Found problems: 2265
2020 AIME Problems, 7
Two congruent right circular cones each with base radius $3$ and height $8$ have axes of symmetry that intersect at right angles at a point in the interior of the cones a distance $3$ from the base of each cone. A sphere with radius $r$ lies inside both cones. The maximum possible value for $r^2$ is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
1972 Polish MO Finals, 4
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.
2010 Stanford Mathematics Tournament, 8
A sphere of radius $1$ is internally tangent to all four faces of a regular tetrahedron. Find the tetrahedron's volume.
1984 Brazil National Olympiad, 3
Given a regular dodecahedron of side $a$. Take two pairs of opposite faces: $E, E' $ and $F, F'$. For the pair $E, E'$ take the line joining the centers of the faces and take points $A$ and $C$ on the line each a distance $m$ outside one of the faces. Similarly, take $B$ and $D$ on the line joining the centers of $F, F'$ each a distance $m$ outside one of the faces. Show that $ABCD$ is a rectangle and find the ratio of its side lengths.
1974 Spain Mathematical Olympiad, 2
In a metallic disk, a circular sector is removed, so that with the remaining can form a conical glass of maximum volume. Calculate, in radians, the angle of the sector that is removed.
[hide=original wording]En un disco metalico se quita un sector circular, de modo que con la parte restante se pueda formar un vaso c´onico de volumen maximo. Calcular, en radianes, el angulo del sector que se quita.[/hide]
1996 All-Russian Olympiad, 1
Which are there more of among the natural numbers from 1 to 1000000, inclusive: numbers that can be represented as the sum of a perfect square and a (positive) perfect cube, or numbers that cannot be?
[i]A. Golovanov[/i]
2001 Balkan MO, 4
A cube side 3 is divided into 27 unit cubes. The unit cubes are arbitrarily labeled 1 to 27 (each cube is given a different number). A move consists of swapping the cube labeled 27 with one of its 6 neighbours. Is it possible to find a finite sequence of moves at the end of which cube 27 is in its original position, but cube $n$ has moved to the position originally occupied by $27-n$ (for each $n = 1, 2, \ldots , 26$)?
2000 Harvard-MIT Mathematics Tournament, 32
How many (nondegenerate) tetrahedrons can be formed from the vertices of an $n$-dimensional hypercube?
2014 Online Math Open Problems, 16
Let $OABC$ be a tetrahedron such that $\angle AOB = \angle BOC = \angle COA = 90^\circ$ and its faces have integral surface areas. If $[OAB] = 20$ and $[OBC] = 14$, find the sum of all possible values of $[OCA][ABC]$. (Here $[\triangle]$ denotes the area of $\triangle$.)
[i]Proposed by Robin Park[/i]
2010 Middle European Mathematical Olympiad, 11
For a nonnegative integer $n$, define $a_n$ to be the positive integer with decimal representation
\[1\underbrace{0\ldots0}_{n}2\underbrace{0\ldots0}_{n}2\underbrace{0\ldots0}_{n}1\mbox{.}\]
Prove that $\frac{a_n}{3}$ is always the sum of two positive perfect cubes but never the sum of two perfect squares.
[i](4th Middle European Mathematical Olympiad, Team Competition, Problem 7)[/i]
2008 Purple Comet Problems, 8
A container is shaped like a square-based pyramid where the base has side length $23$ centimeters and the height is $120$ centimeters. The container is open at the base of the pyramid and stands in an open field with its vertex pointing down. One afternoon $5$ centimeters of rain falls in the open field partially filling the previously empty container. Find the depth in centimeters of the rainwater in the bottom of the container after the rain.
1995 National High School Mathematics League, 11
Color the vertexes of a quadrangular pyramid in one color, satisfying that two end points of any edge are in different colors. We have only 5 colors, then the number of ways coloring the quadrangular pyramid is________.
1968 Poland - Second Round, 5
The tetrahedrons $ ABCD $ and $ A_1B_1C_1D_1 $ are situated so that the midpoints of the segments $ AA_1 $, $ BB_1 $, $ CC_1 $, $ DD_1 $ are the centroids of the triangles $BCD$, $ ACD $, $ A B D $ and $ ABC $, respectively. What is the ratio of the volumes of these tetrahedrons?
1968 Bulgaria National Olympiad, Problem 5
The point $M$ is inside the tetrahedron $ABCD$ and the intersection points of the lines $AM,BM,CM$ and $DM$ with the opposite walls are denoted with $A_1,B_1,C_1,D_1$ respectively. It is given also that the ratios $\frac{MA}{MA_1}$, $\frac{MB}{MB_1}$, $\frac{MC}{MC_1}$, and $\frac{MD}{MD_1}$ are equal to the same number $k$. Find all possible values of $k$.
[i]K. Petrov[/i]
2000 Harvard-MIT Mathematics Tournament, 13
Determine the remainder when $(x^4-1)(x^2-1)$ is divided by $1+x+x^2$.
2003 AMC 10, 10
The polygon enclosed by the solid lines in the figure consists of $ 4$ congruent squares joined edge-to-edge. One more congruent square is attached to an edge at one of the nine positions indicated. How many of the nine resulting polygons can be folded to form a cube with one face missing?
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$ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6$
1999 IberoAmerican, 1
Find all the positive integers less than 1000 such that the cube of the sum of its digits is equal to the square of such integer.
2009 Tournament Of Towns, 3
Every edge of a tetrahedron is tangent to a given sphere. Prove that the three line segments joining the points of tangency of the three pairs of opposite edges of the tetrahedron are concurrent.
[i](7 points)[/i]
1995 National High School Mathematics League, 8
Consider the maximum value of circular cone inscribed to a sphere, the ratio of it to the volume of the sphere is________.
1988 Flanders Math Olympiad, 2
A 3-dimensional cross is made up of 7 cubes, one central cube and 6 cubes that share a face with it.
The cross is inscribed in a circle with radius 1. What's its volume?
1963 IMO Shortlist, 2
Point $A$ and segment $BC$ are given. Determine the locus of points in space which are vertices of right angles with one side passing through $A$, and the other side intersecting segment $BC$.
2005 AMC 12/AHSME, 16
Eight spheres of radius 1, one per octant, are each tangent to the coordinate planes. What is the radius of the smallest sphere, centered at the origin, that contains these eight spheres?
$ \textbf{(A)}\ \sqrt 2\qquad
\textbf{(B)}\ \sqrt 3\qquad
\textbf{(C)}\ 1 \plus{} \sqrt 2\qquad
\textbf{(D)}\ 1 \plus{} \sqrt 3\qquad
\textbf{(E)}\ 3$
1993 Polish MO Finals, 3
Find out whether it is possible to determine the volume of a tetrahedron knowing the areas of its faces and its circumradius.
1999 Romania National Olympiad, 4
Let $SABC$ be a regular pyramid, $O$ the center of basis $ABC$, and $M$ the midpoint of $[BC]$. If $N \in [SA]$ such that $SA = 25 \cdot NS$ and $SO \cap MN=\{P\}$, $AM=2\cdot SO$, prove that the planes $(ABP)$ and $(SBC)$ are perpendicular.
2022 Chile Junior Math Olympiad, 5
In a right circular cone of wood, the radius of the circumference $T$ of the base circle measures $10$ cm, while every point on said circumference is $20$ cm away. from the apex of the cone. A red ant and a termite are located at antipodal points of $T$. A black ant is located at the midpoint of the segment that joins the vertex with the position of the termite. If the red ant moves to the black ant's position by the shortest possible path, how far does it travel?