MathDB

2

Part of 2004 Moldova Team Selection Test

Problems(1)

\frac{1}{\sqrt{r_A^2-r_Ar_B+r_B^2}}+\frac{1}{\sqrt{r_B^2-r_Br_C+r_C^2}}

Source: Moldova TST 2004

3/8/2023
In the tetrahedron ABCDABCD the radius of its inscribed sphere is rr and the radiuses of the exinscribed spheres (each tangent with a face of the tetrahedron and with the planes of the other faces) are rA,rB,rC,rD.r_A, r_B, r_C, r_D. Prove the inequality 1rA2−rArB+rB2+1rB2−rBrC+rC2+1rC2−rCrD+rD2+1rD2−rDrA+rA2≤2r.\frac{1}{\sqrt{r_A^2-r_Ar_B+r_B^2}}+\frac{1}{\sqrt{r_B^2-r_Br_C+r_C^2}}+\frac{1}{\sqrt{r_C^2-r_Cr_D+r_D^2}}+\frac{1}{\sqrt{r_D^2-r_Dr_A+r_A^2}}\leq\frac{2}{r}.
geometry3D geometrytetrahedronsphereinequalities