Question 96
Let $a \neq b$ be positive real numbers. Consider the equation
$$
\lfloor a x + b \rfloor = \lfloor b x + a \rfloor
$$
where $\lfloor y \rfloor$ denotes the largest integer not exceeding $y$. Prove that the set of real solutions $x$ to this equation contains an interval of length at least
$$
\frac{1}{\max\{a, b\}} \qquad (\text{Patrik Bak})
$$
$$
\lfloor a x + b \rfloor = \lfloor b x + a \rfloor
$$
where $\lfloor y \rfloor$ denotes the largest integer not exceeding $y$. Prove that the set of real solutions $x$ to this equation contains an interval of length at least
$$
\frac{1}{\max\{a, b\}} \qquad (\text{Patrik Bak})
$$