Olympiad problems

1990 Czech and Slovak Olympiad III A, 3

Tags: geometry , 3d geometry , tetrahedron , obtuse , triangle , plane

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.

2022 Princeton University Math Competition, B2

Tags: geometry , 3d geometry

Three spheres are all externally tangent to a plane and to each other. Suppose that the radii of these spheres are $6$, $8$, and, $10$. The tangency points of these spheres with the plane form the vertices of a triangle. Determine the largest integer that is smaller than the perimeter of this triangle.

VI Soros Olympiad 1999 - 2000 (Russia), 11.4

Tags: geometry , 3d geometry , sphere

Let the line $L$ be perpendicular to the plane $P$. Three spheres touch each other in pairs so that each sphere touches the plane $P$ and the line $L$. The radius of the larger sphere is $1$. Find the minimum radius of the smallest sphere.

Kyiv City MO 1984-93 - geometry, 1990.11.1

Tags: geometric inequality , geometry , 3d geometry , cube

Prove that the sum of the distances from any point in space from the vertices of a cube with edge $a$ is not less than $4\sqrt3 a$.

2008 ITest, 91

Tags: geometry , 3d geometry

Find the sum of all positive integers $n$ such that \[x^3+y^3+z^3=nx^2y^2z^2\] is satisfied by at least one ordered triplet of positive integers $(x,y,z)$.

1959 AMC 12/AHSME, 1

Tags: cube , geometry , 3d geometry , percent , percent increase

Each edge of a cube is increased by $50 \%$. The percent of increase of the surface area of the cube is: $ \textbf{(A)}\ 50 \qquad\textbf{(B)}\ 125\qquad\textbf{(C)}\ 150\qquad\textbf{(D)}\ 300\qquad\textbf{(E)}\ 750 $

2023 AMC 10, 18

Tags: 3d geometry , dodecahedron

A rhombic dodecahedron is a solid with $12$ congruent rhombus faces. At every vertex, $3$ or $4$ edges meet, depending on the vertex. How many vertices have exactly $3$ edges meet? $\textbf{(A) }5\qquad\textbf{(B) }6\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9$

I Soros Olympiad 1994-95 (Rus + Ukr), 11.4

Tags: geometry , 3d geometry

The wire is bent in the form of a square with side $2$. Find the volume of the body consisting of all points in space located at a distance not exceeding $1$ from at least one point of the wire.

2016 Saint Petersburg Mathematical Olympiad, 2

Tags: geometry , combinatorics , max , cube , 3-dimensional geometry , combinatorial geometry , 3d geometry

The rook, standing on the surface of the checkered cube, beats the cells, located in the same row as well as on the continuations of this series through one or even several edges. (The picture shows an example for a $4 \times 4 \times 4$ cube,visible cells that some beat the rook, shaded gray.) What is the largest number do not beat each other rooks can be placed on the surface of the cube $50 \times 50 \times 50$?

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________.

2012 Online Math Open Problems, 23

Tags: geometry , function , 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]

2009 ISI B.Math Entrance Exam, 8

Tags: 3d geometry , geometry

Suppose you are given six colours and, are asked to colour each face of a cube by a different colour. Determine the different number of colouring possible.

MathLinks Contest 4th, 2.2

Tags: geometry , 3d geometry

Prove that the six sides of any tetrahedron can be the sides of a convex hexagon.

2002 AMC 10, 16

Tags: geometry , 3d geometry , prism

Two walls and the ceiling of a room meet at right angles at point $P$. A fly is in the air one meter from one wall, eight meters from the other wall, and $9$ meters from point $P$. How many meters is the fly from the ceiling? $\textbf{(A) }\sqrt{13}\qquad\textbf{(B) }\sqrt{14}\qquad\textbf{(C) }\sqrt{15}\qquad\textbf{(D) }4\qquad\textbf{(E) }\sqrt{17}$

2008 Harvard-MIT Mathematics Tournament, 9

Tags: geometry , 3d geometry , tetrahedron , pyramid , similar triangles

Consider a circular cone with vertex $ V$, and let $ ABC$ be a triangle inscribed in the base of the cone, such that $ AB$ is a diameter and $ AC \equal{} BC$. Let $ L$ be a point on $ BV$ such that the volume of the cone is 4 times the volume of the tetrahedron $ ABCL$. Find the value of $ BL/LV$.