The product.

Level 2

$x$ ; $y$ ; $z$ are three distinct positive integers that sastify: $x^3 + y^3 + z^3 = (x + y + z)^2$ Calculate the value of $xyz$ .

1 solution

Henry U
Jan 13, 2019

It is known that the sum of the cubes of the integers from 1 to $n$ is equal to the sum of the integers up to $n$ , but this sum squared.

In other words,

$\displaystyle \sum_{i=1}^n i^3 = \left( \sum_{i=1}^n i \right) ^2$

With this known, the solution $(x,y,z)=(1,2,3)$ is obvious and the product $xyz=\boxed{6}$ .

I don't know if there are other solutions though.