If $x$ is the cube of a positive integer and $d$ is the number of positive integers that are divisors of $x$, then $d$ could be $ \textbf{(A)}\ 200\qquad\textbf{(B)}\ 201\qquad\textbf{(C)}\ 202\qquad\textbf{(D)}\ 203\qquad\textbf{(E)}\ 204 $

We consider a cube with sides of length 1. Prove that a tetrahedron with vertices in the set of the vertices of the cube has the volume $\dfrac 16$ if and only if 3 of the vertices of the tetrahedron are vertices on the same face of the cube. [i]Dinu Serbanescu[/i]

16. Create a cube $C_1$ with edge length $1$. Take the centers of the faces and connect them to form an octahedron $O_1$. Take the centers of the octahedron’s faces and connect them to form a new cube $C_2$. Continue this process infinitely. Find the sum of all the surface areas of the cubes and octahedrons. 17. Let $p(x) = x^2 - x + 1$. Let $\alpha$ be a root of $p(p(p(p(x)))$. Find the value of $$(p(\alpha) - 1)p(\alpha)p(p(\alpha))p(p(p(\alpha))$$ 18. An $8$ by $8$ grid of numbers obeys the following pattern: 1) The first row and first column consist of all $1$s. 2) The entry in the $i$th row and $j$th column equals the sum of the numbers in the $(i - 1)$ by $(j - 1)$ sub-grid with row less than i and column less than $j$. What is the number in the 8th row and 8th column?

Can a regular octahedron be inscribed in a cube in such a way that all vertices of the octahedron are on cube's edges? (4)

Let $ l,d,k$ be natural numbers. We want to prove that for large numbers $ n$, for each $ k$-coloring of the $ n$-dimensional cube with side length $ l$, there is a $ d$-dimensional subspace that all of its vertices have the same color. Let $ H(l,d,k)$ be the least number such that for $ n\geq H(l,d,k)$ the previus statement holds. a) Prove that: \[ H(l,d \plus{} 1,k)\leq H(l,1,k) \plus{} H(l,d,k^l)^{H(l,1,k)} \] b) Prove that \[ H(l \plus{} 1,1,k \plus{} 1)\leq H(l,1 \plus{} H(l \plus{} 1,1,k),k \plus{} 1) \] c) Prove the statement of problem. d) Prove Van der Waerden's Theorem.

Three cubes are each formed from the pattern shown. They are then stacked on a table one on top of another so that the $ 13$ visible numbers have the greatest possible sum. What is that sum? [asy]unitsize(.8cm); pen p = linewidth(.8pt); draw(shift(-2,0)*unitsquare,p); label("1",(-1.5,0.5)); draw(shift(-1,0)*unitsquare,p); label("2",(-0.5,0.5)); label("32",(0.5,0.5)); draw(shift(1,0)*unitsquare,p); label("16",(1.5,0.5)); draw(shift(0,1)*unitsquare,p); label("4",(0.5,1.5)); draw(shift(0,-1)*unitsquare,p); label("8",(0.5,-0.5));[/asy]$ \textbf{(A)}\ 154 \qquad \textbf{(B)}\ 159 \qquad \textbf{(C)}\ 164 \qquad \textbf{(D)}\ 167 \qquad \textbf{(E)}\ 189$

Recall that a regular icosahedron is a convex polyhedron having 12 vertices and 20 faces; the faces are congruent equilateral triangles. On each face of a regular icosahedron is written a nonnegative integer such that the sum of all $20$ integers is $39.$ Show that there are two faces that share a vertex and have the same integer written on them.

Starting from a pyramid $T_0$ whose edges are all of length $2019$, we construct the Figure $T_1$ when considering the triangles formed by the midpoints of the edges of each face of $T_0$, building in each of these new pyramid triangles with faces identical to base. Then the bases of these new pyramids are removed. Figure $T_2$ is constructed by applying the same process from $T_1$ on each triangular face resulting from $T_1$, and so on for $T_3, T_4, ...$ Let $D_0= \max \{d(x,y)\}$, where $x$ and $y$ are vertices of $T_0$ and $d(x,y)$ is the distance between $x$ and $y$. Then we define $D_{n + 1} = \max \{d (x, y) |d (x, y) \notin \{D_0, D_1,...,D_n\}$, where $x, y$ are vertices of $T_{n+1}$. Find the value of $D_n$ for all $n$.

Given an acute triangle $T$ with sides $a,b,c$, find the tetrahedra with base $T$ whose all faces are acute triangles of the same area.

Which of all planes, the one and the same body diagonal of a cube with the edge length $a$, cuts out a cut figure with the smallest area from the cube? Calculate the area of such a cut figure. [hide=original wording]Welche von allen Ebenen, die eine und dieselbe Korperdiagonale eines Wurfels mit der Kantenlange a enthalten, schneiden aus den W¨urfel eine Schnittfigur kleinsten Flacheninhaltes heraus? Berechnen Sie den Fl¨acheninhalt solch einer Schnittfigur![/hide]

There are $ n$ points ($ n \geq 4$) on a sphere with radius $ R$, and not all of them lie on the same semi-sphere. Prove that among all the angles formed by any two of the $ n$ points and the sphere centre $ O$ ($ O$ is the vertex of the angle), there is at least one that is not less than $ \displaystyle 2 \arcsin{\frac{\sqrt{6}}{3}}$.

Let $D_{1}, D_{2}, . . . , D_{n}$ be rectangular parallelepipeds in space, with edges parallel to the $x, y, z$ axes. For each $D_{i}$, let $x_{i} , y_{i} , z_{i}$ be the lengths of its projections onto the $x, y, z$ axes, respectively. Suppose that for all pairs $D_{i}$ , $D_{j}$, if at least one of $x_{i} < x_{j}$ , $y_{i} < y_{j}$, $z_{i} < z_{j}$ holds, then $x_{i} \leq x_{j}$ , $y_{i} \leq y_{j}$, and $z_{i} < z_{j}$ . If the volume of the region $\bigcup^{n}_{i=1}{D_{i}}$ equals 1997, prove that there is a subset $\{D_{i_{1}}, D_{i_{2}}, . . . , D_{i_{m}}\}$ of the set $\{D_{1}, . . . , D_{n}\}$ such that $(i)$ if $k \not= l $ then $D_{i_{k}} \cap D_{i_{l}} = \emptyset $, and $(ii)$ the volume of $\bigcup^{m}_{k=1}{D_{i_{k}}}$ is at least 73.

The figure shows a glass prism which is partially filled with liquid. The surface of the prism consists of two isosceles right triangles, two squares with side length $10$ cm and a rectangle. The prism can lie in three different ways. If the prism lies as shown in figure $1$, the height of the liquid is $5$ cm. [img]https://cdn.artofproblemsolving.com/attachments/4/2/cda98a00f8586132fe519855df123534516b50.png[/img] a) What is the height of the liquid when it lies as shown in figure $2$? b) What is the height of the liquid when it lies as shown in figure$ 3$?

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].

A solid in the shape of a right circular cone is 4 inches tall and its base has a 3-inch radius. The entire surface of the cone, including its base, is painted. A plane parallel to the base of the cone divides the cone into two solids, a smaller cone-shaped solid $C$ and a frustum-shaped solid $F$, in such a way that the ratio between the areas of the painted surfaces of $C$ and $F$ and the ratio between the volumes of $C$ and $F$ are both equal to $k$. Given that $k=m/n$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.

A triangular pyramid $ O.ABC$ with base $ ABC$ has the property that the lengths of the altitudes from $ A$, $ B$ and $ C$ are not less than $ \frac{OB \plus{}OC}{2}$, $ \frac{OC \plus{} OA}{2}$ and $ \frac{OA \plus{} OB}{2}$, respectively. Given that the area of $ ABC$ is $ S$, calculate the volume of the pyramid.

A cube is cut into 99 smaller cubes, exactly 98 of which are unit cubes. Find the volume of the original cube. (V Proizvolov)

Given a triangular pyramid $ABCD$. Sphere $S_1$ passing through points $A$, $B$, $C$, intersects edges $AD$, $BD$, $CD$ at points $K$, $L$, $M$, respectively; sphere $S_2$ passing through points $A$, $B$, $D$ intersects the edges $AC$, $BC$, $DC$ at points $P$, $Q$, $M$ respectively. It turned out that $KL \parallel PQ$. Prove that the bisectors of plane angles $KMQ$ and $LMP$ are the same.

$ABC$ is an acute-angled triangle. $D$ is a variable point in space such that all faces of the tetrahedron $ABCD$ are acute-angled. $P$ is the foot of the perpendicular from $D$ to the plane $ABC$. Find the locus of $P$ as $D$ varies.

Two real numbers are selected independently at random from the interval [-20, 10]. What is the probability that the product of those numbers is greater than zero? $ \textbf{(A)}\ \frac{1}{9} \qquad \textbf{(B)}\ \frac{1}{3} \qquad \textbf{(C)}\ \frac{4}{9} \qquad \textbf{(D)}\ \frac{5}{9} \qquad \textbf{(E)}\ \frac{2}{3} $

At the base of the pyramid $SABCD$ lies a convex quadrilateral $ABCD$, such that $BC * AD = BD * AC$. Also $ \angle ADS =\angle BDS ,\angle ACS =\angle BCS$. Prove that the plane $SAB$ is perpendicular to the plane of the base.

Ten tiles numbered $1$ through $10$ are turned face down. One tile is turned up at random, and a die is rolled. What is the probability that the product of the numbers on the tile and the die will be a square? $\textbf{(A)}\ \frac{1}{10}\qquad \textbf{(B)}\ \frac{1}{6}\qquad \textbf{(C)}\ \frac{11}{60}\qquad \textbf{(D)}\ \frac{1}{5}\qquad \textbf{(E)}\ \frac{7}{30}$

For what values of $n$ is it possible to paint the edges of a prism whose base is an $n$-gon so that there are edges of all three colours at each vertex and all the faces (including the upper and lower bases) have edges of all three colours? (AV Shapovelov)

A right circular cone has base radius $ r$ and height $ h$. The cone lies on its side on a flat table. As the cone rolls on the surface of the table without slipping, the point where the cone's base meets the table traces a circular arc centered at the point where the vertex touches the table. The cone first returns to its original position on the table after making $ 17$ complete rotations. The value of $ h/r$ can be written in the form $ m\sqrt {n}$, where $ m$ and $ n$ are positive integers and $ n$ is not divisible by the square of any prime. Find $ m \plus{} n$.

Figure 1 is called a "stack map." The numbers tell how many cubes are stacked in each position. Fig. 2 shows these cubes, and Fig. 3 shows the view of the stacked cubes as seen from the front. Which of the following is the front view for the stack map in Fig. 4? [asy] unitsize(24); draw((0,0)--(2,0)--(2,2)--(0,2)--cycle); draw((1,0)--(1,2)); draw((0,1)--(2,1)); draw((5,0)--(7,0)--(7,1)--(20/3,4/3)--(20/3,13/3)--(19/3,14/3)--(16/3,14/3)--(16/3,11/3)--(13/3,11/3)--(13/3,2/3)--cycle); draw((20/3,13/3)--(17/3,13/3)--(17/3,10/3)--(14/3,10/3)--(14/3,1/3)); draw((20/3,10/3)--(17/3,10/3)--(17/3,7/3)--(20/3,7/3)); draw((17/3,7/3)--(14/3,7/3)); draw((7,1)--(6,1)--(6,2)--(5,2)--(5,0)); draw((5,1)--(6,1)--(6,0)); draw((20/3,4/3)--(6,4/3)); draw((17/3,13/3)--(16/3,14/3)); draw((17/3,10/3)--(16/3,11/3)); draw((14/3,10/3)--(13/3,11/3)); draw((5,2)--(13/3,8/3)); draw((5,1)--(13/3,5/3)); draw((6,2)--(17/3,7/3)); draw((9,0)--(11,0)--(11,4)--(10,4)--(10,3)--(9,3)--cycle); draw((11,3)--(10,3)--(10,0)); draw((11,2)--(9,2)); draw((11,1)--(9,1)); draw((13,0)--(16,0)--(16,2)--(13,2)--cycle); draw((13,1)--(16,1)); draw((14,0)--(14,2)); draw((15,0)--(15,2)); label("Figure 1",(1,0),S); label("Figure 2",(17/3,0),S); label("Figure 3",(10,0),S); label("Figure 4",(14.5,0),S); label("$1$",(1.5,.2),N); label("$2$",(.5,.2),N); label("$3$",(.5,1.2),N); label("$4$",(1.5,1.2),N); label("$1$",(13.5,.2),N); label("$3$",(14.5,.2),N); label("$1$",(15.5,.2),N); label("$2$",(13.5,1.2),N); label("$2$",(14.5,1.2),N); label("$4$",(15.5,1.2),N);[/asy] [asy] unitsize(18); draw((0,0)--(3,0)--(3,2)--(1,2)--(1,4)--(0,4)--cycle); draw((0,3)--(1,3)); draw((0,2)--(1,2)--(1,0)); draw((0,1)--(3,1)); draw((2,0)--(2,2)); draw((5,0)--(8,0)--(8,4)--(7,4)--(7,3)--(6,3)--(6,2)--(5,2)--cycle); draw((8,3)--(7,3)--(7,0)); draw((8,2)--(6,2)--(6,0)); draw((8,1)--(5,1)); draw((10,0)--(12,0)--(12,4)--(11,4)--(11,3)--(10,3)--cycle); draw((12,3)--(11,3)--(11,0)); draw((12,2)--(10,2)); draw((12,1)--(10,1)); draw((14,0)--(17,0)--(17,4)--(16,4)--(16,2)--(14,2)--cycle); draw((17,3)--(16,3)); draw((17,2)--(16,2)--(16,0)); draw((17,1)--(14,1)); draw((15,0)--(15,2)); draw((19,0)--(22,0)--(22,4)--(20,4)--(20,1)--(19,1)--cycle); draw((22,3)--(20,3)); draw((22,2)--(20,2)); draw((22,1)--(20,1)--(20,0)); draw((21,0)--(21,4)); label("(A)",(1.5,0),S); label("(B)",(6.5,0),S); label("(C)",(11,0),S); label("(D)",(15.5,0),S); label("(E)",(20.5,0),S);[/asy]