In the adjoining figure, $PQ$ is a diagonal of the cube. If $PQ$ has length $a$, then the surface area of the cube is $\text{(A)}\ 2a^2 \qquad \text{(B)}\ 2\sqrt{2}a^2 \qquad \text{(C)}\ 2\sqrt{3}a^2 \qquad \text{(D)}\ 3\sqrt{3}a^2 \qquad \text{(E)}\ 6a^2$

A sphere inscribed in a tetrahedron touches one face at the intersection of its angle bisectors, a second face at the intersection of its altitudes, and a third face at the intersection of its medians. Show that the tetrahedron is regular. [i]N. Agakhanov[/i]

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 $A\cup B$?

Can one cover a cube by three paper triangles (without overlapping)?

Erin the ant starts at a given corner of a cube and crawls along exactly $7$ edges in such a way that she visits every corner exactly once and then finds that she is unable to return along an edge to her starting point. How many paths are there meeting these conditions? $ \textbf{(A) }\text{6}\qquad\textbf{(B) }\text{9}\qquad\textbf{(C) }\text{12}\qquad\textbf{(D) }\text{18}\qquad\textbf{(E) }\text{24} $

G. H. Hardy once went to visit Srinivasa Ramanujan in the hospital, and he started the conversation with: “I came here in taxi-cab number $1729$. That number seems dull to me, which I hope isn’t a bad omen.” “Nonsense,” said Ramanujan. “The number isn’t dull at all. It’s quite interesting. It’s the smallest number that can be expressed as the sum of two cubes in two different ways.” Ramujan had immediately seen that $1729=12^3+1^3=10^3+9^3$. What is the smallest positive integer representable as the sum of the cubes of [i]three[/i] positive integers in two different ways?

The net of 20 triangles shown below can be folded to form a regular icosahedron. Inside each of the triangular faces, write a number from 1 to 20 with each number used exactly once. Any pair of numbers that are consecutive must be written on faces sharing an edge in the folded icosahedron, and additionally, 1 and 20 must also be on faces sharing an edge. Some numbers have been given to you. (No proof is necessary.) [asy] unitsize(1cm); pair c(int a, int b){return (a-b/2,sqrt(3)*b/2);} draw(c(0,0)--c(0,1)--c(-1,1)--c(1,3)--c(1,1)--c(2,2)--c(3,2)--c(4,3)--c(4,2)--c(3,1)--c(2,1)--c(2,-1)--c(1,-1)--c(1,-2)--c(0,-3)--c(0,-2)--c(-1,-2)--c(1,0)--cycle); draw(c(0,0)--c(1,1)--c(0,1)--c(1,2)--c(0,2)--c(0,1),linetype("4 4")); draw(c(4,2)--c(3,2)--c(3,1),linetype("4 4")); draw(c(3,2)--c(1,0)--c(1,1)--c(2,1)--c(2,2),linetype("4 4")); draw(c(1,-2)--c(0,-2)--c(0,-1)--c(1,-1)--c(1,0)--c(2,0)--c(0,-2),linetype("4 4")); label("2",(c(0,2)+c(1,2))/2,S); label("15",(c(1,1)+c(2,1))/2,S); label("6",(c(0,1)+c(1,1))/2,N); label("14",(c(0,0)+c(1,0))/2,N);[/asy]

Consider a system of infinitely many spheres made of metal, with centers at points $(a, b, c) \in \mathbb Z^3$. We say that the system is stable if the temperature of each sphere equals the average temperature of the six closest spheres. Assuming that all spheres in a stable system have temperatures between $0^\circ C$ and $1^\circ C$, prove that all the spheres have the same temperature.

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 $ABCD$ be a tetrahedron in which the opposite sides are equal and form equal angles. Prove that it is regular.

Is it possible to disassemble and reassemble a $4\times 4\times 4$ Rubik's Cuble in at least $577,800$ non-equivalent ways? Notes: 1. When we reassemble the cube, a corner cube has to go to a corner cube, an edge cube must go to an edge cube and a central cube must go to a central cube. 2. Two positions of the cube are called equivalent if they can be obtained from one two another by rotating the faces or layers which are parallel to the faces.

Suppose you are given six colours and, are asked to colour each face of a cube by a different colour. Determine the different number of colouring possible.

In $\mathbb{R}^3$, for a rectangular box $\Delta$, let $10\Delta$ be the box with the same center as $\Delta$ but dilated by $10$. For example, if $\Delta$ is an $1 \times 1 \times 10$ box (hence with Lebesgue measure $10$), then $10\Delta$ is the $10 \times 10 \times 100$ box with the same center and orientation as $\Delta$. \medskip If two rectangular boxes $\Delta_1$ and $\Delta_2$ satisfy $\Delta_1 \subset 10\Delta_2$ and $\Delta_2 \subset 10 \Delta_1$, we say that they are [i]almost identical[/i]. Find the largest real number $a$ such that the following holds for some $C=C(a)>0$: For every positive integer $N$ and every collection $S$ of $1 \times 1 \times N$ boxes in $\mathbb{R}^3$, assuming that (i) $\vert S\vert=N$, (ii) every pair of boxes $(\Delta_1,\Delta_2)$ taken from $S$ are not almost identical, and (iii) the long edge of each box in $S$ forms an angle $\frac{\pi}{4}$ against the $xy$-plane. Then the volume \[\left\vert \bigcup_{\Delta \in S} \Delta\right\vert \ge CN^a.\]

The segments $[AD], [BE]$ and $[CF]$ are the side edges of the right triangle prism. (the equilateral triangle is a base) Find all the points in its base $ABC$, situated on the equal distances from the $(AE), (BF)$ and $(CD)$ lines.

Consider all planes through the center of a $2\times2\times2$ cube that create cross sections that are regular polygons. The sum of the cross sections for each of these planes can be written in the form $a\sqrt b+c$, where $b$ is a square-free positive integer. Find $a+b+c$.

Let be given an arbitrary tetrahedron $ABCD$ with volume $V$. Consider all lines which pass through the barycenter $S$ of the tetrahedron and intersect the edges $AD,BD,CD$ at points $A',B',C$ respectively. It is known that among the obtained tetrahedra there exists one with the minimal volume. Express this minimal volume in terms of $V$

A regular tetrahedron with an edge of $30$ cm rests on one of its faces. Assuming it is hollow, $2$ liters of water are poured into it. Find the height of the ''upper'' liquid and the area of the ''free'' surface of the water.

Let $\,S\,$ be a finite set of points in three-dimensional space. Let $\,S_{x},\,S_{y},\,S_{z}\,$ be the sets consisting of the orthogonal projections of the points of $\,S\,$ onto the $yz$-plane, $zx$-plane, $xy$-plane, respectively. Prove that \[ \vert S\vert^{2}\leq \vert S_{x} \vert \cdot \vert S_{y} \vert \cdot \vert S_{z} \vert, \] where $\vert A \vert$ denotes the number of elements in the finite set $A$. [hide="Note"] Note: The orthogonal projection of a point onto a plane is the foot of the perpendicular from that point to the plane. [/hide]

We say that a finite set of points is [i]well scattered[/i] on the surface of a sphere if every open hemisphere (half the surface of the sphere without its boundary) contains at least one of the points. The set $\{ (1,0,0), (0,1,0), (0,0,1) \}$ is not well scattered on the unit sphere (the sphere of radius $1$ centered at the origin), but if you add the correct point $P$ it becomes well scattered. Find, with proof, all possible points $P$ that would make the set well scattered.

A right-triangulated prism made of benzene sits on a table. The hypotenuse makes an angle of $30^\circ$ with the horizontal table. An incoming ray of light hits the hypotenuse horizontally, and leaves the prism from the vertical leg at an acute angle of $ \gamma $ with respect to the vertical leg. Find $\gamma$, in degrees, to the nearest integer. The index of refraction of benzene is $1.50$. [i](Proposed by Ahaan Rungta)[/i]

Three of the edges of a cube are $\overline{AB}, \overline{BC},$ and $\overline{CD},$ and $\overline{AD}$ is an interior diagonal. Points $P, Q,$ and $R$ are on $\overline{AB}, \overline{BC},$ and $\overline{CD},$ respectively, so that $AP=5, PB=15, BQ=15,$ and $CR=10.$ What is the area of the polygon that is the intersection of plane $PQR$ and the cube?

Three cubes of volume $1, 8$ and $27$ are glued together at their faces. The smallest possible surface area of the resulting configuration is $ \textbf{(A)}\ 36 \qquad\textbf{(B)}\ 56 \qquad\textbf{(C)}\ 70 \qquad\textbf{(D)}\ 72 \qquad\textbf{(E)}\ 74 $

In a regular tetrahedron the centers of the four faces are the vertices of a smaller tetrahedron. The ratio of the volume of the smaller tetrahedron to that of the larger is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Let $ ABCD$ be a regular tetrahedron and $ M,N$ distinct points in the planes $ ABC$ and $ ADC$ respectively. Show that the segments $ MN,BN,MD$ are the sides of a triangle.

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?