Question 136
Determine the greatest real number $a$ such that the inequality
$$
x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2} \geq a\left(x_{1} x_{2}+x_{2} x_{3}+x_{3} x_{4}+x_{4} x_{5}\right)
$$
holds for every five real numbers $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}$.
$$
x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2} \geq a\left(x_{1} x_{2}+x_{2} x_{3}+x_{3} x_{4}+x_{4} x_{5}\right)
$$
holds for every five real numbers $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}$.