Found problems: 2265
A tetrahedron is such that the center of the its circumscribed sphere is inside the tetrahedron.
Show that at least one of its edges has a size larger than or equal to the size of the edge of a regular tetrahedron inscribed in this same sphere.
Consider three edges $a, b, c$ of a cube such that no two of these edges lie in one plane. Find the locus of points inside the cube which are equidistant from $a$, $b$ and $c$.
(V Proizvolov,)
Is it possible that two cross-sections of a tetrahedron by two different cutting planes are two squares, one with a side of length no greater than $1$ and another with a side of length at least $100$?
Mikhail Evdokimov
A tetrahedron $T$ is inside a cube $C$. Prove that the volume of $T$ is at most one-third the volume of $C$.
Two uniform solid spheres of equal radii are so placed that one is directly above the other. The bottom sphere is fixed, and the top sphere, initially at rest, rolls off. At what point will contact between the two spheres be "lost"? Assume the coefficient of friction is such that no slipping occurs.
Given a triangle, its midpoint triangle is obtained by joining the midpoints of its sides. A sequence of polyhedra $P_{i}$ is defined recursively as follows: $P_{0}$ is a regular tetrahedron whose volume is 1. To obtain $P_{i+1}$, replace the midpoint triangle of every face of $P_{i}$ by an outward-pointing regular tetrahedron that has the midpoint triangle as a face. The volume of $P_{3}$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Find the smallest positive integer whose cube ends in 888.
A cone is constructed with a semicircular piece of paper, with radius 10. Find the
height of the cone.
Let $ABCDEFGH$ be a parallelepiped with $AE \parallel BF \parallel CG \parallel DH$. Prove the inequality
\[AF + AH + AC \leq AB + AD + AE + AG.\]
In what cases does equality hold?
[i]Proposed by France.[/i]
A castaway is building a rectangular raft $ABCD$. He fixes a mast perpendicular to the raft, with ropes passing from the top of the mast (point $S$ in the figure) to the four corners of the raft. The rope $SA$ measures $8$ meters, the rope $SB$ measures $2$ meters and the rope $SC$ measures $14$ meters. Compute the length of the rope $SD$.
[asy]
size(250);
// Coordinates for the parallelogram ABCD
pair A = (0, 0);
pair B = (8, 0);
pair C = (10, 5);
pair D = (2, 5);
// Position of point S (outside the parallelogram)
pair S = (5, 8);
pair T = (5, 3);
// Draw the parallelogram ABCD
filldraw(A--B--C--D--cycle, lightgray, black);
// Draw the ropes from point S to each corner of the parallelogram
draw(S--A, blue);
draw(S--B, blue);
draw(S--C, blue);
draw(S--D, blue);
draw(S--T, black);
// Mark the points
dot(A);
dot(B);
dot(C);
dot(D);
dot(S);
dot(T);
// Label the points
label("A", A, SW);
label("B", B, SE);
label("C", C, NE);
label("D", D, NW);
label("S", S, N);
[/asy]
From a point in space, $n$ rays are issuing, whereas the angle among any two of these rays is at least $30^{\circ}$. Prove that $n < 59$.
For a positive real number $a$, let $C$ be the cube with vertices at $(\pm a, \pm a, \pm a)$ and let $T$ be the tetrahedron with vertices at $(2a,2a,2a),(2a, -2a, -2a),(-2a, 2a, -2a),(-2a, -2a, -2a)$. If the intersection of $T$ and $C$ has volume $ka^3$ for some $k$, find $k$.
A tetrahedron is called a [i]MEMO-tetrahedron[/i] if all six sidelengths are different positive integers where one of them is $ 2$ and one of them is $ 3$. Let $ l(T)$ be the sum of the sidelengths of the tetrahedron $ T$.
(a) Find all positive integers $ n$ so that there exists a MEMO-Tetrahedron $ T$ with $ l(T)\equal{}n$.
(b) How many pairwise non-congruent MEMO-tetrahedrons $ T$ satisfying $ l(T)\equal{}2007$ exist? Two tetrahedrons are said to be non-congruent if one cannot be obtained from the other by a composition of reflections in planes, translations and rotations. (It is not neccessary to prove that the tetrahedrons are not degenerate, i.e. that they have a positive volume).
[b]7.[/b] Consider four real numbers $t_{1},t_{2},t_{3},t_{4}$ such that each is less than the sum of the others. Show that there exists a tetrahedron whose faces have areas $t_{1},t_{2}, t_{3}$ and $t_{4},$ respectively. [b](G. 9)[/b]
A ring is the area between two circles with the same center, and width of a ring is the difference between the radii of two circles.
[img]http://i18.tinypic.com/6cdmvi8.png[/img]
a) Can we put uncountable disjoint rings of width 1(not necessarily same) in the space such that each two of them can not be separated.
[img]http://i19.tinypic.com/4qgx30j.png[/img]
b) What's the answer if 1 is replaced with 0?
Consider a polyhedron with at least five faces such that exactly three edges emerge from each of its vertices. Two players play the following game: Each, in turn, signs his or her name on a previously unsigned face. The winner is the player who first succeeds in signing three faces that share a common vertex. Show that the player who signs first will always win by playing as well as possible.
In a regular $2n$-gonal prism, bases $A_1A_2\cdots A_{2n}$ and $B_1B_2\cdots B_{2n}$ have circumradii equal to $R$. If the length of the lateral edge $A_1B_1$ varies, the angle between the line $A_1B_{n+1}$ and the plane $A_1A_3B_{n+2}$ is maximal for $A_1B_1=2R\cos\frac\pi{2n}$.
Vertices $A, B, C$ of a equilateral triangle of side $1$ are in the surface of a sphere with radius $1$ and center $O$. Let $D$ be the orthogonal projection of $A$ on the plane $\alpha$ determined by points $B, C, O$. Let $N$ be one of the intersections of the line perpendicular to $\alpha$ passing through $O$ with the sphere. Find the angle $\angle DNO$.
A unit cube $C$ is rotated around one of its diagonals for the angle $\pi /3$ to form a cube $C'$. Find the volume of the intersection of $C$ and $C'$.
Let $S$ be the set of all tetrahedra which satisfy:
(1) the base has area $1$,
(2) the total face area is $4$, and
(3) the angles between the base and the other three faces are all equal.
Find the element of $S$ which has the largest volume.
There are two shawls, one in the shape of a square, the other in the shape of a regular triangle, and their perimeters are the same. Is there a polyhedron that can be completely pasted over with these two shawls without overlap (shawls can be bent, but not cut)?
(S. Markelov).
$E_1$ and $E_2$ are parallel planes and their distance is $p$.
(a) How long is the seitenkante of the regular octahedron such that a side lies in $E_1$ and another in $E_2$?
(b) $E$ is a plane between $E_1$ and $E_2$, parallel to $E_1$ and $E_2$, so that its distances from $E_1$ and $E_2$ are in ratio $1:2$
Draw the intersection figure of $E$ and the octahedron for $P=4\sqrt{\frac32}$ cm and justifies, why the that figure must look in such a way
The length of edge of cube $ABCD-A_1B_1C_1D_1$ is $1$, then the distance between lines $A_1C_1$ and $BD_1$ is________.
Let $ABCD$ be a tetrahedron and let $H_{a},H_{b},H_{c},H_{d}$ be the orthocenters of triangles $BCD,CDA,DAB,ABC$, respectively.
Prove that lines $AH_{a},BH_{b},CH_{c}, DH_{d}$ are concurrent
if and only if
$AB^2 + CD^2 = AC^2 + BD^2 = AD^2 + BC^2$
A set $ E$ of points in the 3D space let $ L(E)$ denote the set of all those points which lie on lines composed of two distinct points of $ E.$ Let $ T$ denote the set of all vertices of a regular tetrahedron. Which points are in the set $ L(L(T))?$