Olympiad problems

1967 IMO Shortlist, 2

In the space $n \geq 3$ points are given. Every pair of points determines some distance. Suppose all distances are different. Connect every point with the nearest point. Prove that it is impossible to obtain (closed) polygonal line in such a way.

1977 Czech and Slovak Olympiad III A, 6

Tags: cube , 3d geometry , volume , geometry , tetrahedron

A cube $ABCDA'B'C'D',AA'\parallel BB'\parallel CC'\parallel DD'$ is given. Denote $S$ the center of square $ABCD.$ Determine all points $X$ lying on some edge such that the volumes of tetrahedrons $ABDX$ and $CB'SX$ are the same.

1969 IMO Longlists, 32

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

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

2012 Online Math Open Problems, 18

Tags: geometric transformation , 3d geometry , geometry , reflection

The sum of the squares of three positive numbers is $160$. One of the numbers is equal to the sum of the other two. The difference between the smaller two numbers is $4.$ What is the difference between the cubes of the smaller two numbers? [i]Author: Ray Li[/i] [hide="Clarification"]The problem should ask for the positive difference.[/hide]

2017 Sharygin Geometry Olympiad, P24

Tags: geometry , similarity , 3d geometry , tetrahedron

Two tetrahedrons are given. Each two faces of the same tetrahedron are not similar, but each face of the first tetrahedron is similar to some face of the second one. Does this yield that these tetrahedrons are similar?

LMT Speed Rounds, 22

Tags: geometry , 3d geometry , algebra

Consider all pairs of points $(a,b,c)$ and $(d,e, f )$ in the $3$-D coordinate system with $ad +be +c f = -2023$. What is the least positive integer that can be the distance between such a pair of points? [i]Proposed by William Hua[/i]

1991 IMTS, 5

Tags: 3d geometry , number theory , geometry

Prove that there are infinitely many positive integers $n$ such that $n \times n \times n$ can not be filled completely with 2 x 2 x 2 and 3 x 3 x 3 solid cubes.

2020 Adygea Teachers' Geometry Olympiad, 3

Tags: 3d geometry , tetrahedron , triangle , geometry

Is it true that of the four heights of an arbitrary tetrahedron, three can be selected from which a triangle can be made?

2004 Estonia Team Selection Test, 6

Tags: polyhedron , pentagon , 3d geometry , hexagon , geometry , combinatorial geometry , combinatorics

Call a convex polyhedron a [i]footballoid [/i] if it has the following properties. (1) Any face is either a regular pentagon or a regular hexagon. (2) All neighbours of a pentagonal face are hexagonal (a [i]neighbour [/i] of a face is a face that has a common edge with it). Find all possibilities for the number of pentagonal and hexagonal faces of a footballoid.

2013 F = Ma, 10

Tags: geometry , 3d geometry , sphere

Which of the following can be used to distinguish a solid ball from a hollow sphere of the same radius and mass? $\textbf{(A)}$ Measurements of the orbit of a test mass around the object. $\textbf{(B)}$ Measurements of the time it takes the object to roll down an inclined plane. $\textbf{(C)}$ Measurements of the tidal forces applied by the object to a liquid body. $\textbf{(D)}$ Measurements of the behavior of the object as it oats in water. $\textbf{(E)}$ Measurements of the force applied to the object by a uniform gravitational field.

1968 Spain Mathematical Olympiad, 6

Tags: geometry , 3d geometry , tetrahedron , concurrency

Check and justify , if in every tetrahedron are concurrent: a) The perpendiculars to the faces at their circumcenters. b) The perpendiculars to the faces at their orthocenters. c) The perpendiculars to the faces at their incenters. If so, characterize with some simple geometric property the point in that attend If not, show an example that clearly shows the not concurrency.

1983 Federal Competition For Advanced Students, P2, 6

Tags: 3d geometry , geometry , combinatorics

Planes $ \pi _1$ and $ \pi _2$ in Euclidean space $ \mathbb{R} ^3$ partition $ S\equal{}\mathbb{R} ^3 \setminus (\pi _1 \cup \pi _2)$ into several components. Show that for any cube in $ \mathbb{R} ^3$, at least one of the components of $ S$ meets at least three faces of the cube.

1993 Poland - First Round, 12

Tags: 3d geometry , tetrahedron , geometry

Prove that the sums of the opposite dihedral angles of a tetrahedron are equal if and only if the sums of the opposite edges of this tetrahedron are equal.

2023 All-Russian Olympiad, 6

Tags: 3d geometry , geometry , tetrahedron

The plane $\alpha$ intersects the edges $AB$, $BC$, $CD$ and $DA$ of the tetrahedron $ABCD$ at points $X, Y, Z$ and $T$, respectively. It turned out, that points $Y$ and $T$ lie on a circle $\omega$ constructed with segment $XZ$ as the diameter. Point $P$ is marked in the plane $\alpha$ so that the lines $P Y$ and $P T$ are tangent to the circle $\omega$.Prove that the midpoints of the edges are $AB$, $BC$, $CD,$ $DA$ and the point $P$ lie in the same plane.

1998 Croatia National Olympiad, Problem 2

Tags: cone , 3d geometry , geometry

A hemisphere is inscribed in a cone so that its base lies on the base of the cone. The ratio of the area of the entire surface of the cone to the area of the hemisphere (without the base) is $\frac{18}5$. Compute the angle at the vertex of the cone.

2017 Swedish Mathematical Competition, 3

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

Given the segments $AB$ and $CD$ not necessarily on the same plane. Point $X$ is the midpoint of the segment $AB$, and the point $Y$ is the midpoint of $CD$. Given that point $X$ is not on line $CD$, and that point $Y$ is not on line $AB$, prove that $2 | XY | \le | AD | + | BC |$. When is equality achieved?

1979 IMO Shortlist, 4

Tags: 3d geometry , prism , combinatorics , pentagon , geometry

We consider a prism which has the upper and inferior basis the pentagons: $A_{1}A_{2}A_{3}A_{4}A_{5}$ and $B_{1}B_{2}B_{3}B_{4}B_{5}$. Each of the sides of the two pentagons and the segments $A_{i}B_{j}$ with $i,j=1,\ldots,5$ is colored in red or blue. In every triangle which has all sides colored there exists one red side and one blue side. Prove that all the 10 sides of the two basis are colored in the same color.

1990 IMO Longlists, 44

Tags: pyramid , symmetry , 3d geometry , pascal triangle , binomial coefficient , geometry

Prove that for any positive integer $n$, the number of odd integers among the binomial coefficients $\binom nh \ ( 0 \leq h \leq n)$ is a power of 2.

MIPT student olimpiad spring 2023, 3

Tags: 3d geometry , sphere , geometry

Prove that if a set $X\subset S^n$ takes up more than half a Riemannian volume of a unit sphere $S^n$, then the set of all possible geodesic segments length less than $\pi$ with endpoints in the set $X$ covers the entire sphere. Geodetic on sphere $S^n$ is a curve lying on some circle of intersection of the sphere $S^n\subset R^{n+1}$ two-dimensional linear subspace $L \subset R^{n+1}$

2003 Tournament Of Towns, 1

Tags: 3d geometry , pyramid , geometry , sphere

A triangular pyramid $ABCD$ is given. Prove that $\frac Rr > \frac ah$, where $R$ is the radius of the circumscribed sphere, $r$ is the radius of the inscribed sphere, $a$ is the length of the longest edge, $h$ is the length of the shortest altitude (from a vertex to the opposite face).

1993 Poland - Second Round, 3

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

A tetrahedron $OA_1B_1C_1$ is given. Let $A_2,A_3 \in OA_1, A_2,A_3 \in OA_1, A_2,A_3 \in OA_1$ be points such that the planes $A_1B_1C_1,A_2B_2C_2$ and $A_3B_3C_3$ are parallel and $OA_1 > OA_2 > OA_3 > 0$. Let $V_i$ be the volume of the tetrahedron $OA_iB_iC_i$ ($i = 1,2,3$) and $V$ be the volume of $OA_1B_2C_3$. Prove that $V_1 +V_2 +V_3 \ge 3V$.

1999 National Olympiad First Round, 17

Tags: pyramid , geometry , 3d geometry , rectangle

In a regular pyramid with top point $ T$ and equilateral base $ ABC$, let $ P$, $ Q$, $ R$, $ S$ be the midpoints of $ \left[AB\right]$, $ \left[BC\right]$, $ \left[CT\right]$ and $ \left[TA\right]$, respectively. If $ \left|AB\right| \equal{} 6$ and the altitude of pyramid is equal to $ 2\sqrt {15}$, then area of $ PQRS$ will be $\textbf{(A)}\ 4\sqrt {15} \qquad\textbf{(B)}\ 8\sqrt {2} \qquad\textbf{(C)}\ 8\sqrt {3} \qquad\textbf{(D)}\ 6\sqrt {5} \qquad\textbf{(E)}\ 9\sqrt {2}$

1999 Gauss, 22

Tags: geometry , perimeter , gauss , 3d geometry

Forty-two cubes with 1 cm edges are glued together to form a solid rectangular block. If the perimeter of the base of the block is 18 cm, then the height, in cm, is $\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ \dfrac{7}{3} \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$

2011 Canadian Open Math Challenge, 3

Tags: 3d geometry , 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?

2005 District Olympiad, 3

Tags: circumcircle , 3d geometry , tetrahedron , geometry

Let $O$ be a point equally distanced from the vertices of the tetrahedron $ABCD$. If the distances from $O$ to the planes $(BCD)$, $(ACD)$, $(ABD)$ and $(ABC)$ are equal, prove that the sum of the distances from a point $M \in \textrm{int}[ABCD]$, to the four planes, is constant.