## A Problem with Parabolas

Let $P$ be the parabola with equation $y=x^2$ and let $Q$ be a parabola with a horizontal axis of symmetry. Suppose that $P$ and $Q$ intersect at $(1,1)$, $(3,9)$, and at one other point. Find the equation of the axis… Continue Reading

Astute geometry students know that when a problem involves a 3 and a 4, the answer will involve a 5 and the solution will make use of a right triangle somehow. The triple of numbers $(3,4,5)$ is called a Pythagorean… Continue Reading

## Certified

The National Board of Professional Teaching Standards tells its candidates when they submit their portfolios each spring that they will get their results by the end of the year. On December 30, 2017, it crossed my mind that I hadn’t… Continue Reading

## Limits of Desmos, Part 2

A few years ago, I posted about some limitations of the Desmos online graphing software. Last week, I was playing with Desmos again and found a couple more. Scaling a Line Starting from any true equation, manipulating both sides in… Continue Reading

## “Do You Know What I Like About Binary Code?”

“Do you know what I like about binary code?”, asked my 4th grader out of nowhere. “What?” “Each number has only one way. For example, 20 is 16+4. But if you used 8+4+2+1, that would only be 15, which isn’t… Continue Reading

## Three Circles in a Triangle: A Solution

In my last post, I posed the problem of showing that if triangles $ABC$ and $PQR$ (as shown in the diagram below) are similar, then $\Delta ABC$ is equilateral. We will solve this problem in two steps: Show that the… Continue Reading

## Three Circles in a Triangle: A Problem

This week, I was playing around with some geometry and came up with a fun problem. Suppose there is a triangle, $\Delta ABC$. Inside the triangle are three circles, each of which is tangent to the other circles and to… Continue Reading

## Math Kangaroo

The movie Mean Girls took some liberties with its Mathletes scene, but it mostly got the essentials of math competitions right. In a typical competition, a handful of students from each school face off to solve some hard problems. I started… Continue Reading

## Maximizing the Product: An Explanation

In my last two posts, I asked what the maximum product of integers adding up to 2017 was and then showed that the answer was $2^2\cdot 3^{671}$. It was not clear at the beginning that the largest possible product would… Continue Reading

## Maximizing the Product: A Solution

In my last post, I asked What is the largest possible product you can make with a group of positive integers if the sum of those integers is 2017? Before we rush to the numerical answer, let’s follow my usual… Continue Reading