# Problem 4.

The integer part $[x]$ of a real number $x$ is defined as the greatest integer $M$ such that $M \leq x$. For example, $[\sqrt{2}]=1,[2]=2,[\pi]=3$. Find all positive real numbers $x$ for which

$$
x[x[x[x]]]<2018
$$
# Solution.

Note that for $x>0$, the left side of the equation is a monotonically non-decreasing function, and that for integer values of $x$ it equals $x^{4}$. Since $2401=7^{4}$, the inequality does not hold for $x \geq 7$.

We will show that it is satisfied for $0<x<7$. Indeed, in this case,

$$
\begin{aligned}
0 \leq[x] \leq 6 & \Rightarrow 0 \leq x[x]<7 \cdot 6=42 \Rightarrow 0 \leq[x[x]] \leq 41 \Rightarrow \\
& \Rightarrow 0 \leq x[x[x]]<287 \Rightarrow 0 \leq x[x[x[x]]]<286 \cdot 7=2002<2018
\end{aligned}
$$

Answer: $0<x<7$.