The orthogonal projections of the vertices $A, B, C$ of the tetrahedron $ABCD$ on the opposite faces are denoted by $A', B', C'$ respectively. Suppose that point $A'$ is the circumcenter of the triangle $BCD$, point $B'$ is the incenter of the triangle $ACD$ and $C'$ is the centroid of the triangle $ABD$. Prove that tetrahedron $ABCD$ is regular.

Find all positive integer bases $b \ge 9$ so that the number \[ \frac{{\overbrace{11 \cdots 1}^{n-1 \ 1's}0\overbrace{77 \cdots 7}^{n-1\ 7's}8\overbrace{11 \cdots 1}^{n \ 1's}}_b}{3} \] is a perfect cube in base 10 for all sufficiently large positive integers $n$. [i]Proposed by Yang Liu[/i]

Consider the rectangular parallelepiped $ABCDA'B'C'D' $ as such the measure of the dihedral angle formed by the planes $(A'BD)$ and $(C'BD)$ is $90^o$ and the measure of the dihedral angle formed by the planes $(AB'C)$ and $(D'B'C)$ is $60^o$. Determine and measure the dihedral angle formed by the planes $(BC'D)$ and $(A'C'D)$.

We say that a prism is [i]binary[/i] if there exists a labelling of the vertices of the prism with integers from the set $\{-1,1\}$ such that the product of the numbers assigned to the vertices of each face (base or lateral face) is equal to $-1$. a) Prove that any [i]binary[/i] prism has the number of total vertices divisible by 8; b) Prove that any prism with 2000 vertices is [i]binary[/i].

Suppose we climb a mountain that is a cone with radius $100$ and height $4$. We start at the bottom of the mountain (on the perimeter of the base of the cone), and our destination is the opposite side of the mountain, halfway up (height $z = 2$). Our climbing speed starts at $v_0=2$ but gets slower at a rate inversely proportional to the distance to the mountain top (so at height $z$ the speed $v$ is $(h-z)v_0/h$). Find the minimum time needed to get to the destination.

In a tetrahedron, the six triangles determined by an edge of the tetrahedron and the midpoint of the opposite edge all have equal area. Prove that the tetrahedron is regular.

$n$ points are given on the surface of a sphere. Show that the surface can be divided into $n$ congruent regions such that each of them contains exactly one of the given points.

Prove that for an arbitrary pair of vectors $f$ and $g$ in the space the inequality \[af^2 + bfg +cg^2 \geq 0\] holds if and only if the following conditions are fulfilled: \[a \geq 0, \quad c \geq 0, \quad 4ac \geq b^2.\]

Given is a regular tetrahedron of volume $1$. We obtain a second regular tetrahedron by reflecting the given one through its center. What is the volume of their intersection?

Two congruent right circular cones each with base radius $3$ and height $8$ have axes of symmetry that intersect at right angles at a point in the interior of the cones a distance $3$ from the base of each cone. A sphere with radius $r$ lies inside both cones. The maximum possible value for $r^2$ is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

The height of a truncated cone is equal to the radius of its base. The perimeter of a regular hexagon circumscribing its top is equal to the perimeter of an equilateral triangle inscribed in its base. Find the angle $\phi$ between the cone’s generating line and its base.

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$

Suppose two regular pyramids with the same base $ABC$: $P-ABC$ and $Q-ABC$ are circumscribed by the same sphere. If the angle formed by one of the lateral face and the base of pyramid $P-ABC$ is $\frac{\pi}{4}$, find the tangent value of the angle formed by one of the lateral face and the base of the pyramid $Q-ABC$.

Let $D$ be a dodecahedron which can be inscribed in a sphere with radius $R$. Let $I$ be an icosahedron which can also be inscribed in a sphere of radius $R$. Which has the greater volume, and why? Note: A regular [i]polyhedron [/i] is a geometric solid, all of whose faces are congruent regular polygons, in which the same number of polygons meet at each vertex. A regular dodecahedron is a polyhedron with $12$ faces which are regular pentagons and a regular icosahedron is a polyhedron with $20$ faces which are equilateral triangles. A polyhedron is inscribed in a sphere if all of its vertices lie on the surface of the sphere. The illustration below shows a dodecahdron and an icosahedron, not necessarily to scale. [img]https://cdn.artofproblemsolving.com/attachments/7/5/9873b42aacf04bb5daa0fe70d4da3bf0b7be38.png[/img]

At the base of the triangular pyramid $ABCD$ lies a regular triangle $ABC$ such that $AD = BC$. All plane angles at vertex $B$ are equal to each other. What might these angles be equal to?

Given a cuboid $Q$ with dimensions $a, b, c$, $a < b < c$. Find the length of the edge of a cube $K$ , which has parallel faces and a common center with the given cuboid so that the volume of the difference of the sets $Q \cup K$ and $Q \cap K$ is minimal.

Which regular polygon can be obtained (and how) by cutting a cube with a plane ?

Three disks of diameter $d$ are touching a sphere in their centers. Besides, every disk touches the other two disks. How to choose the radius $R$ of the sphere in order that axis of the whole figure has an angle of $60^\circ$ with the line connecting the center of the sphere with the point of the disks which is at the largest distance from the axis ? (The axis of the figure is the line having the property that rotation of the figure of $120^\circ$ around that line brings the figure in the initial position. Disks are all on one side of the plane, passing through the center of the sphere and orthogonal to the axis).

Let $ ABCD$ be a triangular pyramid such that no face of the pyramid is a right triangle and the orthocenters of triangles $ ABC$, $ ABD$, and $ ACD$ are collinear. Prove that the center of the sphere circumscribed to the pyramid lies on the plane passing through the midpoints of $ AB$, $ AC$ and $ AD$.

What is the largest possible height of a right cylinder with radius $3$ that can fit in a cube with side length $12$?

A pyramid has a square base with sides of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube? $ \textbf{(A)}\ 5\sqrt{2}-7 \qquad \textbf{(B)}\ 7-4\sqrt{3} \qquad \textbf{(C)}\ \frac{2\sqrt{2}}{27} \qquad \textbf{(D)}\ \frac{\sqrt{2}}{9} \qquad \textbf{(E)}\ \frac{\sqrt{3}}{9} $

A right circular cone stands on plane $P$. The radius of the cone’s base is $r$, its height is $h$. A source of light is placed at distance $H$ from the plane, and distance $1$ from the axis of the cone. What is the illuminated part of the disc of radius $R$, that belongs to $P$ and is concentric with the disc forming the base of the cone?

A closed pentagonal line is inscribed in a sphere of the diameter $1$, and has all edges of length $\ell$. Prove that $\ell \le \sin \frac{2\pi}{5}$ .

A positive integer $n$ is [i]nice[/i] if there is a positive integer $m$ with exactly four positive divisors (including $1$ and $m$) such that the sum of the four divisors is equal to $n$. How many numers in the set $\{2010, 2011, 2012,\ldots,2019\}$ are nice? ${ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D}}\ 4\qquad\textbf{(E)}\ 5 $

A set $ S$ of points from the space will be called [b]completely symmetric[/b] if it has at least three elements and fulfills the condition that for every two distinct points $ A$ and $ B$ from $ S$, the perpendicular bisector plane of the segment $ AB$ is a plane of symmetry for $ S$. Prove that if a completely symmetric set is finite, then it consists of the vertices of either a regular polygon, or a regular tetrahedron or a regular octahedron.