Let $ABCD$ be a convex quadrilateral whose diagonals intersect at $O$ at an angle $\theta$. Let us set $OA = a, OB = b, OC = c$, and $OD = d, c > a > 0$, and $d > b > 0.$ Show that if there exists a right circular cone with vertex $V$, with the properties: [b](1)[/b] its axis passes through $O$, and [b](2)[/b] its curved surface passes through $A,B,C$ and $D,$ then \[OV^2=\frac{d^2b^2(c + a)^2 - c^2a^2(d + b)^2}{ca(d - b)^2 - db(c - a)^2}.\] Show also that if $\frac{c+a}{d+b}$ lies between $\frac{ca}{db}$ and $\sqrt{\frac{ca}{db}},$ and $\frac{c-a}{d-b}=\frac{ca}{db},$ then for a suitable choice of $\theta$, a right circular cone exists with properties [b](1) [/b]and [b](2).[/b]

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]

A number is written in the each vertex of a cube. It is allowed to add one to two numbers written in the ends of one edge. Is it possible to obtain the cube with all equal numbers if the numbers were initially as on the pictures:

The four circumcircles of the four faces of a tetrahedron have equal radii. Prove that the four faces of the tetrahedron are congruent triangles.

Let $I$ be the incenter of a tetrahedron $ABCD$, and $J$ be the center of the exsphere touching the face $BCD$ containing three remaining faces (outside these faces). The segment $IJ$ meets the circumsphere of the tetrahedron at point $K$. Which of two segments $IJ$ and $JK$ is longer?

Consider the cube $ABCDA'B'C'D'$ ($ABCD$ and $A'B'C'D'$ are the upper and lower bases, repsectively, and edges $AA', BB', CC', DD'$ are parallel). The point $X$ moves at a constant speed along the perimeter of the square $ABCD$ in the direction $ABCDA$, and the point $Y$ moves at the same rate along the perimiter of the square $B'C'CB$ in the direction $B'C'CBB'$. Points $X$ and $Y$ begin their motion at the same instant from the starting positions $A$ and $B'$, respectively. Determine and draw the locus of the midpionts of the segments $XY$.

One dimension of a cube is increased by $ 1$, another is decreased by $ 1$, and the third is left unchanged. The volume of the new rectangular solid is $ 5$ less than that of the cube. What was the volume of the cube? $ \textbf{(A)}\ 8 \qquad \textbf{(B)}\ 27 \qquad \textbf{(C)}\ 64 \qquad \textbf{(D)}\ 125 \qquad \textbf{(E)}\ 216$

Let $ABCD$ be a tetrahedron with $BA \perp AC,DB \perp (BAC)$. Denote by $O$ the midpoint of $AB$, and $K$ the foot of the perpendicular from $O$ to $DC$. Suppose that $AC = BD$. Prove that $\frac{V_{KOAC}}{V_{KOBD}}=\frac{AC}{BD}$ if and only if $2AC \cdot BD = AB^2$.

Prove that the volume of a tetrahedron inscribed in a right circular cylinder of volume $1$ does not exceed $\frac{2}{3 \pi}.$

Let $r_1, \ldots , r_n$ be the radii of $n$ spheres. Call $S_1, S_2, \ldots , S_n$ the areas of the set of points of each sphere from which one cannot see any point of any other sphere. Prove that \[\frac{S_1}{r_1^2} + \frac{S_2}{r_2^2}+\cdots+\frac{S_n}{r_n^2} = 4 \pi.\]

The polynomial $ f(x) \equal{} x^{4} \plus{} ax^{3} \plus{} bx^{2} \plus{} cx \plus{} d$ has real coefficients, and $ f(2i) \equal{} f(2 \plus{} i) \equal{} 0.$ What is $ a \plus{} b \plus{} c \plus{} d?$ $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ 16$

$L, L'$ are two skew lines in space and $p$ is a plane not containing either line. $M$ is a variable line parallel to $p$ which meets $L$ at $X$ and $L'$ at $Y$. Find the position of $M$ which minimises the distance $XY$. $L''$ is another fixed line. Find the line $M$ which is also perpendicular to $L''$ .

An ant is on the bottom edge of a right circular cone with base area $\pi$ and slant length $6$. What is the shortest distance that the ant has to travel to loop around the cone and come back to its starting position?

Prove that the sum of dihedral angles in an arbitrary tetrahedron is greater than $2\pi$

Let $ x,y,z$ be real numbers. Find the minimum value of $ x^2\plus{}y^2\plus{}z^2$ if $ x^3\plus{}y^3\plus{}z^3\minus{}3xyz\equal{}1$.

$ABCDA_{1}B_{1}C_{1}D_{1}$ is a cube with side length $4a$. Points $E$ and $F$ are taken on $(AA_{1})$ and $(BB_{1})$ such that $AE=B_{1}F=a$. $G$ and $H$ are midpoints of $(A_{1}B_{1})$ and $(C_{1}D_{1})$, respectively. Find the minimum value of the $CP+PQ$, where $P\in[GH]$ and $Q\in[EF]$.

Fiona had a solid rectangular block of cheese that measured $6$ centimeters from left to right, $5$ centimeters from front to back, and $4$ centimeters from top to bottom. Fiona took a sharp knife and sliced off a $1$ centimeter thick slice from the left side of the block and a $1$ centimeter slice from the right side of the block. After that, she sliced off a $1$ centimeter thick slice from the front side of the remaining block and a $1$ centimeter slice from the back side of the remaining block. Finally, Fiona sliced off a $1$ centimeter slice from the top of the remaining block and a $1$ centimeter slice from the bottom of the remaining block. Fiona now has $7$ blocks of cheese. Find the total surface area of those seven blocks of cheese measured in square centimeters.

Find all primes $p$ for which $p^2 - p + 1$ is a perfect cube.

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.

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 $m + n$

A given truncated pyramid has triangular bases. The areas of the bases are $B_1$ and $B_2$ and the area of the surface is $S$. Prove that if there exists a plane parallel to the bases whose intersection divides the pyramid to two truncated pyramids in which may be inscribed by spheres then $$S=(\sqrt{B_1}+\sqrt{B_2})(\sqrt[4]{B_1}+\sqrt[4]{B_2})^2$$ [i]G. Gantchev[/i]

The altitudes $AA_1,BB_1,CC_1,DD_1$ of a tetrahedron $ABCD$ intersect in the center $H$ of the sphere inscribed in the tetrahedron $A_1B_1C_1D_1$. Prove that the tetrahedron $ABCD$ is regular. (D. Tereshin)

$ (a)$ Prove that every multiple of $ 6$ can be written as a sum of four cubes. $ (b)$ Prove that every integer can be written as a sum of five cubes.

For each vertex of a solid cube, consider the tetrahedron determined by the vertex and the midpoints of the three edges that meet at that vertex. The portion of the cube that remains when these eight tetrahedra are cut away is called a [i]cuboctahedron[/i]. The ratio of the volume of the cuboctahedron to the volume of the original cube is closest to which of these? $ \textbf{(A)}\ 75\%\qquad\textbf{(B)}\ 78\%\qquad\textbf{(C)}\ 81\%\qquad\textbf{(D)}\ 84\%\qquad\textbf{(E)}\ 87\% $

Given a cube $ABCD-A'B'C'D'$, in the $12$ lines:$AB',BA',CD',DC',AD',DA',BC',CB',AC,BD,A'C',B'D'$, how many sets of lines are skew lines? $\text{(A)}30\qquad\text{(B)}60\qquad\text{(C)}24\qquad\text{(D)}48$