Let $VABC$ be a regular triangular pyramid with base $ABC$, of center $O$. Points $I$ and $H$ are the center of the inscribed circle, respectively the orthocenter $\vartriangle VBC$. Knowing that $AH = 3 OI$, determine the measure of the angle between the lateral edge of the pyramid and the plane of the base.

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.

A cuboctahedron is a solid with 6 square faces and 8 equilateral triangle faces, with each edge adjacent to both a square and a triangle (see picture). Suppose the ratio of the volume of an octahedron to a cuboctahedron with the same side length is $r$. Find $100r^2$. [asy] // dragon96, replacing // [img]http://i.imgur.com/08FbQs.png[/img] size(140); defaultpen(linewidth(.7)); real alpha=10, x=-0.12, y=0.025, r=1/sqrt(3); path hex=rotate(alpha)*polygon(6); pair A = shift(x,y)*(r*dir(330+alpha)), B = shift(x,y)*(r*dir(90+alpha)), C = shift(x,y)*(r*dir(210+alpha)); pair X = (-A.x, -A.y), Y = (-B.x, -B.y), Z = (-C.x, -C.y); int i; pair[] H; for(i=0; i<6; i=i+1) { H[i] = dir(alpha+60*i);} fill(X--Y--Z--cycle, rgb(204,255,255)); fill(H[5]--Y--Z--H[0]--cycle^^H[2]--H[3]--X--cycle, rgb(203,153,255)); fill(H[1]--Z--X--H[2]--cycle^^H[4]--H[5]--Y--cycle, rgb(255,203,153)); fill(H[3]--X--Y--H[4]--cycle^^H[0]--H[1]--Z--cycle, rgb(153,203,255)); draw(hex^^X--Y--Z--cycle); draw(H[1]--B--H[2]^^H[3]--C--H[4]^^H[5]--A--H[0]^^A--B--C--cycle, linewidth(0.6)+linetype("5 5")); draw(H[0]--Z--H[1]^^H[2]--X--H[3]^^H[4]--Y--H[5]);[/asy]

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.

Prove that for any positive integer $n$, the number of odd integers among the binomial coefficients $\binom nh \ ( 0 \leq h \leq n)$ is a power of 2.

Let $VA_1A_2A_3A_4$ be a pyramid with the vertex at $V$. Let $M,\, N,\, P$ be the midpoints of the segments $VA_1$, $VA_3$, and $A_2A_4$. Show that the plane $(MNP)$ cuts the pyramid into two parts with the same volume. [i]Radu Gologan[/i]

Let $TABCD$ be a pyramid with top vertex $T$, such that its base $ABCD$ is a square of side length 4. It is given that, among the triangles $TAB$, $TBC$, $TCD$ and $TDA$, one can find an isosceles triangle and a right-angled triangle. Find all possible values for the volume of the pyramid.

A regular hexagon with sides of length $6$ has an isosceles triangle attached to each side. Each of these triangles has two sides of length $8$. The isosceles triangles are folded to make a pyramid with the hexagon as the base of the pyramid. What is the volume of the pyramid? $\textbf{(A) }18\qquad\textbf{(B) }162\qquad\textbf{(C) }36\sqrt{21}\qquad\textbf{(D) }18\sqrt{138}\qquad\textbf{(E) }54\sqrt{21}$

Prove the following statement: There does not exist a pyramid with square base and congruent lateral faces for which the measures of all edges, total area, and volume are integers.

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 $$R_1R_2h^2 = (R_1^2-O_1T_1^2)(R_2^2-O_2T_2^2).$$

Let $ABCDS$ be a pyramid with four faces and with $ABCD$ as a base, and let a plane $\alpha$ through the vertex $A$ meet its edges $SB$ and $SD$ at points $M$ and $N$, respectively. Prove that if the intersection of the plane $\alpha$ with the pyramid $ABCDS$ is a parallelogram, then $SM \cdot SN > BM \cdot DN$.

In the pyramid $SA_1A_2 \cdots A_n$, all sides are equal. Let point $X_i$ be the midpoint of arc $A_iA_{i+1}$ in the circumcircle of $\triangle SA_iA_{i+1}$ for $1 \le i \le n$ with indices taken mod $n$. Prove that the circumcircles of $X_1A_2X_2, X_2A_3X_3, \cdots, X_nA_1X_1$ have a common point.

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.

Let $E$ be a finite set of points such that $E$ is not contained in a plane and no three points of $E$ are collinear. Show that at least one of the following alternatives holds: (i) $E$ contains five points that are vertices of a convex pyramid having no other points in common with $E;$ (ii) some plane contains exactly three points from $E.$

How many planes of symmetry can a triangular pyramid have?