Question 111
The real numbers $a_{1}, a_{2}, \ldots, a_{2003}$ satisfy simultaneously the relations: $a_{i} \geq 0$ for all $i = 1, 2, \ldots, 2003$; $a_{1} + a_{2} + \ldots + a_{2003} = 2$; $a_{1} a_{2} + a_{2} a_{3} + \ldots + a_{2003} a_{1} = 1$. Find the smallest value of the sum $a_{1}^{2} + a_{2}^{2} + \ldots + a_{2003}^{2}$.