Found problems: 2265
The faces of a cube contain the number 1, 2, 3, 4, 5, 6 such that the sum of the numbers on each pair of opposite faces is 7. For each of the cube’s eight corners, we multiply the three numbers on the faces incident to that corner, and write down its value. (In the diagram, the value of the indicated corner is 1 x 2 x 3 = 6.) What is the sum of the eight values assigned to the cube’s corners?
$ABCD$ is a rectangle. The segment $MA$ is perpendicular to plane $ABC$ . $MB= 15$ , $MC=24$ , $MD=20$. Find the length of $MA$ .
Two planes, $P$ and $Q$, intersect along the line $p$. The point $A$ is given in the plane $P$, and the point $C$ in the plane $Q$; neither of these points lies on the straight line $p$. Construct an isosceles trapezoid $ABCD$ (with $AB \parallel CD$) in which a circle can be inscribed, and with vertices $B$ and $D$ lying in planes $P$ and $Q$ respectively.
The plane $\alpha$ intersects the edges $AB$, $BC$, $CD$ and $DA$ of the tetrahedron $ABCD$ at points $X, Y, Z$ and $T$, respectively. It turned out, that points $Y$ and $T$ lie on a circle $\omega$ constructed with segment $XZ$ as the diameter. Point $P$ is marked in the plane $\alpha$ so that the lines $P Y$ and $P T$ are tangent to the circle $\omega$.Prove that the midpoints of the edges are $AB$, $BC$, $CD,$ $DA$ and the point $P$ lie in the same plane.
The segments $[AD], [BE]$ and $[CF]$ are the side edges of the right triangle prism. (the equilateral triangle is a base) Find all the points in its base $ABC$, situated on the equal distances from the $(AE), (BF)$ and $(CD)$ lines.
Let skew lines $PM, QN$ be given such that $PM\perp PQ\perp QN$. Let a plane $\sigma\perp PQ$ containing the midpoint $O$ of segment $PQ$ be given and in it a circle $k$ with center $O$ and given radius $r$. Consider all segments $XY$ with endpoint $X, Y$ on lines $PM, QN$, respectively, which contain a point of $k$. Show that segments $XY$ have the same length. Find the locus of all such points $X$.
Some unit cubes are stacked atop a flat 4 by 4 square. The figures show views of the stacks from two different sides. Find the maximum and minimum number of cubes that could be in the stacks. Also give top views of a maximum arrangement and a minimum arrangement with each stack marked with its height.
[asy]
string s = "1010101010111111";
defaultpen(linewidth(0.7));
for(int x=0;x<4;++x) {
for(int y=0;y<4;++y) {
if(hex(substr(s,4*(3-y)+x,1))==1) {
draw((x,y)--(x,y+1)--(x+1,y+1)--(x+1,y)--cycle);
}
}}
label("South View",(2,4),N);
s = "0101110111111111";
for(int x=0;x<4;++x) {
for(int y=0;y<4;++y) {
if(hex(substr(s,4*(3-y)+x,1))==1) {
x=x+5;
draw((x,y)--(x,y+1)--(x+1,y+1)--(x+1,y)--cycle);
x=x-5;
}
}}
label("East View",(7,4),N);[/asy]
Consider a pyramid whose base is an equilateral triangle $BCD$ and whose other faces are triangles isosceles, right at the common vertex $A$. An ant leaves the vertex $B$ arrives at a point $P$ of the $CD$ edge, from there goes to a point $Q$ of the edge $AC$ and returns to point $B$. If the path you made is minimal, how much is the angle $PQA$ ?
A right circular cone has for its base a circle having the same radius as a given sphere. The volume of the cone is one-half that of the sphere. The ratio of the altitude of the cone to the radius of its base is:
$ \textbf{(A)}\ \frac{1}{1} \qquad
\textbf{(B)}\ \frac{1}{2} \qquad
\textbf{(C)}\ \frac{2}{3} \qquad
\textbf{(D)}\ \frac{2}{1} \qquad
\textbf{(E)}\ \sqrt{\frac{5}{4}}$
The base of the quadrilateral pyramid $SABCD$ lies the $ABCD$ rectangle with the sides $AB = 1$ and $AD =
10$. The edge $SA$ of the pyramid is perpendicular to the base, $SA = 4$. On the edge of $AD$, find a point $M$ such that the perimeter of the triangle of $SMC$ was minimal.
A polygonal line connects two opposite vertices of a cube with side $2$. Each segment of the line has length $3$ and each vertex lies on the faces (or edges) of the cube. What is the smallest number of segments the line can have?
Prove that we can put in any arbitrary triangle with sidelengths $a,b,c$ such that $0\le a,b,c \le \sqrt2$ into a unit cube.
Six segments $S_1, S_2, S_3, S_4, S_5,$ and $S_6$ are given in a plane. These are congruent to the edges $AB, AC, AD, BC, BD,$ and $CD$, respectively, of a tetrahedron $ABCD$. Show how to construct a segment congruent to the altitude of the tetrahedron from vertex $A$ with straight-edge and compasses.
$605$ spheres of same radius are divided in two parts. From one part, upright "pyramid" is made with square base. From the other part, upright "pyramid" is made with equilateral triangle base. Both "pyramids" are put together from equal numbers of sphere rows. Find number of spheres in every "pyramid"
As shown in the figure below a regular dodecahedron (the polyhedron consisting of 12 congruent regular pentagonal faces) floats in space with two horizontal faces. Note that there is a ring of five slanted faces adjacent to the top face, and a ring of five slanted faces adjacent to the bottom face. How many ways are there to move from the top face to the bottom face via a sequence of adjacent faces so that each face is visited at most once and moves are not permitted from the bottom ring to the top ring?
[asy]
import graph;
unitsize(4.5cm);
pair A = (0.082, 0.378);
pair B = (0.091, 0.649);
pair C = (0.249, 0.899);
pair D = (0.479, 0.939);
pair E = (0.758, 0.893);
pair F = (0.862, 0.658);
pair G = (0.924, 0.403);
pair H = (0.747, 0.194);
pair I = (0.526, 0.075);
pair J = (0.251, 0.170);
pair K = (0.568, 0.234);
pair L = (0.262, 0.449);
pair M = (0.373, 0.813);
pair N = (0.731, 0.813);
pair O = (0.851, 0.461);
path[] f;
f[0] = A--B--C--M--L--cycle;
f[1] = C--D--E--N--M--cycle;
f[2] = E--F--G--O--N--cycle;
f[3] = G--H--I--K--O--cycle;
f[4] = I--J--A--L--K--cycle;
f[5] = K--L--M--N--O--cycle;
draw(f[0]);
axialshade(f[1], white, M, gray(0.5), (C+2*D)/3);
draw(f[1]);
filldraw(f[2], gray);
filldraw(f[3], gray);
axialshade(f[4], white, L, gray(0.7), J);
draw(f[4]);
draw(f[5]);
[/asy]
$\textbf{(A) } 125 \qquad \textbf{(B) } 250 \qquad \textbf{(C) } 405 \qquad \textbf{(D) } 640 \qquad \textbf{(E) } 810$
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]
The edges of a regular tetrahedron with vertices $A ,~ B,~ C$, and $D$ each have length one. Find the least possible distance between a pair of points $P$ and $Q$, where $P$ is on edge $AB$ and $Q$ is on edge $CD$.
$\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{3}{4}\qquad\textbf{(C) }\frac{\sqrt{2}}{2}\qquad\textbf{(D) }\frac{\sqrt{3}}{2}\qquad\textbf{(E) }\frac{\sqrt{3}}{3}$
[asy]
size(150);
import patterns;
pair D=(0,0),C=(1,-1),B=(2.5,-0.2),A=(1,2),AA,BB,CC,DD,P,Q,aux;
add("hatch",hatch());
//AA=new A and etc.
draw(rotate(100,D)*(A--B--C--D--cycle));
AA=rotate(100,D)*A;
BB=rotate(100,D)*D;
CC=rotate(100,D)*C;
DD=rotate(100,D)*B;
aux=midpoint(AA--BB);
draw(BB--DD);
P=midpoint(AA--aux);
aux=midpoint(CC--DD);
Q=midpoint(CC--aux);
draw(AA--CC,dashed);
dot(P);
dot(Q);
fill(DD--BB--CC--cycle,pattern("hatch"));
label("$A$",AA,W);
label("$B$",BB,S);
label("$C$",CC,E);
label("$D$",DD,N);
label("$P$",P,S);
label("$Q$",Q,E);
//Credit to TheMaskedMagician for the diagram
[/asy]
An artist has $14$ cubes, each with an edge of $1$ meter. She stands them on the ground to form a sculpture as shown. She then paints the exposed surface of the sculpture. How many square meters does she paint?
$\text{(A)}\ 21 \qquad \text{(B)}\ 24 \qquad \text{(C)}\ 33 \qquad \text{(D)}\ 37 \qquad \text{(E)}\ 42$
[asy]
draw((0,0)--(2.35,-.15)--(2.44,.81)--(.09,.96)--cycle);
draw((.783333333,-.05)--(.873333333,.91)--(1.135,1.135));
draw((1.566666667,-.1)--(1.656666667,.86)--(1.89,1.1));
draw((2.35,-.15)--(4.3,1.5)--(4.39,2.46)--(2.44,.81));
draw((3,.4)--(3.09,1.36)--(2.61,1.4));
draw((3.65,.95)--(3.74,1.91)--(3.29,1.94));
draw((.09,.96)--(.76,1.49)--(.71,1.17)--(2.2,1.1)--(3.6,2.2)--(3.62,2.52)--(4.39,2.46));
draw((.76,1.49)--(.82,1.96)--(2.28,1.89)--(2.2,1.1));
draw((2.28,1.89)--(3.68,2.99)--(3.62,2.52));
draw((1.455,1.135)--(1.55,1.925)--(1.89,2.26));
draw((2.5,2.48)--(2.98,2.44)--(2.9,1.65));
draw((.82,1.96)--(1.55,2.6)--(1.51,2.3)--(2.2,2.26)--(2.9,2.8)--(2.93,3.05)--(3.68,2.99));
draw((1.55,2.6)--(1.59,3.09)--(2.28,3.05)--(2.2,2.26));
draw((2.28,3.05)--(2.98,3.59)--(2.93,3.05));
draw((1.59,3.09)--(2.29,3.63)--(2.98,3.59));
[/asy]
Let a tetrahedron $ ABCD$ and $ A',B',C',D'$ be the circumcenters of triangles $ BCD,CDA,DAB,ABC$ respectively. Denote planes $ (P_A),(P_B),(P_C),(P_D)$ be the planes which pass through $ A,B,C,D$ and perpendicular to $ C'D',D'A',A'B',B'C'$ respectively. Prove that these planes have a common point called $ I.$ If $ P$ is the center of the circumsphere of the tetrahedron, must this tetrahedron be regular?
Ayase randomly picks a number $x \in (0, 1]$ with uniform probability. He then draws the six points $(0, 0, 0),(x, 0, 0),(2x, 3x, 0),(5, 5, 2),(7, 3, 0),(9, 1, 4)$. If the expected value of the volume of the convex polyhedron formed by these six points can be written as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, find $m + n$
The planet Caramelo is a cube with a $1$ km edge. This planet is going to be wrapped with foam anti-gluttons in order to prevent the presence of greedy ships less than $500$ meters from the planet. What the minimum volume of foam that must surround the planet?
Using white cubes of side $1$, a prism (without holes) was assembled. The faces of the prism were painted black. It is known that the cubes left with exactly $4$ white faces are $20$ in total. Determine what the dimensions of the prism can be. Give all the possibilities.
Which pattern of identical squares could NOT be folded along the lines shown to form a cube?
[asy]
unitsize(12);
draw((0,0)--(0,-1)--(1,-1)--(1,-2)--(2,-2)--(2,-3)--(4,-3)--(4,-2)--(3,-2)--(3,-1)--(2,-1)--(2,0)--cycle);
draw((1,0)--(1,-1)--(2,-1)--(2,-2)--(3,-2)--(3,-3));
draw((7,0)--(8,0)--(8,-1)--(11,-1)--(11,-2)--(8,-2)--(8,-3)--(7,-3)--cycle);
draw((7,-1)--(8,-1)--(8,-2)--(7,-2));
draw((9,-1)--(9,-2));
draw((10,-1)--(10,-2));
draw((14,-1)--(15,-1)--(15,0)--(16,0)--(16,-1)--(18,-1)--(18,-2)--(17,-2)--(17,-3)--(16,-3)--(16,-2)--(14,-2)--cycle);
draw((15,-2)--(15,-1)--(16,-1)--(16,-2)--(17,-2)--(17,-1));
draw((21,-1)--(22,-1)--(22,0)--(23,0)--(23,-2)--(25,-2)--(25,-3)--(22,-3)--(22,-2)--(21,-2)--cycle);
draw((23,-1)--(22,-1)--(22,-2)--(23,-2)--(23,-3));
draw((24,-2)--(24,-3));
draw((28,-1)--(31,-1)--(31,0)--(32,0)--(32,-2)--(31,-2)--(31,-3)--(30,-3)--(30,-2)--(28,-2)--cycle);
draw((32,-1)--(31,-1)--(31,-2)--(30,-2)--(30,-1));
draw((29,-1)--(29,-2));
label("(A)",(0,-0.5),W);
label("(B)",(7,-0.5),W);
label("(C)",(14,-0.5),W);
label("(D)",(21,-0.5),W);
label("(E)",(28,-0.5),W);
[/asy]
Three times Dick's age plus Tom's age equals twice Harry's age. Double the cube of Harry's age is equal to three times the cube of Dick's age added to the cube of Tom's age. Their respective ages are relatively prime to each other. The sum of the squares of their ages is
$\textbf{(A) }42\qquad\textbf{(B) }46\qquad\textbf{(C) }122\qquad\textbf{(D) }290\qquad \textbf{(E) }326$
Prove that if $n$ can be written as $n=a^2+ab+b^2$, then also $7n$ can be written that way.