for the International Baccalaureate program.
Hypervolume of a Hypersphere
In this paper, we discuss an interesting manner to find the \(n\)-dimensional hypervolume of a hypersphere, \(V_n\). We exploit the fact that \(V_n\propto r^n\) to avoid a complicated integral.
2017 FAMAT Fall Interschool #23
My solution to an intimidating calculus problem. Tread carefully – this one requires a careful read.
2018 AMSP Test B Solutions
These are my solutions to the 2018 AwesomeMath Summer Program Admission Test B. I present full solutions to 9 out of 10 problems.
The Vandermonde Convolution
A general discussion of the Vandermonde Convolution and its relationship to other combinatorial identities.
Solutions to 100 Geometry Problems
A running solution compilation to David Altizio's 100 Geometry Problems.
A Second Order ODE
We find the solutions to the ODE \(\ddot{x}-Kx=0\). There is particular emphasis on complex exponentials to show the relationship between the \(K>0\) and \(K<0\) cases. This equation shows up in physics, most commonly as simple harmonic motion.
Putnam 2019
Solutions to A1 and B1, which are problems that I managed to solve during the contest.
The Weak Fundamental Theorem of Algebra
Here we present a proof that a polynomial with degree \(n\) can have no more than \(n\) distinct roots using just linear algebra. From this, we can ascertain that this property is no more than a direct consequence of the underlying structure of the "linear combination of terms"-nature of polynomials.
The Contraction Mapping Theorem
We prove the contraction mapping theorem, building a lot of intuitive ground on the topic along the way. In particular, we show that homotheties are just a type of contraction mapping, and we heuristically argue for the uniqueness of the fixed-point by visualizing a vector field with "competing sinks".
The Lebesgue Integral
We introduce the Lebesgue integral, avoiding measure theory (except the concept of measure zero), and instead defining the Lebesgue integral as an infinite sum of Riemann integral. We show the utility of the Lebesgue integral in various contexts, such as proving Feynman's trick. We solve four problems with the Lebesgue integral and related concepts.
Trajectory of a Falling Object
The motion of an object in free fall can be parameterized in the \(x\) and \(y\) dimensions. By implicitizing the equations, we can obtain a standard quadratic describing \(y\) in terms of \(x\).
Complementary Launch Angles
Using the parabolic trajectory equation, we can show that an object launched at an angle \(\theta\) lands in the same spot as an object launched at an angle of \(90^{\circ}-\theta\).
Bounce Times for Successive Bounces
Each bounce starts with an inelastic collision with the ground. What is the recursive relationship between the bounce times of the \(n^{\textrm{th}}\) bounce and the \((n-1)^{\textrm{th}}\) bounce?
Bounce Angles for Successive Bounces
In the spirit of the previous paper, we find the recursive relationship between the bounce angles of the \(n^{\textrm{th}}\) bounce and the \((n-1)^{\textrm{th}}\) bounce. Then, we find an explicit formula for \(\theta_n\).
Bounce Velocities for Successive Bounces
Similar to the previous paper, we find an explicit formula for the \(n^{\textrm{th}}\) initial velocity.
The Bounce Equations
This is the culmination of the previous papers. These equations, when graphed in the first quadrant, describe all bounces of an object given a starting velocity, starting launch angle, gravitational acceleration, and a coefficient of restitution.
Linear Drag
At sufficiently low velocities, drag force obeys \(F\propto v\). We solve a differential equation to obtain the function \(v(t)\).
Center of Mass of a Fluid-Filled Cylinder
Imagine a cylinder with uniform mass distribution. Obviously, such a cylinder must have a center of mass at a height that is half of its total height. As a fluid is poured into the cylinder, the bottom of the cylinder becomes heavier than the top, pulling the center of mass downwards. However, once the cylinder is completely filled with fluid, both halves of the cylinder are equally massive, implying that the center of mass is back where it started: right in the middle. By the extreme value theorem, the height of the center of mass must have attained a minimum value. How much fluid minimizes this height?
Gauss' Law
A discussion of Gauss' Law and the solutions to some problems using it.
Collision Time
How much time does it take for a small object, starting at rest from some substantial distance away, to impact a (much more massive) planet if the only forces on the system are gravitational?
Select Problems from Introductory Classical Mechanics
Solutions to some problems from Introductory Classical Mechanics by David Morin.
Inertial Frames of Reference and Motion in Polar Coordinates
We derive the equations for the radial and angular forces on a particle in polar coordinates. We then briefly explain the existence of the Coriolis term which arises in the equations by imagining our frame to be non-inertial while an inertial frame dances around with the particle, such that the particle is always stationary in that frame. We conclude by solving two problems, including showing that \(\dot{r}=\sqrt{Ar^4+B}\) blows up in finite time.
Gravitational Field of a Thin Wire on the z-axis
We analyze the gravitational field in the \(xy\)-plane of a thin wire along the \(z\)-axis. We also extend the definition of the potential function of a force field to account for force fields that do not enjoy the conditions found in Newtonian gravity or Coulombian electrostatic force.
Corrections to the Pendulum
The period of a pendulum is approximately \(2\pi\sqrt{\frac{\ell}{g}}\). This is found using the small-angle approximation, \(\sin{\theta}\approx\theta\). We derive an integral expression for the true period of a pendulum. Then, we convert the integral to a complete elliptic integral of the first kind and then using the binomial series and sine reduction formula, we obtain an infinite series expression for the period.
Problems on Magnetic Fields
Solutions to a few interesting problems that use the Biot-Savart law and Ampere's law.
A brief problem with a telescoping product.
How many times must you go around a circle one radian at a time to end up at the same point?
A brief application of modular arithmetic.
An Exercise in Symmetric Sums
A problem requiring the manipulation of the symmetric sums of the roots of a polynomial.
What is the sum of the reciprocals of all integers divisible by primes that are only in a given set?
This problem is solved using infinite geometric series. Incidentally, it comes in handy for the 2018 AMC 12A Problem #19.
A Problem on a Parametrically Defined Parabola
This problem was first posed on the Khan Academy fora from an unknown textbook. I use a fundamental property of parabolas to simplify the final step of problem.
A Problem Applying the Number-Theoretic Properties of Primitive Roots of Unity
It is quite stunning that the \(n^{\textrm{th}}\) primitive roots of unity are given by \(\exp{\frac{2\pi k}{n}}\) for positive integers \(k<n\) such that \(\gcd{\left(k,n\right)}=1\). Here, we exploit this fact to solve a problem.
Parity of a Derivative
For nonconstant differentiable functions with parity, \(f\) and \(f'\) have opposite parity. Why is this so?
Canada 2001 #1
A quick solution to the first problem on the 2001 Canadian Olympiad.
Two Problems Using Cyclic Quadrilateral
These two problems are solved using the properties of cyclic quadrilaterals.
2016 AIME I #6
Happy Birthday Will :)
Ellipse to Line Distance via Non-Isometric Axis Scaling
A novel approach to the problem of finding the shortest distance between an ellipse and a line by reducing the problem to a simpler one. This paper will also discuss an algorithm that can be implemented to find this distance.
Ellipse to Line Distance, Part 2
This continues the previous paper by delving into the more theoretical aspects of the technique. We prove that the technique is valid. This paper also happens to be my HL Mathematics Internal Assessment for the International Baccalaureate program.
Hypervolume of a Hypersphere
In this paper, we discuss an interesting manner to find the \(n\)-dimensional hypervolume of a hypersphere, \(V_n\). We exploit the fact that \(V_n\propto r^n\) to avoid a complicated integral.
2017 FAMAT Fall Interschool #23
My solution to an intimidating calculus problem. Tread carefully – this one requires a careful read.
2018 AMSP Test B Solutions
These are my solutions to the 2018 AwesomeMath Summer Program Admission Test B. I present full solutions to 9 out of 10 problems.
The Vandermonde Convolution
A general discussion of the Vandermonde Convolution and its relationship to other combinatorial identities.
Solutions to 100 Geometry Problems
A running solution compilation to David Altizio's 100 Geometry Problems.
A Second Order ODE
We find the solutions to the ODE \(\ddot{x}-Kx=0\). There is particular emphasis on complex exponentials to show the relationship between the \(K>0\) and \(K<0\) cases. This equation shows up in physics, most commonly as simple harmonic motion.
Putnam 2019
Solutions to A1 and B1, which are problems that I managed to solve during the contest.
The Weak Fundamental Theorem of Algebra
Here we present a proof that a polynomial with degree \(n\) can have no more than \(n\) distinct roots using just linear algebra. From this, we can ascertain that this property is no more than a direct consequence of the underlying structure of the "linear combination of terms"-nature of polynomials.
The Contraction Mapping Theorem
We prove the contraction mapping theorem, building a lot of intuitive ground on the topic along the way. In particular, we show that homotheties are just a type of contraction mapping, and we heuristically argue for the uniqueness of the fixed-point by visualizing a vector field with "competing sinks".
The Lebesgue Integral
We introduce the Lebesgue integral, avoiding measure theory (except the concept of measure zero), and instead defining the Lebesgue integral as an infinite sum of Riemann integral. We show the utility of the Lebesgue integral in various contexts, such as proving Feynman's trick. We solve four problems with the Lebesgue integral and related concepts.
Trajectory of a Falling Object
The motion of an object in free fall can be parameterized in the \(x\) and \(y\) dimensions. By implicitizing the equations, we can obtain a standard quadratic describing \(y\) in terms of \(x\).
Complementary Launch Angles
Using the parabolic trajectory equation, we can show that an object launched at an angle \(\theta\) lands in the same spot as an object launched at an angle of \(90^{\circ}-\theta\).
Bounce Times for Successive Bounces
Each bounce starts with an inelastic collision with the ground. What is the recursive relationship between the bounce times of the \(n^{\textrm{th}}\) bounce and the \((n-1)^{\textrm{th}}\) bounce?
Bounce Angles for Successive Bounces
In the spirit of the previous paper, we find the recursive relationship between the bounce angles of the \(n^{\textrm{th}}\) bounce and the \((n-1)^{\textrm{th}}\) bounce. Then, we find an explicit formula for \(\theta_n\).
Bounce Velocities for Successive Bounces
Similar to the previous paper, we find an explicit formula for the \(n^{\textrm{th}}\) initial velocity.
The Bounce Equations
This is the culmination of the previous papers. These equations, when graphed in the first quadrant, describe all bounces of an object given a starting velocity, starting launch angle, gravitational acceleration, and a coefficient of restitution.
Linear Drag
At sufficiently low velocities, drag force obeys \(F\propto v\). We solve a differential equation to obtain the function \(v(t)\).
Center of Mass of a Fluid-Filled Cylinder
Imagine a cylinder with uniform mass distribution. Obviously, such a cylinder must have a center of mass at a height that is half of its total height. As a fluid is poured into the cylinder, the bottom of the cylinder becomes heavier than the top, pulling the center of mass downwards. However, once the cylinder is completely filled with fluid, both halves of the cylinder are equally massive, implying that the center of mass is back where it started: right in the middle. By the extreme value theorem, the height of the center of mass must have attained a minimum value. How much fluid minimizes this height?
Gauss' Law
A discussion of Gauss' Law and the solutions to some problems using it.
Collision Time
How much time does it take for a small object, starting at rest from some substantial distance away, to impact a (much more massive) planet if the only forces on the system are gravitational?
Select Problems from Introductory Classical Mechanics
Solutions to some problems from Introductory Classical Mechanics by David Morin.
Inertial Frames of Reference and Motion in Polar Coordinates
We derive the equations for the radial and angular forces on a particle in polar coordinates. We then briefly explain the existence of the Coriolis term which arises in the equations by imagining our frame to be non-inertial while an inertial frame dances around with the particle, such that the particle is always stationary in that frame. We conclude by solving two problems, including showing that \(\dot{r}=\sqrt{Ar^4+B}\) blows up in finite time.
Gravitational Field of a Thin Wire on the z-axis
We analyze the gravitational field in the \(xy\)-plane of a thin wire along the \(z\)-axis. We also extend the definition of the potential function of a force field to account for force fields that do not enjoy the conditions found in Newtonian gravity or Coulombian electrostatic force.
Corrections to the Pendulum
The period of a pendulum is approximately \(2\pi\sqrt{\frac{\ell}{g}}\). This is found using the small-angle approximation, \(\sin{\theta}\approx\theta\). We derive an integral expression for the true period of a pendulum. Then, we convert the integral to a complete elliptic integral of the first kind and then using the binomial series and sine reduction formula, we obtain an infinite series expression for the period.
Problems on Magnetic Fields
Solutions to a few interesting problems that use the Biot-Savart law and Ampere's law.