## 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

## 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

## 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

## Maximizing the Product: A Problem

This week I offer a problem: What is the largest possible product you can make with a group of positive integers if the sum of those integers is 2017? Here we do not assume that the integers are distinct from… Continue Reading

My last post put forward the challenge to find numbers whose binary representations end with their decimal representations, like how $11 = 1011_2.$ It turns out that there are lots of them. Infinity of them, in fact. If we’re going to… Continue Reading

## Binary Patterns: A Puzzle

Imagine a world in which there are only 2 digits. We are used to having 10 digits, from 0 to 9, but with just 2 of them, we’d be limited to 0 and 1. We could count in that world… Continue Reading

## Puffy Triangles

A few months ago, I was working on a lesson on sectors of circles. A sector of a circle is a piece of it cut off by slicing along two radiuses from the center. There is a formula for the… Continue Reading