The base of a right pyramid is a quadrilateral whose sides are each of length $a$. The planar angles at the vertex of the pyramid are equal to the angles between the lateral edges and the base. Find the volume of the pyramid.

A pyramid has square base and equal sides. It is cut into two parts by a plane parallel to the base. The lower part (which has square top and square base) is such that the circumcircle of the base is smaller than the circumcircles of the lateral faces. Show that the shortest path on the surface joining the two endpoints of a spatial diagonal lies entirely on the lateral faces. [img]https://cdn.artofproblemsolving.com/attachments/c/8/170bec826d5e40308cfd7360725d2aba250bf6.png[/img]

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 $PQA$ ?

There is an infinitely tall tetrahedral stack of spheres where each row of the tetrahedron consists of a triangular arrangement of spheres, as shown below. There is $1$ sphere in the top row (which we will call row $0$), $3$ spheres in row $1$, $6$ spheres in row $2$, $10$ spheres in row $3$, etc. The top-most sphere in row $0$ is assigned the number $1$. We then assign each other sphere the sum of the number(s) assigned to the sphere(s) which touch it in the row directly above it. Find a simplified expression in terms of $n$ for the sum of the numbers assigned to each sphere from row $0$ to row $n$. [asy] import three; import solids; size(8cm); //currentprojection = perspective(1, 1, 10); triple backright = (-2, 0, 0), backleft = (-1, -sqrt(3), 0), backup = (-1, -sqrt(3) / 3, 2 * sqrt(6) / 3); draw(shift(2 * backleft) * surface(sphere(1,20)), white); //2 draw(shift(backleft + backright) * surface(sphere(1,20)), white); //2 draw(shift(2 * backright) * surface(sphere(1,20)), white); //3 draw(shift(backup + backleft) * surface(sphere(1,20)), white); draw(shift(backup + backright) * surface(sphere(1,20)), white); draw(shift(2 * backup) * surface(sphere(1,20)), white); draw(shift(backleft) * surface(sphere(1,20)), white); draw(shift(backright) * surface(sphere(1,20)), white); draw(shift(backup) * surface(sphere(1,20)), white); draw(surface(sphere(1,20)), white); label("Row 0", 2 * backup, 15 * dir(20)); label("Row 1", backup, 25 * dir(20)); label("Row 2", O, 35 * dir(20)); dot(-backup); dot(-7 * backup / 8); dot(-6 * backup / 8); dot((backleft - backup) + backleft * 2); dot(5 * (backleft - backup) / 4 + backleft * 2); dot(6 * (backleft - backup) / 4 + backleft * 2); dot((backright - backup) + backright * 2); dot(5 * (backright - backup) / 4 + backright * 2); dot(6 * (backright - backup) / 4 + backright * 2); [/asy]

The pyramid $VABCD$ has base the rectangle ABCD, and the side edges are congruent. Prove that the plane $(VCD)$ forms congruent angles with the planes $(VAC)$ and $(BAC)$ if and only if $\angle VAC = \angle BAC $.

A regular $n$-gonal truncated pyramid is circumscribed around a sphere. Denote the areas of the base and the lateral surfaces of the pyramid by $S_1, S_2$, and $S$, respectively. Let $\sigma$ be the area of the polygon whose vertices are the tangential points of the sphere and the lateral faces of the pyramid. Prove that \[\sigma S = 4S_1S_2 \cos^2 \frac{\pi}{n}.\]

The height $SO$ of a regular quadrangular pyramid $SABCD$ forms an angle $60^o$ with a side edge , the volume of this pyramid is equal to $18$ cm$^3$ . The vertex of the second regular quadrangular pyramid is at point $S$, the center of the base is at point $C$, and one of the vertices of the base lies on the line $SO$. Find the volume of the common part of these pyramids. (The common part of the pyramids is the set of all such points in space that lie inside or on the surface of both pyramids).

Points $P_1,P_2,P_3,P_4$ are vertexes of a regular triangular pyramid, and $P_5,P_6,P_7,P_8,P_9,P_{10}$ midpoints of edges. The number of groups $(P_1,P_i,P_j,P_k)(1<i<j<k\leq10)$ that $P_1,P_i,P_j,P_k$ are coplane is________.

Show that if all lateral edges of a pentagonal pyramid are of equal length and all the angles between neighboring lateral faces are equal, then the pyramid is regular.

Given a triangular pyramid in which all the plane angles at one of the vertices are right. It is known that there is a point in space located at a distance of $3$ from the indicated vertex and at distances $\sqrt5, \sqrt6, \sqrt7$ from three other vertices. Find the radius of the sphere circumscribed around this pyramid. (The circumscribed sphere for a pyramid is the sphere containing all its vertices.)

Consider a pyramid $P-ABCD$ whose base $ABCD$ is a square and whose vertex $P$ is equidistant from $A$, $B$, $C$, and $D$. If $AB=1$ and $\angle APD=2\theta$ then the volume of the pyramid is $\text{(A)} \ \frac{\sin \theta}{6} \qquad \text{(B)} \ \frac{\cot \theta}{6} \qquad \text{(C)} \ \frac1{6\sin \theta} \qquad \text{(D)} \ \frac{1-\sin 2\theta}{6} \qquad \text{(E)} \ \frac{\sqrt{\cos 2\theta}}{6\sin \theta}$

The four faces of a right triangular pyramid are equilateral triangles whose edge measures $3$ dm. Suppose the pyramid is hollow, resting on one of its faces at a horizontal surface (see attached figure) and that there is $2$ dm$^3$ of water inside. Determine the height that the liquid reaches inside the pyramid. [img]https://cdn.artofproblemsolving.com/attachments/0/7/6cd6e1077620371e56ed57d19fd3d05a58904e.png[/img]

The lateral edges of a right square pyramid are of length $a$. Let $ABCD$ be the base of the pyramid, $E$ its top vertex and $F$ the midpoint of $CE$. Assuming that $BDF$ is an equilateral triangle, compute the volume of the pyramid.

Given is a pyramid $SA_1A_2A_3\ldots A_n$ whose base is convex polygon $A_1A_2A_3\ldots A_n$. For every $i=1,2,3,\ldots ,n$ there is a triangle $X_iA_iA_{i+1} $ congruent to triangle $SA_iA_{i+1}$ that lies on the same side from $A_iA_{i+1}$ as the base of that pyramid. (You can assume $a_1$ is the same as $a_{n+1}$.) Prove that these triangles together cover the entire base.

Consider a regular triangular pyramid with base $\vartriangle ABC$ and apex $D$. If we have $AB = BC =AC = 6$ and $AD = BD = CD = 4$, calculate the surface area of the circumsphere of the pyramid.

A square piece of paper has sides of length $ 100$. From each corner a wedge is cut in the following manner: at each corner, the two cuts for the wedge each start at distance $ \sqrt {17}$ from the corner, and they meet on the diagonal at an angle of $ 60^\circ$ (see the figure below). The paper is then folded up along the lines joining the vertices of adjacent cuts. When the two edges of a cut meet, they are taped together. The result is a paper tray whose sides are not at right angles to the base. The height of the tray, that is, the perpendicular distance between the plane of the base and the plane formed by the upper edges, can be written in the form $ \sqrt [n]{m}$, where $ m$ and $ n$ are positive integers, $ m < 1000$, and $ m$ is not divisible by the $ n$th power of any prime. Find $ m \plus{} n$. [asy]import math; unitsize(5mm); defaultpen(fontsize(9pt)+Helvetica()+linewidth(0.7)); pair O=(0,0); pair A=(0,sqrt(17)); pair B=(sqrt(17),0); pair C=shift(sqrt(17),0)*(sqrt(34)*dir(75)); pair D=(xpart(C),8); pair E=(8,ypart(C)); draw(O--(0,8)); draw(O--(8,0)); draw(O--C); draw(A--C--B); draw(D--C--E); label("$\sqrt{17}$",(0,2),W); label("$\sqrt{17}$",(2,0),S); label("cut",midpoint(A--C),NNW); label("cut",midpoint(B--C),ESE); label("fold",midpoint(C--D),W); label("fold",midpoint(C--E),S); label("$30^\circ$",shift(-0.6,-0.6)*C,WSW); label("$30^\circ$",shift(-1.2,-1.2)*C,SSE);[/asy]

(I.Bogdanov) A section of a regular tetragonal pyramid is a regular pentagon. Find the ratio of its side to the side of the base of the pyramid.

Let $ABCDE$ be a right quadrangular pyramid with vertex $E$ and height $EO$. Point $S$ divides this height in the ratio $ES: SO=m$. In which ratio does the plane $(ABC)$ divide the lateral area of the pyramid.

A truncated trigonal pyramid is circumscribed around a sphere touching its bases at points $T_1, T_2$. Let $h$ be the altitude of the pyramid, $R_1, R_2$ be the circumradii of its bases, and $O_1, O_2$ be the circumcenters of the bases. Prove that $$R_1R_2h^2 = (R_1^2-O_1T_1^2)(R_2^2-O_2T_2^2).$$

The base of a pyramid is a regular triangle having side of size $1$. Two of three angles at the vertex of the pyramid are right. Find the maximum value of the volume of the pyramid.

The base $A_{1}A_{2}\ldots A_{n}$ of the pyramid $VA_{1}A_{2}\ldots A_{n}$ is a regular polygon. Prove that if \[\angle VA_{1}A_{2}\equiv \angle VA_{2}A_{3}\equiv \cdots \equiv \angle VA_{n-1}A_{n}\equiv \angle VA_{n}A_{1},\] then the pyramid is regular.

Assume that a face of a convex polyhedron $ P$ has a common edge with every other face. Show that there exists a simple closed polygon that consists of edges of $ P$ and passes through all vertices. [i]L .Lovasz[/i]

Find all convex polyhedra which have no diagonals (that is, for which every segment connecting two vertices lies on the boundary of the polyhedron).

A convex polyhedron $ Q$ has vertices $ V_1,V_2,\ldots,V_n$, and $ 100$ edges. The polyhedron is cut by planes $ P_1,P_2,\ldots,P_n$ in such a way that plane $ P_k$ cuts only those edges that meet at vertex $ V_k$. In addition, no two planes intersect inside or on $ Q$. The cuts produce $ n$ pyramids and a new polyhedron $ R$. How many edges does $ R$ have? $ \textbf{(A)}\ 200\qquad \textbf{(B)}\ 2n\qquad \textbf{(C)}\ 300\qquad \textbf{(D)}\ 400\qquad \textbf{(E)}\ 4n$

Let $ABCD$ be a regular triangular pyramid with base $ABC$ (this means that $ABC$ is a regular triangle, and edges $AD$, $BD$ and $CD$ are equal) and plane angles at the opposite vertex equal to $a$. A plane parallel to $ABC$ intersects $AD$, $BD$ and $CD$, respectively, at points $A_1$, $B_1$ and $C_1$. The surface of the polyhedron $ABCA_1B_1C_1$ is cut along five edges: $A_1B_1$, $B_1C_1$, $C_1C$, $CA$ and $AB$, after which this surface is turned onto a plane. At what values of $a$ will the resulting scan necessarily cover itself?