$33 \cdot 25$ For every real number $x$, let $[x]$ be the greatest integer not exceeding $x$ (this is the integer $n$ such that $n \leqslant x < n+1$). There are the following statements:
I. $[x+1]=[x]+1$, for all $x$.
II. $[x+y]=[x]+[y]$, for all $x$ and $y$.
III. $[x y]=[x][y]$, for all $x$ and $y$.

Of the above statements, the correct ones are
(A) None.
(B) Only I.
(C) Only I and II.
(D) Only III.
(E) All.
(28th American High School Mathematics Examination, 1977)
[Solution] If $n \leqslant x < n+1$, then $n+1 \leqslant x+1 < n+2$. Thus, $[x]=n, [x+1]=n+1$, so $[x+1]=[x]+1$. By choosing $x=y=2.5$, it can be verified that conclusions II and III do not hold. Therefore, the answer is $(B)$.