Olympiad problems

1952 Moscow Mathematical Olympiad, 220

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 $OS$.

1994 Poland - Second Round, 3

Tags: Plane , Cyclic , hexagon , Regular , geometry , 3D geometry

A plane passing through the center of a cube intersects the cube in a cyclic hexagon. Show that this hexagon is regular.

1979 Chisinau City MO, 182

Tags: geometry , cube , Plane , pentagon , 3D geometry

Prove that a section of a cube by a plane cannot be a regular pentagon.

1950 Moscow Mathematical Olympiad, 186

Tags: geometry , sphere , Plane , 3D geometry , tangent

A spatial quadrilateral is circumscribed around a sphere. Prove that all the tangent points lie in one plane.

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?

2018 Bosnia And Herzegovina - Regional Olympiad, 5

Tags: Plane , combinatorial geometry , combinatorics , covering

It is given $2018$ points in plane. Prove that it is possible to cover them with circles such that: $i)$ sum of lengths of all diameters of all circles is not greater than $2018$ $ii)$ distance between any two circles is greater than $1$

1999 Spain Mathematical Olympiad, 6

Tags: combinatorial geometry , Parallel Lines , minimum , Plane , combinatorics

A plane is divided into $N$ regions by three families of parallel lines. No three lines pass through the same point. What is the smallest number of lines needed so that $N > 1999$?

1956 Putnam, B2

Tags: Putnam , Plane , Subsets

Suppose that each set $X$ of points in the plane has an associated set $\overline{X}$ of points called its cover. Suppose further that (1) $\overline{X\cup Y} \supset \overline{\overline{X}} \cup \overline{Y} \cup Y$ for all sets $X,Y$ . Show that i) $\overline{X} \supset X$, ii) $\overline{\overline{X}}=\overline{X}$ and iii) $X\supset Y \Rightarrow \overline{X} \supset \overline{Y}.$ Prove also that these three statements imply (1).

1989 Tournament Of Towns, (237) 1

Tags: geometry , 3D geometry , sphere , pyramid , Plane , equal areas

Is it possible to choose a sphere, a triangular pyramid and a plane so that every plane, parallel to the chosen one, intersects the sphere and the pyramid in sections of equal area? (Problem from Latvia)

2008 District Olympiad, 1

Tags: geometry , rhombus , Plane , 3D geometry , tetrahedron , square

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

1987 Tournament Of Towns, (142) 2

Tags: 3D geometry , geometry , Plane , parallelogram

In $3$ dimensional space we are given a parallelogram $ABCD$ and plane $M$. The distances from vertices $A, B$ and $C$ to plane $M$ are $a, b$ and $c$ respectively. Find the distance $d$ from vertex $D$ to the plane $M$ .

EGMO 2017, 3

Tags: Plane , Line , combinatorics , EGMO , combinatorial geometry , EGMO 2017

There are $2017$ lines in the plane such that no three of them go through the same point. Turbo the snail sits on a point on exactly one of the lines and starts sliding along the lines in the following fashion: she moves on a given line until she reaches an intersection of two lines. At the intersection, she follows her journey on the other line turning left or right, alternating her choice at each intersection point she reaches. She can only change direction at an intersection point. Can there exist a line segment through which she passes in both directions during her journey?