Baltic Way 2017 · Problem 2

Answer: no.

Assume that such a set of numbers $\left\{a_{1}, \ldots, a_{n}\right\}$ exists. Summing up the inequalities

$$2 a_{i}^{3} \leq a_{i}^{2}+a_{i}^{4}$$

for all $i$ we obtain the inequality $8 \leq 8$. Therefore all the inequalities are in fact equalities. This is possible for the cases $a_{i}=0$ or $a_{i}=1$ only, but the elements of a set are all different.

(Remark: Even if the $a_{i}$ 's are allowed to be equal it is clear that only 0's or only l's do not satisfy the problem conditions.)