Olympiad problems

2006 Kazakhstan National Olympiad, 6

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) $.

1937 Moscow Mathematical Olympiad, 036

* 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?

dodecahedron geometry 3d geometry plane combinatorics combinatorial geometry regular hexagon solid geometry

1993 Bundeswettbewerb Mathematik, 2

Let $M$ be a finite subset of the plane such that for any two different points $A,B\in M$ there is a point $C\in M$ such that $ABC$ is equilateral. What is the maximal number of points in $M?$

geometry equilateral plane combinatorics combinatorial geometry

IV Soros Olympiad 1997 - 98 (Russia), 10.11

A plane intersecting a unit cube divides it into two polyhedra. It is known that for each polyhedron the distance between any two points of it does not exceeds $\frac32$ m. What can be the cross-sectional area of a cube drawn by a plane?

geometric inequality cube geometry 3d geometry plane geometric inequalities solid geometry

2011 IFYM, Sozopol, 4

There are $n$ points in a plane. Prove that there exist a point $O$ (not necessarily from the given $n$) such that on each side of an arbitrary line, through $O$, lie at least $\frac{n}{3}$ points (including the points on the line).

geometry point plane line

1992 All Soviet Union Mathematical Olympiad, 575

A plane intersects a sphere in a circle $C$. The points $A$ and $B$ lie on the sphere on opposite sides of the plane. The line joining $A$ to the center of the sphere is normal to the plane. Another plane $p$ intersects the segment $AB$ and meets $C$ at $P$ and $Q$. Show that $BP\cdot BQ$ is independent of the choice of $p$.

fixed geometry 3d geometry plane sphere product solid geometry

1990 Czech and Slovak Olympiad III A, 3

Let $ABCDEFGH$ be a cube. Consider a plane whose intersection with the tetrahedron $ABDE$ is a triangle with an obtuse angle $\varphi.$ Determine all $\varphi>\pi/2$ for which there is such a plane.

tetrahedron obtuse triangle geometry 3d geometry plane solid geometry triangles

1946 Moscow Mathematical Olympiad, 111

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 $\beta$.

geometry 3d geometry plane angle maximum solid geometry

2011 Oral Moscow Geometry Olympiad, 2

Line $\ell $ intersects the plane $a$. It is known that in this plane there are $2011$ straight lines equidistant from $\ell$ and not intersecting $\ell$. Is it true that $\ell$ is perpendicular to $a$?

geometry 3d geometry plane line solid geometry