An ant is at one corner of a unit cube. If the ant must travel on the box’s surface, the shortest distance the ant must crawl to reach the opposite corner of the cube can be written in the form $\sqrt{a}$, where $a$ is a positive integer. Compute $a$.

The six edges of a tetrahedron $ABCD$ measure $7$, $13$, $18$, $27$, $36$ and $41$ units. If the length of edge $AB$ is $41$, then the length of edge $CD$ is $ \textbf{(A)}\ 7\qquad\textbf{(B)}\ 13\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 36 $

Let $s_1$ be the sum of the first $n$ terms of the arithmetic sequence $8,12,\cdots$ and let $s_2$ be the sum of the first $n$ terms of the arithmetic sequence $17,19\cdots$. Assume $n\ne 0$. Then $s_1=s_2$ for: $\text{(A)} \ \text{no value of n} \qquad \text{(B)} \ \text{one value of n} \qquad \text{(C)} \ \text{two values of n}$ $\text{(D)} \ \text{four values of n} \qquad \text{(E)} \ \text{more than four values of n}$

Let $M$ be an interior point of tetrahedron $V ABC$. Denote by $A_1,B_1, C_1$ the points of intersection of lines $MA,MB,MC$ with the planes $VBC,V CA,V AB$, and by $A_2,B_2, C_2$ the points of intersection of lines $V A_1, VB_1, V C_1$ with the sides $BC,CA,AB$. [b](a)[/b] Prove that the volume of the tetrahedron $V A_2B_2C_2$ does not exceed one-fourth of the volume of $V ABC$. [b](b)[/b] Calculate the volume of the tetrahedron $V_1A_1B_1C_1$ as a function of the volume of $V ABC$, where $V_1$ is the point of intersection of the line $VM$ with the plane $ABC$, and $M$ is the barycenter of $V ABC$.

A cube is constructed from $4$ white unit cubes and $4$ black unit cubes. How many different ways are there to construct the $2 \times 2 \times 2$ cube using these smaller cubes? (Two constructions are considered the same if one can be rotated to match the other.) $\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ 11$

A tetrahedron $ABCD$ satisfies $AB=6$, $CD=8$, and $BC=DA=5$. Let $V$ be the maximum value of $ABCD$ possible. If we can write $V^4=2^n3^m$ for some integers $m$ and $n$, find $mn$.

Consider a container of a hollow cube $ABGCDEPF$ (where $ABGC$, $DEPF$ are squares and $AD\parallel BE\parallel GP\parallel CF$). The cube is placed on a table in a way that the space diagonal $AP=1$ is perpendicular to the table. Then, water is poured into the cube. Denote $x$ the length of part of $AP$ submerged in water. Determine the volume of water $y$ in terms of $x$ when a) $0 < x \leq\frac13$, b) $\frac13 < x \leq\frac12$.

Let $ABCDS$ be a pyramid with four faces and with $ABCD$ as a base, and let a plane $\alpha$ through the vertex $A$ meet its edges $SB$ and $SD$ at points $M$ and $N$, respectively. Prove that if the intersection of the plane $\alpha$ with the pyramid $ABCDS$ is a parallelogram, then $SM \cdot SN > BM \cdot DN$.

Let $P$ be a set of $n$ points and $S$ a set of $l$ segments. It is known that: $(i)$ No four points of $P$ are coplanar. $(ii)$ Any segment from $S$ has its endpoints at $P$. $(iii)$ There is a point, say $g$, in $P$ that is the endpoint of a maximal number of segments from $S$ and that is not a vertex of a tetrahedron having all its edges in $S$. Prove that $l \leq \frac{n^2}{3}$

Title is self explanatory. Pick two points on the unit sphere. What is the expected distance between them?

Let $P_1$ be a convex polyhedron with vertices $A_1,A_2,\ldots,A_9$. Let $P_i$ be the polyhedron obtained from $P_1$ by a translation that moves $A_1$ to $A_i$. Prove that at least two of the polyhedra $P_1,P_2,\ldots,P_9$ have an interior point in common.

The edges of a cube are labeled from $1$ to $12$. Show that there must exist at least eight triples $(i, j, k)$ with $1 \leq i < j < k \leq 12$ so that the edges $i, j, k$ are consecutive edges of a path. Also show that there exists labeling in which we cannot find nine such triples.

Prove that a tetrahedron with just one edge length greater than $1$ has volume at most $ \frac{1}{8}.$

Let $a_{ij}$ $i=1,2,3$; $j=1,2,3$ be real numbers such that $a_{ij}$ is positive for $i=j$ and negative for $i\neq j$. Prove the existence of positive real numbers $c_{1}$, $c_{2}$, $c_{3}$ such that the numbers \[a_{11}c_{1}+a_{12}c_{2}+a_{13}c_{3},\qquad a_{21}c_{1}+a_{22}c_{2}+a_{23}c_{3},\qquad a_{31}c_{1}+a_{32}c_{2}+a_{33}c_{3}\] are either all negative, all positive, or all zero. [i]Proposed by Kiran Kedlaya, USA[/i]

Let $S=\{(x,y,z)\in \mathbb{Z}^3\}$ the set of points with integer coordinates in the space. Gugu has infinitely many solid spheres. All with radii $\ge (\frac{\pi}2)^3$. Is it possible for Gugu to cover all points of $S$ with his spheres?

Let $O$ and $O'$ designate two dìagonally opposite vertices of a cube. Bisect those edges of the cube that contain neither of the points $O$ and $O'$. Prove that these midpoints of edges lie in a plane and form the vertices of a regular hexagon

Points $A$ and $B$ are given on a line having no common points with a sphere $K$. The feet $P$ of the perpendicular from the center of $K$ to the line $AB$ is positioned between $A$ and $B$, and the lengths of segments $AP$ and $BP$ both exceed the radius of $K$. Consider the set $Z$ of all triangles $ABC$ whose sides $AC$ and $BC$ are tangent to $K$. Prove that among all triangles in $Z$, a triangle $T$ with a maximum perimeter also has a maximum area.

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.

What is the largest volume of a sphere which touches to a unit sphere internally and touches externally to a regular tetrahedron whose corners are over the unit sphere? $\textbf{(A)}\ \frac13 \qquad\textbf{(B)}\ \frac14 \qquad\textbf{(C)}\ \frac12\left ( 1 - \frac1{\sqrt3} \right ) \qquad\textbf{(D)}\ \frac12\left ( \frac{2\sqrt2}{\sqrt3} - 1 \right ) \qquad\textbf{(E)}\ \text{None}$

A sphere is inscribed in a regular tetrahedron. Another regular tetrahedron is inscribed in the sphere. Find the ratio of the volumes of these two tetrahedra.

An artist has $14$ cubes, each with an edge of $1$ meter. She stands them on the ground to form a sculpture as shown. She then paints the exposed surface of the sculpture. How many square meters does she paint? $\text{(A)}\ 21 \qquad \text{(B)}\ 24 \qquad \text{(C)}\ 33 \qquad \text{(D)}\ 37 \qquad \text{(E)}\ 42$ [asy] draw((0,0)--(2.35,-.15)--(2.44,.81)--(.09,.96)--cycle); draw((.783333333,-.05)--(.873333333,.91)--(1.135,1.135)); draw((1.566666667,-.1)--(1.656666667,.86)--(1.89,1.1)); draw((2.35,-.15)--(4.3,1.5)--(4.39,2.46)--(2.44,.81)); draw((3,.4)--(3.09,1.36)--(2.61,1.4)); draw((3.65,.95)--(3.74,1.91)--(3.29,1.94)); draw((.09,.96)--(.76,1.49)--(.71,1.17)--(2.2,1.1)--(3.6,2.2)--(3.62,2.52)--(4.39,2.46)); draw((.76,1.49)--(.82,1.96)--(2.28,1.89)--(2.2,1.1)); draw((2.28,1.89)--(3.68,2.99)--(3.62,2.52)); draw((1.455,1.135)--(1.55,1.925)--(1.89,2.26)); draw((2.5,2.48)--(2.98,2.44)--(2.9,1.65)); draw((.82,1.96)--(1.55,2.6)--(1.51,2.3)--(2.2,2.26)--(2.9,2.8)--(2.93,3.05)--(3.68,2.99)); draw((1.55,2.6)--(1.59,3.09)--(2.28,3.05)--(2.2,2.26)); draw((2.28,3.05)--(2.98,3.59)--(2.93,3.05)); draw((1.59,3.09)--(2.29,3.63)--(2.98,3.59)); [/asy]

Calculate the volume $ V $ of tetrahedron $ ABCD $ given the length $ d $ of edge $ AB $ and the area $ S $ of the projection of the tetrahedron on the plane perpendicular to the line $ AB $.

In tetrahedron $ABCD$, edge $AB$ has length 3 cm. The area of face $ABC$ is 15 $\text{cm}^2$ and the area of face $ABD$ is 12 $\text{cm}^2$. These two faces meet each other at a $30^\circ$ angle. Find the volume of the tetrahedron in $\text{cm}^3$.

Each face of a cube is painted with a different color. How many distinct cubes can be created this way? (*Observation: The ways to color the cube are $6!$, since each time a color is used on one face, there is one fewer available for the others. However, this does not determine $6!$ different cubes, since colorings that differ only by rotation should be considered the same.*)

Given $ x,y,z\in (0,1)$ satisfying that $ \sqrt{\frac{1 \minus{} x}{yz}} \plus{} \sqrt{\frac{1 \minus{} y}{xz}} \plus{} \sqrt{\frac{1 \minus{} z}{xy}} \equal{} 2$. Find the maximum value of $ xyz$.