Slični zadaci

IMO Shortlist 1973 problem 1

Let a tetrahedron $ABCD$ be inscribed in a sphere $S$ . Find the locus of points $P$ inside the sphere $S$ for which the equality
$\frac{AP}{PA_1}+\frac{BP}{PB_1}+\frac{CP}{PC_1}+\frac{DP}{PD_1}=4$
holds, where $A_1,B_1, C_1$ , and $D_1$ are the intersection points of $S$ with the lines $AP,BP,CP$ , and $DP$ , respectively.

Slični zadaci

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Zadaci

IMO Shortlist 1966 problem 56

In a tetrahedron, all three pairs of opposite (skew) edges are mutually perpendicular. Prove that the midpoints of the six edges of the tetrahedron lie on one sphere.

IMO Shortlist 1967 problem 2

Prove this proposition: Center the sphere circumscribed around a tetrahedron which coincides with the center of a sphere inscribed in that tetrahedron if and only if the skew edges of the tetrahedron are equal.

IMO Shortlist 1973 problem 13

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

IMO Shortlist 1981 problem 2

A sphere $S$ is tangent to the edges $AB,BC,CD,DA$ of a tetrahedron $ABCD$ at the points $E,F,G,H$ respectively. The points $E,F,G,H$ are the vertices of a square. Prove that if the sphere is tangent to the edge $AC$ , then it is also tangent to the edge $BD.$

IMO Shortlist 1985 problem 9

Determine the radius of a sphere $S$ that passes through the centroids of each face of a given tetrahedron $T$ inscribed in a unit sphere with center $O$ . Also, determine the distance from $O$ to the center of $S$ as a function of the edges of $T.$

IMO Shortlist 1997 problem 23

Let $ABCD$ be a convex quadrilateral. The diagonals $AC$ and $BD$ intersect at $K$ . Show that $ABCD$ is cyclic if and only if $AK \sin A + CK \sin C = BK \sin B + DK \sin D$ .