Olympiad problems

1996 ITAMO, 3

Given a cube of unit side. Let $A$ and $B$ be two opposite vertex. Determine the radius of the sphere, with center inside the cube, tangent to the three faces of the cube with common point $A$ and tangent to the three sides with common point $B$.

2007 Princeton University Math Competition, 4

Tags: geometry , 3D geometry , probability , expected value

A cube is formed from $n^3$ ($n \ge 2$) unit cubes, each painted white on five randomly selected sides. This cube is dipped into paint remover and broken into the original unit cubes. What is the expected number of these unit cubes with exactly four sides painted white?

1992 Bundeswettbewerb Mathematik, 3

Tags: geometry , 3D geometry , sphere

Given is a triangle $ABC$ with side lengths $a, b,c$. Three spheres touch each other in pairs and also touch the plane of the triangle at points $A,B$ and $C$, respectively. Determine the radii of these spheres.

2015 JHMT, 10

Tags: 3D geometry , geometry

A unit sphere is centered at $(0, 0, 1)$. There is a point light source located at $(1, 0, 4)$ that sends out light uniformly in every direction but is blocked by the sphere. What is the area of the sphere’s shadow on the $x-y$ plane? (A point $(a, b, c)$ denotes the point in three dimensions with $x$-coordinate $a$, $y$-coordinate $b$, and $z$-coordinate $c$)

1997 All-Russian Olympiad, 3

Tags: geometry , 3D geometry , tetrahedron , sphere

A sphere inscribed in a tetrahedron touches one face at the intersection of its angle bisectors, a second face at the intersection of its altitudes, and a third face at the intersection of its medians. Show that the tetrahedron is regular. [i]N. Agakhanov[/i]

1988 French Mathematical Olympiad, Problem 3

Tags: geometry , 3D geometry , sphere

Consider two spheres $\Sigma_1$ and $\Sigma_2$ and a line $\Delta$ not meeting them. Let $C_i$ and $r_i$ be the center and radius of $\Sigma_i$, and let $H_i$ and $d_i$ be the orthogonal projection of $C_i$ onto $\Delta$ and the distance of $C_i$ from $\Delta~(i=1,2)$. For a point $M$ on $\Delta$, let $\delta_i(M)$ be the length of a tangent $MT_i$ to $\Sigma_i$, where $T_i\in\Sigma_i~(i=1,2)$. Find $M$ on $\Delta$ for which $\delta_1(M)+\delta_2(M)$ is minimal.

2005 Miklós Schweitzer, 10

Tags: vector , angle , 3D geometry , linear algebra

Given 5 nonzero vectors in three-dimensional Euclidean space, prove that the sum of their pairwise angles is at most $6\pi$.

1990 Bundeswettbewerb Mathematik, 4

Tags: sphere , midpoints , 3D geometry , tetrahedron , geometry

Suppose that every two opposite edges of a tetrahedron are orthogonal. Show that the midpoints of the six edges lie on a sphere.

2005 Vietnam Team Selection Test, 3

Tags: function , geometry , geometric transformation , 3D geometry , induction , calculus , integration

Find all functions $f: \mathbb{Z} \mapsto \mathbb{Z}$ satisfying the condition: $f(x^3 +y^3 +z^3 )=f(x)^3+f(y)^3+f(z)^3.$

2000 Harvard-MIT Mathematics Tournament, 1

Tags: geometry , 3D geometry , tetrahedron , rotation

How many different ways are there to paint the sides of a tetrahedron with exactly $4$ colors? Each side gets its own color, and two colorings are the same if one can be rotated to get the other.

2008 ITest, 25

Tags: geometry , 3D geometry

A cube has edges of length $120\text{ cm}$. The cube gets chopped up into some number of smaller cubes, all of equal size, such that each edge of one of the smaller cubes has an integer length. One of those smaller cubes is then chopped up into some number of $\textit{even smaller}$ cubes, all of equal size. If the edge length of one of those $\textit{even smaller}$ cubes is $n\text{ cm}$, where $n$ is an integer, find the number of possible values of $n$.

2017 Yasinsky Geometry Olympiad, 5

Tags: 3D geometry , 3-Dimensional Geometry , cube , area , geometry

Find the area of the section of a unit cube $ABCDA_1B_1C_1D_1$, when a plane passes through the midpoints of the edges $AB, AD$ and $CC_1$.

1985 IMO Longlists, 95

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

Prove that for every point $M$ on the surface of a regular tetrahedron there exists a point $M'$ such that there are at least three different curves on the surface joining $M$ to $M'$ with the smallest possible length among all curves on the surface joining $M$ to $M'$.

2008 ITest, 14

Tags: geometry , 3D geometry

The sum of the two perfect cubes that are closest to $500$ is $343+512=855$. Find the sum of the two perfect cubes that are closest to $2008$.

2007 Moldova Team Selection Test, 1

Tags: geometry , 3D geometry , algebra , factorization , sum of cubes , number theory proposed , number theory

Find the least positive integers $m,k$ such that a) There exist $2m+1$ consecutive natural numbers whose sum of cubes is also a cube. b) There exist $2k+1$ consecutive natural numbers whose sum of squares is also a square. The author is Vasile Suceveanu

1973 IMO Longlists, 1

Tags: geometry , 3D geometry , sphere , tetrahedron

Find the maximal positive number $r$ with the following property: If all altitudes of a tetrahedron are $\geq 1$, then a sphere of radius $r$ fits into the tetrahedron.

2007 ITest, 27

Tags: geometry , 3D geometry

The face diagonal of a cube has length $4$. Find the value of $n$ given that $n\sqrt2$ is the $\textit{volume}$ of the cube.

1999 IMO Shortlist, 3

Tags: geometry , symmetry , 3D geometry , reflection , IMO , IMO 1999 , Convex hull

A set $ S$ of points from the space will be called [b]completely symmetric[/b] if it has at least three elements and fulfills the condition that for every two distinct points $ A$ and $ B$ from $ S$, the perpendicular bisector plane of the segment $ AB$ is a plane of symmetry for $ S$. Prove that if a completely symmetric set is finite, then it consists of the vertices of either a regular polygon, or a regular tetrahedron or a regular octahedron.

1971 IMO Longlists, 14

Tags: geometry , 3D geometry , Diophantine equation , number theory unsolved , number theory

Note that $8^3 - 7^3 = 169 = 13^2$ and $13 = 2^2 + 3^2.$ Prove that if the difference between two consecutive cubes is a square, then it is the square of the sum of two consecutive squares.

1979 Miklós Schweitzer, 4

Tags: group theory , abstract algebra , geometry , 3D geometry , sphere , superior algebra , superior algebra unsolved

For what values of $ n$ does the group $ \textsl{SO}(n)$ of all orthogonal transformations of determinant $ 1$ of the $ n$-dimensional Euclidean space possess a closed regular subgroup?($ \textsl{G}<\textsl{SO}(n)$ is called $ \textit{regular}$ if for any elements $ x,y$ of the unit sphere there exists a unique $ \varphi \in \textsl{G}$ such that $ \varphi(x)\equal{}y$.) [i]Z. Szabo[/i]

2016 Chile National Olympiad, 6

Tags: geometry , Locus , 3D geometry , Locus problems , collinear

Let $P_1$ and $P_2$ be two non-parallel planes in space, and $A$ a point that does not It is in none of them. For each point $X$, let $X_1$ denote its reflection with respect to $P_1$, and $X_2$ its reflection with respect to $P_2$. Determine the locus of points $X$ for the which $X_1, X_2$ and $A$ are collinear.

2006 Sharygin Geometry Olympiad, 24

Tags: geometry , Locus , 3D geometry , projections , Plane , circle , sphere

a) Two perpendicular rays are drawn through a fixed point $P$ inside a given circle, intersecting the circle at points $A$ and $B$. Find the geometric locus of the projections of $P$ on the lines $AB$. b) Three pairwise perpendicular rays passing through the fixed point $P$ inside a given sphere intersect the sphere at points $A, B, C$. Find the geometrical locus of the projections $P$ on the $ABC$ plane

2002 Belarusian National Olympiad, 4

Tags: geometry , 3D geometry , prism , geometric transformation , rotation , geometry proposed

This requires some imagination and creative thinking: Prove or disprove: There exists a solid such that, for all positive integers $n$ with $n \geq 3$, there exists a "parallel projection" (I hope the terminology is clear) such that the image of the solid under this projection is a convex $n$-gon.

1990 IMO Shortlist, 10

Tags: geometry , 3D geometry , ellipse , sphere , Volume , IMO Shortlist

A plane cuts a right circular cone of volume $ V$ into two parts. The plane is tangent to the circumference of the base of the cone and passes through the midpoint of the altitude. Find the volume of the smaller part. [i]Original formulation:[/i] A plane cuts a right circular cone into two parts. The plane is tangent to the circumference of the base of the cone and passes through the midpoint of the altitude. Find the ratio of the volume of the smaller part to the volume of the whole cone.

1987 IMO Longlists, 35

Tags: algebra , polynomial , analytic geometry , Intersection , 3D geometry , IMO Shortlist

Does there exist a set $M$ in usual Euclidean space such that for every plane $\lambda$ the intersection $M \cap \lambda$ is finite and nonempty ? [i]Proposed by Hungary.[/i] [hide="Remark"]I'm not sure I'm posting this in a right Forum.[/hide]