Question

If $S = \\{ a \in \mathbb{R} : |2a - 1| = 3[a] + 2\\{a\\} \\}$, where $[t]$ denotes the greatest integer less than or equal to $t$ and $\\{t\\}$ represents the fractional part of $t$, then $72 \sum_{a \in S} a$ is equal to _________.

Answer 1

Given:

$|2a - 1| = 3[a] + 2\\{a\\}$

Rewrite $|2a - 1|$ in two forms depending on the value of $a$:

In this case:

$2a - 1 = [a] + 2a$

Since $[a] = -1$, we find that $a \in [-1, 0)$, which is a contradiction because $a > \frac{1}{2}$. Therefore, this case is rejected.

Case 2: $a < \frac{1}{2}$ In this case:

$-2a + 1 = [a] + 2a$

Let $a = I + f$ where $I$ is the integer part and $f$ is the fractional part, so $[a] = 0$ and $\\{a\\} = f$.

Then we have:

$-2(I + f) + 1 = I + 2f$

Substituting $I = 0$, we get:

$1 = 2f \implies f = \frac{1}{4}$

Thus, $a = \frac{1}{4}$.

Now, calculating $72 \sum_{a \in S} a$: $72 \times \frac{1}{4} = 18$