This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 33

2007 Sharygin Geometry Olympiad, 5
Tags: geometry , combinatorial geometry , convex , polyhedron , 3-dimensional geometry
Each edge of a convex polyhedron is shifted such that the obtained edges form the frame of another convex polyhedron. Are these two polyhedra necessarily congruent?

2000 Austrian-Polish Competition, 2
Tags: 3-dimensional geometry , geometry , distance , minimum , 3d geometry
In a unit cube, $CG$ is the edge perpendicular to the face $ABCD$. Let $O_1$ be the incircle of square $ABCD$ and $O_2$ be the circumcircle of triangle $BDG$. Determine min$\{XY|X\in O_1,Y\in O_2\}$.

1979 Vietnam National Olympiad, 6
Tags: geometry , 3-dimensional geometry , construction , tetrahedron , 3d geometry
$ABCD$ is a rectangle with $BC / AB = \sqrt2$. $ABEF$ is a congruent rectangle in a different plane. Find the angle $DAF$ such that the lines $CA$ and $BF$ are perpendicular. In this configuration, find two points on the line $CA$ and two points on the line $BF$ so that the four points form a regular tetrahedron.

1988 Mexico National Olympiad, 8
Tags: geometry , 3-dimensional geometry , octahedron , sphere
Compute the volume of a regular octahedron circumscribed about a sphere of radius $1$.

1991 Mexico National Olympiad, 3
Tags: sphere , geometry , 3-dimensional geometry
Four balls of radius $1$ are placed in space so that each of them touches the other three. What is the radius of the smallest sphere containing all of them?

2014 Sharygin Geometry Olympiad, 24
Tags: geometry , 3-dimensional geometry , pyramid
A circumscribed pyramid $ABCDS$ is given. The opposite sidelines of its base meet at points $P$ and $Q$ in such a way that $A$ and $B$ lie on segments $PD$ and $PC$ respectively. The inscribed sphere touches faces $ABS$ and $BCS$ at points $K$ and $L$. Prove that if $PK$ and $QL$ are complanar then the touching point of the sphere with the base lies on $BD$.

2017 Yasinsky Geometry Olympiad, 2
Tags: 3-dimensional geometry , 3d geometry , tetrahedron , geometry , equal angles , perpendicular
In the tetrahedron $DABC, AB=BC, \angle DBC =\angle DBA$. Prove that $AC \perp DB$.

2015 Sharygin Geometry Olympiad, P24
Tags: sphere , tetrahedron , 3-dimensional geometry , geometry , 3d geometry
The insphere of a tetrahedron ABCD with center $O$ touches its faces at points $A_1,B_1,C_1$ and $D_1$. a) Let $P_a$ be a point such that its reflections in lines $OB,OC$ and $OD$ lie on plane $BCD$. Points $P_b, P_c$ and $P_d$ are defined similarly. Prove that lines $A_1P_a,B_1P_b,C_1P_c$ and $D_1P_d$ concur at some point $ P$. b) Let $I$ be the incenter of $A_1B_1C_1D_1$ and $A_2$ the common point of line $A_1I $ with plane $B_1C_1D_1$. Points $B_2, C_2, D_2$ are defined similarly. Prove that $P$ lies inside $A_2B_2C_2D_2$.