Found problems: 2265
2007 AMC 12/AHSME, 16
Each face of a regular tetrahedron is painted either red, white or blue. Two colorings are considered indistinguishable if two congruent tetrahedra with those colorings can be rotated so that their appearances are identical. How many distinguishable colorings are possible?
$ \textbf{(A)}\ 15 \qquad \textbf{(B)}\ 18 \qquad \textbf{(C)}\ 27 \qquad \textbf{(D)}\ 54 \qquad \textbf{(E)}\ 81$
2021 HMNT, 10
Three faces $X , Y, Z$ of a unit cube share a common vertex. Suppose the projections of $X , Y, Z$ onto a fixed plane $P$ have areas $x, y, z$, respectively. If $x : y : z = 6 : 10 : 15$, then $x + y + z$ can be written as $m/n$ , where $m, n$ are positive integers and $gcd(m, n) = 1$. Find $100m + n$.
2000 Harvard-MIT Mathematics Tournament, 7
A regular tetrahedron of volume $1$ is filled with water of total volume $\frac{7}{16}$. Is it possible that the center of the tetrahedron lies on the surface of the water? How about in a cube of volume $1$?
III Soros Olympiad 1996 - 97 (Russia), 9.3
Let $ABCD$ be a three-link broken line in space, all links of which are equal and $\angle BCD=90^o$. Find the distance from $A$ to the midpoint of $BD$, if $AD = a$.
2022 Oral Moscow Geometry Olympiad, 6
In a tetrahedron, segments connecting the midpoints of heights with the orthocenters of the faces to which these heights are drawn intersect at one point. Prove that in such a tetrahedron all faces are equal or there are perpendicular edges.
(Yu. Blinkov)
2012 National Olympiad First Round, 33
Let $ABCDA'B'C'D'$ be a rectangular prism with $|AB|=2|BC|$. $E$ is a point on the edge $[BB']$ satisfying $|EB'|=6|EB|$. Let $F$ and $F'$ be the feet of the perpendiculars from $E$ at $\triangle AEC$ and $\triangle A'EC'$, respectively. If $m(\widehat{FEF'})=60^{\circ}$, then $|BC|/|BE| = ? $
$ \textbf{(A)}\ \sqrt\frac53 \qquad \textbf{(B)}\ \sqrt\frac{15}2 \qquad \textbf{(C)}\ \frac32\sqrt{15} \qquad \textbf{(D)}\ 5\sqrt\frac53 \qquad \textbf{(E)}\ \text{None}$
2009 Today's Calculation Of Integral, 501
Find the volume of the uion $ A\cup B\cup C$ of the three subsets $ A,\ B,\ C$ in $ xyz$ space such that:
\[ A\equal{}\{(x,\ y,\ z)\ |\ |x|\leq 1,\ y^2\plus{}z^2\leq 1\}\]
\[ B\equal{}\{(x,\ y,\ z)\ |\ |y|\leq 1,\ z^2\plus{}x^2\leq 1\}\]
\[ C\equal{}\{(x,\ y,\ z)\ |\ |z|\leq 1,\ x^2\plus{}y^2\leq 1\}\]
2014 Online Math Open Problems, 16
Let $OABC$ be a tetrahedron such that $\angle AOB = \angle BOC = \angle COA = 90^\circ$ and its faces have integral surface areas. If $[OAB] = 20$ and $[OBC] = 14$, find the sum of all possible values of $[OCA][ABC]$. (Here $[\triangle]$ denotes the area of $\triangle$.)
[i]Proposed by Robin Park[/i]
2008 China Team Selection Test, 3
Determine the greatest positive integer $ n$ such that in three-dimensional space, there exist n points $ P_{1},P_{2},\cdots,P_{n},$ among $ n$ points no three points are collinear, and for arbitary $ 1\leq i < j < k\leq n$, $ P_{i}P_{j}P_{k}$ isn't obtuse triangle.
1985 National High School Mathematics League, 2
$PQ$ is a chord of parabola $y^2=2px(p>0)$ and $PQ$ pass its focus $F$. Line $l$ is its directrix. Projection of $PQ$ on $l$ is $MN$. The area of curved surface that $PQ$ rotate around $l$ is $S_1$, the area of spherical surface of the ball with diameter of $MN$ is $S_2$, then
$\text{(A)}S_1>S_2\qquad\text{(B)}S_1<S_2\qquad\text{(C)}S_1\geq S_2\qquad\text{(D)}$ Not sure
2015 AoPS Mathematical Olympiad, 2
In tetrahedron $ABCD$, let $V$ be the volume of the tetrahedron and $R$ the radius of the sphere that it tangent to all four faces of the tetrahedron. Let $P$ be the surface area of the tetrahedron. Prove that $$r=\frac{3V}{P}.$$
[i]Proposed by CaptainFlint.[/i]
2005 QEDMO 1st, 8 (Z2)
Prove that if $n$ can be written as $n=a^2+ab+b^2$, then also $7n$ can be written that way.
1937 Moscow Mathematical Olympiad, 036
* Given a regular dodecahedron. Find how many ways are there to draw a plane through it so that its section of the dodecahedron is a regular hexagon?
2009 Stanford Mathematics Tournament, 2
Factor completely the expression $(a-b)^3+(b-c)^3+(c-a)^3$
1986 Bulgaria National Olympiad, Problem 3
A regular tetrahedron of unit edge is given. Find the volume of the maximal cube contained in the tetrahedron, whose one vertex lies in the feet of an altitude of the tetrahedron.
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?
1982 Miklós Schweitzer, 7
Let $ V$ be a bounded, closed, convex set in $ \mathbb{R}^n$, and denote by $ r$ the radius of its circumscribed sphere (that is, the radius of the smallest sphere that contains $ V$). Show that $ r$ is the only real number with the following property: for any finite number of points in $ V$, there exists a point in $ V$ such that the arithmetic mean of its distances from the other points is equal to $ r$.
[i]Gy. Szekeres[/i]
1985 IMO Longlists, 28
[i]a)[/i] Let $M$ be the set of the lengths of the edges of an octahedron whose sides are congruent quadrangles. Prove that $M$ has at most three elements.
[i]b)[/i] Let an octahedron whose sides are congruent quadrangles be given. Prove that each of these quadrangles has two equal sides meeting at a common vertex.
2012 Poland - Second Round, 2
Prove that for tetrahedron $ABCD$; vertex $D$, center of insphere and centroid of $ABCD$ are collinear iff areas of triangles $ABD,BCD,CAD$ are equal.
2015 CCA Math Bonanza, I1
Michael the Mouse finds a block of cheese in the shape of a regular tetrahedron (a pyramid with equilateral triangles for all faces). He cuts some cheese off each corner with a sharp knife. How many faces does the resulting solid have?
[i]2015 CCA Math Bonanza Individual Round #1[/i]
1990 AMC 8, 18
Each corner of a rectangular prism is cut off. Two (of the eight) cuts are shown. How many edges does the new figure have?
[asy]
draw((0,0)--(3,0)--(3,3)--(0,3)--cycle);
draw((3,0)--(5,2)--(5,5)--(2,5)--(0,3));
draw((3,3)--(5,5));
draw((2,0)--(3,1.8)--(4,1)--cycle,linewidth(1));
draw((2,3)--(4,4)--(3,2)--cycle,linewidth(1));[/asy]
$ \text{(A)}\ 24\qquad\text{(B)}\ 30\qquad\text{(C)}\ 36\qquad\text{(D)}\ 42\qquad\text{(E)}\ 48 $
[i]Assume that the planes cutting the prism do not intersect anywhere in or on the prism.[/i]
Champions Tournament Seniors - geometry, 2011.4
The height $SO$ of a regular quadrangular pyramid $SABCD$ forms an angle $60^o$ with a side edge , the volume of this pyramid is equal to $18$ cm$^3$ . The vertex of the second regular quadrangular pyramid is at point $S$, the center of the base is at point $C$, and one of the vertices of the base lies on the line $SO$. Find the volume of the common part of these pyramids. (The common part of the pyramids is the set of all such points in space that lie inside or on the surface of both pyramids).
2007 IMO Shortlist, 7
Let $ n$ be a positive integer. Consider
\[ S \equal{} \left\{ (x,y,z) \mid x,y,z \in \{ 0, 1, \ldots, n\}, x \plus{} y \plus{} z > 0 \right \}
\]
as a set of $ (n \plus{} 1)^{3} \minus{} 1$ points in the three-dimensional space. Determine the smallest possible number of planes, the union of which contains $ S$ but does not include $ (0,0,0)$.
[i]Author: Gerhard Wöginger, Netherlands [/i]
1989 IMO Shortlist, 21
Prove that the intersection of a plane and a regular tetrahedron can be an obtuse-angled triangle and that the obtuse angle in any such triangle is always smaller than $ 120^{\circ}.$
2005 Brazil National Olympiad, 3
A square is contained in a cube when all of its points are in the faces or in the interior of the cube. Determine the biggest $\ell > 0$ such that there exists a square of side $\ell$ contained in a cube with edge $1$.