IMO 1986 SL 21

Origin: TUR

Problem

Let ABCD be a tetrahedron having each sum of opposite sides equal to 1. Prove that rA + rB + rC + rD \leq \sqrt 3 , where rA, rB, rC, rD are the inradii of the faces, equality holding only if ABCD is regular.

Solution

Since the sum of all edges of ABCD is 3, the statement of the problem is an immediate consequence of the following statement: Lemma. Let r be the inradius of a triangle with sides a, b, c. Then a + b + c \geq6 \sqrt 3 \cdot r, with equality if and only if the triangle is equilateral. Proof. If S and p denotes the area and semiperimeter of the triangle, by Heron’s formula and the AM–GM inequality we have pr = S =

p(p −a)(p −b)(p −c) \leq ! p (p −a) + (p −b) + (p −c) 3

 p4 27 = p2 \sqrt 3, i.e., p \geq3 \sqrt 3 \cdot r, which is equivalent to the claim.