Found problems: 2265
The medians from $ A $ to the faces $ ABC,ABD,ACD $ of a tetahedron $ ABCD $ are pairwise perpendicular.
Show that the edges from $ A $ have equal lengths.
[i]Dinu Șerbănescu[/i]
Points $E$ and $F$ are taken on the diagonals $AB_1$ and $CA_1$ of the lateral faces $ABB_1A_1$ and $CAA_1C_1$ of a triangular prism $ABCA_1B_1C_1$ so that $EF\parallel BC_1$. Find the ratio of the lengths of $EF$ and $BC_1$.
Consider a regular octahedron $ABCDEF$ with lower vertex $E$, upper vertex $F$, middle cross-section $ABCD$, midpoint $M$ and circumscribed sphere $k$. Further, let $X$ be an arbitrary point inside the face $ABF$. Let the line $EX$ intersect $k$ in $E$ and $Z$, and the plane $ABCD$ in $Y$.
Show that $\sphericalangle{EMZ}=\sphericalangle{EYF}$.
Chords $AA^{\prime}$, $BB^{\prime}$, $CC^{\prime}$ of a sphere meet at an interior point $P$ but are not contained in a plane. The sphere through $A$, $B$, $C$, $P$ is tangent to the sphere through $A^{\prime}$, $B^{\prime}$, $C^{\prime}$, $P$. Prove that $\, AA' = BB' = CC'$.
Let $C$ be a cube. By connecting the centres of the faces of $C$ with lines we form an octahedron $O$. By connecting the centers of each face of $O$ with lines we get a smaller cube $C'$. What is the ratio between the side length of $C$ and the side length of $C'$?
What is the volume of a cube whose surface area is twice that of a cube with volume $ 1$?
$ \textbf{(A)}\ \sqrt{2} \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ 2\sqrt{2} \qquad
\textbf{(D)}\ 4 \qquad
\textbf{(E)}\ 8$
Given a cube $ABCDA_1B_1C_1D_1$ with edge $5$. On the edge $BB_1$ of the cube , point $K$ such thath $BK=4$.
a) Construct a cube section with the plane $a$ passing through the points $K$ and $C_1$ parallel to the diagonal $BD_1$.
b) Find the angle between the plane $a$ and the plane $BB_1C_1$.
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 $V(P_t) =V_P + S_Pt + \frac{\pi}{4} L_P t^2 + \frac{4\pi}{3} t^3$.
Call two circles in three-dimensional space pairwise tangent at a point $ P$ if they both pass through $ P$ and lines tangent to each circle at $ P$ coincide. Three circles not all lying in a plane are pairwise tangent at three distinct points. Prove that there exists a sphere which passes through the three circles.
Divide a cube into three equal pyramids.
Find all natural numbers $n$ for which there exists a convex polyhedron with $n$ edges, with exactly one vertex having four edges and all other vertices having $3$ edges.
Determine positive integers $p, q$, and $r$ such that the diagonal of a block consisting of $p\times q\times r$ unit cubes passes through exactly $1984$ of the unit cubes, while its length is minimal. (The diagonal is said to pass through a unit cube if it has more than one point in common with the unit cube.)
Study the derivatives of the function
\[y=\sqrt{x^3-4x}\]
and sketch its graph on the real line.
Describe how to place the vertices of a triangle in the faces of a cube in such a way that the shortest side of the triangle is the biggest possible.
Four rays spread out from point $O$ in a $3$-dimensional space in a way that the angle between every two rays is $a$. Find $\cos a$.
Each of the $12$ edges of a cube is labeled $0$ or $1$. Two labelings are considered different even if one can be obtained from the other by a sequence of one or more rotations and/or reflections. For how many such labelings is the sum of the labels on the edges of each of the $6$ faces of the cube equal to $2?$
$\textbf{(A) }8\qquad\textbf{(B) }10\qquad\textbf{(C) }12\qquad\textbf{(D) }16\qquad\textbf{(E) }20$
Let $\,S\,$ be a finite set of points in three-dimensional space. Let $\,S_{x},\,S_{y},\,S_{z}\,$ be the sets consisting of the orthogonal projections of the points of $\,S\,$ onto the $yz$-plane, $zx$-plane, $xy$-plane, respectively. Prove that \[ \vert S\vert^{2}\leq \vert S_{x} \vert \cdot \vert S_{y} \vert \cdot \vert S_{z} \vert, \] where $\vert A \vert$ denotes the number of elements in the finite set $A$.
[hide="Note"] Note: The orthogonal projection of a point onto a plane is the foot of the perpendicular from that point to the plane. [/hide]
There are given 2050 points in a unit cube. Show that there are 5 points lying in an (open) ball with the radius 1/9.
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.
Given a sphere, a great circle of the sphere is a circle on the sphere whose diameter is also a diameter of the sphere. For a given positive integer $n,$ the surface of a sphere is divided into several regions by $n$ great circles, and each region is colored black or white. We say that a coloring is good if any two adjacent regions (that share an arc as boundary, not just a finite number of points) have different colors. Find, with proof, all positive integers $n$ such that in every good coloring with $n$ great circles, the sum of the areas of the black regions is equal to the sum of the areas of the white regions.
[b]p1.[/b] Consider the system of simultaneous equations
$$(x+y)(x+z)=a^2$$
$$(x+y)(y+z)=b^2$$
$$(x+z)(y+z)=c^2$$
, where $abc \ne 0$. Find all solutions $(x,y,z)$ in terms of $a$,$b$, and $c$.
[b]p2.[/b] Shown in the figure is a triangle $PQR$ upon whose sides squares of areas $13$, $25$, and $36$ sq. units have been constructed. Find the area of the hexagon $ABCDEF$ .
[img]https://cdn.artofproblemsolving.com/attachments/b/6/ab80f528a2691b07430d407ff19b60082c51a1.png[/img]
[b]p3.[/b] Suppose $p,q$, and $r$ are positive integers no two of which have a common factor larger than $1$. Suppose $P,Q$, and $R$ are positive integers such that $\frac{P}{p}+\frac{Q}{q}+\frac{R}{r}$ is an integer. Prove that each of $P/p$, $Q/q$, and $R/r$ is an integer.
[b]p4.[/b] An isosceles tetrahedron is a tetrahedron in which opposite edges are congruent. Prove that all face angles of an isosceles tetrahedron are acute angles.
[img]https://cdn.artofproblemsolving.com/attachments/7/7/62c6544b7c3651bba8a9d210cd0535e82a65bd.png[/img]
[b]p5.[/b] Suppose that $p_1$, $p_2$, $p_3$ and $p_4$ are the centers of four non-overlapping circles of radius $1$ in a plane and that, $p$ is any point in that plane. Prove that $$\overline{p_1p}^2+\overline{p_2p}^2+\overline{p_3p}^2+\overline{p_4p}^2 \ge 6.$$
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
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]
If an acute-angled triangle $ABC$ is given, construct an equilateral triangle $A'B'C'$ in space such that lines $AA',BB', CC'$ pass through a given point.
Eight spheres of radius 1, one per octant, are each tangent to the coordinate planes. What is the radius of the smallest sphere, centered at the origin, that contains these eight spheres?
$ \textbf{(A)}\ \sqrt 2\qquad
\textbf{(B)}\ \sqrt 3\qquad
\textbf{(C)}\ 1 \plus{} \sqrt 2\qquad
\textbf{(D)}\ 1 \plus{} \sqrt 3\qquad
\textbf{(E)}\ 3$
All faces of a tetrahedron are right-angled triangles. It is known that three of its edges have the same length $s$. Find the volume of the tetrahedron.