Solution of problem 11306 in AMM

Let a,b, and c be the lengths of the sides of a nondegenerate triangle, let p=(a+b+c)/2, and let r and R be the inradius and circumradius of the triangle, respectively. Show that \dfrac{a}{2}\cdot \dfrac{4r-R}{R}\leq\sqrt{(p-b)(p-c)}\leq \dfrac{a}{2},

and determine the cases of equality.

My solution.

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A famous functional equation

That is following problem: Find all functions f:\mathbb{R}\to\mathbb{R} such that
f(x^2 + y + f(y)) = (f(x))^2 + 2\cdot y\; \forall x,y\in\mathbb{R}.(*)
It is famous! Why? Because, it is from AMM(problem 10908, posted by Wu Wei Chao) and it is one in problems from Bulgarian TST 2003, Vietnam TST 2004 and Iran TST 2007. However, in Vietnam TST 2004 it has form:

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