Post 2 of Wallly's Math Books: Fundamental Laws and Operations

This is the second post of a series of blogs devoted to my grandfather Wally's college freshmen math textbooks. One of the books Wally used was called Advanced Algebra. It was published in 1925 and in the preface it states the book is designed for students who have had one or two years of college algebra.  In addition, it states "The syllabi of the College Entrance Board, of the Regents of the State of New York, and of practically all states have been covered." This makes me wonder if there are college math texts today that take the New York Regents or Common Core Standards into account when designing their textbook.

In the previous blog post, I encourgaed people to try the following problem.  In addition to posting on twitter, I linked the blog post to the "Higher Education Learning Collective" Facebook group and was told by some that this problem was too easy to be a college level problem.  Here was that problem.

Problem 1: Prove that "The sum of the squares of any two unequal real numbers is greater than twice their product.  That is if a ≠ b, a² + b² > 2ab."  

Here is essentially the algebraic proof as given in the textbook: 

Given: a ≠ b, we know that a - b ≠ 0.  Squaring both sides gives  a² - 2ab+ b² > 0 and that means that a² + b²  > 2ab.

Here is a way to visually see that the sum of the areas of two squares is more than double the area of a rectangle formed by the sides of the squares by using Desmos - a much more dynamic and visually appealing proof.

And here's a link to the original Desmos page that I used to create the video above: https://www.desmos.com/calculator/r6p2hpdplf

Problem 2: This problem comes from pg. 28 of the Chapter 1 Fundamental Laws and Operations. Although it doesn't appear to have been assigned to my grandfather and his classmates, it seemed like a interesting problem to solve. In fact, it is problem that I could see myself giving to my MATH 120 (Math for Teaching) students on the first day of class just to see what problem solving strategies they might use.

How would you solve this problem?  Are there any generalizations you can draw?  Feel free to post in the comments of this blog post.  We'll look at at least two approaches to this problem in my next post.