Olympiad problems

2000 Bundeswettbewerb Mathematik, 3

For each vertex of a given tetrahedron, a sphere passing through that vertex and the midpoints of the edges outgoing from this vertex is constructed. Prove that these four spheres pass through a single point.

2021 Yasinsky Geometry Olympiad, 3

Tags: geometry , 3d geometry , parallelepiped

Given a rectangular parallelepiped $ABCDA_1B_1C_1D_1$, which has $AD= DC = 3\sqrt2$ cm, and $DD_1 = 8$ cm. Through the diagonal $B_1D$ of the parallelepiped $m$ parallel to line $A_1C_1$ is drawn on the plane $\gamma$. a) Draw a section of a parallelepiped with plane $\gamma$. b) Justify what geometric figure is this section, and find its area. (Alexander Shkolny)

1974 Polish MO Finals, 1

Tags: tetrahedron , 3d geometry , geometry

In a tetrahedron $ABCD$ the edges $AB$ and $CD$ are perpendicular and $\angle ACB =\angle ADB$. Prove that the plane through $AB$ and the midpoint of the edge $CD$, is perpendicular to $CD$.

2013 AMC 10, 22

Tags: geometry , 3d geometry , pythagorean theorem , sphere

Six spheres of radius $1$ are positioned so that their centers are at the vertices of a regular hexagon of side length $2$. The six spheres are internally tangent to a larger sphere whose center is the center of the hexagon. An eighth sphere is externally tangent to the six smaller spheres and internally tangent to the larger sphere. What is the radius of this eighth sphere? $ \textbf{(A)} \ \sqrt{2} \qquad \textbf{(B)} \ \frac{3}{2} \qquad \textbf{(C)} \ \frac{5}{3} \qquad \textbf{(D)} \ \sqrt{3} \qquad \textbf{(E)} \ 2$

2007 ITest, 38

Tags: logarithm , floor function , 3d geometry , inequalities , geometry

Find the largest positive integer that is equal to the cube of the sum of its digits.

1973 USAMO, 1

Tags: 3d geometry , tetrahedron , geometry

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

2002 Putnam, 2

Tags: 3d geometry , sphere , pigeonhole principle , putnam pigeonhole , geometry

Given any five points on a sphere, show that some four of them must lie on a closed hemisphere.

1964 IMO, 5

Tags: combinatorial geometry , 3d geometry , counting , intersection , geometry

Supppose five points in a plane are situated so that no two of the straight lines joining them are parallel, perpendicular, or coincident. From each point perpendiculars are drawn to all the lines joining the other four points. Determine the maxium number of intersections that these perpendiculars can have.

2021 Durer Math Competition (First Round), 5

Tags: combinatorics , combinatorial geometry , line , 3d geometry , plane , geometry

There are $n$ distinct lines in three-dimensional space such that no two lines are parallel and no three lines meet at one point. What is the maximal possible number of planes determined by these $n$ lines? We say that a plane is determined if it contains at least two of the lines.

2002 All-Russian Olympiad Regional Round, 11.2

Tags: geometry , 3d geometry , pyramid

The altitude of a quadrangular pyramid $SABCD$ passes through the intersection point of the diagonals of its base $ABCD$. From the tops of the base perpendiculars $AA_1$, $BB_1$, $CC_1$, $DD_1$ are dropped onto lines $SC$, $SD,$ $SA$ and $SB$ respectively. It turned out that the points $S$, $A_1$, $B_1$, $C_1$, $D_1$ are different and lie on the same sphere. Prove that lines $AA_1$, $ BB_1$, $CC_1$, $DD_1$ pass through one point.

2021 AMC 10 Fall, 24

Tags: 3d geometry , geometry

A cube is constructed from $4$ white unit cubes and $4$ black unit cubes. How many different ways are there to construct the $2 \times 2 \times 2$ cube using these smaller cubes? (Two constructions are considered the same if one can be rotated to match the other.) $\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ 11$

2022 Adygea Teachers' Geometry Olympiad, 4

Tags: geometry , 3d geometry , pyramid

In a regular hexagonal pyramid $SABCDEF$, a plane is drawn through vertex $A$ and the midpoints of edges $SC$ and $CE$. Find the ratio in which this plane divides the volume of the pyramid.

2014 NIMO Problems, 3

Tags: perimeter , 3d geometry , sphere , geometry , ratio

A square and equilateral triangle have the same perimeter. If the triangle has area $16\sqrt3$, what is the area of the square? [i]Proposed by Evan Chen[/i]

1973 AMC 12/AHSME, 32

Tags: 3d geometry , geometry , pyramid

The volume of a pyramid whose base is an equilateral triangle of side length 6 and whose other edges are each of length $ \sqrt{15}$ is $ \textbf{(A)}\ 9 \qquad \textbf{(B)}\ 9/2 \qquad \textbf{(C)}\ 27/2 \qquad \textbf{(D)}\ \frac{9\sqrt3}{2} \qquad \textbf{(E)}\ \text{none of these}$

1947 Putnam, B6

Tags: 3d geometry , orthogonal

Let $OX, OY, OZ$ be mutually orthogonal lines in space. Let $C$ be a fixed point on $OZ$ and $U,V$ variable points on $OX, OY,$ respectively. Find the locus of a point $P$ such that $PU, PV, PC$ are mutually orthogonal.

1997 Vietnam National Olympiad, 3

Tags: pigeonhole principle , geometry , 3d geometry

In the unit cube, given 75 points, no three of which are collinear. Prove that there exits a triangle whose vertices are among the given points and whose area is not greater than 7/72.

MathLinks Contest 6th, 1.2

Tags: 3d geometry , sphere , rectangle , geometry

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.

1966 IMO Longlists, 23

Tags: tetrahedron , geometry , 3d geometry

Three faces of a tetrahedron are right triangles, while the fourth is not an obtuse triangle. [i](a) [/i]Prove that a necessary and sufficient condition for the fourth face to be a right triangle is that at some vertex exactly two angles are right. [i](b)[/i] Prove that if all the faces are right triangles, then the volume of the tetrahedron equals one -sixth the product of the three smallest edges not belonging to the same face.

2008 District Olympiad, 1

Tags: tetrahedron , rhombus , geometry , square , plane , 3d geometry

A regular tetrahedron is sectioned with a plane after a rhombus. Prove that the rhombus is square.

1997 French Mathematical Olympiad, Problem 2

Tags: 3d geometry , geometry

A region in space is determined by a sphere with center $O$ and radius $R$, and a cone with vertex $O$ which intersects the sphere in a circle of radius $r$. Find the maximum volume of a cylinder contained in this region, having the same axis as the cone.

Ukrainian TYM Qualifying - geometry, 2013.17

Tags: geometry , concurrency , tetrahedron , 3d geometry

Through the point of intersection of the medians of each of the faces a tetrahedron is drawn perpendicular to this face. Prove that all these four lines intersect at one point if and only if the four lines containing the heights of this tetrahedron intersect at one point .

2006 Kazakhstan National Olympiad, 6

Tags: tetrahedron , perpendicular , geometry , 3d geometry , plane

In the tetrahedron $ ABCD $ from the vertex $ A $, the perpendiculars $ AB '$, $ AC' $ are drawn, $ AD '$ on planes dividing dihedral angles at edges $ CD $, $ BD $, $ BC $ in half. Prove that the plane $ (B'C'D ') $ is parallel to the plane $ (BCD) $.

1995 May Olympiad, 4

Tags: minimum , distance , 3d geometry , pyramid , geometry

Consider a pyramid whose base is an equilateral triangle $BCD$ and whose other faces are triangles isosceles, right at the common vertex $A$. An ant leaves the vertex $B$ arrives at a point $P$ of the $CD$ edge, from there goes to a point $Q$ of the edge $AC$ and returns to point $B$. If the path you made is minimal, how much is the angle $PQA$ ?

2012 Puerto Rico Team Selection Test, 2

Tags: geometry , 3d geometry

A cone is constructed with a semicircular piece of paper, with radius 10. Find the height of the cone.

2007 IMO Shortlist, 7

Tags: algebra , 3d geometry , nullstellensatz

Let $ n$ be a positive integer. Consider \[ S \equal{} \left\{ (x,y,z) \mid x,y,z \in \{ 0, 1, \ldots, n\}, x \plus{} y \plus{} z > 0 \right \} \] as a set of $ (n \plus{} 1)^{3} \minus{} 1$ points in the three-dimensional space. Determine the smallest possible number of planes, the union of which contains $ S$ but does not include $ (0,0,0)$. [i]Author: Gerhard Wöginger, Netherlands [/i]