Found problems: 2265
2013-2014 SDML (High School), 3
The $48$ faces of $8$ unit cubes are painted white. What is the smallest number of these faces that can be repainted black so that it becomes impossible to arrange the $8$ unit cubes into a two by two by two cube, each of whose $6$ faces is totally white?
2006 Oral Moscow Geometry Olympiad, 2
Six segments are such that any three can form a triangle. Is it true that these segments can be used to form a tetrahedron?
(S. Markelov)
2012-2013 SDML (Middle School), 15
Three faces of a rectangular prism have diagonal lengths of $7$, $8$, and $9$ inches. How many cubic inches are in the volume of the rectangular prism?
$\text{(A) }48\sqrt{11}\qquad\text{(B) }160\qquad\text{(C) }14\sqrt{95}\qquad\text{(D) }35\sqrt{15}\qquad\text{(E) }504$
1997 Austrian-Polish Competition, 9
Given a parallelepiped $P$, let $V_P$ be its volume, $S_P$ the area of its surface and $L_P$ the sum of the lengths of its edges. For a real number $t \ge 0$, let $P_t$ be the solid consisting of all points $X$ whose distance from some point of $P$ is at most $t$. Prove that the volume of the solid $P_t$ is given by the formula $V(P_t) =V_P + S_Pt + \frac{\pi}{4} L_P t^2 + \frac{4\pi}{3} t^3$.
2012 Stanford Mathematics Tournament, 3
Express $\frac{2^3-1}{2^3+1}\times\frac{3^3-1}{3^3+1}\times\frac{4^3-1}{4^3+1}\times\dots\times\frac{16^3-1}{16^3+1}$ as a fraction in lowest terms.
2023 Belarusian National Olympiad, 9.8
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.
1985 Vietnam National Olympiad, 3
A triangular pyramid $ O.ABC$ with base $ ABC$ has the property that the lengths of the altitudes from $ A$, $ B$ and $ C$ are not less than $ \frac{OB \plus{}OC}{2}$, $ \frac{OC \plus{} OA}{2}$ and $ \frac{OA \plus{} OB}{2}$, respectively. Given that the area of $ ABC$ is $ S$, calculate the volume of the pyramid.
1955 Miklós Schweitzer, 8
[b]8.[/b] Show that on any tetrahedron there can be found three acute bihedral angles such that the faces including these angles count among them all faces of tetrahedron. [b](G. 10)[/b]
2004 AMC 10, 23
Each face of a cube is painted either red or blue, each with probability $ 1/2$. The color of each face is determined independently. What is the probability that the painted cube can be placed on a horizontal surface so that the four vertical faces are all the same color?
$ \textbf{(A)}\ \frac14 \qquad
\textbf{(B)}\ \frac{5}{16} \qquad
\textbf{(C)}\ \frac38 \qquad
\textbf{(D)}\ \frac{7}{16} \qquad
\textbf{(E)}\ \frac12$
2002 Flanders Junior Olympiad, 3
Is it possible to number the $8$ vertices of a cube from $1$ to $8$ in such a way that the value of the sum on every edge is different?
1997 Tournament Of Towns, (539) 4
All edges of a tetrahedron $ABCD$ are equal. The tetrahedron $ABCD$ is inscribed in a sphere. $CC'$ and $DD'$ are diameters. Find the angle between the planes $ABC$' and $ACD'$.
(A Zaslavskiy)
2006 All-Russian Olympiad Regional Round, 10.8
A convex polyhedron has $2n$ faces ($n\ge 3$), and all faces are triangles. What is the largest number of vertices at which converges exactly $3$ edges at a such a polyhedron ?
1969 IMO Longlists, 27
$(GBR 4)$ The segment $AB$ perpendicularly bisects $CD$ at $X$. Show that, subject to restrictions, there is a right circular cone whose axis passes through $X$ and on whose surface lie the points $A,B,C,D.$ What are the restrictions?
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 $|CD|/|AB|$.
2014 Online Math Open Problems, 24
Let $\mathcal A = A_0A_1A_2A_3 \cdots A_{2013}A_{2014}$ be a [i]regular 2014-simplex[/i], meaning the $2015$ vertices of $\mathcal A$ lie in $2014$-dimensional Euclidean space and there exists a constant $c > 0$ such that $A_iA_j = c$ for any $0 \le i < j \le 2014$. Let $O = (0,0,0,\dots,0)$, $A_0 = (1,0,0,\dots,0)$, and suppose $A_iO$ has length $1$ for $i=0,1,\dots,2014$. Set $P=(20,14,20,14,\dots,20,14)$. Find the remainder when \[PA_0^2 + PA_1^2 + \dots + PA_{2014}^2 \] is divided by $10^6$.
[i]Proposed by Robin Park[/i]
1989 Dutch Mathematical Olympiad, 4
Given is a regular $n$-sided pyramid with top $T$ and base $A_1A_2A_3... A_n$. The line perpendicular to the ground plane through a point $B$ of the ground plane within $A_1A_2A_3... A_n$ intersects the plane $TA_1A_2$ at $C_1$, the plane $TA_2A_3$ at $C_2$, and so on, and finally the plane $TA_nA_1$ at $C_n$. Prove that $BC_1 + BC_2 + ... + BC_n$ is independent of choice of $B$'s.
1999 Baltic Way, 2
Determine all positive integers $n$ with the property that the third root of $n$ is obtained by removing its last three decimal digits.
PEN R Problems, 2
Show there do not exist four points in the Euclidean plane such that the pairwise distances between the points are all odd integers.
2010 Brazil National Olympiad, 3
What is the biggest shadow that a cube of side length $1$ can have, with the sun at its peak?
Note: "The biggest shadow of a figure with the sun at its peak" is understood to be the biggest possible area of the orthogonal projection of the figure on a plane.
2020 Sharygin Geometry Olympiad, 24
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?
2010 AMC 12/AHSME, 7
Logan is constructing a scaled model of his town. The city's water tower stands $ 40$ meters high, and the top portion is a sphere that holds $ 100,000$ liters of water. Logan's miniature water tower holds $ 0.1$ liters. How tall, in meters, should Logan make his tower?
$ \textbf{(A)}\ 0.04\qquad \textbf{(B)}\ \frac{0.4}{\pi}\qquad \textbf{(C)}\ 0.4\qquad \textbf{(D)}\ \frac{4}{\pi}\qquad \textbf{(E)}\ 4$
2004 Oral Moscow Geometry Olympiad, 6
In the tetrahedron $DABC$ : $\angle ACB = \angle ADB$, $(CD) \perp (ABC)$. In triangle $ABC$, the altitude $h$ drawn to the side $AB$ and the distance $d$ from the center of the circumscribed circle to this side are given. Find the length of the $CD$.
2011 Olympic Revenge, 2
Let $p$ be a fixed prime. Determine all the integers $m$, as function of $p$, such that there exist $a_1, a_2, \ldots, a_p \in \mathbb{Z}$ satisfying
\[m \mid a_1^p + a_2^p + \cdots + a_p^p - (p+1).\]
2008 Harvard-MIT Mathematics Tournament, 19
Let $ ABCD$ be a regular tetrahedron, and let $ O$ be the centroid of triangle $ BCD$. Consider the point $ P$ on $ AO$ such that $ P$ minimizes $ PA \plus{} 2(PB \plus{} PC \plus{} PD)$. Find $ \sin \angle PBO$.
1957 Moscow Mathematical Olympiad, 365
(a) Given a point $O$ inside an equilateral triangle $\vartriangle ABC$. Line $OG$ connects $O$ with the center of mass $G$ of the triangle and intersects the sides of the triangle, or their extensions, at points $A', B', C'$ . Prove that $$\frac{A'O}{A'G} + \frac{B'O}{B'G} + \frac{C'O}{C'G} = 3.$$
(b) Point $G$ is the center of the sphere inscribed in a regular tetrahedron $ABCD$. Straight line $OG$ connecting $G$ with a point $O$ inside the tetrahedron intersects the faces at points $A', B', C', D'$. Prove that $$\frac{A'O}{A'G} + \frac{B'O}{B'G} + \frac{C'O}{C'G}+ \frac{D'O}{D'G} = 4.$$