A box whose shape is a parallelepiped can be completely filled with cubes of side $1.$ If we put in it the maximum possible number of cubes, each of volume $2$, with the sides parallel to those of the box, then exactly $40$ percent of the volume of the box is occupied. Determine the possible dimensions of the box.

Consider a triangular pyramid $ABCD$ with equilateral base $ABC$ of side length $1$. $AD = BD =CD$ and $\angle ADB = \angle BDC = \angle ADC = 90^o$ . Find the volume of $ABCD$.

We have two concentric circles. A polygon is circumscribed around the smaller circle and is contained entirely inside the greater circle. Perpendiculars from the common center of the circles to the sides of the polygon are extended till they intersect the greater circle. Each of the points obtained is connected with the endpoints of the corresponding side of the polygon . When is the resulting star-shaped polygon the unfolding of a pyramid?

Prove that every cube's cross-section, containing its centre, has the area not less then its face's area.

The four bottom corners of a cube are colored red, green, blue, and purple. How many ways are there to color the top four corners of the cube so that every face has four different colored corners? Prove that your answer is correct.

There are $8436$ steel balls, each with radius $1$ centimeter, stacked in a tetrahedral pile, with one ball on top, $3$ balls in the second layer, $6$ in the third layer, $10$ in the fourth, and so on. Determine the height of the pile in centimeters.

From a regular octahedron with edge $1$, cut off a pyramid about each vertex. The base of each pyramid is a square with edge $\frac 13$. Can copies of the polyhedron so obtained, whose faces are either regular hexagons or squares, be used to tile space?

Dagan has a wooden cube. He paints each of the six faces a different color. He then cuts up the cube to get eight identically-sized smaller cubes, each of which now has three painted faces and three unpainted faces. He then puts the smaller cubes back together into one larger cube such that no unpainted face is visible. Compute the number of different cubes that Dagan can make this way. Two cubes are considered the same if one can be rotated to obtain the other. You may express your answer either as an integer or as a product of prime numbers.

Consider the cube $ABCDEFGH$ and denote by, respectively, $M$ and $N$ the midpoints of $[AB]$ and $[CD]$. Let $P$ be a point on the line defined by $[AE]$ and $Q$ the point of intersection of the lines defined by $[PM]$ and $[BF]$. Prove that the triangle $[PQN]$ is isosceles. [img]https://cdn.artofproblemsolving.com/attachments/0/0/57559efbad87903d087c738df279b055b4aefd.png[/img]

A three-quarter sector of a circle of radius $4$ inches together with its interior can be rolled up to form the lateral surface area of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches? [asy] draw(Arc((0,0), 4, 0, 270)); draw((0,-4)--(0,0)--(4,0)); label("$4$", (2,0), S); [/asy] $\textbf{(A)}\ 3\pi \sqrt5 \qquad\textbf{(B)}\ 4\pi \sqrt3 \qquad\textbf{(C)}\ 3 \pi \sqrt7 \qquad\textbf{(D)}\ 6\pi \sqrt3 \qquad\textbf{(E)}\ 6\pi \sqrt7$

Given a regular tetrahedron $OABC$. Take points $P,\ Q,\ R$ on the sides $OA,\ OB,\ OC$ respectively. Note that $P,\ Q,\ R$ are different from the vertices of the tetrahedron $OABC$. If $\triangle{PQR}$ is an equilateral triangle, then prove that three sides $PQ,\ QR,\ RP$ are pararell to three sides $AB,\ BC,\ CA$ respectively. 30 points

Is it possible to tile space using a combination of regular tetrahedra and regular octahedra? (A Belov)

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 $4\times4\times4$ cube is composed of $64$ unit cubes. The faces of $16$ unit cubes are to be coloured red. A colouring is called interesting if there is exactly $1$ red unit cube in every $1\times1\times 4$ rectangular box composed of $4$ unit cubes. Determine the number of interesting colourings.

For $n \in \mathbb{N}$, consider in $\mathbb{R}^3$ the regular tetrahedron with vertices $O(0, 0, 0)$, $A(n, 9n, 4n)$, $B(9n, 4n, n)$ and $C(4n, n, 9n)$. Show that the number $N$ of points $(x, y, z)$, $[x, y, z \in \mathbb{Z}]$ inside or on the boundary of the tetrahedron $OABC$ is given by$$N=\frac{343n^3}{3}+\frac{35n^2}{2}+\frac{7n}{6}+1$$

We have twenty-seven $1$ by $1$ cubes. Each face of every cube is marked with a natural number so that two opposite faces (top and bottom, front and back, left and right) are always marked with an even number and an odd number where the even number is twice that of the odd number. The twenty-seven cubes are put together to form one $3$ by $3$ cube as shown. When two cubes are placed face-to-face, adjoining faces are always marked with an odd number and an even number where the even number is one greater than the odd number. Find the sum of all of the numbers on all of the faces of all the $1$ by $1$ cubes. [asy] import graph; size(7cm); real labelscalefactor = 0.5; pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); draw((-1,7)--(-1,4)); draw((-1,9.15)--(-3.42,8.21)); draw((-1,9.15)--(1.42,8.21)); draw((-1,7)--(1.42,8.21)); draw((1.42,7.21)--(-1,6)); draw((1.42,6.21)--(-1,5)); draw((1.42,5.21)--(-1,4)); draw((1.42,8.21)--(1.42,5.21)); draw((-3.42,8.21)--(-3.42,5.21)); draw((-3.42,7.21)--(-1,6)); draw((-3.42,8.21)--(-1,7)); draw((-1,4)--(-3.42,5.21)); draw((-3.42,6.21)--(-1,5)); draw((-2.61,7.8)--(-2.61,4.8)); draw((-1.8,4.4)--(-1.8,7.4)); draw((-0.2,7.4)--(-0.2,4.4)); draw((0.61,4.8)--(0.61,7.8)); label("2",(-1.07,9.01),SE*labelscalefactor); label("9",(-1.88,8.65),SE*labelscalefactor); label("1",(-2.68,8.33),SE*labelscalefactor); label("3",(-0.38,8.72),SE*labelscalefactor); draw((-1.8,7.4)--(0.63,8.52)); draw((-0.27,8.87)--(-2.61,7.8)); draw((-2.65,8.51)--(-0.2,7.4)); draw((-1.77,8.85)--(0.61,7.8)); label("7",(-1.12,8.33),SE*labelscalefactor); label("5",(-1.9,7.91),SE*labelscalefactor); label("1",(0.58,8.33),SE*labelscalefactor); label("18",(-0.36,7.89),SE*labelscalefactor); label("1",(-1.07,7.55),SE*labelscalefactor); label("1",(-0.66,6.89),SE*labelscalefactor); label("5",(-0.68,5.8),SE*labelscalefactor); label("1",(-0.68,4.83),SE*labelscalefactor); label("2",(0.09,7.27),SE*labelscalefactor); label("1",(0.15,6.24),SE*labelscalefactor); label("2",(0.11,5.26),SE*labelscalefactor); label("1",(0.89,7.61),SE*labelscalefactor); label("3",(0.89,6.63),SE*labelscalefactor); label("9",(0.92,5.62),SE*labelscalefactor); label("18",(-3.18,7.63),SE*labelscalefactor); label("2",(-3.07,6.61),SE*labelscalefactor); label("2",(-3.09,5.62),SE*labelscalefactor); label("1",(-2.29,7.25),SE*labelscalefactor); label("3",(-2.27,6.22),SE*labelscalefactor); label("5",(-2.29,5.2),SE*labelscalefactor); label("7",(-1.49,6.89),SE*labelscalefactor); label("34",(-1.52,5.81),SE*labelscalefactor); label("1",(-1.41,4.86),SE*labelscalefactor); [/asy]

One thousand unit cubes are fastened together to form a large cube with edge length 10 units; this is painted and then separated into the original cubes. The number of these unit cubes which have at least one face painted is $ \textbf{(A)}\ 600 \qquad \textbf{(B)}\ 520 \qquad \textbf{(C)}\ 488 \qquad \textbf{(D)}\ 480 \qquad \textbf{(E)}\ 400$

Let $\mathcal{B}$ be the set of rectangular boxes that have volume $23$ and surface area $54$. Suppose $r$ is the least possible radius of a sphere that can fit any element of $\mathcal{B}$ inside it. Then $r^{2}$ can be expressed as $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

In a circular tower with an internal diameter of $ 2$ m, there is a spiral staircase with a height of $ 6$ m. The height of each stair step is $ 0.15$ m. In the horizontal projection, the steps form adjacent circular sections with an angle of $ 18^\circ $. The narrower ends of the steps are mounted in a round pillar with a diameter of $ 0.64$ m, the axis of which coincides with the axis of the tower. Calculate the greatest length of a straight rod that can be moved up these stairs from the bottom to the top (do not take into account the thickness of the rod or the thickness of the boards from which the stairs are made).

A right rectangular prism has integer side lengths $a$, $b$, and $c$. If $\text{lcm}(a,b)=72$, $\text{lcm}(a,c)=24$, and $\text{lcm}(b,c)=18$, what is the sum of the minimum and maximum possible volumes of the prism? [i]Proposed by Deyuan Li and Andrew Milas[/i]

We are given a cone with height 6, whose base is a circle with radius $\sqrt{2}$. Inside the cone, there is an inscribed cube: Its bottom face on the base of the cone, and all of its top vertices lie on the cone. What is the length of the cube's edge? [img]https://i.imgur.com/AHqHHP6.png[/img]

Let $ r$ and $ s$ two orthogonal lines that does not lay on the same plane. Let $ AB$ be their common perpendicular, where $ A\in{}r$ and $ B\in{}s$(*).Consider the sphere of diameter $ AB$. The points $ M\in{r}$ and $ N\in{s}$ varies with the condition that $ MN$ is tangent to the sphere on the point $ T$. Find the locus of $ T$. Note: The plane that contains $ B$ and $ r$ is perpendicular to $ s$.

$S$ is a sphere center $O. G$ and $G'$ are two perpendicular great circles on $S$. Take $A, B, C$ on $G$ and $D$ on $G'$ such that the altitudes of the tetrahedron $ABCD$ intersect at a point. Find the locus of the intersection.

Prove that the sum of the six angles subtended at an interior point of a tetrahedron by its six edges is greater than 540°.

The base of the pyramid is a convex quadrangle. Is there necessarily a section of this pyramid that does not intersect the base and is an inscribed quadrangle? (M. Volchkevich)