Found problems: 619
2011 AMC 10, 24
Two distinct regular tetrahedra have all their vertices among the vertices of the same unit cube. What is the volume of the region formed by the intersection of the tetrahedra?
$ \textbf{(A)}\ \frac{1}{12}\qquad\textbf{(B)}\ \frac{\sqrt{2}}{12}\qquad\textbf{(C)}\ \frac{\sqrt{3}}{12}\qquad\textbf{(D)}\ \frac{1}{6}\qquad\textbf{(E)}\ \frac{\sqrt{2}}{6} $
1972 Bulgaria National Olympiad, Problem 6
It is given a tetrahedron $ABCD$ for which two points of opposite edges are mutually perpendicular. Prove that:
(a) the four altitudes of $ABCD$ intersects at a common point $H$;
(b) $AH+BH+CH+DH<p+2R$, where $p$ is the sum of the lengths of all edges of $ABCD$ and $R$ is the radii of the sphere circumscribed around $ABCD$.
[i]H. Lesov[/i]
1977 IMO Longlists, 9
Let $ABCD$ be a regular tetrahedron and $\mathbf{Z}$ an isometry mapping $A,B,C,D$ into $B,C,D,A$, respectively. Find the set $M$ of all points $X$ of the face $ABC$ whose distance from $\mathbf{Z}(X)$ is equal to a given number $t$. Find necessary and sufficient conditions for the set $M$ to be nonempty.
1999 Czech And Slovak Olympiad IIIA, 2
In a tetrahedron $ABCD, E$ and $F$ are the midpoints of the medians from $A$ and $D$. Find the ratio of the volumes of tetrahedra $BCEF$ and $ABCD$.
Note: Median in a tetrahedron connects a vertex and the centroid of the opposite side.
1963 German National Olympiad, 6
Consider a pyramid $ABCD$ whose base $ABC$ is a triangle. Through a point $M$ of the edge $DA$, the lines $MN$ and $MP$ on the plane of the surfaces $DAB$ and $DAC$ are drawn respectively, such that $N$ is on $DB$ and $P$ is on $DC$ and $ABNM$ , $ACPM$ are cyclic quadrilaterals.
a) Prove that $BCPN$ is also a cyclic quadrilateral.
b) Prove that the points $A,B,C,M,N, P$ lie on a sphere.
1967 IMO Longlists, 36
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.
1985 IMO Shortlist, 10
Prove that for every point $M$ on the surface of a regular tetrahedron there exists a point $M'$ such that there are at least three different curves on the surface joining $M$ to $M'$ with the smallest possible length among all curves on the surface joining $M$ to $M'$.
1978 Romania Team Selection Test, 2
Points $ A’,B,C’ $ are arbitrarily taken on edges $ SA,SB, $ respectively, $ SC $ of a tetrahedron $ SABC. $ Plane forrmed by $ ABC $ intersects the plane $ \rho , $ formed by $ A’B’C’, $ in a line $ d. $ Prove that, meanwhile the plane $ \rho $ rotates around $ d, $ the lines $ AA’,BB’ $ and $ CC’ $ are, and remain concurrent. Find de locus of the respective intersections.
1960 Polish MO Finals, 2
A plane is drawn through the height of a regular tetrahedron, which intersects the planes of the lateral faces along $ 3 $ lines that form angles $ \alpha $, $ \beta $, $ \gamma $ with the plane of the tetrahedron's base. Prove that
$$ tg^2 \alpha + tg^2 \beta + tg^2 \gamma =12.$$
2018 Moscow Mathematical Olympiad, 2
There is tetrahedron and square pyramid, both with all edges equal $1$. Show how to cut them into several parts and glue together from these parts a cube (without voids and cracks, all parts must be used)
2012 Poland - Second Round, 2
Prove that for tetrahedron $ABCD$; vertex $D$, center of insphere and centroid of $ABCD$ are collinear iff areas of triangles $ABD,BCD,CAD$ are equal.
2001 Polish MO Finals, 2
Given a regular tetrahedron $ABCD$ with edge length $1$ and a point $P$ inside it.
What is the maximum value of $\left|PA\right|+\left|PB\right|+\left|PC\right|+\left|PD\right|$.
2006 All-Russian Olympiad, 6
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 $S^{\prime}A^{\prime}=S^{\prime}B^{\prime}=S^{\prime}C^{\prime}$.
1997 Croatia National Olympiad, Problem 3
The areas of the faces $ABD,ACD,BCD,BCA$ of a tetrahedron $ABCD$ are $S_1,S_2,Q_1,Q_2$, respectively. The angle between the faces $ABD$ and $ACD$ equals $\alpha$, and the angle between $BCD$ and $BCA$ is $\beta$. Prove that
$$S_1^2+S_2^2-2S_1S_2\cos\alpha=Q_1^2+Q_2^2-2Q_1Q_2\cos\beta.$$
1967 Czech and Slovak Olympiad III A, 2
Let $ABCD$ be a tetrahedron such that \[AB^2+CD^2=AC^2+BD^2=AD^2+BC^2.\] Show that at least one of its faces is an acute triangle.
2010 Contests, 2
The orthogonal projections of the vertices $A, B, C$ of the tetrahedron $ABCD$ on the opposite faces are denoted by $A', B', C'$ respectively. Suppose that point $A'$ is the circumcenter of the triangle $BCD$, point $B'$ is the incenter of the triangle $ACD$ and $C'$ is the centroid of the triangle $ABD$. Prove that tetrahedron $ABCD$ is regular.
1969 IMO Shortlist, 58
$(SWE 1)$ Six points $P_1, . . . , P_6$ are given in $3-$dimensional space such that no four of them lie in the same plane. Each of the line segments $P_jP_k$ is colored black or white. Prove that there exists one triangle $P_jP_kP_l$ whose edges are of the same color.
2004 Romania National Olympiad, 2
Let $ABCD$ be a tetrahedron in which the opposite sides are equal and form equal angles.
Prove that it is regular.
2018 Harvard-MIT Mathematics Tournament, 4
A paper equilateral triangle of side length $2$ on a table has vertices labeled $A,B,C.$ Let $M$ be the point on the sheet of paper halfway between $A$ and $C.$ Over time, point $M$ is lifted upwards, folding the triangle along segment $BM,$ while $A,B,$ and $C$ on the table. This continues until $A$ and $C$ touch. Find the maximum volume of tetrahedron $ABCM$ at any time during this process.
2013 Poland - Second Round, 6
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 $T'$.
2003 IMO Shortlist, 1
Let $a_{ij}$ $i=1,2,3$; $j=1,2,3$ be real numbers such that $a_{ij}$ is positive for $i=j$ and negative for $i\neq j$.
Prove the existence of positive real numbers $c_{1}$, $c_{2}$, $c_{3}$ such that the numbers \[a_{11}c_{1}+a_{12}c_{2}+a_{13}c_{3},\qquad a_{21}c_{1}+a_{22}c_{2}+a_{23}c_{3},\qquad a_{31}c_{1}+a_{32}c_{2}+a_{33}c_{3}\] are either all negative, all positive, or all zero.
[i]Proposed by Kiran Kedlaya, USA[/i]
1988 Polish MO Finals, 3
Find the largest possible volume for a tetrahedron which lies inside a hemisphere of radius $1$.
1986 IMO Longlists, 70
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.
1986 IMO Shortlist, 20
Prove that the sum of the face angles at each vertex of a tetrahedron is a straight angle if and only if the faces are congruent triangles.
1990 All Soviet Union Mathematical Olympiad, 532
If every altitude of a tetrahedron is at least $1$, show that the shortest distance between each pair of opposite edges is more than $2$.