Found problems: 2265
2013 Purple Comet Problems, 20
The diagram below shows a $1\times2\times10$ duct with $2\times2\times2$ cubes attached to each end. The resulting object is empty, but the entire surface is solid sheet metal. A spider walks along the inside of the duct between the two marked corners. There are positive integers $m$ and $n$ so that the shortest path the spider could take has length $\sqrt{m}+\sqrt{n}$. Find $m + n$.
[asy]
size(150);
defaultpen(linewidth(1));
draw(origin--(43,0)--(61,20)--(18,20)--cycle--(0,-43)--(43,-43)--(43,0)^^(43,-43)--(61,-23)--(61,20));
draw((43,-43)--(133,57)--(90,57)--extension((90,57),(0,-43),(61,20),(18,20)));
draw((0,-43)--(0,-65)--(43,-65)--(43,-43)^^(43,-65)--(133,35)--(133,57));
draw((133,35)--(133,5)--(119.5,-10)--(119.5,20)^^(119.5,-10)--extension((119.5,-10),(100,-10),(43,-65),(133,35)));
dot(origin^^(133,5));
[/asy]
2006 VTRMC, Problem 7
Three spheres each of unit radius have centers $P,Q,R$ with the property that the center of each sphere lies on the surface of the other two spheres. Let $C$ denote the cylinder with cross-section $PQR$ (the triangular lamina with vertices $P,Q,R$) and axis perpendicular to $PQR$. Let $M$ denote the space which is common to the three spheres and the cylinder $C$, and suppose the mass density of $M$ at a given point is the distance of the point from $PQR$. Determine the mass of $M$.
2008 AMC 10, 17
An equilateral triangle has side length $ 6$. What is the area of the region containing all points that are outside the triangle and not more than $ 3$ units from a point of the triangle?
$ \textbf{(A)}\ 36\plus{}24\sqrt{3} \qquad
\textbf{(B)}\ 54\plus{}9\pi \qquad
\textbf{(C)}\ 54\plus{}18\sqrt{3}\plus{}6\pi \qquad
\textbf{(D)}\ \left(2\sqrt{3}\plus{}3\right)^2\pi \\
\textbf{(E)}\ 9\left(\sqrt{3}\plus{}1\right)^2\pi$
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.$
1997 IMO Shortlist, 5
Let $ ABCD$ be a regular tetrahedron and $ M,N$ distinct points in the planes $ ABC$ and $ ADC$ respectively. Show that the segments $ MN,BN,MD$ are the sides of a triangle.
2024 JHMT HS, 10
One triangular face $F$ of a tetrahedron $\mathcal{T}$ has side lengths $\sqrt{5}$, $\sqrt{65}$, and $2\sqrt{17}$. The other three faces of $\mathcal{T}$ are right triangles whose hypotenuses coincide with the sides of $F$. There exists a sphere inside $\mathcal{T}$ tangent to all four of its faces. Compute the radius of this sphere.
2007 All-Russian Olympiad Regional Round, 9.7
An infinite increasing arithmetical progression consists of positive integers and contains a perfect cube. Prove that this progression also contains a term which is a perfect cube but not a perfect square.
1985 National High School Mathematics League, 2
$PQ$ is a chord of parabola $y^2=2px(p>0)$ and $PQ$ pass its focus $F$. Line $l$ is its directrix. Projection of $PQ$ on $l$ is $MN$. The area of curved surface that $PQ$ rotate around $l$ is $S_1$, the area of spherical surface of the ball with diameter of $MN$ is $S_2$, then
$\text{(A)}S_1>S_2\qquad\text{(B)}S_1<S_2\qquad\text{(C)}S_1\geq S_2\qquad\text{(D)}$ Not sure
1995 China National Olympiad, 1
Given four spheres with their radii equal to $2,2,3,3$ respectively, each sphere externally touches the other spheres. Suppose that there is another sphere that is externally tangent to all those four spheres, determine the radius of this sphere.
1955 Moscow Mathematical Olympiad, 301
Given a trihedral angle with vertex $O$. Find whether there is a planar section $ABC$ such that the angles $\angle OAB$, $\angle OBA$, $\angle OBC$, $\angle OCB$, $\angle OAC$, $\angle OCA$ are acute.
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 $O_1O_2O_3.$
2014-2015 SDML (Middle School), 8
Two regular square pyramids have all edges $12$ cm in length. The pyramids have parallel bases and those bases have parallel edges, and each pyramid has its apex at the center of the other pyramid's base. What is the total number of cubic centimeters in the volume of the solid of intersection of the two pyramids?
2010 Paenza, 6
In space are given two tetrahedra with the same barycenter such that one of them contains the other.
For each tetrahedron, we consider the octahedron whose vertices are the midpoints of the tetrahedron's edges.
Prove that one of this octahedra contains the other.
2007 IMO, 6
Let $ n$ be a positive integer. Consider
\[ S \equal{} \left\{ (x,y,z) \mid x,y,z \in \{ 0, 1, \ldots, n\}, x \plus{} y \plus{} z > 0 \right \}
\]
as a set of $ (n \plus{} 1)^{3} \minus{} 1$ points in the three-dimensional space. Determine the smallest possible number of planes, the union of which contains $ S$ but does not include $ (0,0,0)$.
[i]Author: Gerhard Wöginger, Netherlands [/i]
1987 AMC 12/AHSME, 27
A cube of cheese $C=\{(x, y, z)| 0 \le x, y, z \le 1\}$ is cut along the planes $x=y$, $y=z$ and $z=x$. How many pieces are there? (No cheese is moved until all three cuts are made.)
$ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ 9 $
1995 All-Russian Olympiad, 7
The altitudes of a tetrahedron intersect in a point. Prove that this point, the foot of one of the altitudes, and the points dividing the other three altitudes in the ratio $2 : 1$ (measuring from the vertices) lie on a sphere.
[i]D. Tereshin[/i]
2019 BMT Spring, 3
A cylinder with radius $5$ and height $1$ is rolling on the (unslanted) floor. Inside the cylinder, there is water that has constant height $\frac{15}{2}$ as the cylinder rolls on the floor. What is the volume of the water?
2004 USAMTS Problems, 2
For the equation \[ (3x^2+y^2-4y-17)^3-(2x^2+2y^2-4y-6)^3=(x^2-y^2-11)^3, \]
determine its solutions $(x, y)$ where both $x$ and $y$ are integers. Prove that your answer lists all the integer solutions.
1996 Vietnam National Olympiad, 2
Given a trihedral angle Sxyz. A plane (P) not through S cuts Sx,Sy,Sz respectively at A,B,C. On the plane (P), outside triangle ABC, construct triangles DAB,EBC,FCA which are confruent to the triangles SAB,SBC,SCA respectively. Let (T) be the sphere lying inside Sxyz, but not inside the tetrahedron SABC, toucheing the planes containing the faces of SABC. Prove that (T) touches the plane (P) at the circumcenter of triangle DEF.
Oliforum Contest IV 2013, 6
Let $P$ be a polyhedron whose faces are colored black and white so that there are more black faces and no two black faces are adjacent. Show that $P$ is not circumscribed about a sphere.
1991 Bulgaria National Olympiad, Problem 2
Let $K$ be a cube with edge $n$, where $n>2$ is an even integer. Cube $K$ is divided into $n^3$ unit cubes. We call any set of $n^2$ unit cubes lying on the same horizontal or vertical level a layer. We dispose of $\frac{n^3}4$ colors, in each of which we paint exactly $4$ unit cubes. Prove that we can always select $n$ unit cubes of distinct colors, no two of which lie on the same layer.
2015 BMT Spring, Tie 1
Compute the surface area of a rectangular prism with side lengths $2, 3, 4$.
2005 Swedish Mathematical Competition, 6
A regular tetrahedron of edge length $1$ is orthogonally projected onto a plane. Find the largest possible area of its image.
1949 Moscow Mathematical Olympiad, 157
a) Prove that if a planar polygon has several axes of symmetry, then all of them intersect at one point.
b) A finite solid body is symmetric about two distinct axes. Describe the position of the symmetry planes of the body.
2005 China Girls Math Olympiad, 3
Determine if there exists a convex polyhedron such that
(1) it has 12 edges, 6 faces and 8 vertices;
(2) it has 4 faces with each pair of them sharing a common edge of the polyhedron.