Olympiad problems

1958 February Putnam, B4

Tags: geometry , probability , 3d geometry , sphere , calculus

Title is self explanatory. Pick two points on the unit sphere. What is the expected distance between them?

2023 Olympic Revenge, 4

Tags: geometry , 3d geometry , sphere , combinatorial geometry , combinatorics , lattice points

Let $S=\{(x,y,z)\in \mathbb{Z}^3\}$ the set of points with integer coordinates in the space. Gugu has infinitely many solid spheres. All with radii $\ge (\frac{\pi}2)^3$. Is it possible for Gugu to cover all points of $S$ with his spheres?

1972 Polish MO Finals, 4

Tags: geometry , perimeter , geometric inequalities , 3d geometry , sphere

Points $A$ and $B$ are given on a line having no common points with a sphere $K$. The feet $P$ of the perpendicular from the center of $K$ to the line $AB$ is positioned between $A$ and $B$, and the lengths of segments $AP$ and $BP$ both exceed the radius of $K$. Consider the set $Z$ of all triangles $ABC$ whose sides $AC$ and $BC$ are tangent to $K$. Prove that among all triangles in $Z$, a triangle $T$ with a maximum perimeter also has a maximum area.

2011 National Olympiad First Round, 33

Tags: geometry , 3d geometry , sphere , tetrahedron

What is the largest volume of a sphere which touches to a unit sphere internally and touches externally to a regular tetrahedron whose corners are over the unit sphere? $\textbf{(A)}\ \frac13 \qquad\textbf{(B)}\ \frac14 \qquad\textbf{(C)}\ \frac12\left ( 1 - \frac1{\sqrt3} \right ) \qquad\textbf{(D)}\ \frac12\left ( \frac{2\sqrt2}{\sqrt3} - 1 \right ) \qquad\textbf{(E)}\ \text{None}$

1997 Estonia National Olympiad, 3

Tags: 3d geometry , tetrahedron , volume , geometry , sphere , inscribed , circumscribed

A sphere is inscribed in a regular tetrahedron. Another regular tetrahedron is inscribed in the sphere. Find the ratio of the volumes of these two tetrahedra.

2014 Harvard-MIT Mathematics Tournament, 9

Tags: geometry , 3d geometry , sphere

Compute the side length of the largest cube contained in the region \[ \{(x, y, z) : x^2+y^2+z^2 \le 25 \text{ and } x, y \ge 0 \} \] of three-dimensional space.

Denmark (Mohr) - geometry, 1994.1

Tags: geometry , cylinder , 3d geometry , sphere

A wine glass with a cross section as shown has the property of an orange in shape as a sphere with a radius of $3$ cm just can be placed in the glass without protruding above glass. Determine the height $h$ of the glass. [img]https://1.bp.blogspot.com/-IuLm_IPTvTs/XzcH4FAjq5I/AAAAAAAAMYY/qMi4ng91us8XsFUtnwS-hb6PqLwAON_jwCLcBGAsYHQ/s0/1994%2BMohr%2Bp1.png[/img]

1990 National High School Mathematics League, 15

Tags: geometry , 3d geometry , pyramid , sphere

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.

1986 Flanders Math Olympiad, 4

Tags: geometry , 3d geometry , sphere , algebra

Given a cube in which you can put two massive spheres of radius 1. What's the smallest possible value of the side - length of the cube? Prove that your answer is the best possible.

1976 Bundeswettbewerb Mathematik, 4

Tags: geometry , 3d geometry , sphere , geometric transformation , vector , combinatorics

Each vertex of the 3-dimensional Euclidean space either is coloured red or blue. Prove that within those squares being possible in this space with edge length 1 there is at least one square either with three red vertices or four blue vertices !

1996 Austrian-Polish Competition, 5

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

A sphere $S$ divides every edge of a convex polyhedron $P$ into three equal parts. Show that there exists a sphere tangent to all the edges of $P$.

Champions Tournament Seniors - geometry, 2008.4

Tags: geometry , 3d geometry , sphere , pyramid

Given a quadrangular pyramid $SABCD$, the basis of which is a convex quadrilateral $ABCD$. It is known that the pyramid can be tangent to a sphere. Let $P$ be the point of contact of this sphere with the base $ABCD$. Prove that $\angle APB + \angle CPD = 180^o$.

1997 National High School Mathematics League, 10

Tags: geometry , 3d geometry , pyramid , sphere

Bottom surface of triangular pyramid $S-ABC$ is an isosceles right triangle (hypotenuse is $AB$). $SA=SB=SC=AB=2$, and $S,A,B,C$ are on a sphere with center of $O$. The distance of $O$ to plane $ABC$ is________.

2006 Harvard-MIT Mathematics Tournament, 9

Tags: geometry , 3d geometry , sphere , tetrahedron

Four spheres, each of radius $r$, lie inside a regular tetrahedron with side length $1$ such that each sphere is tangent to three faces of the tetrahedron and to the other three spheres. Find $r$.

2011 USAMTS Problems, 4

Tags: geometry , 3d geometry , sphere

Let $ABCDEF$ and $ABC'D'E'F'$ be regular planar hexagons in three-dimensional space with side length $1$, such that $\angle EAE'=60^{\circ}$. Let $P$ be the convex polyhedron whose vertices are $A$, $B$, $C$, $C'$, $D$, $D'$, $E$, $E'$, $F$, and $F'$. (a) Find the radius $r$ of the largest sphere that can be enclosed in polyhedron $P$. (b) Let $S$ be a sphere enclosed in polyhedron $P$ with radius $r$ (as derived in part (a)). The set of possible centers of $S$ is a line segment $\overline{XY}$. Find the length $XY$.

MIPT Undergraduate Contest 2019, 1.4

Tags: geometry , 3d geometry , sphere , topology

Suppose that in a unit sphere in Euclidean space, there are $2m$ points $x_1, x_2, ..., x_{2m}.$ Prove that it's possible to partition them into two sets of $m$ points in such a way that the centers of mass of these sets are at a distance of at most $\frac{2}{\sqrt{m}}$ from one another.

1983 AMC 12/AHSME, 27

Tags: trigonometry , geometry , 3d geometry , sphere , trig identities , power of a point

A large sphere is on a horizontal field on a sunny day. At a certain time the shadow of the sphere reaches out a distance of $10$ m from the point where the sphere touches the ground. At the same instant a meter stick (held vertically with one end on the ground) casts a shadow of length $2$ m. What is the radius of the sphere in meters? (Assume the sun's rays are parallel and the meter stick is a line segment.) $ \textbf{(A)}\ \frac{5}{2}\qquad\textbf{(B)}\ 9 - 4\sqrt{5}\qquad\textbf{(C)}\ 8\sqrt{10} - 23\qquad\textbf{(D)}\ 6 - \sqrt{15}\qquad\textbf{(E)}\ 10\sqrt{5} - 20 $

1960 IMO, 6

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

Consider a cone of revolution with an inscribed sphere tangent to the base of the cone. A cylinder is circumscribed about this sphere so that one of its bases lies in the base of the cone. let $V_1$ be the volume of the cone and $V_2$ be the volume of the cylinder. a) Prove that $V_1 \neq V_2$; b) Find the smallest number $k$ for which $V_1=kV_2$; for this case, construct the angle subtended by a diamter of the base of the cone at the vertex of the cone.

VII Soros Olympiad 2000 - 01, 11.8

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

Three spheres are tangent to one plane, to a straight line perpendicular to this plane, and in pairs to each other. The radius of the largest sphere is $1$. Within what limits can the radius of the smallest sphere vary?

1985 Bundeswettbewerb Mathematik, 4

Tags: 3d geometry , sphere , geometry

Each point of the 3-dimensional space is coloured with exactly one of the colours red, green and blue. Let $R$, $G$ and $B$, respectively, be the sets of the lengths of those segments in space whose both endpoints have the same colour (which means that both are red, both are green and both are blue, respectively). Prove that at least one of these three sets includes all non-negative reals.

1973 IMO Longlists, 1

Tags: geometry , 3d geometry , sphere , tetrahedron

Find the maximal positive number $r$ with the following property: If all altitudes of a tetrahedron are $\geq 1$, then a sphere of radius $r$ fits into the tetrahedron.

1991 Arnold's Trivium, 17

Tags: geometry , 3d geometry , sphere , integration , calculus , trigonometry , function

Find the distance of the centre of gravity of a uniform $100$-dimensional solid hemisphere of radius $1$ from the centre of the sphere with $10\%$ relative error.

2008 Bundeswettbewerb Mathematik, 3

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

Through a point in the interior of a sphere we put three pairwise perpendicular planes. Those planes dissect the surface of the sphere in eight curvilinear triangles. Alternately the triangles are coloured black and wide to make the sphere surface look like a checkerboard. Prove that exactly half of the sphere's surface is coloured black.

2004 Moldova Team Selection Test, 2

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

In the tetrahedron $ABCD$ the radius of its inscribed sphere is $r$ and the radiuses of the exinscribed spheres (each tangent with a face of the tetrahedron and with the planes of the other faces) are $r_A, r_B, r_C, r_D.$ Prove the inequality $$\frac{1}{\sqrt{r_A^2-r_Ar_B+r_B^2}}+\frac{1}{\sqrt{r_B^2-r_Br_C+r_C^2}}+\frac{1}{\sqrt{r_C^2-r_Cr_D+r_D^2}}+\frac{1}{\sqrt{r_D^2-r_Dr_A+r_A^2}}\leq\frac{2}{r}.$$

2012 Putnam, 2

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

Let $P$ be a given (non-degenerate) polyhedron. Prove that there is a constant $c(P)>0$ with the following property: If a collection of $n$ balls whose volumes sum to $V$ contains the entire surface of $P,$ then $n>c(P)/V^2.$