Found problems: 2265
2013 AMC 10, 14
A solid cube of side length $1$ is removed from each corner of a solid cube of side length $3$. How many edges does the remaining solid have?
$\textbf{(A) }36\qquad
\textbf{(B) }60\qquad
\textbf{(C) }72\qquad
\textbf{(D) }84\qquad
\textbf{(E) }108\qquad$
2015 BAMO, 4
Let $A$ be a corner of a cube. Let $B$ and $C$ the midpoints of two edges in the positions shown on the figure below:
[center][img]http://i.imgur.com/tEODnV0.png[/img][/center]
The intersection of the cube and the plane containing $A,B,$ and $C$ is some polygon, $P$.
[list=a]
[*] How many sides does $P$ have? Justify your answer.
[*] Find the ratio of the area of $P$ to the area of $\triangle{ABC}$ and prove that your answer is correct.
1993 All-Russian Olympiad, 4
Prove that any two rectangular prisms with equal volumes can be placed in a space such that any horizontal plain that intersects one of the prisms will intersect the other forming a polygon with the same area.
1989 National High School Mathematics League, 4
Three points of a triangle are among 8 vertex of a cube. So the number of such acute triangles is
$\text{(A)}0\qquad\text{(B)}6\qquad\text{(C)}8\qquad\text{(D)}24$
2012 Sharygin Geometry Olympiad, 19
Two circles with radii 1 meet in points $X, Y$, and the distance between these points also is equal to $1$. Point $C$ lies on the first circle, and lines $CA, CB$ are tangents to the second one. These tangents meet the first circle for the second time in points $B', A'$. Lines $AA'$ and $BB'$ meet in point $Z$. Find angle $XZY$.
2000 Swedish Mathematical Competition, 4
The vertices of a triangle are three-dimensional lattice points. Show that its area is at least $\frac12$.
2016 Harvard-MIT Mathematics Tournament, 3
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 $x$?
2007 AMC 10, 11
The numbers from $ 1$ to $ 8$ are placed at the vertices of a cube in such a manner that the sum of the four numbers on each face is the same. What is this common sum?
$ \textbf{(A)}\ 14 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 18 \qquad \textbf{(D)}\ 20 \qquad \textbf{(E)}\ 24$
1954 Moscow Mathematical Olympiad, 271
Do there exist points $A, B, C, D$ in space, such that $AB = CD = 8, AC = BD = 10$, and $AD = BC = 13$?
2015 BAMO, 5
We are given $n$ identical cubes, each of size $1\times 1\times 1$. We arrange all of these $n$ cubes to produce one or more congruent rectangular solids, and let $B(n)$ be the number of ways to do this.
For example, if $n=12$, then one arrangement is twelve $1\times1\times1$ cubes, another is one $3\times 2\times2$ solid, another is three $2\times 2\times1$ solids, another is three $4\times1\times1$ solids, etc. We do not consider, say, $2\times2\times1$ and $1\times2\times2$ to be different; these solids are congruent. You may wish to verify, for example, that $B(12) =11$.
Find, with proof, the integer $m$ such that $10^m<B(2015^{100})<10^{m+1}$.
2011 Today's Calculation Of Integral, 759
Given a regular tetrahedron $PQRS$ with side length $d$. Find the volume of the solid generated by a rotation around the line passing through $P$ and the midpoint $M$ of $QR$.
1980 All Soviet Union Mathematical Olympiad, 302
The edge $[AC]$ of the tetrahedron $ABCD$ is orthogonal to $[BC]$, and $[AD]$ is orthogonal to $[BD]$. Prove that the cosine of the angle between lines $(AC)$ and $(BD)$ is less than $|CD|/|AB|$.
2020 German National Olympiad, 6
The insphere and the exsphere opposite to the vertex $D$ of a (not necessarily regular) tetrahedron $ABCD$ touch the face $ABC$ in the points $X$ and $Y$, respectively. Show that $\measuredangle XAB=\measuredangle CAY$.
1994 Flanders Math Olympiad, 3
Two regular tetrahedrons $A$ and $B$ are made with the 8 vertices of a unit cube. (this way is unique)
What's the volume of $A\cup B$?
2018 Polish Junior MO First Round, 7
Square $ABCD$ with sides of length $4$ is a base of a cuboid $ABCDA'B'C'D'$. Side edges $AA'$, $BB'$, $CC'$, $DD'$ of this cuboid have length $7$. Points $K, L, M$ lie respectively on line segments $AA'$, $BB'$, $CC'$, and $AK = 3$, $BL = 2$, $CM = 5$. Plane passing through points $K, L, M$ cuts cuboid on two blocks. Calculate volumes of these blocks.
2010 Kyrgyzstan National Olympiad, 8
Solve in none-negative integers ${x^3} + 7{x^2} + 35x + 27 = {y^3}$.
2023 BMT, 7
A tetrahedron has three edges of length $2$ and three edges of length $4$, and one of its faces is an equilateral triangle. Compute the radius of the sphere that is tangent to every edge of this tetrahedron.
2008 Romania National Olympiad, 1
A tetrahedron has the side lengths positive integers, such that the product of any two opposite sides equals 6. Prove that the tetrahedron is a regular triangular pyramid in which the lateral sides form an angle of at least 30 degrees with the base plane.
2007 Princeton University Math Competition, 2
In how many distinguishable ways can $10$ distinct pool balls be formed into a pyramid ($6$ on the bottom, $3$ in the middle, one on top), assuming that all rotations of the pyramid are indistinguishable?
2023 AMC 10, 18
A rhombic dodecahedron is a solid with $12$ congruent rhombus faces. At every vertex, $3$ or $4$ edges meet, depending on the vertex. How many vertices have exactly $3$ edges meet?
$\textbf{(A) }5\qquad\textbf{(B) }6\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9$
2015 Math Prize for Girls Olympiad, 2
A tetrahedron $T$ is inside a cube $C$. Prove that the volume of $T$ is at most one-third the volume of $C$.
1988 All Soviet Union Mathematical Olympiad, 486
Prove that for any tetrahedron the radius of the inscribed sphere $r <\frac{ ab}{ 2(a + b)}$, where $a$ and $b$ are the lengths of any pair of opposite edges.
2007 ITest, 28
The space diagonal (interior diagonal) of a cube has length $6$. Find the $\textit{surface area}$ of the cube.
2011 China Second Round Olympiad, 6
In a tetrahedral $ABCD$, given that $\angle ADB=\angle BDC =\angle CDA=\frac{\pi}{3}$, $AD=BD=3$, and $CD=2$. Find the radius of the circumsphere of $ABCD$.
2016 Sharygin Geometry Olympiad, P24
A sphere is inscribed into a prism $ABCA'B'C'$ and touches its lateral faces $BCC'B', CAA'C', ABB'A' $ at points $A_o, B_o, C_o$ respectively. It is known that $\angle A_oBB' = \angle B_oCC' =\angle C_oAA'$.
a) Find all possible values of these angles.
b) Prove that segments $AA_o, BB_o, CC_o$ concur.
c) Prove that the projections of the incenter to $A'B', B'C', C'A'$ are the vertices of a regular triangle.