Problem 3. (4 points) Do there exist such natural numbers $a, b, c$ and $d$, for which

$$
a^{3}+b^{3}+c^{3}+d^{3}=100^{100} ?
$$
Solution. Such numbers do exist, for example $\left(100^{33}\right)^{3}+\left(2 \cdot 100^{33}\right)^{3}+\left(3 \cdot 100^{33}\right)^{3}+\left(4 \cdot 100^{33}\right)^{3}=\left(1^{3}+2^{3}+3^{3}+4^{3}\right) \cdot 100^{99}=100 \cdot 100^{99}=100^{100}$.