Found problems: 2265
Two regular tetrahedrons $A$ and $B$ are made with the 8 vertices of a unit cube. (this way is unique)
What's the volume of $A\cup B$?
[asy]
import olympiad;
import cse5;
size(5cm);
pointpen = black;
pair A = Drawing((10,17.32));
pair B = Drawing((0,0));
pair C = Drawing((20,0));
draw(A--B--C--cycle);
pair X = 0.85*A + 0.15*B;
pair Y = 0.82*A + 0.18*C;
pair W = (-11,0) + X;
pair Z = (19, 9);
draw(W--X, EndArrow);
draw(X--Y, EndArrow);
draw(Y--Z, EndArrow);
anglepen=black; anglefontpen=black;
MarkAngle("\theta", C,Y,Z, 3);
[/asy]
The cross-section of a prism with index of refraction $1.5$ is an equilateral triangle, as shown above. A ray of light comes in horizontally from air into the prism, and has the opportunity to leave the prism, at an angle $\theta$ with respect to the surface of the triangle. Find $\theta$ in degrees and round to the nearest whole number.
[i](Ahaan Rungta, 5 points)[/i]
A steel cube has edges of length $3$ cm, and a cone has a diameter of $8$ cm and a height of $24$ cm. The cube is placed in the cone so that one of its interior diagonals coincides with the axis of the cone. What is the distance, in cm, between the vertex of the cone and the closest vertex of the cube?
[NEEDS DIAGRAM]
$ \mathrm{(A) \ } \frac{12\sqrt6-3\sqrt3}{4} \qquad \mathrm{(B) \ } \frac{9\sqrt6-3\sqrt3}{2} \qquad \mathrm {(C) \ } 5\sqrt3 \qquad \mathrm{(D) \ } 6\sqrt6 - \sqrt3 \qquad \mathrm{(E) \ } 6\sqrt6$
Find all integers $n \ge 3$ with the following property:
it is possible to assign pairwise different positive integers to the vertices of an $n$-gonal prism in such a way that vertices with labels $a$ and $b$ are connected by an edge if and only if $a | b$ or $b | a$.
Poland
Bottom surface of triangular pyramid $S-ABC$ is an isosceles right triangle (hypotenuse is $AB$). $SA=SB=SC=AB=2$, and $S,A,B,C$ are on a sphere with center of $O$. The distance of $O$ to plane $ABC$ is________.
Is it true that for each $ n$, the regular $ 2n$-gon is a projection of some polyhedron having not greater than $ n \plus{} 2$ faces?
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]
A right circular cone has base of radius 1 and height 3. A cube is inscribed in the cone so that one face of the cube is contained in the base of the cone. What is the side-length of the cube?
There are $ n$ points ($ n \geq 4$) on a sphere with radius $ R$, and not all of them lie on the same semi-sphere. Prove that among all the angles formed by any two of the $ n$ points and the sphere centre $ O$ ($ O$ is the vertex of the angle), there is at least one that is not less than $ \displaystyle 2 \arcsin{\frac{\sqrt{6}}{3}}$.
Given any triangle $ABC$ and any positive integer $n$, we say that $n$ is a [i]decomposable[/i] number for triangle $ABC$ if there exists a decomposition of the triangle $ABC$ into $n$ subtriangles with each subtriangle similar to $\triangle ABC$. Determine the positive integers that are decomposable numbers for every triangle.
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.
On two intersecting lines $\ell_1$ and $\ell_2$, segments $AB$ and $CD$ of a given length are selected, respectively. Prove that the volume of the tetrahedron $ABCD$ does not depend on the position of the segments $AB$ and $CD$ on the lines $\ell_1$ and $\ell_2$.
Show there do not exist four points in the Euclidean plane such that the pairwise distances between the points are all odd integers.
[b]p1.[/b] Each point in the interior and on the boundary of a square of side $2$ inches is colored either red or blue. Prove that there exists at least one pair of points of the same color whose distance apart is not less than $-\sqrt5$ inches.
[b]p2.[/b] $ABC$ is an equilateral triangle of altitude $h$. A circle with center $0$ and radius $h$ is tangent to side $AB$ at $Z$ and intersects side $AC$ in point $X$ and side $BC$ in point $Y$. Prove that the circular arc $XZY$ has measure $60^o$.
[img]https://cdn.artofproblemsolving.com/attachments/b/e/ac70942f7a14cd0759ac682c3af3551687dd69.png[/img]
[b]p3.[/b] Find all of the real and complex solutions (if any exist) of the equation $x^7 + 7^7 = (x + 7)^7$
[b]p4.[/b] The four points $A, B, C$, and $D$ are not in the same plane. Given that the three angles, angle $ABC$, angle $BCD$, and angle $CDA$, are all right angles, prove that the fourth angle, angle $DAB$, of this skew quadrilateral is acute.
[b]p5.[/b] $A, B, C$ and $D$ are four positive whole numbers with the following properties:
(i) each is less than the sum of the other three, and
(ii) each is a factor of the sum of the other three.
Prove that at least two of the numbers must be equal.
(An example of four such numbers: $A = 4$, $B = 4$, $C = 2$, $D = 2$.)
[b]p6.[/b] $S$ is a set of six points and $L$ is a set of straight line segments connecting certain pairs of points in $S$ so that each point of $S$ is connected with at least four of the other points. Let $A$ and $B$ denote two arbitrary points of $S$. Show that among the triangles having sides in $L$ and vertices in $S$ there are two with the properties:
(i) The two triangles have no common vertex.
(ii) $A$ is a vertex of one of the triangles, and $B$ is a vertex of the other.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
A concrete sewer pipe fitting is shaped like a cylinder with diameter $48$ with a cone on top. A cylindrical hole of diameter $30$ is bored all the way through the center of the fitting as shown. The cylindrical portion has height $60$ while the conical top portion has height $20$. Find $N$ such that the volume of the concrete is $N \pi$.
[asy]
import three;
size(250);
defaultpen(linewidth(0.7)+fontsize(10)); pen dashes = linewidth(0.7) + linetype("2 2");
currentprojection = orthographic(0,-15,5);
draw(circle((0,0,0), 15),dashes);
draw(circle((0,0,80), 15));
draw(scale3(24)*((-1,0,0)..(0,-1,0)..(1,0,0)));
draw(shift((0,0,60))*scale3(24)*((-1,0,0)..(0,-1,0)..(1,0,0)));
draw((-24,0,0)--(-24,0,60)--(-15,0,80)); draw((24,0,0)--(24,0,60)--(15,0,80));
draw((-15,0,0)--(-15,0,80),dashes); draw((15,0,0)--(15,0,80),dashes);
draw("48", (-24,0,-20)--(24,0,-20));
draw((-15,0,-20)--(-15,0,-17)); draw((15,0,-20)--(15,0,-17));
label("30", (0,0,-15));
draw("60", (50,0,0)--(50,0,60));
draw("20", (50,0,60)--(50,0,80));
draw((50,0,60)--(47,0,60));[/asy]
Pyramid $EARLY$ is placed in $(x,y,z)$ coordinates so that $E=(10,10,0),A=(10,-10,0)$, $R=(-10,-10,0)$, $L=(-10,10,0)$, and $Y=(0,0,10)$. Tunnels are drilled through the pyramid in such a way that one can move from $(x,y,z)$ to any of the $9$ points $(x,y,z-1)$, $(x\pm 1,y,z-1)$, $(x,y\pm 1, z-1)$, $(x\pm 1, y\pm 1, z-1)$. Sean starts at $Y$ and moves randomly down to the base of the pyramid, choosing each of the possible paths with probability $\dfrac{1}{9}$. What is the probability that he ends up at the point $(8,9,0)$?
In the tetrahedron $ABCD,\angle BDC=90^o$ and the foot of the perpendicular from $D$ to $ABC$ is the intersection of the altitudes of $ABC$. Prove that: \[ (AB+BC+CA)^2\le6(AD^2+BD^2+CD^2). \] When do we have equality?
Each face of a cube is painted either red or blue, each with probability $ 1/2$. The color of each face is determined independently. What is the probability that the painted cube can be placed on a horizontal surface so that the four vertical faces are all the same color?
$ \textbf{(A)}\ \frac14 \qquad
\textbf{(B)}\ \frac{5}{16} \qquad
\textbf{(C)}\ \frac38 \qquad
\textbf{(D)}\ \frac{7}{16} \qquad
\textbf{(E)}\ \frac12$
A fly and an ant are on one corner of a unit cube. They wish to head to the opposite corner of the cube. The fly can fly through the interior of the cube, while the ant has to walk across the faces of the cube. How much shorter is the fly's path if both insects take the shortest path possible?
A $100 \times 100 \times 100$ cube is divided into a million unit cubes and in each small cube there is a light bulb. Three faces $100 \times 100$ of the large cube having a common vertex are painted: one in red, one in blue and the other in green. Call a $\textit{column}$ a set of $100$ cubes forming a block $1 \times 1 \times 100$. Each of the $30 000$ columns have one painted end cell, on which there is a switch. After pressing a switch, the states of all light bulbs of this column are changed. Petya pressed several switches, getting a situation with exactly $k$ lamps on. Prove that Vasya can press several switches so that all lamps are off, but by using no more than $\frac {k} {100}$ switches on the red face.
Given a tetrahedron such that product of the opposite edges is $ 1$. Let the angle between the opposite edges be $ \alpha$, $ \beta$, $ \gamma$, and circumradii of four faces be $ R_1$, $ R_2$, $ R_3$, $ R_4$. Prove that
\[ \sin^2\alpha \plus{} \sin^2\beta \plus{} \sin^2\gamma\ge\frac {1}{\sqrt {R_1R_2R_3R_4}}
\]
Prove this proposition: Center the sphere circumscribed around a tetrahedron which coincides with the center of a sphere inscribed in that tetrahedron if and only if the skew edges of the tetrahedron are equal.
Two circles in space are said to be tangent to each other if they have a corni-non tangent at the same point of tangency. Assume that there are three circles in space which are mutually tangent at three distinct points. Prove that they either alI lie in a plane or all lie on a sphere.
Each diagonal of the base and each lateral edge of a $9$-gonal pyramid is colored either green or red. Show that there must exist a triangle with the vertices at vertices of the pyramid having all three sides of the same color.
Find the maximal positive number $r$ with the following property: If all altitudes of a tetrahedron are $\geq 1$, then a sphere of radius $r$ fits into the tetrahedron.