Olympiad problems

1988 Polish MO Finals, 3

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

Find the largest possible volume for a tetrahedron which lies inside a hemisphere of radius $1$.

1990 IMO Shortlist, 17

Tags: combinatorics , analytic geometry , geometry , 3d geometry , euler

Unit cubes are made into beads by drilling a hole through them along a diagonal. The beads are put on a string in such a way that they can move freely in space under the restriction that the vertices of two neighboring cubes are touching. Let $ A$ be the beginning vertex and $ B$ be the end vertex. Let there be $ p \times q \times r$ cubes on the string $ (p, q, r \geq 1).$ [i](a)[/i] Determine for which values of $ p, q,$ and $ r$ it is possible to build a block with dimensions $ p, q,$ and $ r.$ Give reasons for your answers. [i](b)[/i] The same question as (a) with the extra condition that $ A \equal{} B.$

1986 IMO Longlists, 70

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

Let $ABCD$ be a tetrahedron having each sum of opposite sides equal to $1$. Prove that \[r_A + r_B + r_C + r_D \leq \frac{\sqrt 3}{3}\] where $r_A, r_B, r_C, r_D$ are the inradii of the faces, equality holding only if $ABCD$ is regular.

1997 AMC 8, 22

Tags: geometry , 3d geometry

A two-inch cube $(2\times 2\times 2)$ of silver weighs 3 pounds and is worth \$200. How much is a three-inch cube of silver worth? $\textbf{(A)}\ 300\text{ dollars} \qquad \textbf{(B)}\ 375\text{ dollars} \qquad \textbf{(C)}\ 450\text{ dollars} \qquad \textbf{(D)}\ 560\text{ dollars} \qquad \textbf{(E)}\ 675\text{ dollars}$

1970 Regional Competition For Advanced Students, 3

Tags: ratio , geometry , 3d geometry , octahedron

$E_1$ and $E_2$ are parallel planes and their distance is $p$. (a) How long is the seitenkante of the regular octahedron such that a side lies in $E_1$ and another in $E_2$? (b) $E$ is a plane between $E_1$ and $E_2$, parallel to $E_1$ and $E_2$, so that its distances from $E_1$ and $E_2$ are in ratio $1:2$ Draw the intersection figure of $E$ and the octahedron for $P=4\sqrt{\frac32}$ cm and justifies, why the that figure must look in such a way

Denmark (Mohr) - geometry, 2005.1

Tags: 3d geometry , pyramid , area , geometry

This figure is cut out from a sheet of paper. Folding the sides upwards along the dashed lines, one gets a (non-equilateral) pyramid with a square base. Calculate the area of the base. [img]https://1.bp.blogspot.com/-lPpfHqfMMRY/XzcBIiF-n2I/AAAAAAAAMW8/nPs_mLe5C8srcxNz45Wg-_SqHlRAsAmigCLcBGAsYHQ/s0/2005%2BMohr%2Bp1.png[/img]

1986 IMO Shortlist, 20

Tags: geometry , tetrahedron , trigonometry , 3d geometry , congruent triangles

Prove that the sum of the face angles at each vertex of a tetrahedron is a straight angle if and only if the faces are congruent triangles.

1990 All Soviet Union Mathematical Olympiad, 532

Tags: inequalities , tetrahedron , geometric inequality , 3d geometry , distance , altitude

If every altitude of a tetrahedron is at least $1$, show that the shortest distance between each pair of opposite edges is more than $2$.

2002 Flanders Math Olympiad, 1

Tags: geometry , 3d geometry

Is it possible to number the $8$ vertices of a cube from $1$ to $8$ in such a way that the value of the sum on every edge is different?

1982 Bulgaria National Olympiad, Problem 3

Tags: circumcircle , geometry , 3d geometry , prism

In a regular $2n$-gonal prism, bases $A_1A_2\cdots A_{2n}$ and $B_1B_2\cdots B_{2n}$ have circumradii equal to $R$. If the length of the lateral edge $A_1B_1$ varies, the angle between the line $A_1B_{n+1}$ and the plane $A_1A_3B_{n+2}$ is maximal for $A_1B_1=2R\cos\frac\pi{2n}$.

2008 Flanders Math Olympiad, 3

Tags: geometry , tetrahedron , 3d geometry , pyramid

A quadrilateral pyramid and a regular tetrahedron have edges that are all equal in length. They are glued together so that they have in common $1$ equilateral triangle . Prove that the resulting body has exactly $5$ sides.

2004 Harvard-MIT Mathematics Tournament, 7

Tags: tetrahedron , geometry , 3d geometry , graph theory

We have a polyhedron such that an ant can walk from one vertex to another, traveling only along edges, and traversing every edge exactly once. What is the smallest possible total number of vertices, edges, and faces of this polyhedron?

1990 National High School Mathematics League, 15

Tags: geometry , sphere , 3d geometry , pyramid

In pyramid $M-ABCD$, bottom surface $ABCD$ is a square. $MA=MC,MA\perp AB$. If the area of $\triangle AMD$ is $1$, find the maximum value of radius of sphere that can be put inside the pyramid.

1988 IMO Longlists, 8

Tags: parallelogram , geometry , tetrahedron , 3d geometry

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.

2008 Baltic Way, 14

Tags: geometry , combinatorics , 3d geometry

Is it possible to build a $ 4\times 4\times4$ cube from blocks of the following shape consisting of $ 4$ unit cubes?

2014 Romania National Olympiad, 2

Tags: geometry , 3d geometry , cube , angle , distance

Let $ABCDA'B'C'D'$ be a cube with side $AB = a$. Consider points $E \in (AB)$ and $F \in (BC)$ such that $AE + CF = EF$. a) Determine the measure the angle formed by the planes $(D'DE)$ and $(D'DF)$. b) Calculate the distance from $D'$ to the line $EF$.

1997 Poland - Second Round, 6

Tags: combinatorial geometry , geometry , 3d geometry , cube , distance , max , point

Let eight points be given in a unit cube. Prove that two of these points are on a distance not greater than $1$.

2022 239 Open Mathematical Olympiad, 2

Tags: 3d geometry , sphere , geometry

Five edges of a tetrahedron are tangent to a sphere. Prove that there are another five edges from this tetrahedron that are also tangent to a $($not necessarily the same$)$ sphere.

2005 Swedish Mathematical Competition, 6

Tags: tetrahedron , geometry , 3d geometry

A regular tetrahedron of edge length $1$ is orthogonally projected onto a plane. Find the largest possible area of its image.

2018 Iranian Geometry Olympiad, 4

Tags: geometry , 3-dimensional geometry , 3d geometry , polyhedron , centroid

We have a polyhedron all faces of which are triangle. Let $P$ be an arbitrary point on one of the edges of this polyhedron such that $P$ is not the midpoint or endpoint of this edge. Assume that $P_0 = P$. In each step, connect $P_i$ to the centroid of one of the faces containing it. This line meets the perimeter of this face again at point $P_{i+1}$. Continue this process with $P_{i+1}$ and the other face containing $P_{i+1}$. Prove that by continuing this process, we cannot pass through all the faces. (The centroid of a triangle is the point of intersection of its medians.) Proposed by Mahdi Etesamifard - Morteza Saghafian

2000 AIME Problems, 8

Tags: geometry , 3d geometry

A container in the shape of a right circular cone is 12 inches tall and its base has a 5-inch radius. The liquid that is sealed inside is 9 inches deep when the cone is held with its point down and its base horizontal. When the liquid is held with its point up and its base horizontal, the liquid is $m-n\sqrt[3]{p},$ where $m,$ $n,$ and $p$ are positive integers and $p$ is not divisible by the cube of any prime number. Find $m+n+p.$

1980 IMO, 5

Tags: combinatorics , geometry , function , 3d geometry , sphere

In the Euclidean three-dimensional space, we call [i]folding[/i] of a sphere $S$ every partition of $S \setminus \{x,y\}$ into disjoint circles, where $x$ and $y$ are two points of $S$. A folding of $S$ is called [b]linear[/b] if the circles of the [i]folding[/i] are obtained by the intersection of $S$ with a family of parallel planes or with a family of planes containing a straight line $D$ exterior to $S$. Is every [i]folding[/i] of a sphere $S$ [b]linear[/b]?

2004 District Olympiad, 4

Tags: trapezoid , geometry , trigonometry , 3d geometry , palnes , right angle , angle

In the right trapezoid $ABCD$ with $AB \parallel CD, \angle B = 90^o$ and $AB = 2DC$. At points $A$ and $D$ there is therefore a part of the plane $(ABC)$ perpendicular to the plane of the trapezoid, on which the points $N$ and $P$ are taken, ($AP$ and $PD$ are perpendicular to the plane) such that $DN = a$ and $AP = \frac{a}{2}$ . Knowing that $M$ is the midpoint of the side $BC$ and the triangle $MNP$ is equilateral, determine: a) the cosine of the angle between the planes $MNP$ and $ABC$. b) the distance from $D$ to the plane $MNP$

2005 AMC 10, 11

Tags: geometry , 3d geometry

A wooden cube $ n$ units on a side is painted red on all six faces and then cut into $ n^3$ unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is $ n$? $ \textbf{(A)}\ 3\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 5\qquad \textbf{(D)}\ 6\qquad \textbf{(E)}\ 7$

1976 Bulgaria National Olympiad, Problem 3

Tags: tetrahedron , 3d geometry , geometry

In the space is given a tetrahedron with length of the edge $2$. Prove that distances from some point $M$ to all of the vertices of the tetrahedron are integer numbers if and only if $M$ is a vertex of tetrahedron. [i]J. Tabov[/i]