Maximum Value of Modulus Function

We are given $0 \le a \le x \le 100$. Since $x \ge a$, $|x - a| = x - a$. Since $x \le 100$, $|x - 100| = 100 - x$.

So $f(x) = (x - a) + (100 - x) + |x - a - 50| = 100 - a + |x - a - 50|$.

For $f(x)$ to achieve a maximum value of $100$ over $x \in [a, 100]$: Note that $|x - a - 50|$ varies as $x$ ranges from $a$ to $100$. When $x = a$: $|x - a - 50| = 50$, so $f(a) = 100 - a + 50 = 150 - a$. When $x = a + 50$ (if $a + 50 \le 100$): $f(x) = 100 - a$. When $x = 100$: $f(100) = 100 - a + |50 - a|$.

For the maximum value of $f(x)$ over the allowed range of $x$ to be equal to 100: If $a = 50$: Range of $x$ is $[50, 100]$. Then $f(x) = 100 - 50 + |x - 100| = 50 + 100 - x = 150 - x$. For $x \in [50, 100]$, the maximum value occurs at $x = 50$, where $f(50) = 100$.

Thus, $a = 50$.