Found problems: 619
1978 All Soviet Union Mathematical Olympiad, 266
Prove that for every tetrahedron there exist two planes such that the projection areas on those planes ratio is not less than $\sqrt 2$.
2004 AMC 10, 25
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} $
2006 Sharygin Geometry Olympiad, 25
In the tetrahedron $ABCD$ , the dihedral angles at the $BC, CD$, and $DA$ edges are equal to $\alpha$, and for the remaining edges equal to $\beta$. Find the ratio $AB / CD$.
1980 AMC 12/AHSME, 26
Four balls of radius 1 are mutually tangent, three resting on the floor and the fourth resting on the others. A tetrahedron, each of whose edges have length $s$, is circumscribed around the balls. Then $s$ equals
$\text{(A)} \ 4\sqrt 2 \qquad \text{(B)} \ 4\sqrt 3 \qquad \text{(C)} \ 2\sqrt 6 \qquad \text{(D)} \ 1+2\sqrt 6 \qquad \text{(E)} \ 2+2\sqrt 6$
2015 CCA Math Bonanza, I13
Let $ABCD$ be a tetrahedron such that $AD \perp BD$, $BD \perp CD$, $CD \perp AD$ and $AD=10$, $BD=15$, $CD=20$. Let $E$ and $F$ be points such that $E$ lies on $BC$, $DE \perp BC$, and $ADEF$ is a rectangle. If $S$ is the solid consisting of all points inside $ABCD$ but outside $FBCD$, compute the volume of $S$.
[i]2015 CCA Math Bonanza Individual Round #13[/i]
1976 Vietnam National Olympiad, 3
$P$ is a point inside the triangle $ABC$. The perpendicular distances from $P$ to the three sides have product $p$. Show that $p \le \frac{ 8 S^3}{27abc}$, where $S =$ area $ABC$ and $a, b, c$ are the sides. Prove a similar result for a tetrahedron.
I Soros Olympiad 1994-95 (Rus + Ukr), 11.10
Given a tetrahedron $A_1A_2A_3A_4$ (not necessarily regulart). We shall call a point $N$ in space [i]Serve point[/i], if it's six projection points on the six edges of the tetrahedron lie on one plane. This plane we denote it by $a (N)$ and call the [i]Serve plane[/i] of the point $N$. By $B_{ij}$ denote, respectively, the midpoint of the edges $A_1A_j$, $1\le i <j \le 4$. For each point $M$, denote by $M_{ij}$ the points symmetric to $M$ with respect to $B_{ij},$ $1\le i <j \le 4$. Prove that if all points $M_{ij}$ are Serve points, then the point $M$ belongs to all Serve planes $a (M_{ij})$, $1\le i <j \le 4$.
2011 Polish MO Finals, 2
In a tetrahedron $ABCD$, the four altitudes are concurrent at $H$. The line $DH$ intersects the plane $ABC$ at $P$ and the circumsphere of $ABCD$ at $Q\neq D$. Prove that $PQ=2HP$.
1999 AMC 12/AHSME, 29
A tetrahedron with four equilateral triangular faces has a sphere inscribed within it and a sphere circumscribed about it. For each of the four faces, there is a sphere tangent externally to the face at its center and to the circumscribed sphere. A point $ P$ is selected at random inside the circumscribed sphere. The probability that $ P$ lies inside one of the five small spheres is closest to
$ \textbf{(A)}\ 0\qquad
\textbf{(B)}\ 0.1\qquad
\textbf{(C)}\ 0.2\qquad
\textbf{(D)}\ 0.3\qquad
\textbf{(E)}\ 0.4$
1981 IMO Shortlist, 2
A sphere $S$ is tangent to the edges $AB,BC,CD,DA$ of a tetrahedron $ABCD$ at the points $E,F,G,H$ respectively. The points $E,F,G,H$ are the vertices of a square. Prove that if the sphere is tangent to the edge $AC$, then it is also tangent to the edge $BD.$
1968 IMO Shortlist, 3
Prove that every tetrahedron has a vertex whose three edges have the right lengths to form a triangle.
1981 Bulgaria National Olympiad, Problem 6
Planes $\alpha,\beta,\gamma,\delta$ are tangent to the circumsphere of a tetrahedron $ABCD$ at points $A,B,C,D$, respectively. Line $p$ is the intersection of $\alpha$ and $\beta$, and line $q$ is the intersection of $\gamma$ and $\delta$. Prove that if lines $p$ and $CD$ meet, then lines $q$ and $AB$ lie on a plane.
Ukrainian TYM Qualifying - geometry, 2010.16
Points $A, B, C, D$ lie on the sphere of radius $1$. It is known that $AB\cdot AC\cdot AD\cdot BC\cdot BD\cdot CD=\frac{512}{27}$. Prove that $ABCD$ is a regular tetrahedron.
2011 Tokyo Instutute Of Technology Entrance Examination, 2
For a positive real number $t$, in the coordiante space, consider 4 points $O(0,\ 0,\ 0),\ A(t,\ 0,\ 0),\ B(0,\ 1,\ 0),\ C(0,\ 0,\ 1)$.
Let $r$ be the radius of the sphere $P$ which is inscribed to all faces of the tetrahedron $OABC$.
When $t$ moves, find the maximum value of $\frac{\text{vol[P]}}{\text{vol[OABC]}}.$
2012 District Olympiad, 4
Consider a tetrahedron $ABCD$ in which $AD \perp BC$ and $AC \perp BD$. We denote by $E$ and $F$ the projections of point $B$ on the lines $AD$ and $AC$, respectively. If $M$ and $N$ are the midpoints of the segments $[AB]$ and $[CD]$, respectively, show that $MN \perp EF$
1986 All Soviet Union Mathematical Olympiad, 440
Consider all the tetrahedrons $AXBY$, circumscribed around the sphere. Let $A$ and $B$ points be fixed. Prove that the sum of angles in the non-plane quadrangle $AXBY$ doesn't depend on points $X$ and $Y$ .
2004 AMC 12/AHSME, 22
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$
2012 Kyoto University Entry Examination, 2
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
2010 Princeton University Math Competition, 8
There is a point source of light in an empty universe. What is the minimum number of solid balls (of any size) one must place in space so that any light ray emanating from the light source intersects at least one ball?
1979 Bulgaria National Olympiad, Problem 2
Points $P,Q,R,S$ are taken on respective edges $AC$, $AB$, $BD$, and $CD$ of a tetrahedron $ABCD$ so that $PR$ and $QS$ intersect at point $N$ and $PS$ and $QR$ intersect at point $M$. The line $MN$ meets the plane $ABC$ at point $L$. Prove that the lines $AL$, $BP$, and $CQ$ are concurrent.
1986 IMO Shortlist, 21
Let $ABCD$ be a tetrahedron having each sum of opposite sides equal to $1$. Prove that
\[r_A + r_B + r_C + r_D \leq \frac{\sqrt 3}{3}\]
where $r_A, r_B, r_C, r_D$ are the inradii of the faces, equality holding only if $ABCD$ is regular.
2020 Jozsef Wildt International Math Competition, W25
In the Crelle $[ABCD]$ tetrahedron, we note with $A',B',C',A'',B'',C''$ the tangent points of the hexatangent sphere $\varphi(J,\rho)$, associated with the tetrahedron, with the edges $|BC|,|CA|,|AB|,|DA|,|DB|,|DC|$. Show that these inequalities occur:
a)
$$2\sqrt3R\ge6\rho\ge A'A''+B'B''+C'C''\ge6\sqrt3r$$
b)
$$4R^2\ge12\rho^2\ge(A'A'')^2+(B'B'')^2+(C'C'')^2\ge36r^2$$
c)
$$\frac{8R^3}{3\sqrt3}\ge8\rho^3\ge A'A''\cdot B'B''\cdot C'C''\ge24\sqrt3r^3$$
where $r,R$ is the length of the radius of the sphere inscribed and respectively circumscribed to the tetrahedron.
[i]Proposed by Marius Olteanu[/i]
1990 Brazil National Olympiad, 3
Each face of a tetrahedron is a triangle with sides $a, b,$c and the tetrahedon has circumradius 1. Find $a^2 + b^2 + c^2$.
2011 Sharygin Geometry Olympiad, 25
Three equal regular tetrahedrons have the common center. Is it possible that all faces of the polyhedron that forms their intersection are equal?
1995 Rioplatense Mathematical Olympiad, Level 3, 3
Given a regular tetrahedron with edge $a$, its edges are divided into $n$ equal segments, thus obtaining $n + 1$ points: $2$ at the ends and $n - 1$ inside. The following set of planes is considered:
$\bullet$ those that contain the faces of the tetrahedron, and
$\bullet$ each of the planes parallel to a face of the tetrahedron and containing at least one of the points determined above.
Now all those points $P$ that belong (simultaneously) to four planes of that set are considered. Determine the smallest positive natural $n$ so that among those points $P$ the eight vertices of a square-based rectangular parallelepiped can be chosen.