Olympiad problems

2007 F = Ma, 21

If the rotational inertia of a sphere about an axis through the center of the sphere is $I$, what is the rotational inertia of another sphere that has the same density, but has twice the radius? $ \textbf{(A)}\ 2I \qquad\textbf{(B)}\ 4I \qquad\textbf{(C)}\ 8I\qquad\textbf{(D)}\ 16I\qquad\textbf{(E)}\ 32I $

2015 Israel National Olympiad, 5

Tags: geometry , 3d geometry , tetrahedron , sphere

Let $ABCD$ be a tetrahedron. Denote by $S_1$ the inscribed sphere inside it, which is tangent to all four faces. Denote by $S_2$ the outer escribed sphere outside $ABC$, tangent to face $ABC$ and to the planes containing faces $ABD,ACD,BCD$. Let $K$ be the tangency point of $S_1$ to the face $ABC$, and let $L$ be the tangency point of $S_2$ to the face $ABC$. Let $T$ be the foot of the perpendicular from $D$ to the face $ABC$. Prove that $L,T,K$ lie on one line.

1988 Nordic, 3

Tags: geometry , 3d geometry , sphere

Two concentric spheres have radii $r$ and $R,r < R$. We try to select points $A, B$ and $C$ on the surface of the larger sphere such that all sides of the triangle $ABC$ would be tangent to the surface of the smaller sphere. Show that the points can be selected if and only if $R \le 2r$.

1991 Arnold's Trivium, 100

Tags: geometry , 3d geometry , sphere , symmetry

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.

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 $A_2B_2C_2D_2$.

1967 IMO Longlists, 34

Tags: geometry , 3d geometry , sphere , polyhedron

Faces of a convex polyhedron are six squares and 8 equilateral triangles and each edge is a common side for one triangle and one square. All dihedral angles obtained from the triangle and square with a common edge, are equal. Prove that it is possible to circumscribe a sphere around the polyhedron, and compute the ratio of the squares of volumes of that polyhedron and of the ball whose boundary is the circumscribed sphere.

1980 IMO Longlists, 15

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°.

1978 IMO, 2

Tags: geometry , 3d geometry , sphere , locus problems , locus

We consider a fixed point $P$ in the interior of a fixed sphere$.$ We construct three segments $PA, PB,PC$, perpendicular two by two$,$ with the vertexes $A, B, C$ on the sphere$.$ We consider the vertex $Q$ which is opposite to $P$ in the parallelepiped (with right angles) with $PA, PB, PC$ as edges$.$ Find the locus of the point $Q$ when $A, B, C$ take all the positions compatible with our problem.

1985 Polish MO Finals, 6

Tags: 3d geometry , geometry , polyhedron , sphere , inscribed

There is a convex polyhedron with $k$ faces. Show that if more than $k/2$ of the faces are such that no two have a common edge, then the polyhedron cannot have an inscribed sphere.

2015 Oral Moscow Geometry Olympiad, 5

Tags: geometry , 3d geometry , sphere , fixed point , moscow , plane

A triangle $ABC$ and spheres are given in space $S_1$ and $S_2$, each of which passes through points $A, B$ and $C$. For points $M$ spheres $S_1$ not lying in the plane of triangle $ABC$ are drawn lines $MA, MB$ and $MC$, intersecting the sphere $S_2$ for the second time at points $A_1,B_1$ and $C_1$, respectively. Prove that the planes passing through points $A_1, B_1$ and $C_1$, touch a fixed sphere or pass through a fixed point.

2003 Iran MO (3rd Round), 18

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

In tetrahedron $ ABCD$, radius four circumcircles of four faces are equal. Prove that $ AB\equal{}CD$, $ AC\equal{}BD$ and $ AD\equal{}BC$.

1996 National High School Mathematics League, 6

Tags: geometry , 3d geometry , sphere

Height of a circular truncated cone is $8$. Center of sphere $O_1$ with a radius of $2$ is on the axis of the circular truncated cone. Sphere $O_1$ is tangent to the top surface and the flank. We can put another sphere $O_2$, satisfying that sphere $O_2$ with a radius of $3$ have only one common point with sphere $O_1$, bottom surface and the flank. Besides $O_2$, how many spheres can we put inside the circular truncated cone? $\text{(A)}1\qquad\text{(B)}2\qquad\text{(C)}3\qquad\text{(D)}4$

2009 Today's Calculation Of Integral, 501

Tags: calculus , integration , 3d geometry , sphere , dodecahedron , calculus computations , geometry

Find the volume of the uion $ A\cup B\cup C$ of the three subsets $ A,\ B,\ C$ in $ xyz$ space such that: \[ A\equal{}\{(x,\ y,\ z)\ |\ |x|\leq 1,\ y^2\plus{}z^2\leq 1\}\] \[ B\equal{}\{(x,\ y,\ z)\ |\ |y|\leq 1,\ z^2\plus{}x^2\leq 1\}\] \[ C\equal{}\{(x,\ y,\ z)\ |\ |z|\leq 1,\ x^2\plus{}y^2\leq 1\}\]

KoMaL A Problems 2024/2025, A. 894

Tags: solid geometry , insphere , 3d geometry , sphere , geometry

In convex polyhedron $ABCDE$ line segment $DE$ intersects the plane of triangle $ABC$ inside the triangle. Rotate the point $D$ outward into the plane of triangle $ABC$ around the lines $AB$, $BC$, $CA$; let the resulting points be $D_1$, $D_2$, and $D_3$. Similarly, rotate the point $E$ outward into the plane of triangle $ABC$ around the lines $AB$, $BC$, $CA$; let the resulting points be $E_1$, $E_2$, and $E_3$. Show that if the polyhedron has an inscribed sphere, then the circumcircles of $D_1D_2D_3$ and $E_1E_2E_3$ are concentric. [i]Proposed by: Géza Kós, Budapest[/i]

1991 Arnold's Trivium, 16

Tags: geometry , 3d geometry , sphere , dodecahedron , icosahedron

What fraction of a $5$-dimensional cube is the volume of the inscribed sphere? What fraction is it of a $10$-dimensional cube?

2016 Junior Regional Olympiad - FBH, 5

Tags: geometry , 3d geometry , sphere , pyramid

$605$ spheres of same radius are divided in two parts. From one part, upright "pyramid" is made with square base. From the other part, upright "pyramid" is made with equilateral triangle base. Both "pyramids" are put together from equal numbers of sphere rows. Find number of spheres in every "pyramid"

Denmark (Mohr) - geometry, 2000.2

Tags: geometry , 3d geometry , sphere , prism , hexagon

Three identical spheres fit into a glass with rectangular sides and bottom and top in the form of regular hexagons such that every sphere touches every side of the glass. The glass has volume $108$ cm$^3$. What is the sidelength of the bottom? [img]https://1.bp.blogspot.com/-hBkYrORoBHk/XzcDt7B83AI/AAAAAAAAMXs/P5PGKTlNA7AvxkxMqG-qxqDVc9v9cU0VACLcBGAsYHQ/s0/2000%2BMohr%2Bp2.png[/img]

1964 German National Olympiad, 3

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

Given a (not necessarily regular) tetrahedron, all of its sides are equal in area. Prove that the following points then coincide: a) the center of the inscribed sphere, i.e. all four side surfaces internally touching sphere, b) the center of the surrounding sphere, i.e. the sphere passing through the four vertixes.

2000 Romania Team Selection Test, 3

Tags: geometry , 3d geometry , sphere , function , calculus , topology , analytic geometry

Let $S$ be the set of interior points of a sphere and $C$ be the set of interior points of a circle. Find, with proof, whether there exists a function $f:S\rightarrow C$ such that $d(A,B)\le d(f(A),f(B))$ for any two points $A,B\in S$ where $d(X,Y)$ denotes the distance between the points $X$ and $Y$. [i]Marius Cavachi[/i]

1980 IMO Shortlist, 15

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°.

Kyiv City MO 1984-93 - geometry, 1993.11.3

Tags: 3d geometry , sphere , cube , geometry

Two cubes are inscribed in a sphere of radius $R$. Calculate the sum of squares of all segments connecting the vertices of one cube with the vertices of the other cube

1995 Iran MO (2nd round), 1

Tags: geometry , 3d geometry , sphere , vector , number theory

Prove that for every positive integer $n \geq 3$ there exist two sets $A =\{ x_1, x_2,\ldots, x_n\}$ and $B =\{ y_1, y_2,\ldots, y_n\}$ for which [b]i)[/b] $A \cap B = \varnothing.$ [b]ii)[/b] $x_1+ x_2+\cdots+ x_n= y_1+ y_2+\cdots+ y_n.$ [b]ii)[/b] $x_1^2+ x_2^2+\cdots+ x_n^2= y_1^2+ y_2^2+\cdots+ y_n^2.$

1994 All-Russian Olympiad Regional Round, 11.7

Tags: geometry , 3d geometry , pyramid , sphere

Points $A_1$, $B_1$ and $C_1$ are taken on the respective edges $SA$, $SB$, $SC$ of a regular triangular pyramid $SABC$ so that the planes $A_1B_1C_1$ and $ABC$ are parallel. Let $O$ be the center of the sphere passing through $A$, $B$, $C_1$ and $S$. Prove that the line $SO$ is perpendicular to the plane $A_1B_1C$.