Olympiad problems

2011 Canadian Open Math Challenge, 3

Tags: geometry , 3D geometry

The faces of a cube contain the number 1, 2, 3, 4, 5, 6 such that the sum of the numbers on each pair of opposite faces is 7. For each of the cube’s eight corners, we multiply the three numbers on the faces incident to that corner, and write down its value. (In the diagram, the value of the indicated corner is 1 x 2 x 3 = 6.) What is the sum of the eight values assigned to the cube’s corners?

1984 Tournament Of Towns, (061) O2

Tags: geometry , 3D geometry , tetrahedron , parallel , Plane , altitude

Six altitudes are constructed from the three vertices of the base of a tetrahedron to the opposite sides of the three lateral faces. Prove that all three straight lines joining two base points of the altitudes in each lateral face are parallel to a certain plane. (IF Sharygin, Moscow)

2007 IMO Shortlist, 7

Tags: algebra , 3D geometry , nullstellensatz , IMO , IMO 2007 , IMO Shortlist , Gerhard Woeginger

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]

1984 National High School Mathematics League, 8

Tags: function , geometry , 3D geometry , tetrahedron

Lengths of five edges of a tetrahedron are $1$, while the last one is $x$. Its volume is $F(x)$. On its domain of definition, we have $\text{(A)}$ $F(x)$ is an increasing function, it has no maximum value. $\text{(B)}$ $F(x)$ is an increasing function, it has maximum value. $\text{(C)}$ $F(x)$ is not an increasing function, it has no maximum value. $\text{(D)}$ $F(x)$ is an increasing function, it has maximum value.

1990 AMC 12/AHSME, 9

Tags: geometry , 3D geometry , AMC

Each edge of a cube is colored either red or black. Every face of the cube has at least one black edge. The smallest possible number of black edges is $\textbf{(A) }2\qquad \textbf{(B) }3\qquad \textbf{(C) }4\qquad \textbf{(D) }5\qquad \textbf{(E) }6\qquad$

1969 IMO Shortlist, 32

Tags: geometry , 3D geometry , sphere , dissection , combinatorics , IMO Shortlist , IMO Longlist

$(GDR 4)$ Find the maximal number of regions into which a sphere can be partitioned by $n$ circles.

1973 USAMO, 5

Tags: geometry , 3D geometry , number theory , prime numbers , arithmetic sequence

Show that the cube roots of three distinct prime numbers cannot be three terms (not necessarily consecutive) of an arithmetic progression.

2016 Sharygin Geometry Olympiad, 5

Tags: combinatorics , geometry , 3D geometry

Does there exist a convex polyhedron having equal number of edges and diagonals? [i](A diagonal of a polyhedron is a segment through two vertices not lying on the same face) [/i]

2006 Oral Moscow Geometry Olympiad, 5

Tags: geometry , cyclic quadrilateral , 3D geometry , pyramid

The base of the pyramid is a convex quadrangle. Is there necessarily a section of this pyramid that does not intersect the base and is an inscribed quadrangle? (M. Volchkevich)

1990 Vietnam National Olympiad, 3

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

A tetrahedron is to be cut by three planes which form a parallelepiped whose three faces and all vertices lie on the surface of the tetrahedron. (a) Can this be done so that the volume of the parallelepiped is at least $ \frac{9}{40}$ of the volume of the tetrahedron? (b) Determine the common point of the three planes if the volume of the parallelepiped is $ \frac{11}{50}$ of the volume of the tetrahedron.

2018 NZMOC Camp Selection Problems, 6

Tags: pentagon , cube , 3D geometry , geometry

The intersection of a cube and a plane is a pentagon. Prove the length of at least oneside of the pentagon differs from 1 metre by at least 20 centimetres.

2017 China Team Selection Test, 1

Tags: geometry , Tstst , solid geometry , 3D geometry , octahedron

Find out the maximum value of the numbers of edges of a solid regular octahedron that we can see from a point out of the regular octahedron.(We define we can see an edge $AB$ of the regular octahedron from point $P$ outside if and only if the intersection of non degenerate triangle $PAB$ and the solid regular octahedron is exactly edge $AB$.

May Olympiad L2 - geometry, 1995.4

Tags: geometry , pyramid , minimum , distance , 3D 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 $PQA$ ?

2016 All-Russian Olympiad, 2

Tags: geometry , 3D geometry , sphere

In the space given three segments $A_1A_2, B_1B_2$ and $C_1C_2$, do not lie in one plane and intersect at a point $P$. Let $O_{ijk}$ be center of sphere that passes through the points $A_i, B_j, C_k$ and $P$. Prove that $O_{111}O_{222}, O_{112}O_{221}, O_{121}O_{212}$ and$O_{211}O_{122}$ intersect at one point. (P.Kozhevnikov)

1966 IMO Shortlist, 56

Tags: geometry , 3D geometry , tetrahedron , circumcircle , sphere , IMO Shortlist , IMO Longlist

In a tetrahedron, all three pairs of opposite (skew) edges are mutually perpendicular. Prove that the midpoints of the six edges of the tetrahedron lie on one sphere.

Ukrainian TYM Qualifying - geometry, IX.12

Tags: geometry , 3D geometry , tetrahedron , Ukrainian TYM

Let $AB,AC$ and $AD$ be the edges of a cube, $AB=\alpha$. Point $E$ was marked on the ray $AC$ so that $AE=\lambda \alpha$, and point $F$ was marked on the ray $AD$ so that $AF=\mu \alpha$ ($\mu> 0, \lambda >0$). Find (characterize) pairs of numbers $\lambda$ and $\mu$ such that the cross-sectional area of a cube by any plane parallel to the plane $BCD$ is equal to the cross-sectional area of the tetrahedron $ABEF$ by the same plane.

2005 National High School Mathematics League, 4

Tags: geometry , 3D geometry , perimeter

In cube $ABCD-A_1B_1C_1D_1$, draw a plane $\alpha$ perpendicular to line $AC'$, and $\alpha$ has intersections with any surface of the cube. The area of the cross section is $S$, the perimeter of the cross section is $l$, then $\text{(A)}$ The value of $S$ is fixed, but the value of $l$ is not fixed. $\text{(B)}$ The value of $S$ is not fixed, but the value of $l$ is fixed. $\text{(C)}$ The value of $S$ is fixed, the value of $l$ is fixed as well. $\text{(D)}$ The value of $S$ is not fixed, the value of $l$ is not fixed either.

2013 Stanford Mathematics Tournament, 4

Tags: geometry , 3D geometry , tetrahedron , symmetry , rhombus

$ABCD$ is a regular tetrahedron with side length $1$. Find the area of the cross section of $ABCD$ cut by the plane that passes through the midpoints of $AB$, $AC$, and $CD$.

1999 Harvard-MIT Mathematics Tournament, 7

Tags: quadratics , geometry , 3D geometry

Find an ordered pair $(a,b)$ of real numbers for which $x^2+ax+b$ has a non-real root whose cube is $343$.

2009 All-Russian Olympiad, 3

Tags: geometry , 3D geometry , sphere , pyramid , circumcircle , symmetry , perpendicular bisector

Let $ ABCD$ be a triangular pyramid such that no face of the pyramid is a right triangle and the orthocenters of triangles $ ABC$, $ ABD$, and $ ACD$ are collinear. Prove that the center of the sphere circumscribed to the pyramid lies on the plane passing through the midpoints of $ AB$, $ AC$ and $ AD$.

1999 Czech And Slovak Olympiad IIIA, 2

Tags: 3D geometry , tetrahedron , Volume , geometry

In a tetrahedron $ABCD, E$ and $F$ are the midpoints of the medians from $A$ and $D$. Find the ratio of the volumes of tetrahedra $BCEF$ and $ABCD$. Note: Median in a tetrahedron connects a vertex and the centroid of the opposite side.

1997 Taiwan National Olympiad, 5

Tags: geometry , 3D geometry , tetrahedron , trigonometry , parallelogram , geometry proposed

Let $ABCD$ is a tetrahedron. Show that a)If $AB=CD,AC=DB,AD=BC$ then triangles $ABC,ABD,ACD,BCD$ are acute. b)If the triangles $ABC,ABD,ACD,BCD$ have the same area , then $AB=CD,AC=DB,AD=BC$.

2018 Peru IMO TST, 8

Tags: Coloring , Color problem , dodecahedron , combinatorics , geometry , 3D geometry

You want to paint some edges of a regular dodecahedron red so that each face has an even number of painted edges (which can be zero). Determine from How many ways this coloration can be done. Note: A regular dodecahedron has twelve pentagonal faces and in each vertex concur three edges. The edges of the dodecahedron are all different for the purpose of the coloring . In this way, two colorings are the same only if the painted edges they are the same.

2009 District Olympiad, 3

Tags: geometry , 3D geometry , prism , planes , angle

Consider the regular quadrilateral prism $ABCDA'B'C 'D'$, in which $AB = a,AA' = \frac{a \sqrt {2}}{2}$, and $M$ is the midpoint of $B' C'$. Let $F$ be the foot of the perpendicular from $B$ on line $MC$, Let determine the measure of the angle between the planes $(BDF)$ and $(HBS)$.

2009 AMC 12/AHSME, 22

Tags: geometry , 3D geometry , octahedron , number theory , relatively prime , AMC

A regular octahedron has side length $ 1$. A plane parallel to two of its opposite faces cuts the octahedron into the two congruent solids. The polygon formed by the intersection of the plane and the octahedron has area $ \frac {a\sqrt {b}}{c}$, where $ a$, $ b$, and $ c$ are positive integers, $ a$ and $ c$ are relatively prime, and $ b$ is not divisible by the square of any prime. What is $ a \plus{} b \plus{} c$? $ \textbf{(A)}\ 10\qquad \textbf{(B)}\ 11\qquad \textbf{(C)}\ 12\qquad \textbf{(D)}\ 13\qquad \textbf{(E)}\ 14$