Problem

Source: Iranian TST 2018, third exam day 1, problem 2

Tags: Iran, Iranian TST, maximum value, algebra



Find the maximum possible value of $k$ for which there exist distinct reals $x_1,x_2,\ldots ,x_k $ greater than $1$ such that for all $1 \leq i, j \leq k$, $$x_i^{\lfloor x_j \rfloor }= x_j^{\lfloor x_i\rfloor}.$$ Proposed by Morteza Saghafian