Question 119
Let $a, b, c, d, e, f$ be positive integers, each at least $2$, whose sum is $S$. Prove that
$$
a(a-1)+b(b-1)+c(c-1)+d(d-1)+e(e-1)+f(f-1) \leq (S-10)(S-11)+10.
$$
When is equality achieved?
$$
a(a-1)+b(b-1)+c(c-1)+d(d-1)+e(e-1)+f(f-1) \leq (S-10)(S-11)+10.
$$
When is equality achieved?