Olympiad problems

2014 Bosnia And Herzegovina - Regional Olympiad, 4

Determine the set $S$ with minimal number of points defining $7$ distinct lines

1984 Tournament Of Towns, (061) O2

Tags: tetrahedron , geometry , 3d geometry , plane , altitude , parallel

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)

1998 Tuymaada Olympiad, 4

Tags: plane , dissecting plane , 3d geometry , solid geometry , geometry , tetrahedron

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.

1979 Chisinau City MO, 182

Tags: cube , plane , 3d geometry , pentagon , geometry

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

2018 Polish Junior MO First Round, 7

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

Kvant 2020, M2598

Tags: plane , 3d geometry , square , geometry

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

2002 Switzerland Team Selection Test, 1

Tags: point , plane , combinatorial geometry , combinatorics

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.

1989 Tournament Of Towns, (237) 1

Tags: pyramid , geometry , equal area , 3d geometry , plane , sphere

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)

1983 All Soviet Union Mathematical Olympiad, 358

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

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 $(D_1D_2)$ .

1995 Grosman Memorial Mathematical Olympiad, 5

Tags: combinatorial geometry , combinatorics , plane

For non-coplanar points are given in space. A plane $\pi$ is called [i]equalizing [/i] if all four points have the same distance from $\pi$. Find the number of equilizing planes.

2015 Oral Moscow Geometry Olympiad, 5

Tags: geometry , 3d geometry , sphere , fixed point , moscow , plane

A triangle $ABC$ and spheres are given in space $S_1$ and $S_2$, each of which passes through points $A, B$ and $C$. For points $M$ spheres $S_1$ not lying in the plane of triangle $ABC$ are drawn lines $MA, MB$ and $MC$, intersecting the sphere $S_2$ for the second time at points $A_1,B_1$ and $C_1$, respectively. Prove that the planes passing through points $A_1, B_1$ and $C_1$, touch a fixed sphere or pass through a fixed point.

1998 Israel National Olympiad, 1

Tags: geometry , angle , plane , combinatorial geometry

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.

2011 IFYM, Sozopol, 4

Tags: point , plane , line , geometry

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).

1938 Moscow Mathematical Olympiad, 038

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

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.

2018 Bosnia And Herzegovina - Regional Olympiad, 5

Tags: plane , covering , combinatorial geometry , combinatorics

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 $1$

2012 District Olympiad, 2

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

The pyramid $VABCD$ has base the rectangle ABCD, and the side edges are congruent. Prove that the plane $(VCD)$ forms congruent angles with the planes $(VAC)$ and $(BAC)$ if and only if $\angle VAC = \angle BAC $.

EGMO 2017, 3

Tags: combinatorial geometry , plane , line , combinatorics

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?

2021 Durer Math Competition (First Round), 5

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

There are $n$ distinct lines in three-dimensional space such that no two lines are parallel and no three lines meet at one point. What is the maximal possible number of planes determined by these $n$ lines? We say that a plane is determined if it contains at least two of the lines.

2008 District Olympiad, 1

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

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

2006 Kazakhstan National Olympiad, 6

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

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) $.

1994 ITAMO, 5

Tags: geometry , maximum , cube , plane , area , minimum

Let $OP$ be a diagonal of a unit cube. Find the minimum and the maximum value of the area of the intersection of the cube with a plane through $OP$.

2011 Oral Moscow Geometry Olympiad, 2

Tags: geometry , 3d 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 $a$?

1987 Tournament Of Towns, (142) 2

Tags: 3d geometry , plane , geometry , 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 $M$ .

2017 EGMO, 3

Tags: combinatorics , combinatorial geometry , line , plane

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?

1981 Austrian-Polish Competition, 8

Tags: combinatorics , parallel , combinatorial geometry , plane , line

The plane has been partitioned into $N$ regions by three bunches of parallel lines. What is the least number of lines needed in order that $N > 1981$?