Found problems: 2265
1988 Poland - Second Round, 6
Given is a convex polyhedron with $ k $ faces $ S_1, \ldots, S_k $. Let us denote the vector of length 1 perpendicular to the wall $ S_i $ ($ i = 1, \ldots, k $) directed outside the given polyhedron by $ \overrightarrow{n_i} $, and the surface area of this wall by $ P_i $. Prove that
$$
\sum_{i=1}^k P_i \cdot \overrightarrow{n_i} = \overrightarrow{0}.$$
1985 AMC 8, 11
[asy]size(100);
draw((0,0)--(1,0)--(1,1)--(1,2)--(2,2)--(2,3)--(2,4)--(1,4)--(1,3)--(0,3)--(-1,3)--(-1,2)--(0,2)--(0,1)--cycle);
draw((0,1)--(1,1));
draw((0,2)--(1,2));
draw((0,2)--(0,3));
draw((1,2)--(1,3));
draw((1,3)--(2,3));
label("Z",(0.5,0.2),N);
label("X",(0.5,1.2),N);
label("V",(0.5,2.2),N);
label("U",(-0.5,2.2),N);
label("W",(1.5,2.2),N);
label("Y",(1.5,3.2),N);[/asy]
A piece of paper containing six joined squares labeled as shown in the diagram is folded along the edges of the squares to form a cube. The label of the face opposite the face labeled $ \text{X}$ is:
\[ \textbf{(A)}\ \text{Z} \qquad
\textbf{(B)}\ \text{U} \qquad
\textbf{(C)}\ \text{V} \qquad
\textbf{(D)}\ \text{W} \qquad
\textbf{(E)}\ \text{Y}
\]
2011 Flanders Math Olympiad, 2
The area of the ground plane of a truncated cone $K$ is four times as large as the surface of the top surface. A sphere $B$ is circumscribed in $K$, that is to say that $B$ touches both the top surface and the base and the sides. Calculate ratio volume $B :$ Volume $K$.
2013 Oral Moscow Geometry Olympiad, 3
Is there a polyhedron whose area ratio of any two faces is at least $2$ ?
2014 PUMaC Combinatorics A, 3
You have three colors $\{\text{red}, \text{blue}, \text{green}\}$ with which you can color the faces of a regular octahedron (8 triangle sided polyhedron, which is two square based pyramids stuck together at their base), but you must do so in a way that avoids coloring adjacent pieces with the same color. How many different coloring schemes are possible? (Two coloring schemes are considered equivalent if one can be rotated to fit the other.)
1952 Poland - Second Round, 6
Prove that a plane that passes:
a) through the centers of two opposite edges of the tetrahedron and
b) through the center of one of the other edges of the tetrahedron
divides the tetrahedron into two parts of equal volumes.
Will the thesis remain true if we reject assumption (b) ?
1972 Poland - Second Round, 4
A cube with edge length $ n $ is divided into $ n^3 $ unit cubes by planes parallel to its faces. How many pairs of such unit cubes exist that have no more than two vertices in common?
1964 Polish MO Finals, 3
Given a tetrahedron $ ABCD $ whose edges $ AB, BC, CD, DA $ are tangent to a certain sphere. Prove that the points of tangency lie in the same plane.
1972 IMO Longlists, 13
Given a sphere $K$, determine the set of all points $A$ that are vertices of some parallelograms $ABCD$ that satisfy $AC \le BD$ and whose entire diagonal $BD$ is contained in $K$.
2010 May Olympiad, 1
A closed container in the shape of a rectangular parallelepiped contains $1$ liter of water. If the container rests horizontally on three different sides, the water level is $2$ cm, $4$ cm and $5$ cm. Calculate the volume of the parallelepiped.
2023 AMC 12/AHSME, 21
A lampshade is made in the form of the lateral surface of the frustum of a right circular cone. The height of the frustum is $3\sqrt{3}$ inches, its top diameter is 6 inches, and its bottom diameter is 12 inches. A bug is at the bottom of the lampshade and there is a glob of honey on the top edge of the lampshade at the spot farthest from the bug. The bug wants to crawl to the honey, but it must stay on the surface of the lampshade. What is the length in inches of its shortest path to the honey?
[center]
[img]https://cdn.artofproblemsolving.com/attachments/b/4/23f9bc88ea057cb2676f2b8b373330b0f5df69.png[/img][/center]
$\textbf{(A) } 6 + 3\pi\qquad \textbf{(B) }6 + 6\pi\qquad \textbf{(C) } 6\sqrt3 \qquad \textbf{(D) } 6\sqrt5 \qquad \textbf{(E) } 6\sqrt3 + \pi$
1997 National High School Mathematics League, 2
In regular tetrahedron $ABCD$, $E\in AB,F\in CD$, satisfying: $\frac{|AE|}{|EB|}=\frac{|CF|}{|FD|}=\lambda(\lambda\in R_+)$. Note that $f(\lambda)=\alpha_{\lambda}+\beta_{\lambda}$, where $\alpha_{\lambda}=<EF,AC>,\alpha_{\lambda}=<EF,BD>$.
$\text{(A)}$ $f(\lambda)$ increases in $(0,+\infty)$
$\text{(B)}$ $f(\lambda)$ decreases in $(0,+\infty)$
$\text{(C)}$ $f(\lambda)$ increases in $(0,1)$, decreases in $(1,+\infty)$
$\text{(D)}$ $f(\lambda)$ is a fixed value in $(0,+\infty)$
2019 AMC 8, 12
The faces of a cube are painted in six different colors: red (R), white (W), green (G), brown (B), aqua (A), and purple (P). Three views of the cube are shown below. What is the color of the face opposite the aqua face?
$\text{\textbf{(A) }red}\qquad \text{\textbf{(B) } white}\qquad\text{\textbf{(C) }green}\qquad\text{\textbf{(D) }brown }\qquad\text{\textbf{(E)} purple}$
[asy]
unitsize(2 cm);
pair x, y, z, trans;
int i;
x = dir(-5);
y = (0.6,0.5);
z = (0,1);
trans = (2,0);
for (i = 0; i <= 2; ++i) {
draw(shift(i*trans)*((0,0)--x--(x + y)--(x + y + z)--(y + z)--z--cycle));
draw(shift(i*trans)*((x + z)--x));
draw(shift(i*trans)*((x + z)--(x + y + z)));
draw(shift(i*trans)*((x + z)--z));
}
label(rotate(-3)*"$R$", (x + z)/2);
label(rotate(-5)*slant(0.5)*"$B$", ((x + z) + (y + z))/2);
label(rotate(35)*slant(0.5)*"$G$", ((x + z) + (x + y))/2);
label(rotate(-3)*"$W$", (x + z)/2 + trans);
label(rotate(50)*slant(-1)*"$B$", ((x + z) + (y + z))/2 + trans);
label(rotate(35)*slant(0.5)*"$R$", ((x + z) + (x + y))/2 + trans);
label(rotate(-3)*"$P$", (x + z)/2 + 2*trans);
label(rotate(-5)*slant(0.5)*"$R$", ((x + z) + (y + z))/2 + 2*trans);
label(rotate(-85)*slant(-1)*"$G$", ((x + z) + (x + y))/2 + 2*trans);
[/asy]
1940 Moscow Mathematical Olympiad, 066
* Given an infinite cone. The measure of its unfolding’s angle is equal to $\alpha$. A curve on the cone is represented on any unfolding by the union of line segments. Find the number of the curve’s self-intersections.
2002 AIME Problems, 2
Three vertices of a cube are $P=(7,12,10),$ $Q=(8,8,1),$ and $R=(11,3,9).$ What is the surface area of the cube?
1957 AMC 12/AHSME, 17
A cube is made by soldering twelve $ 3$-inch lengths of wire properly at the vertices of the cube. If a fly alights at one of the vertices and then walks along the edges, the greatest distance it could travel before coming to any vertex a second time, without retracing any distance, is:
$ \textbf{(A)}\ 24\text{ in.}\qquad
\textbf{(B)}\ 12\text{ in.}\qquad
\textbf{(C)}\ 30\text{ in.}\qquad
\textbf{(D)}\ 18\text{ in.}\qquad
\textbf{(E)}\ 36\text{ in.}$
2020 USOJMO, 3
An empty $2020 \times 2020 \times 2020$ cube is given, and a $2020 \times 2020$ grid of square unit cells is drawn on each of its six faces. A [i]beam[/i] is a $1 \times 1 \times 2020$ rectangular prism. Several beams are placed inside the cube subject to the following conditions:
[list=]
[*]The two $1 \times 1$ faces of each beam coincide with unit cells lying on opposite faces of the cube. (Hence, there are $3 \cdot {2020}^2$ possible positions for a beam.)
[*]No two beams have intersecting interiors.
[*]The interiors of each of the four $1 \times 2020$ faces of each beam touch either a face of the cube or the interior of the face of another beam.
[/list]
What is the smallest positive number of beams that can be placed to satisfy these conditions?
[i]Proposed by Alex Zhai[/i]
2011 AMC 12/AHSME, 18
A pyramid has a square base with sides of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube?
$ \textbf{(A)}\ 5\sqrt{2}-7 \qquad
\textbf{(B)}\ 7-4\sqrt{3} \qquad
\textbf{(C)}\ \frac{2\sqrt{2}}{27} \qquad
\textbf{(D)}\ \frac{\sqrt{2}}{9} \qquad
\textbf{(E)}\ \frac{\sqrt{3}}{9} $
2018 AMC 8, 19
In a sign pyramid a cell gets a "+" if the two cells below it have the same sign, and it gets a "-" if the two cells below it have different signs. The diagram below illustrates a sign pyramid with four levels. How many possible ways are there to fill the four cells in the bottom row to produce a "+" at the top of the pyramid?
[asy]
unitsize(2cm);
path box = (-0.5,-0.2)--(-0.5,0.2)--(0.5,0.2)--(0.5,-0.2)--cycle;
draw(box); label("$+$",(0,0));
draw(shift(1,0)*box); label("$-$",(1,0));
draw(shift(2,0)*box); label("$+$",(2,0));
draw(shift(3,0)*box); label("$-$",(3,0));
draw(shift(0.5,0.4)*box); label("$-$",(0.5,0.4));
draw(shift(1.5,0.4)*box); label("$-$",(1.5,0.4));
draw(shift(2.5,0.4)*box); label("$-$",(2.5,0.4));
draw(shift(1,0.8)*box); label("$+$",(1,0.8));
draw(shift(2,0.8)*box); label("$+$",(2,0.8));
draw(shift(1.5,1.2)*box); label("$+$",(1.5,1.2));
[/asy]
$\textbf{(A) } 2 \qquad \textbf{(B) } 4 \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 12 \qquad \textbf{(E) } 16$
2009 Princeton University Math Competition, 6
Consider the solid with 4 triangles and 4 regular hexagons as faces, where each triangle borders 3 hexagons, and all the sides are of length 1. Compute the [i]square[/i] of the volume of the solid. Express your result in reduced fraction and concatenate the numerator with the denominator (e.g., if you think that the square is $\frac{1734}{274}$, then you would submit 1734274).
1994 All-Russian Olympiad, 7
The altitudes $AA_1,BB_1,CC_1,DD_1$ of a tetrahedron $ABCD$ intersect in the center $H$ of the sphere inscribed in the tetrahedron $A_1B_1C_1D_1$. Prove that the tetrahedron $ABCD$ is regular.
(D. Tereshin)
2019 Romania National Olympiad, 3
In the regular hexagonal prism $ABCDEFA_1B_1C_1D_1E_1F_1$, We construct $, Q$, the projections of point $A$ on the lines $A_1B$ and $A_1C$ repsectilvely. We construct $R,S$, the projections of point $D_1$ on the lines $A_1D$ and $C_1D$ respectively.
a) Determine the measure of the angle between the planes $(AQP)$ and $(D_1RS)$.
b) Show that $\angle AQP = \angle D_1RS$.
2009 Tournament Of Towns, 2
Mike has $1000$ unit cubes. Each has $2$ opposite red faces, $2$ opposite blue faces and $2$ opposite white faces. Mike assembles them into a $10 \times 10 \times 10$ cube. Whenever two unit cubes meet face to face, these two faces have the same colour. Prove that an entire face of the $10 \times 10 \times 10$ cube has the same colour.
[i](6 points)[/i]
2011 Today's Calculation Of Integral, 768
Let $r$ be a real such that $0<r\leq 1$. Denote by $V(r)$ the volume of the solid formed by all points of $(x,\ y,\ z)$ satisfying
\[x^2+y^2+z^2\leq 1,\ x^2+y^2\leq r^2\]
in $xyz$-space.
(1) Find $V(r)$.
(2) Find $\lim_{r\rightarrow 1-0} \frac{V(1)-V(r)}{(1-r)^{\frac 32}}.$
(3) Find $\lim_{r\rightarrow +0} \frac{V(r)}{r^2}.$
2012 AMC 12/AHSME, 19
A unit cube has vertices $P_1, P_2, P_3, P_4, P_1', P_2', P_3'$, and $P_4'$. Vertices $P_2, P_3$, and $P_4$ are adjacent to $P_1$, and for $1\leq i\leq 4$, vertices $P_i$ and $P_i'$ are opposite to each other. A regular octahedron has one vertex in each of the segments $P_1P_2, P_1P_3, P_1P_4, P_1'P_2', P_1'P_3'$, and $P_1'P_4'$. What is the octahedron's side length?
[asy]
import three;
size(7.5cm);
triple eye = (-4, -8, 3);
currentprojection = perspective(eye);
triple[] P = {(1, -1, -1), (-1, -1, -1), (-1, 1, -1), (-1, -1, 1), (1, -1, -1)}; // P[0] = P[4] for convenience
triple[] Pp = {-P[0], -P[1], -P[2], -P[3], -P[4]};
// draw octahedron
triple pt(int k){ return (3*P[k] + P[1])/4; }
triple ptp(int k){ return (3*Pp[k] + Pp[1])/4; }
draw(pt(2)--pt(3)--pt(4)--cycle, gray(0.6));
draw(ptp(2)--pt(3)--ptp(4)--cycle, gray(0.6));
draw(ptp(2)--pt(4), gray(0.6));
draw(pt(2)--ptp(4), gray(0.6));
draw(pt(4)--ptp(3)--pt(2), gray(0.6) + linetype("4 4"));
draw(ptp(4)--ptp(3)--ptp(2), gray(0.6) + linetype("4 4"));
// draw cube
for(int i = 0; i < 4; ++i){
draw(P[1]--P[i]); draw(Pp[1]--Pp[i]);
for(int j = 0; j < 4; ++j){
if(i == 1 || j == 1 || i == j) continue;
draw(P[i]--Pp[j]); draw(Pp[i]--P[j]);
}
dot(P[i]); dot(Pp[i]);
dot(pt(i)); dot(ptp(i));
}
label("$P_1$", P[1], dir(P[1]));
label("$P_2$", P[2], dir(P[2]));
label("$P_3$", P[3], dir(-45));
label("$P_4$", P[4], dir(P[4]));
label("$P'_1$", Pp[1], dir(Pp[1]));
label("$P'_2$", Pp[2], dir(Pp[2]));
label("$P'_3$", Pp[3], dir(-100));
label("$P'_4$", Pp[4], dir(Pp[4]));
[/asy]
$ \textbf{(A)}\ \frac{3\sqrt{2}}{4}\qquad\textbf{(B)}\ \frac{7\sqrt{6}}{16}\qquad\textbf{(C)}\ \frac{\sqrt{5}}{2}\qquad\textbf{(D)}\ \frac{2\sqrt{3}}{3}\qquad\textbf{(E)}\ \frac{\sqrt{6}}{2} $