Question 48
Let $x$ be a real number. Prove that $x$ is an integer if and only if
$$
[x] + [2x] + [3x] + \dots + [nx] = \frac{n([x] + [nx])}{2}
$$
holds for all positive integers $n$ (here, $[a]$ denotes the integer part (floor function) of the real number $a$).
$$
[x] + [2x] + [3x] + \dots + [nx] = \frac{n([x] + [nx])}{2}
$$
holds for all positive integers $n$ (here, $[a]$ denotes the integer part (floor function) of the real number $a$).