Writeup: IMO Shortlist 2002 N1

28 Apr, 2025

Problem Statement

What is the smallest positive integer $t$ such that there exist integers $x_{1}, x_{2}, \dots, x_{t}$ with

x_{1}^{3} + x_{2}^{3} + \dots + x_{t}^{3} = 2002^{2002} ?

My Solution

I don't like this problem. I feel like with this problem, you either see the solution right away or you spam random numbers for what feels like forever until you get the solution. It's not hard to see that $2002$ is $1 mod 3$ . We can rewrite $2002^{2002}$ as

2002^{2001} \cdot 2002 = 2002^{2001} \cdot (1 + 1 + 1000 + 1000)

which yields a sum of 4 cubes.

Now we prove $t < 4$ is not possible. Mod 9, $2002^{2002} \equiv 4$ , and since cubic residues mod 9 are $- 1, 0, 1$ , we must have at least 4 cubes to get a residue of $4$ .