1. N1 (UZB) What is the smallest positive integer \( t \) such that there exist integers \( x_{1}, x_{2}, \ldots, x_{t} \) with 
\[ x_{1}^{3} + x_{2}^{3} + \cdots + x_{t}^{3} = 2002^{2002} ? \]
1. Consider the given equation modulo 9. Since each cube is congruent to either $-1,0$ or 1, whereas $2002^{2002} \equiv 4^{2002}=4 \cdot 64^{667} \equiv 4(\bmod 9)$, we conclude that $t \geq 4$. On the other hand, $2002^{2002}=2002 \cdot\left(2002^{667}\right)^{3}=\left(10^{3}+10^{3}+1^{3}+1^{3}\right)\left(2002^{667}\right)^{3}$ is a solution with $t=4$. Hence the answer is 4.