Olympiad problems

VI Soros Olympiad 1999 - 2000 (Russia), 11.6

It is known that a $n$-vertex contains within itself a polyhedron $M$ with a center of symmetry at some point $Q$ and is itself contained in a polyhedron homothetic to $M$ with a homothety center at a point $Q$ and coefficient $k$. Find the smallest value of $k$ if a) $n = 4$, b) $n = 5$.

2008 Romania National Olympiad, 4

Tags: geometry , 3d geometry

Let $ ABCDA'B'C'D'$ be a cube. On the sides $ (A'D')$, $ (A'B')$ and $ (A'A)$ we consider the points $ M_1$, $ N_1$ and $ P_1$ respectively. On the sides $ (CB)$, $ (CD)$ and $ (CC')$ we consider the points $ M_2$, $ N_2$ and $ P_2$ respectively. Let $ d_1$ be the distance between the lines $ M_1N_1$ and $ M_2N_2$, $ d_2$ be the distance between the lines $ N_1P_1$ and $ N_2P_2$, and $ d_3$ be the distance between the lines $ P_1M_1$ and $ P_2M_2$. Suppose that the distances $ d_1$, $ d_2$ and $ d_3$ are pairwise distinct. Prove that the lines $ M_1M_2$, $ N_1N_2$ and $ P_1P_2$ are concurrent.

1991 Baltic Way, 18

Tags: geometry , 3d geometry , sphere

Is it possible to place two non-intersecting tetrahedra of volume $\frac{1}{2}$ into a sphere with radius $1$?

2024-25 IOQM India, 7

Tags: geometry , 3d geometry

Determine the sum of all possible surface area of a cube two of whose vertices are $(1,2,0)$ and $(3,3,2)$.

1992 IMO Longlists, 76

Tags: pyramid , geometry , 3d geometry

Given any triangle $ABC$ and any positive integer $n$, we say that $n$ is a [i]decomposable[/i] number for triangle $ABC$ if there exists a decomposition of the triangle $ABC$ into $n$ subtriangles with each subtriangle similar to $\triangle ABC$. Determine the positive integers that are decomposable numbers for every triangle.

1986 ITAMO, 5

Tags: geometry , 3d geometry , tetrahedron , acute triangle

Given an acute triangle $T$ with sides $a,b,c$, find the tetrahedra with base $T$ whose all faces are acute triangles of the same area.

2002 Iran Team Selection Test, 11

Tags: combinatorics , 3d geometry

A $10\times10\times10$ cube has $1000$ unit cubes. $500$ of them are coloured black and $500$ of them are coloured white. Show that there are at least $100$ unit squares, being the common face of a white and a black unit cube.

2012 Sharygin Geometry Olympiad, 19

Tags: projective geometry , incenter , geometry , 3d geometry , prism , dilation , geometric transformation

Two circles with radii 1 meet in points $X, Y$, and the distance between these points also is equal to $1$. Point $C$ lies on the first circle, and lines $CA, CB$ are tangents to the second one. These tangents meet the first circle for the second time in points $B', A'$. Lines $AA'$ and $BB'$ meet in point $Z$. Find angle $XZY$.

2007 Moldova Team Selection Test, 1

Tags: algebra , geometry , 3d geometry , factorization , sum of cubes , 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

1980 IMO, 3

Tags: geometry , 3d geometry , tetrahedron , sphere , triangle inequality , angle

Prove that the sum of the six angles subtended at an interior point of a tetrahedron by its six edges is greater than 540°.

2023 Portugal MO, 3

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

A crate with a base of $4 \times 2$ and a height of $2$ is open at the top. Tomas wants to completely fill the crate with some of his cubes. It has $16$ equal cubes of volume $1$ and two equal cubes of volume $8$. A cube of volume $1$ can only be placed on the top layer if the cube on the bottom layer has already been placed. In how many ways can Tom'as fill the box with cubes, placing them one by one?

2001 German National Olympiad, 6 (11)

Tags: geometry , pyramid , volume , 3d geometry , ratio

In a pyramid $SABCD$ with the base $ABCD$ the triangles $ABD$ and $BCD$ have equal areas. Points $M,N,P,Q$ are the midpoints of the edges $AB,AD,SC,SD$ respectively. Find the ratio between the volumes of the pyramids $SABCD$ and $MNPQ$.

2019 PUMaC Geometry A, 1

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 $a$?

1987 Traian Lălescu, 1.4

Tags: geometry , perimeter , 3d geometry , tetrahedron

Let $ ABCD $ be a regular tetahedron and $ M,N $ be middlepoints for $ AD, $ respectively, $ BC. $ Through a point $ P $ that is on segment $ MN, $ passes a plane perpendicular on $ MN, $ and meets the sides $ AB,AC,CD,BD $ of the tetahedron at $ E,F,G, $ respectively, $ H. $ [b]a)[/b] Prove that the perimeter of the quadrilateral $ EFGH $ doesn't depend on $ P. $ [b]b)[/b] Determine the maximum area of $ EFGH $ (depending on a side of the tetahedron).

2013 Stanford Mathematics Tournament, 11

Tags: geometry , 3d geometry

Sara has an ice cream cone with every meal. The cone has a height of $2\sqrt2$ inches and the base of the cone has a diameter of $2$ inches. Ice cream protrudes from the top of the cone in a perfect hempisphere. Find the surface area of the ice cream cone, ice cream included, in square inches.

II Soros Olympiad 1995 - 96 (Russia), 11.2

Tags: 3d geometry , tetrahedron , geometry

Is it possible that the heights of a tetrahedron (that is, a triangular pyramid) would be equal to the numbers $1$, $2$, $3$ and $6$?

2006 IMO Shortlist, 7

Tags: 3d geometry , euler , polyhedron , geometry

Consider a convex polyhedron without parallel edges and without an edge parallel to any face other than the two faces adjacent to it. Call a pair of points of the polyhedron [i]antipodal[/i] if there exist two parallel planes passing through these points and such that the polyhedron is contained between these planes. Let $A$ be the number of antipodal pairs of vertices, and let $B$ be the number of antipodal pairs of midpoint edges. Determine the difference $A-B$ in terms of the numbers of vertices, edges, and faces. [i]Proposed by Kei Irei, Japan[/i]

1988 AIME Problems, 9

Tags: modular arithmetic , 3d geometry , geometry

Find the smallest positive integer whose cube ends in 888.

1987 Yugoslav Team Selection Test, Problem 3

Tags: 3d geometry , construction , geometry

Let there be given lines $a,b,c$ in the space, no two of which are parallel. Suppose that there exist planes $\alpha,\beta,\gamma$ which contain $a,b,c$ respectively, which are perpendicular to each other. Construct the intersection point of these three planes. (A space construction permits drawing lines, planes and spheres and translating objects for any vector.)

1987 Traian Lălescu, 1.3

Tags: geometry , 3d geometry , tetrahedron

Let $ ABCD $ be a tetahedron and $ M,N $ the middlepoints of $ AB, $ respectively, $ CD. $ Show that any plane that contains $ M $ and $ N $ cuts the tetrahedron in two polihedra that have same volume.

1995 Poland - Second Round, 5

Tags: 3d geometry , tetrahedron , concyclic , incircle , geometry

The incircles of the faces $ABC$ and $ABD$ of a tetrahedron $ABCD$ are tangent to the edge $AB$ in the same point. Prove that the points of tangency of these incircles to the edges $AC,BC,AD,BD$ are concyclic.

1997 Romania National Olympiad, 4

Tags: geometry , distance , 3d geometry

Let $S$ be a point outside of the plane of the parallelogram $ABCD$, such that the triangles $SAB$, $SBC$, $SCD$ and $SAD$ are equivalent. a) Prove that $ABCD$ is a rhombus. b) If the distance from $S$ to the plane $(A, B, C, D)$ is $12$, $BD = 30$ and $AC = 40$, compute the distance from the projection of the point $S$ on the plane $(A, B, C, D)$ to the plane $(S,B,C)$ .

1977 Bulgaria National Olympiad, Problem 2

Tags: 3d geometry , tetrahedron , geometry

In the space are given $n$ points and no four of them belongs to a common plane. Some of the points are connected with segments. It is known that four of the given points are vertices of tetrahedron which edges belong to the segments given. It is also known that common number of the segments, passing through vertices of tetrahedron is $2n$. Prove that there exists at least two tetrahedrons every one of which have a common face with the first (initial) tetrahedron. [i]N. Nenov, N. Hadzhiivanov[/i]

1991 Arnold's Trivium, 100

Tags: geometry , symmetry , 3d geometry , sphere

Find the mathematical expectation of the area of the projection of a cube with edge of length $1$ onto a plane with an isotropically distributed random direction of projection.

1986 China Team Selection Test, 2

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

Given a tetrahedron $ABCD$, $E$, $F$, $G$, are on the respectively on the segments $AB$, $AC$ and $AD$. Prove that: i) area $EFG \leq$ max{area $ABC$,area $ABD$,area $ACD$,area $BCD$}. ii) The same as above replacing "area" for "perimeter".