Found problems: 619
Let $A_1A_2A_3A_4$ be a tetrahedron. We construct four mutually tangent spheres $S_1,S_2,S_3,S_4$ with centers $A_1,A_2,A_3,A_4$ respectively. Suppose that there exists a point $Q$ such that we can construct two spheres centered at $Q$ satisfying the following conditions:
i) One sphere with radius $r$ is tangent to $S_1,S_2,S_3,S_4$;
ii) One sphere with radius $R$ is tangent to every edges of tetrahedron $A_1A_2A_3A_4$.
Prove that $A_1A_2A_3A_4$ is a regular tetrahedron.
The altitudes of a tetrahedron intersect in a point. Prove that this point, the foot of one of the altitudes, and the points dividing the other three altitudes in the ratio $2 : 1$ (measuring from the vertices) lie on a sphere.
[i]D. Tereshin[/i]
Prove that the sum of the distances of the vertices of a regular tetrahedron from the center of its circumscribed sphere is less than the sum of the distances of these vertices from any other point in space.
A sphere is inscribed in tetrahedron $ ABCD$ and is tangent to faces $ BCD,CAD,ABD,ABC$ at points $ P,Q,R,S$ respectively. Segment $ PT$ is the sphere's diameter, and lines $ TA,TQ,TR,TS$ meet the plane $ BCD$ at points $ A',Q',R',S'$. respectively. Show that $ A$ is the center of a circumcircle on the triangle $ S'Q'R'$.
In tetrahedron $ABCD$, vertex $D$ is connected with $D_0$, the centrod if $\triangle ABC$. Line parallel to $DD_0$ are drawn through $A,B$ and $C$. These lines intersect the planes $BCD, CAD$ and $ABD$ in points $A_2, B_1,$ and $C_1$, respectively. Prove that the volume of $ABCD$ is one third the volume of $A_1B_1C_1D_0$. Is the result if point $D_o$ is selected anywhere within $\triangle ABC$?
(a) $K_1,K_2,..., K_n$ are the feet of the perpendiculars from an arbitrary point $M$ inside a given regular $n$-gon to its sides (or sides produced). Prove that the sum $\overrightarrow{MK_1} + \overrightarrow{MK_2} + ... + \overrightarrow{MK_n}$ equals $\frac{n}{2}\overrightarrow{MO}$, where $O$ is the centre of the $n$-gon.
(b) Prove that the sum of the vectors whose origin is an arbitrary point $M$ inside a given regular tetrahedron and whose endpoints are the feet of the perpendiculars from $M$ to the faces of the tetrahedron equals $\frac43 \overrightarrow{MO}$, where $O$ is the centre of the tetrahedron.
(VV Prasolov, Moscow)
Let $M$ and $N$ are midpoint of edges $AB$ and $CD$ of the tetrahedron $ABCD$, $AN=DM$ and $CM=BN$. Prove that $AC=BD$.
S. Berlov
In space are given two tetrahedra with the same barycenter such that one of them contains the other.
For each tetrahedron, we consider the octahedron whose vertices are the midpoints of the tetrahedron's edges.
Prove that one of this octahedra contains the other.
In space, a point $O$ and a finite set of vectors $ \overrightarrow{v_1},\ldots,\overrightarrow{v_n} $ are given . We consider the set of points $ P $ for which the vector $ \overrightarrow{OP} $can be represented as a sum $ a_1 \overrightarrow{v_1} + \ldots + a_n\overrightarrow{v_n} $with coefficients satisfying the inequalities $ 0 \leq a_i \leq 1 $ $( i = 1, 2, \ldots, n $). Decide whether this set can be a tetrahedron.
The $KLMN$ tetrahedron (triangle pyramid) vertices are situated inside or on the faces or on the edges of the $ABCD$ tetrahedron. Prove that perimeter of $KLMN$ is less than $4/3$ perimeter of $ABCD$.
Let $x_1, x_2, x_3, x_4$ be the distances from an arbitrary point inside the tetrahedron to the planes of its faces, and let $h_1, h_2, h_3, h_4$ be the corresponding heights of the tetrahedron. Prove that $$\sqrt{h_1+h_2+h_3+h_4} \ge \sqrt{x_1}+\sqrt{x_2}+\sqrt{x_3}+\sqrt{x_4}$$
(Dmitry Nomirovsky)
Prove that if in the tetrahedron $ ABCD $ we have $ AB = CD $, $ AC = BD $, $ AD = BC $, then all faces of the tetrahedron are acute-angled triangles.
Through the centre of a cube (tetrahedron, icosahedron) draw a straight line in such a way that the sum of the squares of its distances from the vertices is a) minimal, b) maximal.
Prove that the volume of a tetrahedron inscribed in a right circular cylinder of volume $1$ does not exceed $\frac{2}{3 \pi}.$
The midpoints of all heights of a certain tetrahedron lie on its inscribed sphere. Is this tetrahedron necessarily regular then?
Consider a tetrahedron $SABC$. The incircle of the triangle $ABC$ has the center $I$ and touches its sides $BC$, $CA$, $AB$ at the points $E$, $F$, $D$, respectively. Let $A^{\prime}$, $B^{\prime}$, $C^{\prime}$ be the points on the segments $SA$, $SB$, $SC$ such that $AA^{\prime}=AD$, $BB^{\prime}=BE$, $CC^{\prime}=CF$, and let $S^{\prime}$ be the point diametrically opposite to the point $S$ on the circumsphere of the tetrahedron $SABC$. Assume that the line $SI$ is an altitude of the tetrahedron $SABC$. Show that $S^{\prime}A^{\prime}=S^{\prime}B^{\prime}=S^{\prime}C^{\prime}$.
Prove that if the distance from a point inside a convex polyhedra with $n$ faces to the vertices of the polyhedra is at most 1, then the sum of the distances from this point to the faces of the polyhedra is smaller than $n-2$.
[i]Calin Popescu[/i]
Prove this proposition: Center the sphere circumscribed around a tetrahedron which coincides with the center of a sphere inscribed in that tetrahedron if and only if the skew edges of the tetrahedron are equal.
The rectangle $ABCD$ below has dimensions $AB = 12 \sqrt{3}$ and $BC = 13 \sqrt{3}$. Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at $P$. If triangle $ABP$ is cut out and removed, edges $\overline{AP}$ and $\overline{BP}$ are joined, and the figure is then creased along segments $\overline{CP}$ and $\overline{DP}$, we obtain a triangular pyramid, all four of whose faces are isosceles triangles. Find the volume of this pyramid.
[asy]
pair D=origin, A=(13,0), B=(13,12), C=(0,12), P=(6.5, 6);
draw(B--C--P--D--C^^D--A);
filldraw(A--P--B--cycle, gray, black);
label("$A$", A, SE);
label("$B$", B, NE);
label("$C$", C, NW);
label("$D$", D, SW);
label("$P$", P, N);
label("$13\sqrt{3}$", A--D, S);
label("$12\sqrt{3}$", A--B, E);[/asy]
Prove that the volume of a tetrahedron inscribed in a right circular cylinder of volume $1$ does not exceed $\frac{2}{3 \pi}.$
There are $8436$ steel balls, each with radius $1$ centimeter, stacked in a tetrahedral pile, with one ball on top, $3$ balls in the second layer, $6$ in the third layer, $10$ in the fourth, and so on. Determine the height of the pile in centimeters.
Given three tetrahedrons $A_iB_i C_i D_i$ ($i=1,2,3$), planes $\alpha _i,\beta _i,\gamma _i$ ($i=1,2,3$) are drawn through $B_i ,C_i ,D_i$ respectively, and they are perpendicular to edges $A_i B_i, A_i C_i, A_i D_i$ ($i=1,2,3$) respectively. Suppose that all nine planes $\alpha _i,\beta _i,\gamma _i$ ($i=1,2,3$) meet at a point $E$, and points $A_1,A_2,A_3$ lie on line $l$. Determine the intersection (shape and position) of the circumscribed spheres of the three tetrahedrons.
Let $ ABCD$ is a tetrahedron. Denote by $ A'$, $ B'$ the feet of the perpendiculars from $ A$ and $ B$, respectively to the opposite faces. Show that $ AA'$ and $ BB'$ intersect if and only if $ AB$ is perpendicular to $ CD$. Do they intersect if $ AC \equal{} AD \equal{} BC \equal{} BD$?
On the surface of a regular tetrahedron of edge length $1$ are given finitely many segments such that every two vertices of the tetrahedron can be joined by a polygonal line consisting of given segments. Can the sum of the lengths of the given segments be less than $1+\sqrt3 $?
Prove that if two altitudes of a tetrahedron intersect, then so do the other two altitudes.