Found problems: 2265
A real number $f(X)\neq 0$ is assigned to each point $X$ in the space.
It is known that for any tetrahedron $ABCD$ with $O$ the center of the inscribed sphere, we have :
\[ f(O)=f(A)f(B)f(C)f(D). \]
Prove that $f(X)=1$ for all points $X$.
[i]Proposed by Aleksandar Ivanov[/i]
A set of points has [i]point symmetry[/i] if a reflection in some point maps the set to itself. Let $\cal P$ be a solid convex polyhedron whose orthogonal projections onto any plane have point symmetry. Prove that $\cal P$ has point symmetry.
[i]Proposed by Ethan Tan[/i]
Let it be is a wooden unit cube. We cut along every plane which is perpendicular to the segment joining two distinct vertices and bisects it. How many pieces do we get?
On the faces of a cube several positive integer numbers are written. On every edge the sum of the numbers of it's two faces is written, and in every vertex the sum of numbers on the three faces that have this vertex. It turned out that all the written numbers are different.
Find the smallest possible amount of the sum of all written numbers.
A cube of cheese $C=\{(x, y, z)| 0 \le x, y, z \le 1\}$ is cut along the planes $x=y$, $y=z$ and $z=x$. How many pieces are there? (No cheese is moved until all three cuts are made.)
$ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ 9 $
A sphere $S$ is tangent to the edges $AB,BC,CD,DA$ of a tetrahedron $ABCD$ at the points $E,F,G,H$ respectively. The points $E,F,G,H$ are the vertices of a square. Prove that if the sphere is tangent to the edge $AC$, then it is also tangent to the edge $BD.$
a conic with apotheme 1 slides (varying height and radius, with $r < \frac12$) so that the conic's area is $9$ times that of its inscribed sphere. What's the height of that conic?
We consider a prism which has the upper and inferior basis the pentagons: $A_{1}A_{2}A_{3}A_{4}A_{5}$ and $B_{1}B_{2}B_{3}B_{4}B_{5}$. Each of the sides of the two pentagons and the segments $A_{i}B_{j}$ with $i,j=1,\ldots$,5 is colored in red or blue. In every triangle which has all sides colored there exists one red side and one blue side. Prove that all the 10 sides of the two basis are colored in the same color.
A regular tetrahedron with an edge of $30$ cm rests on one of its faces. Assuming it is hollow, $2$ liters of water are poured into it. Find the height of the ''upper'' liquid and the area of the ''free'' surface of the water.
Determine the sum of all possible surface area of a cube two of whose vertices are $(1,2,0)$ and $(3,3,2)$.
Triangle $\vartriangle ABC$ has side lengths $AB = 39$, $BC = 16$, and $CA = 25$. What is the volume of the solid formed by rotating $\vartriangle ABC$ about line $BC$?
Give two congruent regular triangular pyramids, stick their bottom surfaces together. Then ,it becomes a hexahedron with all dihedral angles equal. The length of the shortest edge of the hexahedron is $2$. Then, the furthest distance between two vertexes is________.
Find all positive integer bases $b \ge 9$ so that the number
\[ \frac{{\overbrace{11 \cdots 1}^{n-1 \ 1's}0\overbrace{77 \cdots 7}^{n-1\ 7's}8\overbrace{11 \cdots 1}^{n \ 1's}}_b}{3} \]
is a perfect cube in base 10 for all sufficiently large positive integers $n$.
[i]Proposed by Yang Liu[/i]
The altitude of a quadrangular pyramid $SABCD$ passes through the intersection point of the diagonals of its base $ABCD$. From the tops of the base perpendiculars $AA_1$, $BB_1$, $CC_1$, $DD_1$ are dropped onto lines $SC$, $SD,$ $SA$ and $SB$ respectively. It turned out that the points $S$, $A_1$, $B_1$, $C_1$, $D_1$ are different and lie on the same sphere. Prove that lines $AA_1$, $ BB_1$, $CC_1$, $DD_1$ pass through one point.
$A,B$, and $C$ are three interior points of a sphere $S$ such that $AB$ and $AC$ are perpendicular to the diameter of $S$ through $A$, and so that two spheres can be constructed through $A$, $B$, and $C$ which are both tangent to $S$. Prove that the sum of their radii is equal to the radius of $S$.
A regular tetrahedron is sectioned with a plane after a rhombus. Prove that the rhombus is square.
Given a rectangular strip of measure $12 \times 1$. Paste this strip in two layers over the cube with edge $1$ (the strip can be bent, but cannot be cut).
(V. Shevyakov)
In space, there are \( 13 \) points, no four of which lie in the same plane. Three of the points are colored blue, and the triangle with these points as vertices will be called a [i]blue triangle[/i]. The remaining \( 10 \) points are colored red. We say that a triangle with three red vertices is [i]attached[/i] to the blue triangle if the boundary of the red triangle intersects the blue triangle (either in its interior or on its boundary) at exactly one point. Is it possible for the number of attached triangles to be \( 33 \)?
Let $M$ be an interior point of the tetrahedron $ABCD$. Prove that
\[ \begin{array}{c}\ \stackrel{\longrightarrow }{MA} \text{vol}(MBCD) +\stackrel{\longrightarrow }{MB} \text{vol}(MACD) +\stackrel{\longrightarrow }{MC} \text{vol}(MABD) + \stackrel{\longrightarrow }{MD} \text{vol}(MABC) = 0 \end{array}\]
($\text{vol}(PQRS)$ denotes the volume of the tetrahedron $PQRS$).
A tetrahedron $ABCD$ is given such that $AD = BC = a; AC = BD = b; AB\cdot CD = c^2$. Let $f(P) = AP + BP + CP + DP$, where $P$ is an arbitrary point in space. Compute the least value of $f(P).$
A cube with side length 10 is suspended above a plane. The vertex closest to the plane is labelled $A$. The three vertices adjacent to vertex $A$ are at heights 10, 11, and 12 above the plane. The distance from vertex $A$ to the plane can be expressed as $\tfrac{r-\sqrt{s}}{t}$, where $r$, $s$, and $t$ are positive integers, and $r+s+t<1000$. Find $r+s+t$.
Prove that one can construct two triangles from six edges of an arbitrary tetrahedron.
(VV Proizvolov)
What is the volume of tetrahedron $ABCD$ with edge lengths $AB=2, AC=3, AD=4, BC=\sqrt{13}, BD=2\sqrt{5},$ and $CD=5$?
$\textbf{(A) }3 \qquad \textbf{(B) }2\sqrt{3} \qquad \textbf{(C) }4 \qquad \textbf{(D) }3\sqrt{3} \qquad \textbf{(E) }6$
A positive integer $n$ is [i]nice[/i] if there is a positive integer $m$ with exactly four positive divisors (including $1$ and $m$) such that the sum of the four divisors is equal to $n$. How many numers in the set $\{2010, 2011, 2012,\ldots,2019\}$ are nice?
${ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D}}\ 4\qquad\textbf{(E)}\ 5 $
Given an arbitrary tetrahedron. Prove that its six edges can be divided into two triplets so that from each triple it was possible to form a triangle.