Olympiad problems

VI Soros Olympiad 1999 - 2000 (Russia), 11.8

Prove that the plane dividing in equal proportions the surface area and volume of the circumscribed polyhedron passes through the center of the sphere inscribed in this polyhedron.

1954 AMC 12/AHSME, 27

Tags: geometry , 3D geometry , ratio

A right circular cone has for its base a circle having the same radius as a given sphere. The volume of the cone is one-half that of the sphere. The ratio of the altitude of the cone to the radius of its base is: $ \textbf{(A)}\ \frac{1}{1} \qquad \textbf{(B)}\ \frac{1}{2} \qquad \textbf{(C)}\ \frac{2}{3} \qquad \textbf{(D)}\ \frac{2}{1} \qquad \textbf{(E)}\ \sqrt{\frac{5}{4}}$

2011 Pre-Preparation Course Examination, 3

Tags: geometry , 3D geometry , tetrahedron , combinatorics proposed , combinatorics

Calculate number of the hamiltonian cycles of the graph below: (15 points)

2023 Junior Balkan Team Selection Tests - Romania, P4

Tags: geometry , 3D geometry , combinatorics , pigeonhole principle

Given is a cube $3 \times 3 \times 3$ with $27$ unit cubes. In each such cube a positive integer is written. Call a $\textit {strip}$ a block $1 \times 1 \times 3$ of $3$ cubes. The numbers are written so that for each cube, its number is the sum of three other numbers, one from each of the three strips it is in. Prove that there are at least $16$ numbers that are at most $60$.

1979 Poland - Second Round, 3

Tags: geometry , 3D geometry , projection , areas

In space there is a line $ k $ and a cube with a vertex $ M $ and edges $ \overline{MA} $, $ \overline{MB} $, $ \overline{MC} $, of length$ 1$. Prove that the length of the orthogonal projection of edge $ MA $ on the line $ k $ is equal to the area of the orthogonal projection of a square with sides $ MB $ and $ MC $ onto a plane perpendicular to the line $ k $. [hide=original wording]W przestrzeni dana jest prosta $ k $ oraz sześcian o wierzchołku $ M $ i krawędziach $ \overline{MA} $, $ \overline{MB} $, $ \overline{MC} $, długości 1. Udowodnić, że długość rzutu prostokątnego krawędzi $ MA $ na prostą $ k $ jest równa polu rzutu prostokątnego kwadratu o bokach $ MB $ i $ MC $ na płaszczyznę prostopadłą do prostej $ k $.[/hide]

1975 Bundeswettbewerb Mathematik, 2

Tags: combinatorial geometry , polyhedron , 3D geometry , edge , geometry

Prove that in each polyhedron there exist two faces with the same number of edges.

1992 Poland - First Round, 10

Tags: geometry , 3D geometry

Let $C$ be a cube and let $f: C \longrightarrow C$ be a surjection with $|PQ| \geq |f(P)f(Q)|$ for all $P,Q \in C$. Prove that $f$ is an isometry.

2010 Moldova Team Selection Test, 4

Tags: geometry , 3D geometry , combinatorics proposed , combinatorics

Let $ n\geq6$ be a even natural number. Prove that any cube can be divided in $ \dfrac{3n(n\minus{}2)}4\plus{}2$ cubes.

2021 Macedonian Balkan MO TST, Problem 4

Tags: combinatorics , graph theory , geometry , 3D geometry

Viktor and Natalia play a colouring game with a 3-dimensional cube taking turns alternatingly. Viktor goes first, and on each of his turns, he selects an unpainted edge, and paints it violet. On each of Natalia's turns, she selects an unpainted edge, or at most once during the game a face diagonal, and paints it neon green. If the player on turn cannot make a legal move, then the turn switches to the other player. The game ends when nobody can make any more legal moves. Natalia wins if at the end of the game every vertex of the cube can be reached from every other vertex by traveling only along neon green segments (edges or diagonal), otherwise Viktor wins. Who has a winning strategy? (Prove your answer.) [i]Authored by Viktor Simjanoski[/i]

2020 BMT Fall, 3

Tags: geometry , 3D geometry , cube

An ant is at one corner of a unit cube. If the ant must travel on the box’s surface, the shortest distance the ant must crawl to reach the opposite corner of the cube can be written in the form $\sqrt{a}$, where $a$ is a positive integer. Compute $a$.

1999 Israel Grosman Mathematical Olympiad, 6

Tags: geometry , parallelogram , 3D geometry , coplanar , midpoints

Let $A,B,C,D,E,F$ be points in space such that the quadrilaterals $ABDE,BCEF, CDFA$ are parallelograms. Prove that the six midpoints of the sides $AB,BC,CD,DE,EF,FA$ are coplanar

1977 Poland - Second Round, 6

Tags: combinatorics , geometry , combinatorial geometry , 3D geometry

What is the greatest number of parts into which the plane can be cut by the edges of $ n $ squares?

1972 AMC 12/AHSME, 12

Tags: geometry , 3D geometry , AMC

The number of cubic feet in the volume of a cube is the same as the number of square inches in its surface area. The length of the edge expressed as a number of feet is $\textbf{(A) }6\qquad\textbf{(B) }864\qquad\textbf{(C) }1728\qquad\textbf{(D) }6\times 1728\qquad \textbf{(E) }2304$

2015 Sharygin Geometry Olympiad, P24

Tags: sphere , tetrahedron , 3-Dimensional Geometry , geometry , 3D geometry

The insphere of a tetrahedron ABCD with center $O$ touches its faces at points $A_1,B_1,C_1$ and $D_1$. a) Let $P_a$ be a point such that its reflections in lines $OB,OC$ and $OD$ lie on plane $BCD$. Points $P_b, P_c$ and $P_d$ are defined similarly. Prove that lines $A_1P_a,B_1P_b,C_1P_c$ and $D_1P_d$ concur at some point $ P$. b) Let $I$ be the incenter of $A_1B_1C_1D_1$ and $A_2$ the common point of line $A_1I $ with plane $B_1C_1D_1$. Points $B_2, C_2, D_2$ are defined similarly. Prove that $P$ lies inside $A_2B_2C_2D_2$.

1981 Czech and Slovak Olympiad III A, 6

Tags: geometry , combinatorial geometry , 3D geometry

There are given 11 distinct points inside a ball with volume $V.$ Show that there are two planes $\varrho,\sigma,$ both containing the center of the ball, such that the resulting spherical wedge has volume $V/8$ and its interior contains none of the given points.

II Soros Olympiad 1995 - 96 (Russia), 11.7

Tags: geometry , 3D geometry , prism , Volume

Three edges of a parallelepiped lie on three intersecting diagonals of the lateral faces of a triangular prism. Find the ratio of the volumes of the parallelepiped and the prism.

1935 Moscow Mathematical Olympiad, 003

Tags: pyramid , geometry , 3D geometry , angles

The base of a pyramid is an isosceles triangle with the vertex angle $\alpha$. The pyramid’s lateral edges are at angle $\phi$ to the base. Find the dihedral angle $\theta$ at the edge connecting the pyramid’s vertex to that of angle $\alpha$.

2014 Polish MO Finals, 3

Tags: contests , geometry , 3D geometry , tetrahedron , sphere

A tetrahedron $ABCD$ with acute-angled faces is inscribed in a sphere with center $O$. A line passing through $O$ perpendicular to plane $ABC$ crosses the sphere at point $D'$ that lies on the opposide side of plane $ABC$ than point $D$. Line $DD'$ crosses plane $ABC$ in point $P$ that lies inside the triangle $ABC$. Prove, that if $\angle APB=2\angle ACB$, then $\angle ADD'=\angle BDD'$.

1996 IberoAmerican, 1

Tags: geometry , 3D geometry , number theory unsolved , number theory

Let $ n$ be a natural number. A cube of edge $ n$ may be divided in 1996 cubes whose edges length are also natural numbers. Find the minimum possible value for $ n$.

1951 AMC 12/AHSME, 6

Tags: geometry , 3D geometry

The bottom, side, and front areas of a rectangular box are known. The product of these areas is equal to: $ \textbf{(A)}\ \text{the volume of the box} \qquad\textbf{(B)}\ \text{the square root of the volume} \qquad\textbf{(C)}\ \text{twice the volume}$ $ \textbf{(D)}\ \text{the square of the volume} \qquad\textbf{(E)}\ \text{the cube of the volume}$

2000 Tournament Of Towns, 3

Tags: 3D geometry , prism , geometry , angles

In each lateral face of a pentagonal prism at least one of the four angles is equal to $f$. Find all possible values of $f$. (A Shapovalov)

Today's calculation of integrals, 891

Tags: calculus , integration , geometry , 3D geometry , conics , hyperbola , rotation

Given a triangle $OAB$ with the vetices $O(0,\ 0,\ 0),\ A(1,\ 0,\ 0),\ B(1,\ 1,\ 0)$ in the $xyz$ space. Let $V$ be the cone obtained by rotating the triangle around the $x$-axis. Find the volume of the solid obtained by rotating the cone $V$ around the $y$-axis.

1984 Czech And Slovak Olympiad IIIA, 1

Tags: geometry , 3D geometry , cube

A cube $A_1A_2A_3A_4A_5A_6A_7A_8$ is given in space. We will mark its center with the letter $S$ (intersection of solid diagonals). Find all natural numbers $k$ for which there exists a plane not containing the point $S$ and intersecting just $k$ of the rays $SA_1, SA_2, .. SA_8$

1960 IMO, 5

Tags: geometry , 3D geometry , cube , Locus , Locus problems , IMO , IMO 1960

Consider the cube $ABCDA'B'C'D'$ (with face $ABCD$ directly above face $A'B'C'D'$). a) Find the locus of the midpoints of the segments $XY$, where $X$ is any point of $AC$ and $Y$ is any piont of $B'D'$; b) Find the locus of points $Z$ which lie on the segment $XY$ of part a) with $ZY=2XZ$.

ICMC 6, 2

Tags: geometry , 3D geometry , tetrahedron , ICMC , college contests

Show that if the distance between opposite edges of a tetrahedron is at least $1$, then its volume is at least $\frac{1}{3}$. [i]Proposed by Simeon Kiflie[/i]