Found problems: 473
1988 IMO Longlists, 37
[b]i.)[/b] Four balls of radius 1 are mutually tangent, three resting on the floor and the fourth resting on the others. A tedrahedron, each of whose edges has length $ s,$ is circumscribed around the balls. Find the value of $ s.$
[b]ii.)[/b] Suppose that $ ABCD$ and $ EFGH$ are opposite faces of a retangular solid, with $ \angle DHC \equal{} 45^{\circ}$ and $ \angle FHB \equal{} 60^{\circ}.$ Find the cosine of $ \angle BHD.$
1991 Vietnam National Olympiad, 3
Three mutually perpendicular rays $O_x,O_y,O_z$ and three points $A,B,C$ on $O_x,O_y,O_z$, respectively. A variable sphere є through $A, B,C$ meets $O_x,O_y,O_z$ again at $A', B',C'$, respectively. Let $M$ and $M'$ be the centroids of triangles $ABC$ and $A'B'C'$. Find the locus of the midpoint of $MM'$.
2016 Miklós Schweitzer, 10
Let $X$ and $Y$ be independent, identically distributed random points on the unit sphere in $\mathbb{R}^3$. For which distribution of $X$ will the expectation of the (Euclidean) distance of $X$ and $Y$ be maximal?
2004 Miklós Schweitzer, 7
Suppose that the closed subset $K$ of the sphere
$$S^2=\{ (x,y,z)\in \mathbb{R}^3\colon x^2+y^2+z^2=1 \}$$
is symmetric with respect to the origin and separates any two antipodal points in $S^2 \backslash K$. Prove that for any positive $\varepsilon$ there exists a homogeneous polynomial $P$ of odd degree such that the Hausdorff distance between
$$Z(P)=\{ (x,y,z)\in S^2 \colon P(x,y,z)=0\}$$
and $K$ is less than $\varepsilon$.
2005 Sharygin Geometry Olympiad, 11.6
The sphere inscribed in the tetrahedron $ABCD$ touches its faces at points $A',B',C',D'$. The segments $AA'$ and $BB'$ intersect, and the point of their intersection lies on the inscribed sphere. Prove that the segments $CC'$ and $DD'$ also intersect on the inscribed sphere.
2022 Sharygin Geometry Olympiad, 24
Let $OABCDEF$ be a hexagonal pyramid with base $ABCDEF$ circumscribed around a sphere $\omega$. The plane passing through the touching points of $\omega$ with faces $OFA$, $OAB$ and $ABCDEF$ meets $OA$ at point $A_1$, points $B_1$, $C_1$, $D_1$, $E_1$ and $F_1$ are defined similarly. Let $\ell$, $m$ and $n$ be the lines $A_1D_1$, $B_1E_1$ and $C_1F_1$ respectively. It is known that $\ell$ and $m$ are coplanar, also $m$ and $n$ are coplanar. Prove that $\ell$ and $n$ are coplanar.
1974 USAMO, 3
Two boundary points of a ball of radius 1 are joined by a curve contained in the ball and having length less than 2. Prove that the curve is contained entirely within some hemisphere of the given ball.
VI Soros Olympiad 1999 - 2000 (Russia), 11.3
Three spheres $s_1$, $s_2$, $s_3$ intersect along one circle $\omega$. Let $A $be an arbitrary point lying on the circle $\omega$. Ray $AB$ intersects spheres $s_1$, $s_2$, $s_3$ at points $B_1$, $B_2$, $B_3$, respectively, ray $AC$ intersects spheres $s_1$, $s_2$, $s_3$ at points $C_1$, $C_2$, $C_3$, respectively ($B_i \ne A_i$, $C_i \ne A_i$, $i=1,2,3$). It is known that $B_2$ is the midpoint of the segment $B_1B_3$. Prove that $C_2$ is the midpoint of the segment $C_1C_3$.
2013 IPhOO, 9
Bob, a spherical person, is floating around peacefully when Dave the giant orange fish launches him straight up 23 m/s with his tail. If Bob has density 100 $\text{kg/m}^3$, let $f(r)$ denote how far underwater his centre of mass plunges underwater once he lands, assuming his centre of mass was at water level when he's launched up. Find $\lim_{r\to0} \left(f(r)\right) $. Express your answer is meters and round to the nearest integer. Assume the density of water is 1000 $\text{kg/m}^3$.
[i](B. Dejean, 6 points)[/i]
1967 IMO Longlists, 32
Determine the volume of the body obtained by cutting the ball of radius $R$ by the trihedron with vertex in the center of that ball, it its dihedral angles are $\alpha, \beta, \gamma.$
1987 AIME Problems, 2
What is the largest possible distance between two points, one on the sphere of radius 19 with center $(-2, -10, 5)$ and the other on the sphere of radius 87 with center $(12, 8, -16)$?
2016 All-Russian Olympiad, 2
In the space given three segments $A_1A_2, B_1B_2$ and $C_1C_2$, do not lie in one plane and intersect at a point $P$. Let $O_{ijk}$ be center of sphere that passes through the points $A_i, B_j, C_k$ and $P$. Prove that $O_{111}O_{222}, O_{112}O_{221}, O_{121}O_{212}$ and$O_{211}O_{122}$ intersect at one point. (P.Kozhevnikov)
1964 Polish MO Finals, 3
Given a tetrahedron $ ABCD $ whose edges $ AB, BC, CD, DA $ are tangent to a certain sphere. Prove that the points of tangency lie in the same plane.
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.
2007 Pre-Preparation Course Examination, 2
a) Prove that center of smallest sphere containing a finite subset of $\mathbb R^{n}$ is inside convex hull of the point that lie on sphere.
b) $A$ is a finite subset of $\mathbb R^{n}$, and distance of every two points of $A$ is not larger than 1. Find radius of the largest sphere containing $A$.
2009 Polish MO Finals, 5
A sphere is inscribed in tetrahedron $ ABCD$ and is tangent to faces $ BCD,CAD,ABD,ABC$ at points $ P,Q,R,S$ respectively. Segment $ PT$ is the sphere's diameter, and lines $ TA,TQ,TR,TS$ meet the plane $ BCD$ at points $ A',Q',R',S'$. respectively. Show that $ A$ is the center of a circumcircle on the triangle $ S'Q'R'$.
1977 AMC 12/AHSME, 27
There are two spherical balls of different sizes lying in two corners of a rectangular room, each touching two walls and the floor. If there is a point on each ball which is $5$ inches from each wall which that ball touches and $10$ inches from the floor, then the sum of the diameters of the balls is
$\textbf{(A) }20\text{ inches}\qquad\textbf{(B) }30\text{ inches}\qquad\textbf{(C) }40\text{ inches}\qquad$
$\textbf{(D) }60\text{ inches}\qquad \textbf{(E) }\text{not determined by the given information}$
1990 National High School Mathematics League, 15
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.
1995 Dutch Mathematical Olympiad, 4
A number of spheres with radius $ 1$ are being placed in the form of a square pyramid. First, there is a layer in the form of a square with $ n^2$ spheres. On top of that layer comes the next layer with $ (n\minus{}1)^2$ spheres, and so on. The top layer consists of only one sphere. Compute the height of the pyramid.
2014 IPhOO, 15
The period of a given pendulum on a planet of radius $R$ is constant (unchanged) as we go from the surface of the planet down to radius $a$, where $R > a$. The planet has mass density evenly distributed at any radius $ r < a $. This density is $\rho_0$. Find the total mass of the planet. Express your answer in terms of $\rho_0$, $a$, $R$, the period of the pendulum, $T$, the length of the pendulum string, $L$, and other constants, as necessary.
[b]Warning[/b]: Your answer may contain some math. So be sure to input this correctly!
[i]Problem proposed by Trung Phan[/i]
2011 Flanders Math Olympiad, 2
The area of the ground plane of a truncated cone $K$ is four times as large as the surface of the top surface. A sphere $B$ is circumscribed in $K$, that is to say that $B$ touches both the top surface and the base and the sides. Calculate ratio volume $B :$ Volume $K$.
1994 Vietnam National Olympiad, 2
$S$ is a sphere center $O. G$ and $G'$ are two perpendicular great circles on $S$. Take $A, B, C$ on $G$ and $D$ on $G'$ such that the altitudes of the tetrahedron $ABCD$ intersect at a point. Find the locus of the intersection.
2016 Miklós Schweitzer, 7
Show that the unit sphere bundle of the $r$-fold direct sum of the tautological (universal) complex line bundle over the space $\mathbb{C}P^{\infty}$ is homotopically equivalent to $\mathbb{C}P^{r-1}$.
1992 Romania Team Selection Test, 3
Let $ABCD$ be a tetrahedron; $B', C', D'$ be the midpoints of the edges $AB, AC, AD$; $G_A, G_B, G_C, G_D$ be the barycentres of the triangles $BCD, ACD, ABD, ABC$, and $G$ be the barycentre of the tetrahedron. Show that $A, G, G_B, G_C, G_D$ are all on a sphere if and only if $A, G, B', C', D'$ are also on a sphere.
[i]Dan Brânzei[/i]
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.