Olympiad problems

1952 Moscow Mathematical Olympiad, 228

How to arrange three right circular cylinders of diameter $a/2$ and height $a$ into an empty cube with side $a$ so that the cylinders could not change position inside the cube? Each cylinder can, however, rotate about its axis of symmetry.

2019 LIMIT Category C, Problem 10

Tags: geometry , 3D geometry , sphere

A right circular cylinder is inscribed in a sphere of radius $\sqrt3$. What is the height of the cylinder when its volume is maximal?

1993 Tournament Of Towns, (361) 4

Tags: combinatorics , cube , combinatorial geometry , geometry , 3D geometry

An ant crawls along the edges of a cube turning only at its vertices. It has visited one of the vertices $25$ times. Is it possible that it has visited each of the other $7$ vertices exactly $20$ times? (S Tokarev)

2003 Tournament Of Towns, 5

Tags: geometry , 3D geometry , tetrahedron , combinatorics unsolved , combinatorics

A paper tetrahedron is cut along some of so that it can be developed onto the plane. Could it happen that this development cannot be placed on the plane in one layer?

1989 AMC 12/AHSME, 26

Tags: geometry , 3D geometry , octahedron , ratio , AMC

A regular octahedron is formed by joining the centers of adjoining faces of a cube. The ratio of the volume of the octahedron to the volume of the cube is $ \textbf{(A)}\ \frac{\sqrt{3}}{12} \qquad\textbf{(B)}\ \frac{\sqrt{6}}{16} \qquad\textbf{(C)}\ \frac{1}{6} \qquad\textbf{(D)}\ \frac{\sqrt{2}}{8} \qquad\textbf{(E)}\ \frac{1}{4} $

2013 Tuymaada Olympiad, 5

Tags: geometry , 3D geometry , combinatorics proposed , combinatorics

Each face of a $7 \times 7 \times 7$ cube is divided into unit squares. What is the maximum number of squares that can be chosen so that no two chosen squares have a common point? [i]A. Chukhnov[/i]

MathLinks Contest 6th, 1.2

Tags: geometry , 3D geometry , sphere , rectangle , 6th edition

Let $ABCD$ be a rectangle of center $O$ in the plane $\alpha$, and let $V \notin\alpha$ be a point in space such that $V O \perp \alpha$. Let $A' \in (V A)$, $B'\in (V B)$, $C'\in (V C)$, $D'\in (V D)$ be four points, and let $M$ and $N$ be the midpoints of the segments $A'C'$ and $B'D'$. .Prove that $MN \parallel \alpha$ if and only if $V , A', B', C', D'$ all lie on a sphere.

2013 Princeton University Math Competition, 7

Tags: geometry , 3D geometry , tetrahedron

A tetrahedron $ABCD$ satisfies $AB=6$, $CD=8$, and $BC=DA=5$. Let $V$ be the maximum value of $ABCD$ possible. If we can write $V^4=2^n3^m$ for some integers $m$ and $n$, find $mn$.

2008 Purple Comet Problems, 9

Tags: geometry , 3D geometry

Find the sum of all the integers $N > 1$ with the properties that the each prime factor of $N $ is either $2, 3,$ or $5,$ and $N$ is not divisible by any perfect cube greater than $1.$

1969 Poland - Second Round, 6

Tags: combinatorics , geometry , 3D geometry , polyhedron , combinatorial geometry

Prove that every polyhedron has at least two faces with the same number of sides.

2024 Sharygin Geometry Olympiad, 24

Tags: pyramid , geometry , 3D geometry

Let $SABC$ be a pyramid with right angles at the vertex $S$. Points $A', B', C'$ lie on the edges $SA, SB, SC$ respectively in such a way that the triangles $ABC$ and $A'B'C'$ are similar. Does this yield that the planes $ABC$ and $A'B'C'$ are parallel?

1971 IMO Longlists, 9

Tags: geometry , 3D geometry , prism , trigonometry , geometry unsolved

The base of an inclined prism is a triangle $ABC$. The perpendicular projection of $B_1$, one of the top vertices, is the midpoint of $BC$. The dihedral angle between the lateral faces through $BC$ and $AB$ is $\alpha$, and the lateral edges of the prism make an angle $\beta$ with the base. If $r_1, r_2, r_3$ are exradii of a perpendicular section of the prism, assuming that in $ABC, \cos^2 A + \cos^2 B + \cos^2 C = 1, \angle A < \angle B < \angle C,$ and $BC = a$, calculate $r_1r_2 + r_1r_3 + r_2r_3.$

1986 IMO Longlists, 80

Tags: geometry , 3D geometry , tetrahedron , incenter , geometry proposed

Let $ABCD$ be a tetrahedron and $O$ its incenter, and let the line $OD$ be perpendicular to $AD$. Find the angle between the planes $DOB$ and $DOC.$

1959 IMO Shortlist, 6

Tags: geometry , trapezoid , 3D geometry , IMO , IMO 1959

Two planes, $P$ and $Q$, intersect along the line $p$. The point $A$ is given in the plane $P$, and the point $C$ in the plane $Q$; neither of these points lies on the straight line $p$. Construct an isosceles trapezoid $ABCD$ (with $AB \parallel CD$) in which a circle can be inscribed, and with vertices $B$ and $D$ lying in planes $P$ and $Q$ respectively.

1989 Spain Mathematical Olympiad, 3

Tags: inequalities , geometry , 3D geometry , induction , limit

Prove $ \frac{1}{10\sqrt2}<\frac{1}{2}\frac{3}{4}\frac{5}{6}...\frac{99}{100}<\frac{1}{10} $

II Soros Olympiad 1995 - 96 (Russia), 11.2

Tags: geometry , 3D geometry , tetrahedron

Is it possible that the heights of a tetrahedron (that is, a triangular pyramid) would be equal to the numbers $1$, $2$, $3$ and $6$?

1986 Poland - Second Round, 3

Tags: geometry , 3D geometry , sphere , tetrahedron , geometric inequality

Let S be a sphere cirucmscribed on a regular tetrahedron with an edge length greater than 1. The sphere $ S $ is represented as the sum of four sets. Prove that one of these sets includes points $ P $, $ Q $ such that the length of the segment $ PQ $ exceeds 1.

2024 Czech-Polish-Slovak Junior Match, 2

Tags: number theory , number theory proposed , Product , perfect cube , geometry , 3D geometry

How many non-empty subsets of $\{1,2,\dots,11\}$ are there with the property that the product of its elements is the cube of an integer?

2003 Iran MO (3rd Round), 18

Tags: geometry , 3D geometry , tetrahedron , circumcircle , sphere , geometry proposed

In tetrahedron $ ABCD$, radius four circumcircles of four faces are equal. Prove that $ AB\equal{}CD$, $ AC\equal{}BD$ and $ AD\equal{}BC$.

1967 IMO Shortlist, 1

Tags: geometry , 3D geometry , tetrahedron , Volume , geometric inequality , IMO , IMO 1967

Prove that a tetrahedron with just one edge length greater than $1$ has volume at most $ \frac{1}{8}.$

2004 Purple Comet Problems, 23

Tags: geometry , 3D geometry , rectangle

A cubic block with dimensions $n$ by $n$ by $n$ is made up of a collection of $1$ by $1$ by $1$ unit cubes. What is the smallest value of $n$ so that if the outer layer of unit cubes are removed from the block, more than half the original unit cubes will still remain?

2012 Online Math Open Problems, 23

Tags: Online Math Open , function , geometry , 3D geometry

For reals $x\ge3$, let $f(x)$ denote the function \[f(x) = \frac {-x + x\sqrt{4x-3} } { 2} .\]Let $a_1, a_2, \ldots$, be the sequence satisfying $a_1 > 3$, $a_{2013} = 2013$, and for $n=1,2,\ldots,2012$, $a_{n+1} = f(a_n)$. Determine the value of \[a_1 + \sum_{i=1}^{2012} \frac{a_{i+1}^3} {a_i^2 + a_ia_{i+1} + a_{i+1}^2} .\] [i]Ray Li.[/i]

1967 All Soviet Union Mathematical Olympiad, 086

Tags: 3D geometry , geometry

a) A lamp of a lighthouse enlights an angle of $90$ degrees. Prove that you can turn the lamps of four arbitrary posed lighthouses so, that all the plane will be enlightened. b) There are eight lamps in eight points of the space. Each can enlighten an octant (three-faced space polygon with three mutually orthogonal edges). Prove that you can turn them so, that all the space will be enlightened.

1969 IMO Longlists, 58

Tags: pigeonhole principle , geometry , 3D geometry , tetrahedron , IMO Shortlist , IMO Longlist

$(SWE 1)$ Six points $P_1, . . . , P_6$ are given in $3-$dimensional space such that no four of them lie in the same plane. Each of the line segments $P_jP_k$ is colored black or white. Prove that there exists one triangle $P_jP_kP_l$ whose edges are of the same color.

2008 Saint Petersburg Mathematical Olympiad, 5

Tags: geometry , 3D geometry , tetrahedron

All faces of the tetrahedron $ABCD $ are acute-angled triangles.$AK$ and $AL$ -are altitudes in faces $ABC$ and $ABD$. Points $C,D,K,L$ lies on circle. Prove, that $AB \perp CD$