1977 AMC 12 Question 11

For each real number $x,$ let $[x]$ be the largest integer not exceeding $x$ (i.e., the integer $n$ such that $n\le x\lt n+1$). Which of the following statements is (are) true?

$\mathrm{I}.$ $[x+1]=[x]+1$ for all $x$

$\mathrm{II}.$ $[x+y]=[x]+[y]$ for all $x$ and $y$

$\mathrm{III}.$ $[xy]=[x][y]$ for all $x$ and $y$