Question 162
Assume $a$ is a real number in $[\frac{1}{2}, \frac{2}{3}]$. Consider two sequences $(u_n), (v_n), (n = 0, 1, \dots)$, defined by:
$$
u_n = \frac{3}{2^{n+1}} \cdot (-1)^{\lfloor 2^{n+1}a \rfloor}, \quad v_n = \frac{3}{2^{n+1}} \cdot (-1)^{n+\lfloor 2^{n+1}a \rfloor}.$$
a. Prove that
$$
\left(\sum_{i=0}^{2018} u_i\right)^2 + \left(\sum_{i=0}^{2018} v_i\right)^2 \le 72a^2 - 48a + 10 + \frac{2}{4^{2019}}.
$$
b. Find all value of $a$ for which equality occurs.
$$
u_n = \frac{3}{2^{n+1}} \cdot (-1)^{\lfloor 2^{n+1}a \rfloor}, \quad v_n = \frac{3}{2^{n+1}} \cdot (-1)^{n+\lfloor 2^{n+1}a \rfloor}.$$
a. Prove that
$$
\left(\sum_{i=0}^{2018} u_i\right)^2 + \left(\sum_{i=0}^{2018} v_i\right)^2 \le 72a^2 - 48a + 10 + \frac{2}{4^{2019}}.
$$
b. Find all value of $a$ for which equality occurs.