Olympiad problems

2019 Jozsef Wildt International Math Competition, W. 58

Tags: insphere , 3d geometry , exsphere , circumsphere , inequalities , tetrahedron

In the $[ABCD]$ tetrahedron having all the faces acute angled triangles, is denoted by $r_X$, $R_X$ the radius lengths of the circle inscribed and circumscribed respectively on the face opposite to the $X \in \{A,B,C,D\}$ peak, and with $R$ the length of the radius of the sphere circumscribed to the tetrahedron. Show that inequality occurs$$8R^2 \geq (r_A + R_A)^2 + (r_B + R_B)^2 + (r_C + R_C)^2 + (r_D + R_D)^2$$

KoMaL A Problems 2024/2025, A. 894

Tags: sphere , solid geometry , insphere , 3d geometry , geometry

In convex polyhedron $ABCDE$ line segment $DE$ intersects the plane of triangle $ABC$ inside the triangle. Rotate the point $D$ outward into the plane of triangle $ABC$ around the lines $AB$, $BC$, $CA$; let the resulting points be $D_1$, $D_2$, and $D_3$. Similarly, rotate the point $E$ outward into the plane of triangle $ABC$ around the lines $AB$, $BC$, $CA$; let the resulting points be $E_1$, $E_2$, and $E_3$. Show that if the polyhedron has an inscribed sphere, then the circumcircles of $D_1D_2D_3$ and $E_1E_2E_3$ are concentric. [i]Proposed by: Géza Kós, Budapest[/i]

I Soros Olympiad 1994-95 (Rus + Ukr), 11.4

Tags: geometry , 3d geometry , insphere , tetrahedron

A tetrahedron $ABCD$ is given, in which each pair of adjacent edges are equal segments. Let $O$ be the center of the sphere inscribed in this tetrahedron . $X$ is an arbitrary point inside the tetrahedron, $X \ne O$. The line $OX$ intersects the planes of the faces of the tetrahedron at the points marked by $A_1$, $B_1$, $C_1$, $D_1$. Prove that $$\frac{A_1X}{A_1O} +\frac{B_1X}{B_1O} +\frac{C_1X}{C_1O}+\frac{D_1X}{D_1O}=4$$