Found problems: 473
1983 Iran MO (2nd round), 4
The point $M$ moves such that the sum of squares of the lengths from $M$ to faces of a cube, is fixed. Find the locus of $M.$
2000 French Mathematical Olympiad, Exercise 2
Let $A,B,C$ be three distinct points in space, $(A)$ the sphere with center $A$ and radius $r$. Let $E$ be the set of numbers $R>0$ for which there is a sphere $(H)$ with center $H$ and radius $R$ such that $B$ and $C$ are outside the sphere, and the points of the sphere $(A)$ are strictly inside it.
(a) Suppose that $B$ and $C$ are on a line with $A$ and strictly outside $(A)$. Show that $E$ is nonempty and bounded, and determine its supremum in terms of the given data.
(b) Find a necessary and sufficient condition for $E$ to be nonempty and bounded
(c) Given $r$, compute the smallest possible supremum of $E$, if it exists.
1976 IMO Longlists, 22
A regular pentagon $A_1A_2A_3A_4A_5$ with side length $s$ is given. At each point $A_i$, a sphere $K_i$ of radius $\frac{s}{2}$ is constructed. There are two spheres $K_1$ and $K_2$ each of radius $\frac{s}{2}$ touching all the five spheres $K_i.$ Decide whether $K_1$ and $K_2$ intersect each other, touch each other, or have no common points.
2011 Purple Comet Problems, 30
Four congruent spheres are stacked so that each is tangent to the other three. A larger sphere, $R$, contains the four congruent spheres so that all four are internally tangent to $R$. A smaller sphere, $S$, sits in the space between the four congruent spheres so that all four are externally tangent to $S$. The ratio of the surface area of $R$ to the surface area of $S$ can be written $m+\sqrt{n}$ where $m$ and $n$ are positive integers. Find $m + n$.
2024 UMD Math Competition Part II, #3
A right triangle $A_1 A_2 A_3$ with side lengths $6,\,8,$ and $10$ on a plane $\mathcal P$ is given. Three spheres $S_1,S_2$ and $S_3$ with centers $O_1, O_2,$ and $O_3,$ respectively, are located on the same side of the plane $\mathcal P$ in such a way that $S_i$ is tangent to $\mathcal P$ at $A_i$ for $i = 1, 2, 3.$ Assume $S_1, S_2, S_3$ are pairwise externally tangent. Find the area of triangle $O_1O_2O_3.$
1990 Brazil National Olympiad, 3
Each face of a tetrahedron is a triangle with sides $a, b,$c and the tetrahedon has circumradius 1. Find $a^2 + b^2 + c^2$.
Denmark (Mohr) - geometry, 2000.2
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]
2001 Federal Math Competition of S&M, Problem 2
Vertices of a square $ABCD$ of side $\frac{25}4$ lie on a sphere. Parallel lines passing through points $A,B,C$ and $D$ intersect the sphere at points $A_1,B_1,C_1$ and $D_1$, respectively. Given that $AA_1=2$, $BB_1=10$, $CC_1=6$, determine the length of the segment $DD_1$.
1962 IMO, 7
The tetrahedron $SABC$ has the following property: there exist five spheres, each tangent to the edges $SA, SB, SC, BC, CA, AB,$ or to their extensions.
a) Prove that the tetrahedron $SABC$ is regular.
b) Prove conversely that for every regular tetrahedron five such spheres exist.
1963 German National Olympiad, 6
Consider a pyramid $ABCD$ whose base $ABC$ is a triangle. Through a point $M$ of the edge $DA$, the lines $MN$ and $MP$ on the plane of the surfaces $DAB$ and $DAC$ are drawn respectively, such that $N$ is on $DB$ and $P$ is on $DC$ and $ABNM$ , $ACPM$ are cyclic quadrilaterals.
a) Prove that $BCPN$ is also a cyclic quadrilateral.
b) Prove that the points $A,B,C,M,N, P$ lie on a sphere.
1979 Romania Team Selection Tests, 4.
Let $A_1A_2A_3A_4$ be a tetrahedron. Consider the sphere centered at $A_1$ which is tangent to the face $A_2A_3A_4$ of the tetrahedron. Show that the surface area of the part of the sphere which is inside the tetrahedron is less than the area of the triangle $A_2A_3A_4$.
[i]Sorin Rădulescu[/i]
1964 German National Olympiad, 3
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.
1965 Miklós Schweitzer, 7
Prove that any uncountable subset of the Euclidean $ n$-space contains an countable subset with the property that the distances between different pairs of points are different (that is, for any points $ P_1 \not\equal{} P_2$ and $ Q_1\not\equal{} Q_2$ of this subset, $ \overline{P_1P_2}\equal{}\overline{Q_1Q_2}$ implies either $ P_1\equal{}Q_1$ and $ P_2\equal{}Q_2$, or $ P_1\equal{}Q_2$ and $ P_2\equal{}Q_1$). Show that a similar statement is not valid if the Euclidean $ n$-space is replaced with a (separable) Hilbert space.
1984 Bundeswettbewerb Mathematik, 4
A sphere is touched by all the four sides of a (space) quadrilateral. Prove that all the four touching points are in the same plane.
1986 Balkan MO, 2
Let $ABCD$ be a tetrahedron and let $E,F,G,H,K,L$ be points lying on the edges $AB,BC,CD$ $,DA,DB,DC$ respectively, in such a way that
\[AE \cdot BE = BF \cdot CF = CG \cdot AG= DH \cdot AH=DK \cdot BK=DL \cdot CL.\]
Prove that the points $E,F,G,H,K,L$ all lie on a sphere.
2010 All-Russian Olympiad, 1
Let $a \neq b a,b \in \mathbb{R}$ such that $(x^2+20ax+10b)(x^2+20bx+10a)=0$ has no roots for $x$. Prove that $20(b-a)$ is not an integer.
2022 JHMT HS, 2
Four mutually externally tangent spherical apples of radius $4$ are placed on a horizontal flat table. Then, a spherical orange of radius $3$ is placed such that it rests on all the apples. Find the distance from the center of the orange to the table.
1987 IMO Shortlist, 10
Let $S_1$ and $S_2$ be two spheres with distinct radii that touch externally. The spheres lie inside a cone $C$, and each sphere touches the cone in a full circle. Inside the cone there are $n$ additional solid spheres arranged in a ring in such a way that each solid sphere touches the cone $C$, both of the spheres $S_1$ and $S_2$ externally, as well as the two neighboring solid spheres. What are the possible values of $n$?
[i]Proposed by Iceland.[/i]
1991 Arnold's Trivium, 70
Calculate the mean value of the solid angle by which the disc $x^2 + y^2 \le 1$ lying in the plane $z = 0$ is seen from points of the sphere $x^2 + y^2 + (z-2)^2 = 1$.
Kyiv City MO 1984-93 - geometry, 1987.10.3
In a right circular cone with the radius of the base $R$ and the height $h$ are $n$ spheres of the same radius $r$ ($n \ge 3$). Each ball touches the base of the cone, its side surface and other two balls. Determine $r$.
1989 Brazil National Olympiad, 5
A tetrahedron is such that the center of the its circumscribed sphere is inside the tetrahedron.
Show that at least one of its edges has a size larger than or equal to the size of the edge of a regular tetrahedron inscribed in this same sphere.
VI Soros Olympiad 1999 - 2000 (Russia), 11.4
Let the line $L$ be perpendicular to the plane $P$. Three spheres touch each other in pairs so that each sphere touches the plane $P$ and the line $L$. The radius of the larger sphere is $1$. Find the minimum radius of the smallest sphere.
1959 Poland - Second Round, 6
From a point $ M $ on the surface of a sphere, three mutually perpendicular chords $ MA $, $ MB $, $ MC $ are drawn. Prove that the segment joining the point $ M $ with the center of the sphere intersects the plane of the triangle $ ABC $ at the center of gravity of this triangle.
1978 IMO Longlists, 46
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.
1982 Brazil National Olympiad, 6
Five spheres of radius $r$ are inside a right circular cone. Four of the spheres lie on the base of the cone. Each touches two of the others and the sloping sides of the cone. The fifth sphere touches each of the other four and also the sloping sides of the cone. Find the volume of the cone.