Olympiad problems

1974 Spain Mathematical Olympiad, 7

A tank has the shape of a regular hexagonal prism, whose bases are $1$ m on a side and its height is $10$ m. The lateral edges are placed in an oblique position and is partially filled with $9$ m$^3$ of water. The plane of the free surface of the water cuts to all lateral edges. One of them is left with a part of $2$ m under water. What part is under water on the opposite side edge of the prism?

2006 AMC 10, 24

Tags: geometry , 3d geometry , octahedron , tetrahedron , pyramid

Centers of adjacent faces of a unit cube are joined to form a regular octahedron. What is the volume of this octahedron? $ \textbf{(A) } \frac 18 \qquad \textbf{(B) } \frac 16 \qquad \textbf{(C) } \frac 14 \qquad \textbf{(D) } \frac 13 \qquad \textbf{(E) } \frac 12$

2002 Flanders Math Olympiad, 1

Tags: 3d geometry , geometry

Is it possible to number the $8$ vertices of a cube from $1$ to $8$ in such a way that the value of the sum on every edge is different?

2016 Harvard-MIT Mathematics Tournament, 3

Tags: hmmt , geometry , 3d geometry , prism

Let $V$ be a rectangular prism with integer side lengths. The largest face has area $240$ and the smallest face has area $48$. A third face has area $x$, where $x$ is not equal to $48$ or $240$. What is the sum of all possible values of $x$?

2023 Stanford Mathematics Tournament, 2

Tags: geometry , 3d geometry

Triangle $\vartriangle ABC$ has side lengths $AB = 39$, $BC = 16$, and $CA = 25$. What is the volume of the solid formed by rotating $\vartriangle ABC$ about line $BC$?

2017 Simon Marais Mathematical Competition, B1

Tags: geometry , combinatorics , 3d geometry , tetrahedron

Maryam labels each vertex of a tetrahedron with the sum of the lengths of the three edges meeting at that vertex. She then observes that the labels at the four vertices of the tetrahedron are all equal. For each vertex of the tetrahedron, prove that the lengths of the three edges meeting at that vertex are the three side lengths of a triangle.

1978 Germany Team Selection Test, 5

Tags: geometry , 3d geometry , pyramid , combinatorics

Let $E$ be a finite set of points such that $E$ is not contained in a plane and no three points of $E$ are collinear. Show that at least one of the following alternatives holds: (i) $E$ contains five points that are vertices of a convex pyramid having no other points in common with $E;$ (ii) some plane contains exactly three points from $E.$

2014 Harvard-MIT Mathematics Tournament, 9

Tags: geometry , 3d geometry , sphere

Compute the side length of the largest cube contained in the region \[ \{(x, y, z) : x^2+y^2+z^2 \le 25 \text{ and } x, y \ge 0 \} \] of three-dimensional space.

1997 Taiwan National Olympiad, 5

Tags: 3d geometry , tetrahedron , trigonometry , parallelogram , geometry

Let $ABCD$ is a tetrahedron. Show that a)If $AB=CD,AC=DB,AD=BC$ then triangles $ABC,ABD,ACD,BCD$ are acute. b)If the triangles $ABC,ABD,ACD,BCD$ have the same area , then $AB=CD,AC=DB,AD=BC$.

2006 District Olympiad, 4

Tags: geometry , 3d geometry , prism

a) Prove that we can assign one of the numbers $1$ or $-1$ to the vertices of a cube such that the product of the numbers assigned to the vertices of any face is equal to $-1$. b) Prove that for a hexagonal prism such a mapping is not possible.

2004 AMC 12/AHSME, 20

Tags: 3d geometry , probability , geometry

Each face of a cube is painted either red or blue, each with probability $ 1/2$. The color of each face is determined independently. What is the probability that the painted cube can be placed on a horizontal surface so that the four vertical faces are all the same color? $ \textbf{(A)}\ \frac14 \qquad \textbf{(B)}\ \frac{5}{16} \qquad \textbf{(C)}\ \frac38 \qquad \textbf{(D)}\ \frac{7}{16} \qquad \textbf{(E)}\ \frac12$

Denmark (Mohr) - geometry, 1994.1

Tags: geometry , cylinder , 3d geometry , sphere

A wine glass with a cross section as shown has the property of an orange in shape as a sphere with a radius of $3$ cm just can be placed in the glass without protruding above glass. Determine the height $h$ of the glass. [img]https://1.bp.blogspot.com/-IuLm_IPTvTs/XzcH4FAjq5I/AAAAAAAAMYY/qMi4ng91us8XsFUtnwS-hb6PqLwAON_jwCLcBGAsYHQ/s0/1994%2BMohr%2Bp1.png[/img]

1990 National High School Mathematics League, 15

Tags: geometry , 3d geometry , pyramid , sphere

In pyramid $M-ABCD$, bottom surface $ABCD$ is a square. $MA=MC,MA\perp AB$. If the area of $\triangle AMD$ is $1$, find the maximum value of radius of sphere that can be put inside the pyramid.

1986 Flanders Math Olympiad, 4

Tags: geometry , 3d geometry , sphere , algebra

Given a cube in which you can put two massive spheres of radius 1. What's the smallest possible value of the side - length of the cube? Prove that your answer is the best possible.

2019 Yasinsky Geometry Olympiad, p2

Tags: geometry , perimeter , 3d geometry , pyramid , rectangle

The base of the quadrilateral pyramid $SABCD$ lies the $ABCD$ rectangle with the sides $AB = 1$ and $AD = 10$. The edge $SA$ of the pyramid is perpendicular to the base, $SA = 4$. On the edge of $AD$, find a point $M$ such that the perimeter of the triangle of $SMC$ was minimal.

1965 IMO Shortlist, 3

Tags: geometry , 3d geometry , tetrahedron , prism

Given the tetrahedron $ABCD$ whose edges $AB$ and $CD$ have lengths $a$ and $b$ respectively. The distance between the skew lines $AB$ and $CD$ is $d$, and the angle between them is $\omega$. Tetrahedron $ABCD$ is divided into two solids by plane $\epsilon$, parallel to lines $AB$ and $CD$. The ratio of the distances of $\epsilon$ from $AB$ and $CD$ is equal to $k$. Compute the ratio of the volumes of the two solids obtained.

1976 Bundeswettbewerb Mathematik, 4

Tags: geometry , 3d geometry , sphere , geometric transformation , vector , combinatorics

Each vertex of the 3-dimensional Euclidean space either is coloured red or blue. Prove that within those squares being possible in this space with edge length 1 there is at least one square either with three red vertices or four blue vertices !

2000 Polish MO Finals, 1

Tags: 3d geometry , pyramid , inequalities , geometry

$PA_1A_2...A_n$ is a pyramid. The base $A_1A_2...A_n$ is a regular n-gon. The apex $P$ is placed so that the lines $PA_i$ all make an angle $60^{\cdot}$ with the plane of the base. For which $n$ is it possible to find $B_i$ on $PA_i$ for $i = 2, 3, ... , n$ such that $A_1B_2 + B_2B_3 + B_3B_4 + ... + B_{n-1}B_n + B_nA_1 < 2A_1P$?

PEN H Problems, 24

Tags: modular arithmetic , diophantine equation , 3d geometry , geometry

Prove that if $n$ is a positive integer such that the equation \[x^{3}-3xy^{2}+y^{3}=n.\] has a solution in integers $(x,y),$ then it has at least three such solutions. Show that the equation has no solutions in integers when $n=2891$.

1996 Austrian-Polish Competition, 5

Tags: geometry , 3d geometry , sphere , polyhedron , convex

A sphere $S$ divides every edge of a convex polyhedron $P$ into three equal parts. Show that there exists a sphere tangent to all the edges of $P$.

1979 Spain Mathematical Olympiad, 7

Tags: geometry , 3d geometry , volume

Prove that the volume of a tire (torus) is equal to the volume of a cylinder whose base is a meridian section of that and whose height is the length of the circumference formed by the centers of the meridian sections.

2016 Israel Team Selection Test, 3

Tags: geometry , 3d geometry , octahedron , conic

Prove that there exists an ellipsoid touching all edges of an octahedron if and only if the octahedron's diagonals intersect. (Here an octahedron is a polyhedron consisting of eight triangular faces, twelve edges, and six vertices such that four faces meat at each vertex. The diagonals of an octahedron are the lines connecting pairs of vertices not connected by an edge).

2024 Euler Olympiad, Round 1, 9

Tags: 3d geometry , geometry

Ants, named Anna and Bob, are located at vertices $A$ and $B$ respectively of a cube $ABCD A_1 B_1 C_1 D_1$, with a sugar cube placed at vertex $C_1$. It is known that Bob can move at a speed of $20$ meters per minute. Determine the minimum speed in integer meters per minute that Anna must be able to travel in order to reach the sugar cube at $C_1$ before Bob. [i]Proposed by Tamar Turashvili, Georgia [/i]

1998 National High School Mathematics League, 12

Tags: geometry , 3d geometry , pyramid

In $\triangle ABC$, $\angle C=90^{\circ},\angle B=30^{\circ}, AC=2$. $M$ is the midpoint of $AB$. Fold up $\triangle ACM$ along $CM$, satisfying that $|AB|=2\sqrt2$. The volume of triangular pyramid $A-BCM$ is________.

Champions Tournament Seniors - geometry, 2008.4

Tags: geometry , 3d geometry , sphere , pyramid

Given a quadrangular pyramid $SABCD$, the basis of which is a convex quadrilateral $ABCD$. It is known that the pyramid can be tangent to a sphere. Let $P$ be the point of contact of this sphere with the base $ABCD$. Prove that $\angle APB + \angle CPD = 180^o$.