One triangular face $F$ of a tetrahedron $\mathcal{T}$ has side lengths $\sqrt{5}$, $\sqrt{65}$, and $2\sqrt{17}$. The other three faces of $\mathcal{T}$ are right triangles whose hypotenuses coincide with the sides of $F$. There exists a sphere inside $\mathcal{T}$ tangent to all four of its faces. Compute the radius of this sphere.

From among $2^{1/2},$ $3^{1/3},$ $8^{1/8},$ $9^{1/9}$ those which have the greatest and the next to the greatest values, in that order, are \[ \begin{array}{rlrlrlrl} \hbox {(A)}& 3^{1/3},\ 2^{1/2} \quad & \hbox {(B)}& 3^{1/3},\ 8^{1/8} \quad & \hbox {(C)}& 3^{1/3},\ 9^{1/9} \quad & \hbox {(D)}& 8^{1/8},\ 9^{1/9} \\ \hbox {(E)}& \multicolumn{3}{l}{\hbox{None of these}} \end{array} \]

If five geometric means are inserted between 8 and 5832, the fifth term in the geometric series: $\textbf{(A)}\ 648 \qquad \textbf{(B)}\ 832 \qquad \textbf{(C)}\ 1168 \qquad \textbf{(D)}\ 1944 \qquad \textbf{(E)}\ \text{None of these}$

A parallelepiped has surface area 216 and volume 216. Show that it is a cube.

[b]a)[/b] Let $ D_1,D_2,D_3 $ be pairwise skew lines. Through every point $ P_2\in D_2 $ there is an unique common secant of these three lines that intersect $ D_1 $ at $ P_1 $ and $ D_3 $ at $ P_3. $ Let coordinate systems be introduced on $ D_2 $ and $ D_3 $ having as origin $ O_2, $ respectively, $ O_3. $ Find a relation between the coordinates of $ P_2 $ and $ P_3. $ [b]b)[/b] Show that there exist four pairwise skew lines with exactly two common secants. Also find examples with exactly one and with no common secants. [b]c)[/b] Let $ F_1,F_2,F_3,F_4 $ be any four secants of $ D_1,D_2, D_3. $ Prove that $ F_1,F_2, F_3, F_4 $ have infinitely many common secants.

Given three lines $ a $, $ b $, $ c $ pairwise skew. Is it possible to construct a parallelepiped whose edges lie on the lines $ a $, $ b $, $ c $?

Prove that every integer can be written as sum of $5$ third powers of integers.

(N.Avilov) Can the surface of a regular tetrahedron be glued over with equal regular hexagons?

A point $P$ in the interior of a convex polyhedron in Euclidean space is called a [i]pivot point[/i] of the polyhedron if every line through $P$ contains exactly $0$ or $2$ vertices of the polyhedron. Determine, with proof, the maximum number of pivot points that a polyhedron can contain.

The corner of a unit cube is chopped off such that the cut runs through the three vertices adjacent to the vertex of the chosen corner. What is the height of the cube when the freshly-cut face is placed on a table?

Find the value of $ x$ that satisfies the equation \[ 25^{\minus{}2}\equal{}\frac{5^{48/x}}{5^{26/x}\cdot25^{17/x}}. \]$ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 9$

In the tetrahedron $DABC, AB=BC, \angle DBC =\angle DBA$. Prove that $AC \perp DB$.

Let a tetrahedron $ABCD$ and its inner point $O$ be given. For any edge $e$ of $ABCD$ consider the segment $f(e)$ containing $O$ such that $f(e)\parallel e$ and the endpoints of $f(e)$ lie on the faces of the tetrahedron. Show that \[\sum_{e\text{ edge}}\,\frac{\,f(e)\,}{e}=3.\]

A spatial quadrilateral is circumscribed around a sphere. Prove that all the tangent points lie in one plane.

The height and radius of the base of the cylinder are equal to $1$. What is the smallest number of balls of radius $1$ that can cover the entire cylinder?

Maryam labels each vertex of a tetrahedron with the sum of the lengths of the three edges meeting at that vertex. She then observes that the labels at the four vertices of the tetrahedron are all equal. For each vertex of the tetrahedron, prove that the lengths of the three edges meeting at that vertex are the three side lengths of a triangle.

$S$ and $S'$ are two intersecting spheres. The line $BXB'$ is parallel to the line of centers, where $B$ is a point on $S, B'$ is a point on $S'$ and $X$ lies on both spheres. $A$ is another point on $S$, and $A'$ is another point on S' such that the line $AA'$ has a point on both spheres. Show that the segments $AB$ and $A'B'$ have equal projections on the line $AA'$.

$480$ $1$ cm unit cubes are used to build a block measuring $6$ cm by $8$ cm by $10$ cm. A tiny ant then chews his way in a straight line from one vertex of the block to the furthest vertex. How many cubes does the ant pass through? The ant is so tiny that he does not "pass through" cubes if he is merely passing through where their edges or vertices meet. [i]2017 CCA Math Bonanza Individual Round #11[/i]

A cube lies on the plane. After being rolled a few times (over its edges), it is brought back to its initial location with the same face up. Could the top face have been rotated by 90 degrees? [i](5 points)[/i]

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

Let $ABCD$ be a tetrahedron with $\angle BAD = 60^{\cdot}$, $\angle BAC = 40^{\cdot}$, $\angle ABD = 80^{\cdot}$, $\angle ABC = 70^{\cdot}$. Prove that the lines $AB$ and $CD$ are perpendicular.

Each face of a cube is of the same size as each square of a chessboard. The cube is coloured black and white, placed on one of the squares of the chessboard and rolled so that each square of the chessboard is visited exactly once. Can this be done in such a way that the colour of the visited square and the colour of the bottom face of the cube are always the same? (A Shapovalov)

Consider a cube $ABCDA'B'C'D$ (with $AB\perp AA'\parallel BB'\parallel CC'\parallel DD$). Let $X$ be an inner point of the segment $AB$ and denote $Y$ the intersection of the edge $AD$ and the plane $B'D'X$. (a) Let $M=B'Y\cap D'X$. Find the locus of all $M$s. (b) Determine whether there is a quadrilateral $B'D'YX$ such that its diagonals divide each other in the ratio 1:2.

After swimming around the ocean with some snorkling gear, Joshua walks back to the beach where Alexis works on a mural in the sand beside where they drew out symbol lists. Joshua walks directly over the mural without paying any attention. "You're a square, Josh." "No, $\textit{you're}$ a square," retorts Joshua. "In fact, you're a $\textit{cube}$, which is $50\%$ freakier than a square by dimension. And before you tell me I'm a hypercube, I'll remind you that mom and dad confirmed that they could not have given birth to a four dimension being." "Okay, you're a cubist caricature of male immaturity," asserts Alexis. Knowing nothing about cubism, Joshua decides to ignore Alexis and walk to where he stashed his belongings by a beach umbrella. He starts thinking about cubes and computes some sums of cubes, and some cubes of sums: \begin{align*}1^3+1^3+1^3&=3,\\1^3+1^3+2^3&=10,\\1^3+2^3+2^3&=17,\\2^3+2^3+2^3&=24,\\1^3+1^3+3^3&=29,\\1^3+2^3+3^3&=36,\\(1+1+1)^3&=27,\\(1+1+2)^3&=64,\\(1+2+2)^3&=125,\\(2+2+2)^3&=216,\\(1+1+3)^3&=125,\\(1+2+3)^3&=216.\end{align*} Josh recognizes that the cubes of the sums are always larger than the sum of cubes of positive integers. For instance, \begin{align*}(1+2+4)^3&=1^3+2^3+4^3+3(1^2\cdot 2+1^2\cdot 4+2^2\cdot 1+2^2\cdot 4+4^2\cdot 1+4^2\cdot 2)+6(1\cdot 2\cdot 4)\\&>1^3+2^3+4^3.\end{align*} Josh begins to wonder if there is a smallest value of $n$ such that \[(a+b+c)^3\leq n(a^3+b^3+c^3)\] for all natural numbers $a$, $b$, and $c$. Joshua thinks he has an answer, but doesn't know how to prove it, so he takes it to Michael who confirms Joshua's answer with a proof. What is the correct value of $n$ that Joshua found?

Let $C$ be a cube. Let $P$, $Q$, and $R$ be random vertices of $C$, chosen uniformly and independently from the set of vertices of $C$. (Note that $P$, $Q$, and $R$ might be equal.) Compute the probability that some face of $C$ contains $P$, $Q$, and $R$.