Let $SABCDE$ be a pyramid whose base $ABCDE$ is a regular pentagon and whose other faces are acute triangles. The altitudes from $S$ to the base sides dissect them into ten triangles, colored red and blue alternatingly. Prove that the sum of the squared areas of the red triangles is equal to the sum of the squared areas of the blue triangles.

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 $E.$

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________.

A quadrilateral pyramid is cut by a plane parallel to the base. Suppose that a sphere $S$ is circumscribed and a sphere $\Sigma$ inscribed in the obtained solid, and moreover that the line through the centers of these two spheres is perpendicular to the base of the pyramid. Show that the pyramid is regular.

A quadrilateral pyramid and a regular tetrahedron have edges that are all equal in length. They are glued together so that they have in common $1$ equilateral triangle . Prove that the resulting body has exactly $5$ sides.

Let $MABCD$ be a pyramid with the square $ABCD$ as the base, in which $MA=MD$, $MA^2+AB^2=MB^2$ and the area of $\triangle ADM$ is equal to $1$. Determine the radius of the largest ball that is contained in the given pyramid.

Circles $C_1$ and $C_2$ intersect at different points $A$ and $B$. The straight lines tangents to $C_1$ that pass through $A$ and $B$ intersect at $T$. Let $M$ be a point on $C_1$ that is out of $C_2$. The $MT$ line intersects $C_1$ at $C$ again, the $MA$ line intersects again to $C_2$ in $K$ and the line $AC$ intersects again to the circumference $C_2$ in $L$. Prove that the $MC$ line passes through the midpoint of the $KL$ segment.

$O$ is the center of the bottom surface of regular triangular pyramid $P-ABC$. A plane passes $O$ intersects line segment $PC$ at $S$, intersects the extended line of $PA,PB$ at $Q,R$, then $\frac{1}{|PQ|}+\frac{1}{|PR|}+\frac{1}{|PS|}$ $\text{(A)}$ has a maximum value, but no minumum value $\text{(B)}$ has a minumum value, but no maximum value $\text{(C)}$ has both minumum value and maximum value (different) $\text{(D)}$ is a fixed value

The centers of the faces of the right rectangular prism shown below are joined to create an octahedron, What is the volume of the octahedron? [asy] import three; size(2inch); currentprojection=orthographic(4,2,2); draw((0,0,0)--(0,0,3),dashed); draw((0,0,0)--(0,4,0),dashed); draw((0,0,0)--(5,0,0),dashed); draw((5,4,3)--(5,0,3)--(5,0,0)--(5,4,0)--(0,4,0)--(0,4,3)--(0,0,3)--(5,0,3)); draw((0,4,3)--(5,4,3)--(5,4,0)); label("3",(5,0,3)--(5,0,0),W); label("4",(5,0,0)--(5,4,0),S); label("5",(5,4,0)--(0,4,0),SE); [/asy] $\textbf{(A) } \dfrac{75}{12} \qquad\textbf{(B) } 10 \qquad\textbf{(C) } 12 \qquad\textbf{(D) } 10\sqrt2 \qquad\textbf{(E) } 15 $

In the rectangular parallelpiped shown, $AB = 3, BC= 1,$ and $CG = 2.$ Point $M$ is the midpoint of $\overline{FG}$. What is the volume of the rectangular pyramid with base $BCHE$ and apex $M$? [asy] size(250); defaultpen(fontsize(10pt)); pair A =origin; pair B = (4.75,0); pair E1=(0,3); pair F = (4.75,3); pair G = (5.95,4.2); pair C = (5.95,1.2); pair D = (1.2,1.2); pair H= (1.2,4.2); pair M = ((4.75+5.95)/2,3.6); draw(E1--M--H--E1--A--B--E1--F--B--M--C--G--H); draw(B--C); draw(F--G); draw(A--D--H--C--D,dashed); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,E); label("$D$",D,W); label("$E$",E1,W); label("$F$",F,SW); label("$G$",G,NE); label("$H$",H,NW); label("$M$",M,N); dot(A); dot(B); dot(E1); dot(F); dot(G); dot(C); dot(D); dot(H); dot(M); label("3",A/2+B/2,S); label("2",C/2+G/2,E); label("1",C/2+B/2,SE);[/asy] $\textbf{(A) } 1 \qquad \textbf{(B) } \frac{4}{3} \qquad \textbf{(C) } \frac{3}{2} \qquad \textbf{(D) } \frac{5}{3} \qquad \textbf{(E) } 2$

(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.

A pyramid with a square base is cut by a plane that is parallel to its base and is $ 2$ units from the base. The surface area of the smaller pyramid that is cut from the top is half the surface area of the original pyramid. What is the altitude of the original pyramid? $ \textbf{(A)}\ 2\qquad \textbf{(B)}\ 2 \plus{} \sqrt{2}\qquad \textbf{(C)}\ 1 \plus{} 2\sqrt{2}\qquad \textbf{(D)}\ 4\qquad \textbf{(E)}\ 4 \plus{} 2\sqrt{2}$

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]

In triangular pyramid $S-ABC$, any two of $SA,SB,SC$ are perpendicular. $M$ is the centre of gravity of $\triangle ABC$. $D$ is the midpoint of $AB$, line $DP//SC$. Prove: [b](a)[/b] $DP$ and $SM$ intersect. [b](b)[/b] $DP\cap SM=D'$, then $D'$ is the center of circumsphere of $S-ABC$.

Triangle $ ABC$, with sides of length $ 5$, $ 6$, and $ 7$, has one vertex on the positive $ x$-axis, one on the positive $ y$-axis, and one on the positive $ z$-axis. Let $ O$ be the origin. What is the volume of tetrahedron $ OABC$? $ \textbf{(A)}\ \sqrt{85} \qquad \textbf{(B)}\ \sqrt{90} \qquad \textbf{(C)}\ \sqrt{95} \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ \sqrt{105}$

A solid tetrahedron is sliced off a solid wooden unit cube by a plane passing through two nonadjacent vertices on one face and one vertex on the opposite face not adjacent to either of the first two vertices. The tetrahedron is discarded and the remaining portion of the cube is placed on a table with the cut surface face down. What is the height of this object? $ \textbf{(A)}\ \dfrac{\sqrt{3}}{3}\qquad\textbf{(B)}\ \dfrac{2\sqrt{2}}{3}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ \dfrac{2\sqrt{3}}{3}\qquad\textbf{(E)}\ \sqrt{2} $

Give two congruent regular triangular pyramids, stick their bottom surfaces together. Then ,it becomes a hexahedron with all dihedral angles equal. The length of the shortest edge of the hexahedron is $2$. Then, the furthest distance between two vertexes is________.