Olympiad problems

1946 Moscow Mathematical Olympiad, 111

Given two intersecting planes $\alpha$ and $\beta$ and a point $A$ on the line of their intersection. Prove that of all lines belonging to $\alpha$ and passing through $A$ the line which is perpendicular to the intersection line of $\alpha$ and $\beta$ forms the greatest angle with $\beta$.

2011 Federal Competition For Advanced Students, Part 1, 4

Tags: geometry , tetrahedron , function , 3d geometry

Inside or on the faces of a tetrahedron with five edges of length $2$ and one edge of lenght $1$, there is a point $P$ having distances $a, b, c, d$ to the four faces of the tetrahedron. Determine the locus of all points $P$ such that $a+b+c+d$ is minimal and the locus of all points $P$ such that $a+b+c+d$ is maximal.

1998 AMC 8, 21

Tags: 3d geometry , geometry

A $4*4*4$ cubical box contains 64 identical small cubes that exactly fill the box. How many of these small cubes touch a side or the bottom of the box? $ \text{(A)}\ 48\qquad\text{(B)}\ 52\qquad\text{(C)}\ 60\qquad\text{(D)}\ 64\qquad\text{(E)}\ 80 $

2000 Harvard-MIT Mathematics Tournament, 1

Tags: geometry , 3d geometry

The sum of $3$ real numbers is known to be zero. If the sum of their cubes is $\pi^e$, what is their product equal to?

2003 Croatia National Olympiad, Problem 4

Tags: 3d geometry , geometry

Given $8$ unit cubes, $24$ of their faces are painted in blue and the remaining $24$ faces in red. Show that it is always possible to assemble these cubes into a cube of edge $2$ on whose surface there are equally many blue and red unit squares.

1954 Poland - Second Round, 5

Tags: parallelogram , geometry , 3d geometry , projection

Given points $ A $, $ B $, $ C $ and $ D $ that do not lie in the same plane. Draw a plane through the point $ A $ such that the orthogonal projection of the quadrilateral $ ABCD $ on this plane is a parallelogram.

2019 PUMaC Geometry B, 2

Tags: geometry , 3d geometry

A right cone in $xyz$-space has its apex at $(0,0,0)$, and the endpoints of a diameter on its base are $(12,13,-9)$ and $(12,-5,15)$. The volume of the cone can be expressed as $a\pi$. What is $a$?

2018 Bundeswettbewerb Mathematik, 4

Tags: combinatorics , geometry , 3d geometry , triangle , distance

We are given six points in space with distinct distances, no three of them collinear. Consider all triangles with vertices among these points. Show that among these triangles there is one such that its longest side is the shortest side in one of the other triangles.

1967 IMO Shortlist, 5

Tags: polyhedron , geometry , 3d geometry , sphere

Faces of a convex polyhedron are six squares and 8 equilateral triangles and each edge is a common side for one triangle and one square. All dihedral angles obtained from the triangle and square with a common edge, are equal. Prove that it is possible to circumscribe a sphere around the polyhedron, and compute the ratio of the squares of volumes of that polyhedron and of the ball whose boundary is the circumscribed sphere.

2009 Princeton University Math Competition, 8

Tags: tetrahedron , geometry , 3d geometry

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).

1988 Czech And Slovak Olympiad IIIA, 3

Tags: geometry , 3d geometry , tetrahedron , geometric inequality

Given a tetrahedron $ABCD$ with edges $|AD|=|BC|= a$, $|AC|=|BD|=b$, $AB=c$ and $|CD| = d$. Determine the smallest value of the sum $|AX|+|BX|+|CX|+|DX|$, where $X$ is any point in space.

1979 Vietnam National Olympiad, 6

Tags: 3-dimensional geometry , tetrahedron , 3d geometry , construction , geometry

$ABCD$ is a rectangle with $BC / AB = \sqrt2$. $ABEF$ is a congruent rectangle in a different plane. Find the angle $DAF$ such that the lines $CA$ and $BF$ are perpendicular. In this configuration, find two points on the line $CA$ and two points on the line $BF$ so that the four points form a regular tetrahedron.

Ukrainian TYM Qualifying - geometry, I.13

Tags: geometry , 3d geometry , reflection

A candle and a man are placed in a dihedral mirror angle. How many reflections can the man see ?

2008 Iran Team Selection Test, 4

Tags: geometry , function , 3d geometry , tetrahedron , vector , inequalities

Let $ P_1,P_2,P_3,P_4$ be points on the unit sphere. Prove that $ \sum_{i\neq j}\frac1{|P_i\minus{}P_j|}$ takes its minimum value if and only if these four points are vertices of a regular pyramid.

1983 Putnam, B1

Tags: geometry , calculus , 3d geometry , volume , integration

Let $v$ be a vertex of a cube $C$ with edges of length $4$. Let $S$ be the largest sphere that can be inscribed in $C$. Let $R$ be the region consisting of all points $p$ between $S$ and $C$ such that $p$ is closer to $v$ than to any other vertex of the cube. Find the volume of $R$.

2012 Today's Calculation Of Integral, 772

Tags: rotation , calculus , geometry , geometric transformation , integration , 3d geometry , analytic geometry

Given are three points $A(2,\ 0,\ 2),\ B(1,\ 1,\ 0),\ C(0,\ 0,\ 3)$ in the coordinate space. Find the volume of the solid of a triangle $ABC$ generated by a rotation about $z$-axis.

1996 Tuymaada Olympiad, 8

Tags: 3d geometry , solid geometry , tetrahedron , geometry , sphere

Given a tetrahedron $ABCD$, in which $AB=CD= 13 , AC=BD=14$ and $AD=BC=15$. Show that the centers of the inscribed sphere and sphere around it coincide, and find the radii of these spheres.

2008 Romania National Olympiad, 1

Tags: pyramid , geometry , tetrahedron , 3d geometry

A tetrahedron has the side lengths positive integers, such that the product of any two opposite sides equals 6. Prove that the tetrahedron is a regular triangular pyramid in which the lateral sides form an angle of at least 30 degrees with the base plane.

2006 Moldova MO 11-12, 4

Tags: geometry , 3d geometry , pyramid , ratio

Let $ABCDE$ be a right quadrangular pyramid with vertex $E$ and height $EO$. Point $S$ divides this height in the ratio $ES: SO=m$. In which ratio does the plane $(ABC)$ divide the lateral area of the pyramid.

1995 Tournament Of Towns, (457) 2

Tags: geometry , combinatorial geometry , 3d geometry , coloring , prism

For what values of $n$ is it possible to paint the edges of a prism whose base is an $n$-gon so that there are edges of all three colours at each vertex and all the faces (including the upper and lower bases) have edges of all three colours? (AV Shapovelov)

1990 ITAMO, 1

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

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.

2018 Israel Olympic Revenge, 2

Tags: combinatorics , geometry , rubik s cube , 3d geometry

Is it possible to disassemble and reassemble a $4\times 4\times 4$ Rubik's Cuble in at least $577,800$ non-equivalent ways? Notes: 1. When we reassemble the cube, a corner cube has to go to a corner cube, an edge cube must go to an edge cube and a central cube must go to a central cube. 2. Two positions of the cube are called equivalent if they can be obtained from one two another by rotating the faces or layers which are parallel to the faces.