Show that the sum of the distances from any point of a regular tetrahedron to its faces is equal to the height of this tetrahedron.

[b]a)[/b] Show that there are $ 4 $ equidistant parallel planes that passes through the vertices of the same tetrahedron. [b]b)[/b] How many such $ \text{4-tuplets} $ of planes does exist, in function of the tetrahedron?

(N.Avilov) Can the surface of a regular tetrahedron be glued over with equal regular hexagons?

(a) Prove the parallelogram law that says that in a parallelogram the sum of the squares of the lengths of the four sides equals the sum of the squares of the lengths of the two diagonals. (b) The edges of a tetrahedron have lengths $a, b, c, d, e$ and $f$. The three line segments connecting the centers of intersecting edges have lengths $x, y$ and $z$. Prove that $$4 (x^2 + y^2 + z^2) = a^2 + b^2 + c^2 + d^2 + e^2 + f^2$$

Decide, whether exist tetrahedrons $T$, $T'$ with walls $S_1$, $S_2$, $S_3$, $S_4$ and $S_1'$, $S_2'$, $S_3'$, $S_4'$, respectively, such that for $i = 1, 2, 3, 4$ triangle $S_i$ is similar to triangle $S_i'$, but despite this, tetrahedron $T$ is not similar to tetrahedron $T'$.

Let $a,b,c,d$ be the areas of the triangular faces of a tetrahedron, and let $h_a, h_b, h_c, h_d$ be the corresponding altitudes of the tetrahedron. If $V$ denotes the volume of tetrahedron, prove that \[ (a+b+c+d)(h_a+h_b+h_c+h_d) \geq 48V \]

Let $ABCD$ be a tetrahedron. (a) Prove that the midpoints of the edges $AB,AC,BD$, and $CD$ lie in a plane. (b) Find the point in that plane, whose sum of distances from the lines $AD$ and $BC$ is minimal.

The edge lengths of a tetrahedron are a, b, c, d, e, f, the areas of its faces are S1, S2, S3, S4, and its volume is V . Prove that 2 [S1 S2 S3 S4](1/2) > 3V [abcdef](1/6) this problem comes from: http://www.imomath.com/othercomp/jkasfvgkusa/MonMO99.pdf I was just wondering if someone could write it in LATEX form. [color=red]_____________________________________ EDIT by moderator: If you type[/color] [code]The edge lengths of a tetrahedron are $a, b, c, d, e, f,$ the areas of its faces are $S_1, S_2, S_3, S_4,$ and its volume is $V.$ Prove that $2 \sqrt{S_1 S_2 S_3 S_4} > 3V \sqrt[6]{abcdef}$[/code] [color=red]it shows up as:[/color] The edge lengths of a tetrahedron are $ a, b, c, d, e, f,$ the areas of its faces are $ S_1, S_2, S_3, S_4,$ and its volume is $ V.$ Prove that $ 2 \sqrt{S_1 S_2 S_3 S_4} > 3V \sqrt[6]{abcdef}$

Let $ABCD$ be a tetrahedron that is not degenerate and not necessarily regular, where sides $AD$ and $BC$ have the same length $a$, sides $BD$ and $AC$ have the same length $b$, side $AB$ has length $c_1$ and the side $CD$ has length $c_2$. There is a point $P$ for which the sum of the distances to the vertices of the tetrahedron is minimal. Determine this sum depending on the quantities $a, b, c_1$ and $c_2$.

In tetrahedron $SABC$, the circumcircles of faces $SAB$, $SBC$, and $SCA$ each have radius $108$. The inscribed sphere of $SABC$, centered at $I$, has radius $35.$ Additionally, $SI = 125$. Let $R$ be the largest possible value of the circumradius of face $ABC$. Given that $R$ can be expressed in the form $\sqrt{\frac{m}{n}}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$. [i]Author: Alex Zhu[/i]

Given is a regular tetrahedron of volume $1$. We obtain a second regular tetrahedron by reflecting the given one through its center. What is the volume of their intersection?

There is an infinitely tall tetrahedral stack of spheres where each row of the tetrahedron consists of a triangular arrangement of spheres, as shown below. There is $1$ sphere in the top row (which we will call row $0$), $3$ spheres in row $1$, $6$ spheres in row $2$, $10$ spheres in row $3$, etc. The top-most sphere in row $0$ is assigned the number $1$. We then assign each other sphere the sum of the number(s) assigned to the sphere(s) which touch it in the row directly above it. Find a simplified expression in terms of $n$ for the sum of the numbers assigned to each sphere from row $0$ to row $n$. [asy] import three; import solids; size(8cm); //currentprojection = perspective(1, 1, 10); triple backright = (-2, 0, 0), backleft = (-1, -sqrt(3), 0), backup = (-1, -sqrt(3) / 3, 2 * sqrt(6) / 3); draw(shift(2 * backleft) * surface(sphere(1,20)), white); //2 draw(shift(backleft + backright) * surface(sphere(1,20)), white); //2 draw(shift(2 * backright) * surface(sphere(1,20)), white); //3 draw(shift(backup + backleft) * surface(sphere(1,20)), white); draw(shift(backup + backright) * surface(sphere(1,20)), white); draw(shift(2 * backup) * surface(sphere(1,20)), white); draw(shift(backleft) * surface(sphere(1,20)), white); draw(shift(backright) * surface(sphere(1,20)), white); draw(shift(backup) * surface(sphere(1,20)), white); draw(surface(sphere(1,20)), white); label("Row 0", 2 * backup, 15 * dir(20)); label("Row 1", backup, 25 * dir(20)); label("Row 2", O, 35 * dir(20)); dot(-backup); dot(-7 * backup / 8); dot(-6 * backup / 8); dot((backleft - backup) + backleft * 2); dot(5 * (backleft - backup) / 4 + backleft * 2); dot(6 * (backleft - backup) / 4 + backleft * 2); dot((backright - backup) + backright * 2); dot(5 * (backright - backup) / 4 + backright * 2); dot(6 * (backright - backup) / 4 + backright * 2); [/asy]

The intersection of a plane with a regular tetrahedron with edge $a$ is a quadrilateral with perimeter $P.$ Prove that $2a \leq P \leq 3a.$

In a tetrahedron, the midpoints of all the edges lie on the same sphere. Prove that it's altitudes intersect at one point.

It is given a tetrahedron $ABCD$ and a plane $\alpha$ intersecting the three edges passing through $D$. Prove that $\alpha$ divides the surface of the tetrahedron into two parts proportional to the volumes of the bodies formed if and only if $\alpha$ is passing through the center of the inscribed tetrahedron sphere.