Is it possible for $\sum_{i=1}^n x_i^3 > \sum_{i=1}^n y_i^3 $ when $\sum_{i=1}^n x_i < \sum_{i=1}^n y_i $ where $x_i,y_i \in \mathbb{N}$

Question

I.e. is it possible for a sum of $n$ positive integers to be strictly less than another sum of $n$ positive integers but be strictly greater when it's the sum of their cubes?

$x_1 + x_2 + ... + x_n < y_1 + y_2 + ... + y_n$

$x_1^3 + x_2^3 + ... + x_n^3 > y_1^3 + y_2^3 + ... + y_n^3 $

I feel like this is trivially wrong but I can't prove it.

Answer 1

It is definitely possible, even with $n=2$ where the smallest solution is $$1+4<3+3$$ $$1^3+4^3>3^3+3^3$$ For higher $n$, simply add 1s to both sides. This works because small differences between the numbers on both sides are amplified by the operation of cubing.