Olympiad problems

2006 Kazakhstan National Olympiad, 6

Tags: geometry , 3d geometry , tetrahedron , perpendicular , 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) $.

2015 Sharygin Geometry Olympiad, P23

Tags: geometry , tetrahedron , 3-dimensional geometry , 3d geometry , plane

A tetrahedron $ABCD$ is given. The incircles of triangles $ ABC$ and $ABD$ with centers $O_1, O_2$, touch $AB$ at points $T_1, T_2$. The plane $\pi_{AB}$ passing through the midpoint of $T_1T_2$ is perpendicular to $O_1O_2$. The planes $\pi_{AC},\pi_{BC}, \pi_{AD}, \pi_{BD}, \pi_{CD}$ are defined similarly. Prove that these six planes have a common point.

2009 District Olympiad, 3

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

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 $(HBS)$.

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.

1980 All Soviet Union Mathematical Olympiad, 293

Tags: vector , plane , combinatorial geometry , geometry

Given $1980$ vectors in the plane, and there are some non-collinear among them. The sum of every $1979$ vectors is collinear to the vector not included in that sum. Prove that the sum of all vectors equals to the zero vector.

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 $a$?

1946 Moscow Mathematical Olympiad, 111

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

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 $\beta$.

2021 Durer Math Competition (First Round), 5

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

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.

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 $OS$.

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?

1956 Putnam, B2

Tags: plane , subset

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).

1992 All Soviet Union Mathematical Olympiad, 575

Tags: 3d geometry , geometry , plane , fixed , product , sphere

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 $p$.

1995 Grosman Memorial Mathematical Olympiad, 5

Tags: combinatorics , combinatorial geometry , 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.

1983 All Soviet Union Mathematical Olympiad, 358

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

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 $(D_1D_2)$ .

Kvant 2020, M2598

Tags: 3d geometry , square , plane , 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

2011 IFYM, Sozopol, 4

Tags: plane , line , point , 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, 040

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

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).

2023 Tuymaada Olympiad, 6

Tags: inequalities , geometry , plane , geometry inequality

In the plane $n$ segments with lengths $a_1, a_2, \dots , a_n$ are drawn. Every ray beginning at the point $O$ meets at least one of the segments. Let $h_i$ be the distance from $O$ to the $i$-th segment (not the line!) Prove the inequality \[\frac{a_1}{h_1}+\frac{a_2}{h_2} + \ldots + \frac{a_i}{h_i} \geqslant 2 \pi.\]

2010 Sharygin Geometry Olympiad, 7

Tags: geometry , polyhedron , cut , plane

Each of two regular polyhedrons $P$ and $Q$ was divided by the plane into two parts. One part of $P$ was attached to one part of $Q$ along the dividing plane and formed a regular polyhedron not equal to $P$ and $Q$. How many faces can it have?

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.

1938 Moscow Mathematical Olympiad, 038

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

EGMO 2017, 3

Tags: plane , combinatorics , combinatorial geometry , line

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?