Found problems: 2265
2021 AMC 10 Fall, 24
A cube is constructed from $4$ white unit cubes and $4$ black unit cubes. How many different ways are there to construct the $2 \times 2 \times 2$ cube using these smaller cubes? (Two constructions are considered the same if one can be rotated to match the other.)
$\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\
10 \qquad\textbf{(E)}\ 11$
LMT Speed Rounds, 7
Isabella is making sushi. She slices a piece of salmon into the shape of a solid triangular prism. The prism is $2$ cm thick, and its triangular faces have side lengths $7$ cm, $ 24$cm, and $25$ cm. Find the volume of this piece of salmon in cm$^3$.
[i]Proposed by Isabella Li[/i]
1962 IMO Shortlist, 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.
1982 Kurschak Competition, 1
A cube of integral dimensions is given in space so that all four vertices of one of the faces are lattice points. Prove that the other four vertices are also lattice points.
2010 German National Olympiad, 6
Let $A,B,C,D,E,F,G$ and $H$ be eight pairwise distinct points on the surface of a sphere. The quadruples $(A,B,C,D), (A,B,F,E),(B,C,G,F),(C,D,H,G)$ and $(D,A,E,H)$ of points are coplanar.
Prove that the quadruple $(E,F,G,H)$ is coplanar aswell.
1960 IMO Shortlist, 6
Consider a cone of revolution with an inscribed sphere tangent to the base of the cone. A cylinder is circumscribed about this sphere so that one of its bases lies in the base of the cone. let $V_1$ be the volume of the cone and $V_2$ be the volume of the cylinder.
a) Prove that $V_1 \neq V_2$;
b) Find the smallest number $k$ for which $V_1=kV_2$; for this case, construct the angle subtended by a diamter of the base of the cone at the vertex of the cone.
2004 Belarusian National Olympiad, 7
A cube $ABCDA_1B_1C_1D_1$ is given. Find the locus of points $E$ on the face $A_1B_1C_1D_1$ for which there exists a line intersecting the lines $AB$, $A_1D_1$, $B_1D$, and $EC$.
2000 National High School Mathematics League, 11
A sphere is tangent to six edges of a regular tetrahedron. If the length of each edge is $a$, then the volume of the sphere is________.
PEN G Problems, 18
Show that the cube roots of three distinct primes cannot be terms in an arithmetic progression.
1989 Romania Team Selection Test, 4
Let $A,B,C$ be variable points on edges $OX,OY,OZ$ of a trihedral angle $OXYZ$, respectively.
Let $OA = a, OB = b, OC = c$ and $R$ be the radius of the circumsphere $S$ of $OABC$.
Prove that if points $A,B,C$ vary so that $a+b+c = R+l$, then the sphere $S$ remains tangent to a fixed sphere.
1986 Traian Lălescu, 2.4
Prove that $ ABCD $ is a rectangle if and only if $ MA^2+MC^2=MB^2+MD^2, $ for all spatial points $ M. $
1976 IMO Shortlist, 6
A box whose shape is a parallelepiped can be completely filled with cubes of side $1.$ If we put in it the maximum possible number of cubes, each of volume $2$, with the sides parallel to those of the box, then exactly $40$ percent of the volume of the box is occupied. Determine the possible dimensions of the box.
1981 National High School Mathematics League, 5
Given a cube $ABCD-A'B'C'D'$, in the $12$ lines:$AB',BA',CD',DC',AD',DA',BC',CB',AC,BD,A'C',B'D'$, how many sets of lines are skew lines?
$\text{(A)}30\qquad\text{(B)}60\qquad\text{(C)}24\qquad\text{(D)}48$
2016 Saint Petersburg Mathematical Olympiad, 2
The rook, standing on the surface of the checkered cube, beats the cells, located in the same row as well as on the
continuations of this series through one or even several edges. (The picture shows an example for a $4 \times 4 \times 4$ cube,visible cells that some beat the rook, shaded gray.) What is the largest number do not beat each other rooks can be placed on the surface of the cube $50 \times 50 \times 50$?
2006 China Second Round Olympiad, 10
Suppose four solid iron balls are placed in a cylinder with the radius of 1 cm, such that every two of the four balls are tangent to each other, and the two balls in the lower layer are tangent to the cylinder base. Now put water into the cylinder. Find, in $\text{cm}^2$, the volume of water needed to submerge all the balls.
1946 Moscow Mathematical Olympiad, 111
Given two intersecting planes $\alpha$ and $\beta$ and a point $A$ on the line of their intersection. Prove that of all lines belonging to $\alpha$ and passing through $A$ the line which is perpendicular to the intersection line of $\alpha$ and $\beta$ forms the greatest angle with $\beta$.
2001 Tournament Of Towns, 2
The decimal expression of the natural number $a$ consists of $n$ digits, while that of $a^3$ consists of $m$ digits. Can $n + m$ be equal to 2001?
2006 Estonia Math Open Junior Contests, 7
A solid figure consisting of unit cubes is shown in the picture. Is it possible to exactly fill a cube with these figures if the side length of the cube is
a) 15;
b) 30?
Ukrainian TYM Qualifying - geometry, VI.1
Find all nonconvex quadrilaterals in which the sum of the distances to the lines containing the sides is the same for any interior point. Try to generalize the result in the case of an arbitrary non-convex polygon, polyhedron.
2006 Bundeswettbewerb Mathematik, 2
Prove that there are no integers $x,y$ for that it is $x^3+y^3=4\cdot(x^2y+xy^2+1)$.
2022 Purple Comet Problems, 29
Sphere $S$ with radius $100$ has diameter $\overline{AB}$ and center $C$. Four small spheres all with radius $17$ have centers that lie in a plane perpendicular to $\overline{AB}$ such that each of the four spheres is internally tangent to $S$ and externally tangent to two of the other small spheres. Find the radius of the smallest sphere that is both externally tangent to two of the four spheres with radius $17$ and internally tangent to $S$ at a point in the plane perpendicular to $\overline{AB}$ at $C$.
2017 Oral Moscow Geometry Olympiad, 2
Given pyramid with base $n-gon$. How many maximum number of edges can be perpendicular to base?
2014 BMT Spring, 13
A cylinder is inscribed within a sphere of radius 10 such that its volume is [i]almost-half[/i] that of the sphere. If [i]almost-half[/i] is defined such that the cylinder has volume $\frac12+\frac{1}{250}$ times the sphere’s volume, find the sum of all possible heights for the cylinder.
1990 AMC 8, 18
Each corner of a rectangular prism is cut off. Two (of the eight) cuts are shown. How many edges does the new figure have?
[asy]
draw((0,0)--(3,0)--(3,3)--(0,3)--cycle);
draw((3,0)--(5,2)--(5,5)--(2,5)--(0,3));
draw((3,3)--(5,5));
draw((2,0)--(3,1.8)--(4,1)--cycle,linewidth(1));
draw((2,3)--(4,4)--(3,2)--cycle,linewidth(1));[/asy]
$ \text{(A)}\ 24\qquad\text{(B)}\ 30\qquad\text{(C)}\ 36\qquad\text{(D)}\ 42\qquad\text{(E)}\ 48 $
[i]Assume that the planes cutting the prism do not intersect anywhere in or on the prism.[/i]
2023 Israel TST, P2
Let $SABCDE$ be a pyramid whose base $ABCDE$ is a regular pentagon and whose other faces are acute triangles. The altitudes from $S$ to the base sides dissect them into ten triangles, colored red and blue alternatingly. Prove that the sum of the squared areas of the red triangles is equal to the sum of the squared areas of the blue triangles.