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: 59

1985 Tournament Of Towns, (093) 1
Tags: projection , cube , perpendicular , plane , 3d geometry , geometry
Prove that the area of a unit cube's projection on any plane equals the length of its projection on the perpendicular to this plane.

2011 Oral Moscow Geometry Olympiad, 2
Tags: 3d geometry , geometry , line , plane
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 $a$?

2000 Tuymaada Olympiad, 2
Tags: combinatorics , combinatorial geometry , coloring , plane , color problem
Is it possible to paint the plane in $4$ colors so that inside any circle are the dots of all four colors?

2018 Adygea Teachers' Geometry Olympiad, 4
Tags: geometry , 3d geometry , cube , angle , plane
Given a cube $ABCDA_1B_1C_1D_1$ with edge $5$. On the edge $BB_1$ of the cube , point $K$ such thath $BK=4$. a) Construct a cube section with the plane $a$ passing through the points $K$ and $C_1$ parallel to the diagonal $BD_1$. b) Find the angle between the plane $a$ and the plane $BB_1C_1$.

1998 Tuymaada Olympiad, 4
Tags: plane , geometry , tetrahedron , dissecting plane , 3d geometry , solid geometry
Given the tetrahedron $ABCD$, whose opposite edges are equal, that is, $AB=CD, AC=BD$ and $BC=AD$. Prove that exist exactly $6$ planes intersecting the triangular angles of the tetrahedron and dividing the total surface and volume of this tetrahedron in half.

1990 Romania Team Selection Test, 4
Tags: plane , 3d geometry , tetrahedron , concurrent planes , incenter , concurrency
Let $M$ be a point on the edge $CD$ of a tetrahedron $ABCD$ such that the tetrahedra $ABCM$ and $ABDM$ have the same total areas. We denote by $\pi_{AB}$ the plane $ABM$. Planes $\pi_{AC},...,\pi_{CD}$ are analogously defined. Prove that the six planes $\pi_{AB},...,\pi_{CD}$ are concurrent in a certain point $N$, and show that $N$ is symmetric to the incenter $I$ with respect to the barycenter $G$.

1997 Mexico National Olympiad, 4
Tags: plane , geometry , combinatorial geometry , combinatorics
What is the minimum number of planes determined by $6$ points in space which are not all coplanar, and among which no three are collinear?

1952 Moscow Mathematical Olympiad, 227
Tags: combinatorial geometry , plane , lines in plane
$99$ straight lines divide a plane into $n$ parts. Find all possible values of $n$ less than $199$.

EGMO 2017, 3
Tags: line , plane , combinatorics , combinatorial geometry
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?

IV Soros Olympiad 1997 - 98 (Russia), 10.11
Tags: geometry , 3d geometry , cube , plane , geometric inequality
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?

1984 Tournament Of Towns, (061) O2
Tags: geometry , 3d geometry , tetrahedron , parallel , plane , altitude
Six altitudes are constructed from the three vertices of the base of a tetrahedron to the opposite sides of the three lateral faces. Prove that all three straight lines joining two base points of the altitudes in each lateral face are parallel to a certain plane. (IF Sharygin, Moscow)

1989 Tournament Of Towns, (237) 1
Tags: equal area , geometry , 3d geometry , sphere , pyramid , plane
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)

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 $N > 1999$?

1950 Moscow Mathematical Olympiad, 186
Tags: sphere , plane , 3d geometry , tangent , geometry
A spatial quadrilateral is circumscribed around a sphere. Prove that all the tangent points lie in one plane.

1992 ITAMO, 1
Tags: combinatorial geometry , 3d geometry , cube , plane , geometry
A cube is divided into $27$ equal smaller cubes. A plane intersects the cube. Find the maximum possible number of smaller cubes the plane can intersect.

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.

2006 Sharygin Geometry Olympiad, 24
Tags: projection , geometry , locus , 3d geometry , plane , sphere , circles
a) Two perpendicular rays are drawn through a fixed point $P$ inside a given circle, intersecting the circle at points $A$ and $B$. Find the geometric locus of the projections of $P$ on the lines $AB$. b) Three pairwise perpendicular rays passing through the fixed point $P$ inside a given sphere intersect the sphere at points $A, B, C$. Find the geometrical locus of the projections $P$ on the $ABC$ plane

2020 Tournament Of Towns, 3
Tags: plane , 3d geometry , geometry , square
Is it possible that two cross-sections of a tetrahedron by two different cutting planes are two squares, one with a side of length no greater than $1$ and another with a side of length at least $100$? Mikhail Evdokimov

1990 Czech and Slovak Olympiad III A, 3
Tags: plane , geometry , 3d geometry , tetrahedron , obtuse , triangle
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.

2002 Switzerland Team Selection Test, 1
Tags: point , plane , combinatorics , combinatorial geometry
In space are given $24$ points, no three of which are collinear. Suppose that there are exactly $2002$ planes determined by three of these points. Prove that there is a plane containing at least six points.

2003 District Olympiad, 1
Tags: 3d geometry , equilateral , plane , angle , geometry
Let $ABC$ be an equilateral triangle. On the plane $(ABC)$ rise the perpendiculars $AA'$ and $BB'$ on the same side of the plane, so that $AA' = AB$ and $BB' =\frac12 AB$. Determine the measure the angle between the planes $(ABC)$ and $(A'B'C')$.

1939 Moscow Mathematical Olympiad, 053
Tags: 3d geometry , sphere , plane , combinatorics , combinatorial geometry , geometry
What is the greatest number of parts that $5$ spheres can divide the space into?

1938 Moscow Mathematical Olympiad, 038
Tags: plane , equidistant , 3d geometry , combinatorial geometry , geometry
In space $4$ points are given. How many planes equidistant from these points are there? Consider separately (a) the generic case (the points given do not lie on a single plane) and (b) the degenerate cases.

1953 Czech and Slovak Olympiad III A, 1
Tags: complex number , plane
Find the locus of all numbers $z\in\mathbb C$ in complex plane satisfying $$z+\bar z=a\cdot|z|,$$ where $a\in\mathbb R$ is given.

1998 Israel National Olympiad, 1
Tags: combinatorial geometry , plane , geometry , angle
In space are given $n$ segments $A_iB_i$ and a point $O$ not lying on any segment, such that the sum of the angles $A_iOB_i$ is less than $180^o$ . Prove that there exists a plane passing through $O$ and not intersecting any of the segments.