Olympiad problems

2000 Croatia National Olympiad, Problem 3

Tags: geometry , 3D geometry

A plane intersects a rectangular parallelepiped in a regular hexagon. Prove that the rectangular parallelepiped is a cube.

2004 AMC 10, 7

Tags: geometry , 3D geometry , pyramid

A grocer stacks oranges in a pyramid-like stack whose rectangular base is $ 5$ oranges by $ 8$ oranges. Each orange above the first level rests in a pocket formed by four oranges in the level below. The stack is completed by a single row of oranges. How many oranges are in the stack? $ \textbf{(A)}\ 96 \qquad \textbf{(B)}\ 98 \qquad \textbf{(C)}\ 100 \qquad \textbf{(D)}\ 101 \qquad \textbf{(E)}\ 134$

2000 Harvard-MIT Mathematics Tournament, 5

Tags: geometry , 3D geometry

Find all natural numbers $n$ such that $n$ equals the cube of the sum of its digits.

1959 IMO, 6

Tags: geometry , trapezoid , 3D geometry , IMO , IMO 1959

Two planes, $P$ and $Q$, intersect along the line $p$. The point $A$ is given in the plane $P$, and the point $C$ in the plane $Q$; neither of these points lies on the straight line $p$. Construct an isosceles trapezoid $ABCD$ (with $AB \parallel CD$) in which a circle can be inscribed, and with vertices $B$ and $D$ lying in planes $P$ and $Q$ respectively.

2005 Abels Math Contest (Norwegian MO), 2a

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

In an aquarium there are nine small fish. The aquarium is cube shaped with a side length of two meters and is completely filled with water. Show that it is always possible to find two small fish with a distance of less than $\sqrt3$ meters.

2015 India PRMO, 4

Tags: combinatorial geometry , geometry , 3D geometry

$4.$ How many line segments have both their endpoints located at the vertices of a given cube $?$

1989 National High School Mathematics League, 14

Tags: geometry , 3D geometry , pyramid

In regular triangular pyramid $S-ABC$, hieght $SO=3$, length of sides of bottom surface is $6$. Projection of $A$ on plane $SBC$ is $O'$. $P\in AO',\frac{AP}{PO'}=8$. Draw a plane parallel to plane $ABC$ and passes $P$. Find the area of the cross section.

2016 Israel National Olympiad, 2

Tags: geometry , 3D geometry , cone , symmetry

We are given a cone with height 6, whose base is a circle with radius $\sqrt{2}$. Inside the cone, there is an inscribed cube: Its bottom face on the base of the cone, and all of its top vertices lie on the cone. What is the length of the cube's edge? [img]https://i.imgur.com/AHqHHP6.png[/img]

1986 IMO Longlists, 17

Tags: geometry , 3D geometry , tetrahedron , geometry unsolved

We call a tetrahedron right-faced if each of its faces is a right-angled triangle. [i](a)[/i] Prove that every orthogonal parallelepiped can be partitioned into six right-faced tetrahedra. [i](b)[/i] Prove that a tetrahedron with vertices $A_1,A_2,A_3,A_4$ is right-faced if and only if there exist four distinct real numbers $c_1, c_2, c_3$, and $c_4$ such that the edges $A_jA_k$ have lengths $A_jA_k=\sqrt{|c_j-c_k|}$ for $1\leq j < k \leq 4.$

TNO 2008 Senior, 11

Tags: Chile , combinatorics , geometry , 3D geometry

Each face of a cube is painted with a different color. How many distinct cubes can be created this way? (*Observation: The ways to color the cube are $6!$, since each time a color is used on one face, there is one fewer available for the others. However, this does not determine $6!$ different cubes, since colorings that differ only by rotation should be considered the same.*)

2008 Harvard-MIT Mathematics Tournament, 7

Tags: geometry , 3D geometry , limit , function , calculus , derivative , calculus computations

([b]5[/b]) Find $ p$ so that $ \lim_{x\rightarrow\infty}x^p\left(\sqrt[3]{x\plus{}1}\plus{}\sqrt[3]{x\minus{}1}\minus{}2\sqrt[3]{x}\right)$ is some non-zero real number.

2000 Romania National Olympiad, 3

Tags: geometry , 3D geometry , tetrahedron

Let be a tetahedron $ ABCD, $ and $ E $ be the projection of $ D $ on the plane formed by $ ABC. $ If $ \mathcal{A}_{\mathcal{R}} $ denotes the area of the region $ \mathcal{R}, $ show that the following affirmations are equivalent: [b]a)[/b] $ C=E\vee CE\parallel AB $ [b]b)[/b] $ M\in\overline{CD}\implies\mathcal{A}_{ABM}^2=\frac{CM^2}{CD^2}\cdot\mathcal{A}_{ABD}^2 +\left( 1-\frac{CM^2}{CD^2}\right)\cdot\mathcal{A}_{ABC}^2 $

2011 Math Prize For Girls Problems, 15

Tags: geometry , 3D geometry , probability

The game of backgammon has a "doubling" cube, which is like a standard 6-faced die except that its faces are inscribed with the numbers 2, 4, 8, 16, 32, and 64, respectively. After rolling the doubling cube four times at random, we let $a$ be the value of the first roll, $b$ be the value of the second roll, $c$ be the value of the third roll, and $d$ be the value of the fourth roll. What is the probability that $\frac{a + b}{c + d}$ is the average of $\frac{a}{c}$ and $\frac{b}{d}$ ?

1991 Arnold's Trivium, 93

Tags: function , geometry , 3D geometry , invariant

Decompose the space of functions defined on the vertices of a cube into invariant subspaces irreducible with respect to the group of a) its symmetries, b) its rotations.

1971 Czech and Slovak Olympiad III A, 6

Tags: geometry , 3D geometry , tetrahedron , ratio

Let a tetrahedron $ABCD$ and its inner point $O$ be given. For any edge $e$ of $ABCD$ consider the segment $f(e)$ containing $O$ such that $f(e)\parallel e$ and the endpoints of $f(e)$ lie on the faces of the tetrahedron. Show that \[\sum_{e\text{ edge}}\,\frac{\,f(e)\,}{e}=3.\]

2019 Tournament Of Towns, 5

Tags: orthogonal projection , projection , 3D geometry , geometry , tetrahedron , trapezoid

The orthogonal projection of a tetrahedron onto a plane containing one of its faces is a trapezoid of area $1$, which has only one pair of parallel sides. a) Is it possible that the orthogonal projection of this tetrahedron onto a plane containing another its face is a square of area $1$? b) The same question for a square of area $1/2019$. (Mikhail Evdokimov)

1959 Czech and Slovak Olympiad III A, 3

Tags: geometry , 3D geometry , frustum

Consider a piece of material in the shape of a right circular conical frustum with radii $R,r,R>r$. A cavity in the shape of another coaxial right circular conical frustum was drilled into the material (see the picture). That way only half of the original volume of material remained. Compute radii $R',r'$ of the cavity. Decide for which ratio $R/r$ the problem has a solution. [img]https://cdn.artofproblemsolving.com/attachments/b/f/12f579458b7cf0fc31849b319e6f58e50b0363.png[/img]

1982 Vietnam National Olympiad, 3

Tags: geometry , 3D geometry , geometry unsolved

Let $ABCDA'B'C'D'$ be a cube (where $ABCD$ and $A'B'C'D'$ are faces and $AA',BB',CC',DD'$ are edges). Consider the four lines $AA', BC, D'C'$ and the line joining the midpoints of $BB'$ and $DD'$. Show that there is no line which cuts all the four lines.

Kyiv City MO 1984-93 - geometry, 1987.10.3

Tags: geometry , 3D geometry , sphere , cone

In a right circular cone with the radius of the base $R$ and the height $h$ are $n$ spheres of the same radius $r$ ($n \ge 3$). Each ball touches the base of the cone, its side surface and other two balls. Determine $r$.

1965 All Russian Mathematical Olympiad, 070

Tags: polyhedron , geometry , 3D geometry

Prove that the sum of the lengths of the polyhedron edges exceeds its tripled diameter (distance between two farest vertices).

2002 Tournament Of Towns, 2

Tags: geometry , 3D geometry , inequalities , geometry proposed

A cube is cut by a plane such that the cross section is a pentagon. Show there is a side of the pentagon of length $\ell$ such that the inequality holds: \[ |\ell-1|>\frac{1}{5} \]

2009 Princeton University Math Competition, 8

Tags: geometry , 3D geometry , tetrahedron

Consider the solid with 4 triangles and 4 regular hexagons as faces, where each triangle borders 3 hexagons, and all the sides are of length 1. Compute the [i]square[/i] of the volume of the solid. Express your result in reduced fraction and concatenate the numerator with the denominator (e.g., if you think that the square is $\frac{1734}{274}$, then you would submit 1734274).

2008 Bundeswettbewerb Mathematik, 3

Tags: geometry , 3D geometry , sphere , geometric transformation , reflection , combinatorics unsolved , combinatorics

Through a point in the interior of a sphere we put three pairwise perpendicular planes. Those planes dissect the surface of the sphere in eight curvilinear triangles. Alternately the triangles are coloured black and wide to make the sphere surface look like a checkerboard. Prove that exactly half of the sphere's surface is coloured black.

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.

1988 IMO Longlists, 8

Tags: geometry , 3D geometry , tetrahedron , parallelogram , IMO Shortlist

In a given tedrahedron $ ABCD$ let $ K$ and $ L$ be the centres of edges $ AB$ and $ CD$ respectively. Prove that every plane that contains the line $ KL$ divides the tedrahedron into two parts of equal volume.