Olympiad problems

1937 Moscow Mathematical Olympiad, 036

Tags: combinatorics , combinatorial geometry , plane , 3d geometry , regular hexagon , dodecahedron , geometry

* Given a regular dodecahedron. Find how many ways are there to draw a plane through it so that its section of the dodecahedron is a regular hexagon?

1983 All Soviet Union Mathematical Olympiad, 358

Tags: plane , 3d geometry , geometry , combinatorial geometry , parallel , projection , tetrahedron

The points $A_1,B_1,C_1,D_1$ and $A_2,B_2,C_2,D_2$ are orthogonal projections of the $ABCD$ tetrahedron vertices on two planes. Prove that it is possible to move one of the planes to provide the parallelness of lines $(A_1A_2), (B_1B_2), (C_1C_2)$ and $(D_1D_2)$ .

1938 Moscow Mathematical Olympiad, 040

Tags: plane , combinatorics , combinatorial geometry , geometry , 3d geometry

What is the largest number of parts into which $n$ planes can divide space? We assume that the set of planes is non-degenerate in the sense that any three planes intersect in one point and no four planes have a common point (and for n=2 it is necessary to require that the planes are not parallel).

1952 Moscow Mathematical Olympiad, 220

Tags: geometry , 3d geometry , trihedral angle , plane , perpendicular

A sphere with center at $O$ is inscribed in a trihedral angle with vertex $S$. Prove that the plane passing through the three tangent points is perpendicular to $OS$.

2008 District Olympiad, 1

Tags: rhombus , plane , 3d geometry , tetrahedron , square , geometry

A regular tetrahedron is sectioned with a plane after a rhombus. Prove that the rhombus is square.

2018 Polish Junior MO First Round, 7

Tags: 3d geometry , square , volume , plane , geometry

Square $ABCD$ with sides of length $4$ is a base of a cuboid $ABCDA'B'C'D'$. Side edges $AA'$, $BB'$, $CC'$, $DD'$ of this cuboid have length $7$. Points $K, L, M$ lie respectively on line segments $AA'$, $BB'$, $CC'$, and $AK = 3$, $BL = 2$, $CM = 5$. Plane passing through points $K, L, M$ cuts cuboid on two blocks. Calculate volumes of these blocks.

2006 Kazakhstan National Olympiad, 6

Tags: plane , geometry , 3d geometry , tetrahedron , perpendicular

In the tetrahedron $ ABCD $ from the vertex $ A $, the perpendiculars $ AB '$, $ AC' $ are drawn, $ AD '$ on planes dividing dihedral angles at edges $ CD $, $ BD $, $ BC $ in half. Prove that the plane $ (B'C'D ') $ is parallel to the plane $ (BCD) $.

1946 Moscow Mathematical Olympiad, 111

Tags: plane , maximum , geometry , 3d geometry , angle

Given two intersecting planes $\alpha$ and $\beta$ and a point $A$ on the line of their intersection. Prove that of all lines belonging to $\alpha$ and passing through $A$ the line which is perpendicular to the intersection line of $\alpha$ and $\beta$ forms the greatest angle with $\beta$.

2009 District Olympiad, 3

Tags: plane , geometry , 3d geometry , prism , angle

Consider the regular quadrilateral prism $ABCDA'B'C 'D'$, in which $AB = a,AA' = \frac{a \sqrt {2}}{2}$, and $M$ is the midpoint of $B' C'$. Let $F$ be the foot of the perpendicular from $B$ on line $MC$, Let determine the measure of the angle between the planes $(BDF)$ and $(HBS)$.