Olympiad problems

2016 BMT Spring, 16

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

What is the radius of the largest sphere that fits inside the tetrahedron whose vertices are the points $(0, 0, 0)$, $(1, 0, 0)$, $(0, 1, 0)$, $(0, 0, 1)$?

2007 AMC 12/AHSME, 20

Tags: geometry , 3D geometry , tetrahedron

Corners are sliced off a unit cube so that the six faces each become regular octagons. What is the total volume of the removed tetrahedra? $ \textbf{(A)}\ \frac {5\sqrt {2} \minus{} 7}{3}\qquad \textbf{(B)}\ \frac {10 \minus{} 7\sqrt {2}}{3}\qquad \textbf{(C)}\ \frac {3 \minus{} 2\sqrt {2}}{3}\qquad \textbf{(D)}\ \frac {8\sqrt {2} \minus{} 11}{3}\qquad \textbf{(E)}\ \frac {6 \minus{} 4\sqrt {2}}{3}$

1985 IMO Longlists, 28

Tags: geometry , 3D geometry , octahedron , geometry unsolved

[i]a)[/i] Let $M$ be the set of the lengths of the edges of an octahedron whose sides are congruent quadrangles. Prove that $M$ has at most three elements. [i]b)[/i] Let an octahedron whose sides are congruent quadrangles be given. Prove that each of these quadrangles has two equal sides meeting at a common vertex.

2003 AMC 8, 1

Tags: geometry , 3D geometry

Jamie counted the number of edges of a cube, Jimmy counted the number of corners, and Judy counted the number of faces. They then added the three numbers. What was the resulting sum? $\textbf{(A)}\ 12 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 20\qquad \textbf{(D)}\ 22 \qquad \textbf{(E)}\ 26$

1969 IMO Shortlist, 27

Tags: geometry , 3D geometry , ratio , sphere , IMO Shortlist , IMO Longlist

$(GBR 4)$ The segment $AB$ perpendicularly bisects $CD$ at $X$. Show that, subject to restrictions, there is a right circular cone whose axis passes through $X$ and on whose surface lie the points $A,B,C,D.$ What are the restrictions?

1999 Baltic Way, 2

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

Determine all positive integers $n$ with the property that the third root of $n$ is obtained by removing its last three decimal digits.

1986 Tournament Of Towns, (115) 3

Tags: vector , tetrahedron , regular tetrahedron , geometry , 3D geometry

Vectors coincide with the edges of an arbitrary tetrahedron (possibly non-regular). Is it possible for the sum of these six vectors to equal the zero vector? (Problem from Leningrad)

1995 Poland - Second Round, 5

Tags: geometry , 3D geometry , tetrahedron , Concyclic , incircle

The incircles of the faces $ABC$ and $ABD$ of a tetrahedron $ABCD$ are tangent to the edge $AB$ in the same point. Prove that the points of tangency of these incircles to the edges $AC,BC,AD,BD$ are concyclic.

1969 IMO Longlists, 27

Tags: geometry , 3D geometry , ratio , sphere , IMO Shortlist , IMO Longlist

$(GBR 4)$ The segment $AB$ perpendicularly bisects $CD$ at $X$. Show that, subject to restrictions, there is a right circular cone whose axis passes through $X$ and on whose surface lie the points $A,B,C,D.$ What are the restrictions?

2019 BMT Spring, Tie 4

Tags: geometry , pyramid , 3D geometry , sphere

Consider a regular triangular pyramid with base $\vartriangle ABC$ and apex $D$. If we have $AB = BC =AC = 6$ and $AD = BD = CD = 4$, calculate the surface area of the circumsphere of the pyramid.

1998 National High School Mathematics League, 5

Tags: geometry , 3D geometry , tetrahedron

In regular tetrahedron $ABCD$, $E,F,G$ are midpoints of $AB,BC,CD$. Dihedral angle $C-FG-E$ is equal to $\text{(A)}\arcsin\frac{\sqrt6}{3}\qquad\text{(B)}\frac{\pi}{2}+\arccos\frac{\sqrt3}{3}\qquad\text{(C)}\frac{\pi}{2}-\arctan{\sqrt2}\qquad\text{(D)}\pi-\text{arccot}\frac{\sqrt2}{2}$

1984 Czech And Slovak Olympiad IIIA, 5

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

Find all natural numbers $n$ for which there exists a convex polyhedron with $n$ edges, with exactly one vertex having four edges and all other vertices having $3$ edges.

1971 Vietnam National Olympiad, 2

Tags: geometry , 3D geometry

$ABCDA'B'C'D'$ is a cube (with $ABCD$ and $A'B'C'D'$ faces, and $AA', BB', CC', DD'$ edges). $L$ is a line which intersects or is parallel to the lines $AA', BC$ and $DB'$. $L$ meets the line $BC$ at $M$ (which may be the point at infinity). Let $m = |BM|$. The plane $MAA'$ meets the line $B'C'$ at $E$. Show that $|B'E| = m$. The plane $MDB'$ meets the line $A'D'$ at $F$. Show that $|D'F| = m$. Hence or otherwise show how to construct the point $P$ at the intersection of $L$ and the plane $A'B'C'D'$. Find the distance between $P$ and the line $A'B'$ and the distance between $P$ and the line $A'D'$ in terms of $m$. Find a relation between these two distances that does not depend on $m$. Find the locus of $M$. Let $S$ be the envelope of the line $L$ as $M$ varies. Find the intersection of $S$ with the faces of the cube.

2016 Junior Regional Olympiad - FBH, 5

Tags: geometry , 3D geometry , sphere , pyramid

$605$ spheres of same radius are divided in two parts. From one part, upright "pyramid" is made with square base. From the other part, upright "pyramid" is made with equilateral triangle base. Both "pyramids" are put together from equal numbers of sphere rows. Find number of spheres in every "pyramid"

1995 ITAMO, 5

Tags: 3D geometry , geometry , sphere , circles , tangent circles , Tangents

Two non-coplanar circles in space are tangent at a point and have the same tangents at this point. Show that both circles lie on some sphere.

2000 Harvard-MIT Mathematics Tournament, 10

Tags: geometry , 3D geometry

What is the total surface area of an ice cream cone, radius $R$, height $H$, with a spherical scoop of ice cream of radius $r$ on top? (Given $R<r$)

2007 ITest, 50

Tags: geometry , 3D geometry , floor function , geometry unsolved , unsolved

A block $Z$ is formed by gluing one face of a solid cube with side length 6 onto one of the circular faces of a right circular cylinder with radius $10$ and height $3$ so that the centers of the square and circle coincide. If $V$ is the smallest convex region that contains Z, calculate $\lfloor\operatorname{vol}V\rfloor$ (the greatest integer less than or equal to the volume of $V$).

2017 NZMOC Camp Selection Problems, 3

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

Find all prime numbers $p$ such that $16p + 1$ is a perfect cube.

1988 ITAMO, 5

Tags: 3D geometry , projections , parallelogram , geometry

Given four non-coplanar points, is it always possible to find a plane such that the orthogonal projections of the points onto the plane are the vertices of a parallelogram? How many such planes are there in general?

2018 AMC 12/AHSME, 23

Tags: AMC , AMC 12 , geometry , 3D geometry , sphere , 2018 AMC 12B , 2018 AMC

Ajay is standing at point $A$ near Pontianak, Indonesia, $0^\circ$ latitude and $110^\circ \text{ E}$ longitude. Billy is standing at point $B$ near Big Baldy Mountain, Idaho, USA, $45^\circ \text{ N}$ latitude and $115^\circ \text{ W}$ longitude. Assume that Earth is a perfect sphere with center $C$. What is the degree measure of $\angle ACB$? $ \textbf{(A) }105 \qquad \textbf{(B) }112\frac{1}{2} \qquad \textbf{(C) }120 \qquad \textbf{(D) }135 \qquad \textbf{(E) }150 \qquad $

2023 Math Prize for Girls Problems, 14

Tags: geometry , 3D geometry , sphere

Five points are chosen uniformly and independently at random on the surface of a sphere. Next, 2 of these 5 points are randomly picked, with every pair equally likely. What is the probability that the 2 points are separated by the plane containing the other 3 points?

1966 IMO Shortlist, 57

Tags: geometry , 3D geometry , symmetry , rotation , IMO Longlist , IMO Shortlist , cube

Is it possible to choose a set of $100$ (or $200$) points on the boundary of a cube such that this set is fixed under each isometry of the cube into itself? Justify your answer.

2012 CHMMC Spring, 2

Tags: geometry , 3D geometry , octahedron , geometric inequality , CHMMC

A convex octahedron in Cartesian space contains the origin in its interior. Two of its vertices are on the $x$-axis, two are on the $y$-axis, and two are on the $z$-axis. One triangular face $F$ has side lengths $\sqrt{17}$, $\sqrt{37}$, $\sqrt{52}$. A second triangular face $F_0$ has side lengths $\sqrt{13}$, $\sqrt{29}$, $\sqrt{34}$. What is the minimum possible volume of the octahedron?

2000 All-Russian Olympiad Regional Round, 11.2

Tags: geometry , 3D geometry , combinatorial geometry , geometric inequality , cylinder , Spheres

The height and radius of the base of the cylinder are equal to $1$. What is the smallest number of balls of radius $1$ that can cover the entire cylinder?

2021 AMC 10 Spring, 13

Tags: AMC 10 , AMC 10 A , geometry , 3D geometry , tetrahedron , probs , amc10A 2021

What is the volume of tetrahedron $ABCD$ with edge lengths $AB=2, AC=3, AD=4, BC=\sqrt{13}, BD=2\sqrt{5},$ and $CD=5$? $\textbf{(A) }3 \qquad \textbf{(B) }2\sqrt{3} \qquad \textbf{(C) }4 \qquad \textbf{(D) }3\sqrt{3} \qquad \textbf{(E) }6$