Each face of a tetrahedron can be placed in a circle of radius $ 1$. Show that the tetrahedron can be placed in a sphere of radius $ \frac{3}{2\sqrt2}$.

Let $M_A, M_B, M_C$ be the midpoints of the sides $BC, CA, AB$ respectively of a non-isosceles triangle $ABC$. Points $H_A, H_B, H_C$ lie on the corresponding sides, different from $M_A, M_B, M_C$ such that $M_AH_B=M_AH_C, $ $M_BH_A=M_BH_C,$ and $M_CH_A=M_CH_B$. Prove that $H_A, H_B, H_C$ are the feet of the corresponding altitudes.

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} $

Let $n$ and $k$ be positive integers and let $$T = \{ (x,y,z) \in \mathbb{N}^3 \mid 1 \leq x,y,z \leq n \}$$ be the length $n$ lattice cube. Suppose that $3n^2 - 3n + 1 + k$ points of $T$ are colored red such that if $P$ and $Q$ are red points and $PQ$ is parallel to one of the coordinate axes, then the whole line segment $PQ$ consists of only red points. Prove that there exists at least $k$ unit cubes of length $1$, all of whose vertices are colored red.

A pyramid has a square base $ ABCD$ and vertex $ E$. The area of square $ ABCD$ is $ 196$, and the areas of $ \triangle{ABE}$ and $ \triangle{CDE}$ are $ 105$ and $ 91$, respectively. What is the volume of the pyramid? $ \textbf{(A)}\ 392 \qquad \textbf{(B)}\ 196\sqrt{6} \qquad \textbf{(C)}\ 392\sqrt2 \qquad \textbf{(D)}\ 392\sqrt3 \qquad \textbf{(E)}\ 784$

A rectangular box $ P$ is inscribed in a sphere of radius $ r$. The surface area of $ P$ is 384, and the sum of the lengths of its 12 edges is 112. What is $ r$? $ \textbf{(A)}\ 8 \qquad \textbf{(B)}\ 10 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 16$

A lampshade is made in the form of the lateral surface of the frustum of a right circular cone. The height of the frustum is $3\sqrt{3}$ inches, its top diameter is 6 inches, and its bottom diameter is 12 inches. A bug is at the bottom of the lampshade and there is a glob of honey on the top edge of the lampshade at the spot farthest from the bug. The bug wants to crawl to the honey, but it must stay on the surface of the lampshade. What is the length in inches of its shortest path to the honey? [center] [img]https://cdn.artofproblemsolving.com/attachments/b/4/23f9bc88ea057cb2676f2b8b373330b0f5df69.png[/img][/center] $\textbf{(A) } 6 + 3\pi\qquad \textbf{(B) }6 + 6\pi\qquad \textbf{(C) } 6\sqrt3 \qquad \textbf{(D) } 6\sqrt5 \qquad \textbf{(E) } 6\sqrt3 + \pi$

Given three points in space forming an acute-angled triangle, show that we can find two further points such that no three of the five points are collinear and the line through any two is normal to the plane through the other three.

Prove that if two segments of a tetrahedron, going from the ends of some edge to the centers of the inscribed circles of opposite faces, intersect, then the segments issued from the ends of the crossing with it edges to the centers of the inscribed circles of the other two faces, also intersect.

Show that the sum of the distances from any point of a regular tetrahedron to its faces is equal to the height of this tetrahedron.

Consider a cylinder and a cone with a common base such that the volume of the part of the cylinder enclosed in the cone equals the volume of the part of the cylinder outside the cone. Find the ratio of the height of the cone to the height of the cylinder.

Three spheres have radii $144$, $225$, and $400$, are pairwise externally tangent to each other, and are all tangent to the same plane at $A$, $B$, and $C$. Compute the area of triangle $ABC$. [i]2021 CCA Math Bonanza Team Round #6[/i]

The base $ABC$ of a tetrahedron $MABC$ is an equilateral triangle, and the lateral edges $MA,MB,MC$ are sides of a triangle of the area $S$. If $R$ is the circumradius and $V$ the volume of the tetrahedron, prove that $RS\ge2V$. When does equality hold?

Prove that if the four faces of a tetrahedron are of the same area they are equal.

Let $ R$, $ r$ be respectively the circumradius and inradius of a regular $ 1986$-gonal pyramid. Prove that \[ \frac{R}{r}\ge 1\plus{}\frac{1}{\cos\frac{\pi}{1986}}\] and find the total area of the surface of the pyramid when the equality occurs.

A $10 \times 10 \times 10$ cube is made up up from $500$ white unit cubes and $500$ black unit cubes, arranged in such a way that every two unit cubes that shares a face are in different colors. A line is a $1 \times 1 \times 10$ portion of the cube that is parallel to one of cube’s edges. From the initial cube have been removed $100$ unit cubes such that $300$ lines of the cube has exactly one missing cube. Determine if it is possible that the number of removed black unit cubes is divisible by $4$.

Let $V$ be a rectangular prism with integer side lengths. The largest face has area $240$ and the smallest face has area $48$. A third face has area $x$, where $x$ is not equal to $48$ or $240$. What is the sum of all possible values of $x$?

Ayase randomly picks a number $x \in (0, 1]$ with uniform probability. He then draws the six points $(0, 0, 0),(x, 0, 0),(2x, 3x, 0),(5, 5, 2),(7, 3, 0),(9, 1, 4)$. If the expected value of the volume of the convex polyhedron formed by these six points can be written as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, find $m + n$

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} $

There is a ring outside of Saturn. In order to distinguish if the ring is actually a part of Saturn or is instead part of the satellites of Saturn, we need to know the relation between the velocity $v$ of each layer in the ring and the distance $R$ of the layer to the center of Saturn. Which of the following statements is correct? $\textbf{(A) }$ If $v \propto R$, then the layer is part of Saturn. $\textbf{(B) }$ If $v^2 \propto R$, then the layer is part of the satellites of Saturn. $\textbf{(C) }$ If $v \propto 1/R$, then the layer is part of Saturn. $\textbf{(D) }$ If $v^2 \propto 1/R$, then the layer is part of Saturn. $\textbf{(E) }$ If $v \propto R^2$, then the layer is part of the satellites of Saturn.

Does there exist a regular triangular prism that can be covered (without overlapping) by different equilateral triangles? (One is allowed to bend the triangles around the edges of the prism.)

A spider and a fly are sitting on a cube. The fly wants to maximize the shortest path to the spider along the surface of the cube. Is it necessarily best for the fly to be at the point opposite to the spider? (“Opposite” means “symmetric with respect to the centre of the cube”.)

We call a tetrahedron divisor of a parallelepiped if the parallelepiped can be divided into $6$ copies of that tetrahedron. Does there exist a parallelepiped that it has at least two different divisor tetrahedrons?

A sphere is tangent to six edges of a regular tetrahedron. If the length of each edge is $a$, then the volume of the sphere is________.

A brick has the shape of a cube of size $2$ with one corner unit cube removed. Given a cube of side $2^{n}$ divided into unit cubes from which an arbitrary unit cube is removed, show that the remaining figure can be built using the described bricks.