Found problems: 2265
The solid shown has a square base of side length $s$. The upper edge is parallel to the base and has length $2s$. All other edges have length $s$. Given that $s = 6 \sqrt{2}$, what is the volume of the solid?
[asy]
import three;
size(170);
pathpen = black+linewidth(0.65);
pointpen = black;
currentprojection = perspective(30,-20,10);
real s = 6 * 2^.5;
triple A=(0,0,0),B=(s,0,0),C=(s,s,0),D=(0,s,0),E=(-s/2,s/2,6),F=(3*s/2,s/2,6);
draw(F--B--C--F--E--A--B);
draw(A--D--E, dashed);
draw(D--C, dashed);
label("$2s$", (s/2, s/2, 6), N);
label("$s$", (s/2, 0, 0), SW);
[/asy]
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.
A cone with semivertical angle $30^{\circ}$ is half filled with water. What is the angle it must be tilted by so that water starts spilling?
Consider a system of infinitely many spheres made of metal, with centers at points $(a, b, c) \in \mathbb Z^3$. We say that the system is stable if the temperature of each sphere equals the average temperature of the six closest spheres. Assuming that all spheres in a stable system have temperatures between $0^\circ C$ and $1^\circ C$, prove that all the spheres have the same temperature.
Consider a sphere and a plane $\pi$. For a variable point $M \in \pi$, exterior to the sphere, one considers the circular cone with vertex in $M$ and tangent to the sphere. Find the locus of the centers of all circles which appear as tangent points between the sphere and the cone.
[i]Octavian Stanasila[/i]
Let $ ABCD $ be a tetahedron and $ M,N $ the middlepoints of $ AB, $ respectively, $ CD. $ Show that any plane that contains $ M $ and $ N $ cuts the tetrahedron in two polihedra that have same volume.
A tetrahedron has the property that the three segments connecting the pairs of midpoints of opposite edges are equal and mutually orthogonal. Prove that this tetrahedron is regular.
A cube consisting of $(2N)^3$ unit cubes is pierced by several needles parallel to the edges of the cube (each needle pierces exactly $2N$ unit cubes). Each unit cube is pierced by at least one needle. Let us call any subset of these needles “regular” if there are no two needles in this subset that pierce the same unit cube.
a) Prove that there exists a regular subset consisting of $2N^2$ needles such that all of them have either the same direction or two different directions.
b) What is the maximum size of a regular subset that does exist for sure?
(Nikita Gladkov, Alexandr Zimin)
Let $A,B,C,D$ be four points in space, and $M$ and $N$ be the midpoints of $AC$ and $BD$, respectively. Show that $$AB^2+BC^2+CD^2+DA^2 = AC^2+BD^2+4MN^2$$
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.
Recall that a plane divides $\mathbb{R}^3$ into two regions, two parallel planes divide it into three regions, and two intersecting planes divide space into four regions. Consider the six planes which the faces of the cube $ABCD-A_1B_1C_1D_1$ lie on, and the four planes that the tetrahedron $ACB_1D_1$ lie on. How many regions do these ten planes split the space into?
In how many distinguishable ways can $10$ distinct pool balls be formed into a pyramid ($6$ on the bottom, $3$ in the middle, one on top), assuming that all rotations of the pyramid are indistinguishable?
Find the largest possible volume for a tetrahedron which lies inside a hemisphere of radius $1$.
Faces of a cube $9\times 9\times 9$ are partitioned onto unit squares. The surface of a cube is pasted over by $243$ strips $2\times 1$ without overlapping. Prove that the number of bent strips is odd.
[i]A. Poliansky[/i]
Let $a,b,c$ be integers such that $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}=3$ prove that $abc$ is a perfect cube!
We are given $n$ mass points of equal mass in space. We define a sequence of points $O_1,O_2,O_3,\ldots $ as follows: $O_1$ is an arbitrary point (within the unit distance of at least one of the $n$ points); $O_2$ is the centre of gravity of all the $n$ given points that are inside the unit sphere centred at $O_1$;$O_3$ is the centre of gravity of all of the $n$ given points that are inside the unit sphere centred at $O_2$; etc. Prove that starting from some $m$, all points $O_m,O_{m+1},O_{m+2},\ldots$ coincide.
The dimensions of a wooden octahedron are natural numbers. We painted all its surface (the six faces), cut it by planes parallel to the cubed faces of an edge unit and observed that exactly half of the cubes did not have any painted faces. Prove that the number of octahedra with such property is finite.
(It may be useful to keep in mind that $\sqrt[3]{\frac{1}{2}}=1,79 ... <1,8$).
[hide=original wording] Las dimensiones de un ortoedro de madera son enteras. Pintamos toda su superficie (las seis caras), lo cortamos mediante planos paralelos a las caras en cubos de una unidad de arista y observamos que exactamente la mitad de los cubos no tienen ninguna cara pintada. Probar que el número de ortoedros con tal propiedad es finito[/hide]
We have an empty equilateral triangle with length of a side $l$. We put the triangle, horizontally, over a sphere of radius $r$. Clearly, if the triangle is small enough, the triangle is held by the sphere. Which is the distance between any vertex of the triangle and the centre of the sphere (as a function of $l$ and $r$)?
Given the pyramid $ABCD$. Let $O$ be the midpoint of edge $AC$. Given that $DO$ is the height of the pyramid, $AB=BC=2DO$ and the angle $ABC$ is right. Cut this pyramid into $8$ equal and similar to it pyramids.
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]
Suppose that in a unit sphere in Euclidean space, there are $2m$ points $x_1, x_2, ..., x_{2m}.$ Prove that it's possible to partition them into two sets of $m$ points in such a way that the centers of mass of these sets are at a distance of at most $\frac{2}{\sqrt{m}}$ from one another.
The net of 20 triangles shown below can be folded to form a regular icosahedron. Inside each of the triangular faces, write a number from 1 to 20 with each number used exactly once. Any pair of numbers that are consecutive must be written on faces sharing an edge in the folded icosahedron, and additionally, 1 and 20 must also be on faces sharing an edge. Some numbers have been given to you. (No proof is necessary.)
[asy]
unitsize(1cm);
pair c(int a, int b){return (a-b/2,sqrt(3)*b/2);}
draw(c(0,0)--c(0,1)--c(-1,1)--c(1,3)--c(1,1)--c(2,2)--c(3,2)--c(4,3)--c(4,2)--c(3,1)--c(2,1)--c(2,-1)--c(1,-1)--c(1,-2)--c(0,-3)--c(0,-2)--c(-1,-2)--c(1,0)--cycle);
draw(c(0,0)--c(1,1)--c(0,1)--c(1,2)--c(0,2)--c(0,1),linetype("4 4"));
draw(c(4,2)--c(3,2)--c(3,1),linetype("4 4"));
draw(c(3,2)--c(1,0)--c(1,1)--c(2,1)--c(2,2),linetype("4 4"));
draw(c(1,-2)--c(0,-2)--c(0,-1)--c(1,-1)--c(1,0)--c(2,0)--c(0,-2),linetype("4 4"));
label("2",(c(0,2)+c(1,2))/2,S);
label("15",(c(1,1)+c(2,1))/2,S);
label("6",(c(0,1)+c(1,1))/2,N);
label("14",(c(0,0)+c(1,0))/2,N);[/asy]
Consider a piece of material in the shape of a right circular conical frustum with radii $R,r,R>r$. A cavity in the shape of another coaxial right circular conical frustum was drilled into the material (see the picture). That way only half of the original volume of material remained. Compute radii $R',r'$ of the cavity. Decide for which ratio $R/r$ the problem has a solution.
[img]https://cdn.artofproblemsolving.com/attachments/b/f/12f579458b7cf0fc31849b319e6f58e50b0363.png[/img]
Suppose you have $27$ identical unit cubes colored such that $3$ faces adjacent to a vertex are red and the other $3$ are colored blue. Suppose further that you assemble these $27$ cubes randomly into a larger cube with $3$ cubes to an edge (in particular, the orientation of each cube is random). The probability that the entire cube is one solid color can be written as $\frac{1}{2^n}$ for some positive integer $n$. Find $n$.
What is the greatest number of parts that $5$ spheres can divide the space into?