Olympiad problems

1983 IMO Shortlist, 13

Tags: geometry , combinatorics , Coloring , counting , 3D geometry , IMO Shortlist

Let $E$ be the set of $1983^3$ points of the space $\mathbb R^3$ all three of whose coordinates are integers between $0$ and $1982$ (including $0$ and $1982$). A coloring of $E$ is a map from $E$ to the set {red, blue}. How many colorings of $E$ are there satisfying the following property: The number of red vertices among the $8$ vertices of any right-angled parallelepiped is a multiple of $4$ ?

1990 ITAMO, 1

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

A cube of edge length $3$ consists of $27$ unit cubes. Find the number of lines passing through exactly three centers of these $27$ cubes, as well as the number of those passing through exactly two such centers.

2002 AMC 12/AHSME, 9

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? $\textbf{(A) }\sqrt{13}\qquad\textbf{(B) }\sqrt{14}\qquad\textbf{(C) }\sqrt{15}\qquad\textbf{(D) }4\qquad\textbf{(E) }\sqrt{17}$

1987 Polish MO Finals, 4

Tags: geometry , 3D geometry , tetrahedron , max , Volume

Let $S$ be the set of all tetrahedra which satisfy: (1) the base has area $1$, (2) the total face area is $4$, and (3) the angles between the base and the other three faces are all equal. Find the element of $S$ which has the largest volume.

2005 District Olympiad, 2

Tags: geometry , 3D geometry , pyramid

Let $ABCD$ and $ABEF$ be two squares situated in two perpendicular planes and let $O$ be the intersection of the lines $AE$ and $BF$. If $AB=4$ compute: a) the distance from $B$ to the line of intersection between the planes $(DOC)$ and $(DAF)$; b) the distance between the lines $AC$ and $BF$.

1973 USAMO, 1

Tags: geometry , 3D geometry , tetrahedron

Two points $ P$ and $ Q$ lie in the interior of a regular tetrahedron $ ABCD$. Prove that angle $ PAQ < 60^\circ$.

2023 Olympic Revenge, 4

Tags: geometry , 3D geometry , sphere , combinatorial geometry , combinatorics , lattice points

Let $S=\{(x,y,z)\in \mathbb{Z}^3\}$ the set of points with integer coordinates in the space. Gugu has infinitely many solid spheres. All with radii $\ge (\frac{\pi}2)^3$. Is it possible for Gugu to cover all points of $S$ with his spheres?

2000 Polish MO Finals, 1

Tags: geometry , 3D geometry , pyramid , inequalities , geometry unsolved

$PA_1A_2...A_n$ 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 $A_1B_2 + B_2B_3 + B_3B_4 + ... + B_{n-1}B_n + B_nA_1 < 2A_1P$?

2004 Switzerland Team Selection Test, 5

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

A brick has the shape of a cube of size $2$ with one corner unit cube removed. Given a cube of side $2^{n}$ divided into unit cubes from which an arbitrary unit cube is removed, show that the remaining figure can be built using the described bricks.

2012 Online Math Open Problems, 28

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

A fly is being chased by three spiders on the edges of a regular octahedron. The fly has a speed of $50$ meters per second, while each of the spiders has a speed of $r$ meters per second. The spiders choose their starting positions, and choose the fly's starting position, with the requirement that the fly must begin at a vertex. Each bug knows the position of each other bug at all times, and the goal of the spiders is for at least one of them to catch the fly. What is the maximum $c$ so that for any $r<c,$ the fly can always avoid being caught? [i]Author: Anderson Wang[/i]

2016 Fall CHMMC, 5

Tags: geometry , 3D geometry , probability

Suppose you have $27$ identical unit cubes colored such that $3$ faces adjacent to a vertex are red and the other $3$ are colored blue. Suppose further that you assemble these $27$ cubes randomly into a larger cube with $3$ cubes to an edge (in particular, the orientation of each cube is random). The probability that the entire cube is one solid color can be written as $\frac{1}{2^n}$ for some positive integer $n$. Find $n$.

1985 AIME Problems, 12

Tags: AMC , AIME , geometry , 3D geometry , tetrahedron , probability , linear algebra

Let $A$, $B$, $C$, and $D$ be the vertices of a regular tetrahedron, each of whose edges measures 1 meter. A bug, starting from vertex $A$, observes the following rule: at each vertex it chooses one of the three edges meeting at that vertex, each edge being equally likely to be chosen, and crawls along that edge to the vertex at its opposite end. Let $p = n/729$ be the probability that the bug is at vertex $A$ when it has crawled exactly 7 meters. Find the value of $n$.

1982 All Soviet Union Mathematical Olympiad, 334

Tags: geometry , trigonometry , 3D geometry , tetrahedron , angles

Given a point $M$ inside a right tetrahedron. Prove that at least one tetrahedron edge is seen from the $M$ in an angle, that has a cosine not greater than $-1/3$. (e.g. if $A$ and $B$ are the vertices, corresponding to that edge, $cos(\widehat{AMB}) \le -1/3$)

2017 Israel Oral Olympiad, 6

Tags: 3D geometry , geometry , polyhedron

What is the maximal number of vertices of a convex polyhedron whose each face is either a regular triangle or a square?

2004 Baltic Way, 6

Tags: geometry , 3D geometry , number theory proposed , number theory

A positive integer is written on each of the six faces of a cube. For each vertex of the cube we compute the product of the numbers on the three adjacent faces. The sum of these products is $1001$. What is the sum of the six numbers on the faces?

2011 District Olympiad, 3

Tags: geometry , 3D geometry , prism , areas , angles

Let $ABCA'B'C'$ a right triangular prism with the bases equilateral triangles. A plane $\alpha$ containing point $A$ intersects the rays $BB'$ and $CC'$ at points E and $F$, so that $S_ {ABE} + S_{ACF} = S_{AEF}$. Determine the measure of the angle formed by the plane $(AEF)$ with the plane $(BCC')$.

2004 Romania National Olympiad, 3

Tags: geometry , 3D geometry , pyramid , trigonometry

Let $ABCD A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ be a truncated regular pyramid in which $BC^{\prime}$ and $DA^{\prime}$ are perpendicular. (a) Prove that $\measuredangle \left( AB^{\prime},DA^{\prime} \right) = 60^{\circ}$; (b) If the projection of $B^{\prime}$ on $(ABC)$ is the center of the incircle of $ABC$, then prove that $d \left( CB^{\prime},AD^{\prime} \right) = \frac12 BC^{\prime}$. [i]Mircea Fianu[/i]

2018 Adygea Teachers' Geometry Olympiad, 4

Tags: geometry , 3D geometry , cube , angle , planes

Given a cube $ABCDA_1B_1C_1D_1$ with edge $5$. On the edge $BB_1$ of the cube , point $K$ such thath $BK=4$. a) Construct a cube section with the plane $a$ passing through the points $K$ and $C_1$ parallel to the diagonal $BD_1$. b) Find the angle between the plane $a$ and the plane $BB_1C_1$.

1986 French Mathematical Olympiad, Problem 1

Tags: geometry , 3D geometry , tetrahedron , inequalities

Let $ABCD$ be a tetrahedron. (a) Prove that the midpoints of the edges $AB,AC,BD$, and $CD$ lie in a plane. (b) Find the point in that plane, whose sum of distances from the lines $AD$ and $BC$ is minimal.

2012 AMC 8, 21

Tags: geometry , 3D geometry , AMC

Marla has a large white cube that has an edge of 10 feet. She also has enough green paint to cover 300 square feet. Marla uses all the paint to create a white square centered on each face, surrounded by a green border. What is the area of one of the white squares, in square feet? $\textbf{(A)}\hspace{.05in}5\sqrt2 \qquad \textbf{(B)}\hspace{.05in}10 \qquad \textbf{(C)}\hspace{.05in}10\sqrt2 \qquad \textbf{(D)}\hspace{.05in}50 \qquad \textbf{(E)}\hspace{.05in}50\sqrt2 $

1985 AMC 12/AHSME, 12

Tags: geometry , 3D geometry , number theory , prime numbers

Let's write p,q, and r as three distinct prime numbers, where 1 is not a prime. Which of the following is the smallest positive perfect cube leaving $ n \equal{} pq^2r^4$ as a divisor? $ \textbf{(A)}\ p^8q^8r^8\qquad \textbf{(B)}\ (pq^2r^2)^3\qquad \textbf{(C)}\ (p^2q^2r^2)^3\qquad \textbf{(D)}\ (pqr^2)^3\qquad \textbf{(E)}\ 4p^3q^3r^3$

1952 Moscow Mathematical Olympiad, 210

Tags: 3D geometry , geometry , parallelogram , rhombus , parallelepiped

Prove that if all faces of a parallelepiped are equal parallelograms, they are rhombuses.

1991 China Team Selection Test, 3

Tags: inequalities , geometry , 3D geometry , tetrahedron , combinatorics unsolved , combinatorics , graph theory

All edges of a polyhedron are painted with red or yellow. For an angle of a facet, if the edges determining it are of different colors, then the angle is called [i]excentric[/i]. The[i] excentricity [/i]of a vertex $A$, namely $S_A$, is defined as the number of excentric angles it has. Prove that there exist two vertices $B$ and $C$ such that $S_B + S_C \leq 4$.

2000 Harvard-MIT Mathematics Tournament, 37

Tags: geometry , 3D geometry

A cone with semivertical angle $30^{\circ}$ is half filled with water. What is the angle it must be tilted by so that water starts spilling?

1983 IMO Longlists, 40

Tags: ratio , geometry , 3D geometry , tetrahedron , sphere , function , congruent triangles

Four faces of tetrahedron $ABCD$ are congruent triangles whose angles form an arithmetic progression. If the lengths of the sides of the triangles are $a < b < c$, determine the radius of the sphere circumscribed about the tetrahedron as a function on $a, b$, and $c$. What is the ratio $c/a$ if $R = a \ ?$