Question 4
Let $a\in[\tfrac12,\tfrac23]$. For $n=0,1,2,\ldots$, define
$$u_n=\frac3{2^{n+1}}(-1)^{\lfloor2^{n+1}a\rfloor},\qquad v_n=\frac3{2^{n+1}}(-1)^{n+\lfloor2^{n+1}a\rfloor}.$$
(a) Prove that
$$\left(\sum_{i=0}^{2018}u_i\right)^2+\left(\sum_{i=0}^{2018}v_i\right)^2\le72a^2-48a+10+\frac2{4^{2019}}.$$
(b) Find all values of $a$ for which equality holds.
$$u_n=\frac3{2^{n+1}}(-1)^{\lfloor2^{n+1}a\rfloor},\qquad v_n=\frac3{2^{n+1}}(-1)^{n+\lfloor2^{n+1}a\rfloor}.$$
(a) Prove that
$$\left(\sum_{i=0}^{2018}u_i\right)^2+\left(\sum_{i=0}^{2018}v_i\right)^2\le72a^2-48a+10+\frac2{4^{2019}}.$$
(b) Find all values of $a$ for which equality holds.