Found problems: 619
In a tetrahedron $ABCD$, the medians of the faces $ABD$, $ACD$, $BCD$ from $D$ make equal angles with the corresponding edges $AB$, $AC$, $BC$. Prove that each of these faces has area less than or equal to the sum of the areas of the other two faces.
[hide="Comment"][i]Equivalent version of the problem:[/i] $ABCD$ is a tetrahedron. $DE$, $DF$, $DG$ are medians of triangles $DBC$, $DCA$, $DAB$. The angles between $DE$ and $BC$, between $DF$ and $CA$, and between $DG$ and $AB$ are equal. Show that: area $DBC$ $\leq$ area $DCA$ + area $DAB$. [/hide]
We say that a tetrahedron is [i]median [/i] if and only if for each vertex the plane that passes through the midpoints of the edges emerging from the vertex is tangent to the inscribed sphere. Also a tetrahedron is called [i]regular [/i] if all its faces are congruent. Prove that a tetrahedron is regular if and only if it is median.
$ABCD$ is a regular tetrahedron. The points $P,Q,R$ and $S$ lie outside this tetrahedron in such a way that $ABCP$, $ABDQ$, $ACDR$ and $BCDS$ are regular tetrahedra. Prove that the volume of the tetrahedron $PQRS$ is less than the sum of the volumes of $ABCP$,$ABDQ$,$ACDR$, $BCDS$ and $ABCD$.
A game consists of pieces of the shape of a regular tetrahedron of side $1$. Each face of each piece is painted in one of $n$ colors, and by this, the faces of one piece are not necessarily painted in different colors. Determine the maximum possible number of pieces, no two of which are identical.
Let $OX, OY$ and $OZ$ be three rays in the space, and $G$ a point "[i]between these rays[/i]" (i. e. in the interior of the part of the space bordered by the angles $Y OZ, ZOX$ and $XOY$). Consider a plane passing through $G$ and meeting the rays $OX, OY$ and $OZ$ in the points $A, B, C$, respectively. There are infinitely many such planes; construct the one which minimizes the volume of the tetrahedron $OABC$.
Three mutually tangent spheres of radius $ 1$ rest on a horizontal plane. A sphere of radius $ 2$ rests on them. What is the distance from the plane to the top of the larger sphere?
$ \textbf{(A)}\ 3 \plus{} \frac {\sqrt {30}}{2} \qquad \textbf{(B)}\ 3 \plus{} \frac {\sqrt {69}}{3} \qquad \textbf{(C)}\ 3 \plus{} \frac {\sqrt {123}}{4}\qquad \textbf{(D)}\ \frac {52}{9}\qquad \textbf{(E)}\ 3 \plus{} 2\sqrt2$
Two tetrahedrons are given. Each two faces of the same tetrahedron are not similar, but each face of the first tetrahedron is similar to some face of the second one. Does this yield that these tetrahedrons are similar?
Three mutually tangent spheres of radius 1 rest on a horizontal plane. A sphere of radius 2 rests on them. What is the distance from the plane to the top of the larger sphere?
$ \textbf{(A)}\; 3+\frac{\sqrt{30}}2\qquad
\textbf{(B)}\; 3+\frac{\sqrt{69}}3\qquad
\textbf{(C)}\; 3+\frac{\sqrt{123}}4\qquad
\textbf{(D)}\; \frac{52}9\qquad
\textbf{(E)}\; 3+2\sqrt{2} $
Let $ A_1A_2A_3A_4$ be a tetrahedron, $ G$ its centroid, and $ A'_1, A'_2, A'_3,$ and $ A'_4$ the points where the circumsphere of $ A_1A_2A_3A_4$ intersects $ GA_1,GA_2,GA_3,$ and $ GA_4,$ respectively. Prove that
\[ GA_1 \cdot GA_2 \cdot GA_3 \cdot GA_ \cdot4 \leq GA'_1 \cdot GA'_2 \cdot GA'_3 \cdot GA'_4\]
and
\[ \frac{1}{GA'_1} \plus{} \frac{1}{GA'_2} \plus{} \frac{1}{GA'_3} \plus{} \frac{1}{GA'_4} \leq \frac{1}{GA_1} \plus{} \frac{1}{GA_2} \plus{} \frac{1}{GA_3} \plus{} \frac{1}{GA_4}.\]
Prove that the sum of dihedral angles in an arbitrary tetrahedron is greater than $2\pi$
Given a regular tetrahedron $OABC$. Take points $P,\ Q,\ R$ on the sides $OA,\ OB,\ OC$ respectively. Note that $P,\ Q,\ R$ are different from the vertices of the tetrahedron $OABC$. If $\triangle{PQR}$ is an equilateral triangle, then prove that three sides $PQ,\ QR,\ RP$ are pararell to three sides $AB,\ BC,\ CA$ respectively.
30 points
Given two regular tetrahedrons with edges of length $\sqrt2$, transforming into one another with central symmetry. Let $\Phi$ be the set the midpoints of segments whose ends belong to different tetrahedrons. Find the volume of the figure $\Phi$.
The sides of $\triangle ABC$ measure 11,20, and 21 units. We fold it along $PQ,QR,RP$ where $P,Q,R$ are the midpoints of its sides until $A,B,C$ coincide. What is the volume of the resulting tetrahedron?
Let $ABCD$ be a tetrahedron and let $S$ be its center of gravity. A line through $S$ intersects the surface of $ABCD$ in the points $K$ and $L$. Prove that \[\frac{1}{3}\leq \frac{KS}{LS}\leq 3\]
A tetrahedron $ABCD$ with acute-angled faces is inscribed in a sphere with center $O$. A line passing through $O$ perpendicular to plane $ABC$ crosses the sphere at point $D'$ that lies on the opposide side of plane $ABC$ than point $D$. Line $DD'$ crosses plane $ABC$ in point $P$ that lies inside the triangle $ABC$. Prove, that if $\angle APB=2\angle ACB$, then $\angle ADD'=\angle BDD'$.
All edges of a polyhedron are painted with red or yellow. For an angle of a facet, if the edges determining it are of different colors, then the angle is called [i]excentric[/i]. The[i] excentricity [/i]of a vertex $A$, namely $S_A$, is defined as the number of excentric angles it has. Prove that there exist two vertices $B$ and $C$ such that $S_B + S_C \leq 4$.
A tetrahedron $ABCD$ is given, in which each pair of adjacent edges are equal segments. Let $O$ be the center of the sphere inscribed in this tetrahedron . $X$ is an arbitrary point inside the tetrahedron, $X \ne O$. The line $OX$ intersects the planes of the faces of the tetrahedron at the points marked by $A_1$, $B_1$, $C_1$, $D_1$. Prove that
$$\frac{A_1X}{A_1O} +\frac{B_1X}{B_1O} +\frac{C_1X}{C_1O}+\frac{D_1X}{D_1O}=4$$
If the sum of the lengths of the six edges of a trirectangular tetrahedron $ PABC$ (i.e., $ \angle APB \equal{} \angle BPC \equal{} \angle CPA \equal{} 90^\circ$) is $ S$, determine its maximum volume.
Let $ABCD$ be an arbitrary tetrahedron. The bisectors of the angles $\angle BDC$, $\angle CDA$ and $\angle ADB$ intersect $BC$, $CA$ and $AB$, in the points $M$, $N$, $P$, respectively.
a) Show that the planes $(ADM)$, $(BDN)$ and $(CDP)$ have a common line $d$.
b) Let the points $A' \in (AD)$, $B' \in (BD)$ and $C' \in (CD)$ be such that $(AA') = (BB') = (CC')$ ; show that if $G$ and $G'$ are the centroids of $ABC$ and $A'B'C'$ then the lines $GG'$ and $d$ are either parallel or identical.
[b]i.)[/b] Four balls of radius 1 are mutually tangent, three resting on the floor and the fourth resting on the others. A tedrahedron, each of whose edges has length $ s,$ is circumscribed around the balls. Find the value of $ s.$
[b]ii.)[/b] Suppose that $ ABCD$ and $ EFGH$ are opposite faces of a retangular solid, with $ \angle DHC \equal{} 45^{\circ}$ and $ \angle FHB \equal{} 60^{\circ}.$ Find the cosine of $ \angle BHD.$
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.
Three 12 cm $\times$ 12 cm squares are each cut into two pieces $A$ and $B$, as shown in the first figure below, by joining the midpoints of two adjacent sides. These six pieces are then attached to a regular hexagon, as shown in the second figure, so as to fold into a polyhedron. What is the volume (in $\text{cm}^3$) of this polyhedron?
[asy]
defaultpen(fontsize(10));
size(250);
draw(shift(0, sqrt(3)+1)*scale(2)*rotate(45)*polygon(4));
draw(shift(-sqrt(3)*(sqrt(3)+1)/2, -(sqrt(3)+1)/2)*scale(2)*rotate(165)*polygon(4));
draw(shift(sqrt(3)*(sqrt(3)+1)/2, -(sqrt(3)+1)/2)*scale(2)*rotate(285)*polygon(4));
filldraw(scale(2)*polygon(6), white, black);
pair X=(2,0)+sqrt(2)*dir(75), Y=(-2,0)+sqrt(2)*dir(105), Z=(2*dir(300))+sqrt(2)*dir(225);
pair[] roots={2*dir(0), 2*dir(60), 2*dir(120), 2*dir(180), 2*dir(240), 2*dir(300)};
draw(roots[0]--X--roots[1]);
label("$B$", centroid(roots[0],X,roots[1]));
draw(roots[2]--Y--roots[3]);
label("$B$", centroid(roots[2],Y,roots[3]));
draw(roots[4]--Z--roots[5]);
label("$B$", centroid(roots[4],Z,roots[5]));
label("$A$", (1+sqrt(3))*dir(90));
label("$A$", (1+sqrt(3))*dir(210));
label("$A$", (1+sqrt(3))*dir(330));
draw(shift(-10,0)*scale(2)*polygon(4));
draw((sqrt(2)-10,0)--(-10,sqrt(2)));
label("$A$", (-10,0));
label("$B$", centroid((sqrt(2)-10,0),(-10,sqrt(2)),(sqrt(2)-10, sqrt(2))));[/asy]
The points $A_1,B_1,C_1,D_1$ and $A_2,B_2,C_2,D_2$ are orthogonal projections of the $ABCD$ tetrahedron vertices on two planes. Prove that it is possible to move one of the planes to provide the parallelness of lines $(A_1A_2), (B_1B_2), (C_1C_2)$ and $(D_1D_2)$ .
$ABC$ is an equilateral triangle. $D$ and$ D'$ are points on opposite sides of the plane $ABC$ such that the two tetrahedra $ABCD$ and $ABCD'$ are congruent (but not necessarily with the vertices in that order). If the polyhedron with the five vertices $A, B, C, D, D'$ is such that the angle between any two adjacent faces is the same, find $DD'/AB$ .
At the vertices $A, B, C, D, E, F, G, H$ of a cube, $2001, 2002, 2003, 2004, 2005, 2008, 2007$ and $2006$ stones respectively are placed. It is allowed to move a stone from a vertex to each of its three neighbours, or to move a stone to a vertex from each of its three neighbours. Which of the following arrangements of stones at $A, B, \ldots , H$ can be obtained?
$(\text{a})\quad 2001, 2002, 2003, 2004, 2006, 2007, 2008, 2005;$
$(\text{b})\quad 2002, 2003, 2004, 2001, 2006, 2005, 2008, 2007;$
$(\text{c})\quad 2004, 2002, 2003, 2001, 2005, 2008, 2007, 2006.$