Question 9
(GDR 3) Let $u_1,u_2,\ldots,u_n,v_1,v_2,\ldots,v_n$ be real numbers. Prove that
$$1+\sum_{i=1}^n(u_i+v_i)^2\le\frac43\left(1+\sum_{i=1}^nu_i^2\right)\left(1+\sum_{i=1}^nv_i^2\right).$$
In what case does equality hold?
$$1+\sum_{i=1}^n(u_i+v_i)^2\le\frac43\left(1+\sum_{i=1}^nu_i^2\right)\left(1+\sum_{i=1}^nv_i^2\right).$$
In what case does equality hold?