IMO 1973 SL 13

Find the sphere of maximal radius that can be placed inside

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IMO 1973 SL 13

Origin: YUG

Problem

Find the sphere of maximal radius that can be placed inside every tetrahedron that has all altitudes of length greater than or equal to 1.

Solution

Let S1, S2, S3, S4 denote the areas of the faces of the tetrahedron, V its volume, h1, h2, h3, h4 its altitudes, and r the radius of its inscribed sphere. Since 3V = S1h1 = S2h2 = S3h3 = S4h4 = (S1 + S2 + S3 + S4)r, it follows that h1

  • 1 h2
  • 1 h3
  • 1 h4 = 1 r . In our case, h1, h2, h3, h4 \geq1, hence r \geq1/4. On the other hand, it is clear that a sphere of radius greater than 1/4 cannot be inscribed in a tetrahedron all of whose altitudes have length equal to 1. Thus the answer is 1/4.