Found problems: 2265
2014 Harvard-MIT Mathematics Tournament, 5
Let $\mathcal{C}$ be a circle in the $xy$ plane with radius $1$ and center $(0, 0, 0)$, and let $P$ be a point in space with coordinates $(3, 4, 8)$. Find the largest possible radius of a sphere that is contained entirely in the slanted cone with base $\mathcal{C}$ and vertex $P$.
1997 Rioplatense Mathematical Olympiad, Level 3, 2
Consider a prism, not necessarily right, whose base is a rhombus $ABCD$ with side $AB = 5$ and diagonal $AC = 8$. A sphere of radius $r$ is tangent to the plane $ABCD$ at $C$ and tangent to the edges $AA_1$ , $BB _1$ and $DD_ 1$ of the prism. Calculate $r$ .
2014 USAMTS Problems, 4:
A point $P$ in the interior of a convex polyhedron in Euclidean space is called a [i]pivot point[/i] of the polyhedron if every line through $P$ contains exactly $0$ or $2$ vertices of the polyhedron. Determine, with proof, the maximum number of pivot points that a polyhedron can contain.
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.
2005 QEDMO 1st, 1 (Z4)
Prove that every integer can be written as sum of $5$ third powers of integers.
1992 Romania Team Selection Test, 10
In a tetrahedron $VABC$, let $I$ be the incenter and $A',B',C'$ be arbitrary points on the edges $AV,BV,CV$, and let $S_a,S_b,S_c,S_v$ be the areas of triangles $VBC,VAC,VAB,ABC$, respectively. Show that points $A',B',C',I$ are coplanar if and only if $\frac{AA'}{A'V}S_a +\frac{BB'}{B'V}S_b +\frac{CC'}{C'V}S_c = S_v$
2009 Tournament Of Towns, 5
Suppose that $X$ is an arbitrary point inside a tetrahedron. Through each vertex of the tetrahedron, draw a straight line that is parallel to the line segment connecting $X$ with the intersection point of the medians of the opposite face. Prove that these four lines meet at the same point.
2008 AMC 12/AHSME, 23
The sum of the base-$ 10$ logarithms of the divisors of $ 10^n$ is $ 792$. What is $ n$?
$ \textbf{(A)}\ 11\qquad
\textbf{(B)}\ 12\qquad
\textbf{(C)}\ 13\qquad
\textbf{(D)}\ 14\qquad
\textbf{(E)}\ 15$
2016 District Olympiad, 4
Let $ ABCDA’B’C’D’ $ a right parallelepiped and $ M,N $ the feet of the perpendiculars of $ BD $ through $ A’, $ respectively, $ C’. $ We know that $ AB=\sqrt 2, BC=\sqrt 3, AA’=\sqrt 2. $
[b]a)[/b] Prove that $ A’M\perp C’N. $
[b]b)[/b] Calculate the dihedral angle between the plane formed by $ A’MC $ and the plane formed by $ ANC’. $
2016 BMT Spring, 19
Regular tetrahedron $P_1P_2P_3P_4$ has side length $1$. Define $P_i$ for $i > 4$ to be the centroid of tetrahedron $P_{i-1}P_{i-2}P_{i-3}P_{i-4}$, and $P_{ \infty} = \lim_{n\to \infty} P_n$. What is the length of $P_5P_{ \infty}$?
1988 IMO Longlists, 66
Let $C$ be a cube with edges of length 2. Construct a solid with fourteen faces by cutting off all eight corners at $C,$ keeping the new faces perpendicular to the diagonals of the cube, and keeping the newly formed faces indentical. If at the conclusion of this process the fourteen faces so have the same area, find the area of each of face of the new solid.
1982 All Soviet Union Mathematical Olympiad, 348
The $KLMN$ tetrahedron (triangle pyramid) vertices are situated inside or on the faces or on the edges of the $ABCD$ tetrahedron. Prove that perimeter of $KLMN$ is less than $4/3$ perimeter of $ABCD$.
1979 Spain Mathematical Olympiad, 7
Prove that the volume of a tire (torus) is equal to the volume of a cylinder whose base is a meridian section of that and whose height is the length of the circumference formed by the centers of the meridian sections.
1979 USAMO, 2
Let $S$ be a great circle with pole $P$. On any great circle through $P$, two points $A$ and $B$ are chosen equidistant from $P$. For any [i] spherical triangle [/i] $ABC$ (the sides are great circles ares), where $C$ is on $S$, prove that the great circle are $CP$ is the angle bisector of angle $C$.
[b] Note. [/b] A great circle on a sphere is one whose center is the center of the sphere. A pole of the great circle $S$ is a point $P$ on the sphere such that the diameter through $P$ is perpendicular to the plane of $S$.
2010 All-Russian Olympiad Regional Round, 11.6
At the base of the quadrangular pyramid $SABCD$ lies the parallelogram $ABCD$. Prove that for any point $O$ inside the pyramid, the sum of the volumes of the tetrahedra $OSAB$ and $OSCD$ is equal to the sum of the volumes of the tetrahedra $OSBC$ and $OSDA$ .
1950 Polish MO Finals, 3
Prove that if the two altitudes of a tetrahedron intersect, then the other two atltitudes intersect also.
1982 Czech and Slovak Olympiad III A, 1
Given a tetrahedron $ABCD$ and inside the tetrahedron points $K, L, M, N$ that do not lie on a plane. Denote also the centroids of $P$, $Q$, $R$, $S$ of the tetrahedrons $KBCD$, $ALCD$, $ABMD$, $ABCN$ do not lie on a plane. Let $T$ be the centroid of the tetrahedron ABCD, $T_o$ be the centroid of the tetrahedron $PQRS$ and $T_1$ be the centroid of the tetrahedron $KLMN$.
a) Prove that the points $T, T_0, T_1$ lie in one straight line.
b) Determine the ratio $|T_0T| : |T_0 T_1|$.
1949 Moscow Mathematical Olympiad, 169
Construct a convex polyhedron of equal “bricks” shown in Figure.
[img]https://cdn.artofproblemsolving.com/attachments/6/6/75681a90478f978665b6874d0c0c9441ea3bd2.gif[/img]
Denmark (Mohr) - geometry, 1994.1
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]
1975 Putnam, A6
Given three points in space forming an acute-angled triangle, show that we can find two further points such that no three of the five points are collinear and the line through any two is normal to the plane through the other three.
2008 Tournament Of Towns, 2
Space is dissected into congruent cubes. Is it necessarily true that for each cube there exists another cube so that both cubes have a whole face in common?
1995 Tournament Of Towns, (457) 2
For what values of $n$ is it possible to paint the edges of a prism whose base is an $n$-gon so that there are edges of all three colours at each vertex and all the faces (including the upper and lower bases) have edges of all three colours?
(AV Shapovelov)
2021 Sharygin Geometry Olympiad, 22
A convex polyhedron and a point $K$ outside it are given. For each point $M$ of a polyhedron construct a ball with diameter $MK$. Prove that there exists a unique point on a polyhedron which belongs to all such balls.
1989 French Mathematical Olympiad, Problem 3
Find the greatest real $k$ such that, for every tetrahedron $ABCD$ of volume $V$, the product of areas of faces $ABC,ABD$ and $ACD$ is at least $kV^2$.
1981 Romania Team Selection Tests, 3.
Determine the lengths of the edges of a right tetrahedron of volume $a^3$ so that the sum of its edges' lengths is minumum.