Found problems: 2265
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 $BD.$
2007 IMAR Test, 2
Denote by $ \mathcal{C}$ the family of all configurations $ C$ of $ N > 1$ distinct
points on the sphere $ S^2,$ and by $ \mathcal{H}$ the family of all closed hemispheres $ H$
of $ S^2.$ Compute:
$ \displaystyle\max_{H\in\mathcal{H}}\displaystyle\min_{C\in\mathcal{C}}|H\cap C|$, $ \displaystyle\min_{H\in\mathcal{H}}\displaystyle\max_{C\in\mathcal{C}}|H\cap C|$
$ \displaystyle\max_{C\in\mathcal{C}}\displaystyle\min_{H\in\mathcal{H}}|H\cap C|$ and $ \displaystyle\min_{C\in\mathcal{C}}\displaystyle\max_{H\in\mathcal{H}}|H\cap C|.$
1993 Austrian-Polish Competition, 2
Consider all tetrahedra $ABCD$ in which the sum of the areas of the faces $ABD, ACD, BCD$ does not exceed $1$. Among such tetrahedra, find those with the maximum volume.
1984 USAMO, 5
$P(x)$ is a polynomial of degree $3n$ such that
\begin{eqnarray*}
P(0) = P(3) = \cdots &=& P(3n) = 2, \\
P(1) = P(4) = \cdots &=& P(3n-2) = 1, \\
P(2) = P(5) = \cdots &=& P(3n-1) = 0, \quad\text{ and }\\
&& P(3n+1) = 730.\end{eqnarray*}
Determine $n$.
1994 National High School Mathematics League, 11
Intersections between a plane and 12 edges of a cube are all $\alpha$, then $\sin\alpha=$________.
1995 AMC 8, 21
A plastic snap-together cube has a protruding snap on one side and receptacle holes on the other five sides as shown. What is the smallest number of these cubes that can be snapped together so that only receptacle holes are showing?
[asy]
draw((0,0)--(4,0)--(4,4)--(0,4)--cycle);
draw(circle((2,2),1));
draw((4,0)--(6,1)--(6,5)--(4,4));
draw((6,5)--(2,5)--(0,4));
draw(ellipse((5,2.5),0.5,1));
fill(ellipse((3,4.5),1,0.25),black);
fill((2,4.5)--(2,5.25)--(4,5.25)--(4,4.5)--cycle,black);
fill(ellipse((3,5.25),1,0.25),black);
[/asy]
$\text{(A)}\ 3 \qquad \text{(B)}\ 4 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 8$
1998 Croatia National Olympiad, Problem 2
A hemisphere is inscribed in a cone so that its base lies on the base of the cone. The ratio of the area of the entire surface of the cone to the area of the hemisphere (without the base) is $\frac{18}5$. Compute the angle at the vertex of the cone.
2012 Puerto Rico Team Selection Test, 2
A cone is constructed with a semicircular piece of paper, with radius 10. Find the
height of the cone.
2023 Stanford Mathematics Tournament, 2
Triangle $\vartriangle ABC$ has side lengths $AB = 39$, $BC = 16$, and $CA = 25$. What is the volume of the solid formed by rotating $\vartriangle ABC$ about line $BC$?
2024 Euler Olympiad, Round 1, 9
Ants, named Anna and Bob, are located at vertices \(A\) and \(B\) respectively of a cube \(ABCD A_1 B_1 C_1 D_1\), with a sugar cube placed at vertex \(C_1\). It is known that Bob can move at a speed of $20$ meters per minute. Determine the minimum speed in integer meters per minute that Anna must be able to travel in order to reach the sugar cube at \(C_1\) before Bob.
[i]Proposed by Tamar Turashvili, Georgia [/i]
2004 AIME Problems, 3
A solid rectangular block is formed by gluing together $N$ congruent 1-cm cubes face to face. When the block is viewed so that three of its faces are visible, exactly 231 of the 1-cm cubes cannot be seen. Find the smallest possible value of $N$.
2013 National Olympiad First Round, 34
How many triples of positive integers $(a,b,c)$ are there such that $a!+b^3 = 18+c^3$?
$
\textbf{(A)}\ 4
\qquad\textbf{(B)}\ 3
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 1
\qquad\textbf{(E)}\ 0
$
1996 All-Russian Olympiad, 5
At the vertices of a cube are written eight pairwise distinct natural numbers, and on each of its edges is written the greatest common divisor of the numbers at the endpoints of the edge. Can the sum of the numbers written at the vertices be the same as the sum of the numbers written at the edges?
[i]A. Shapovalov[/i]
2024 May Olympiad, 4
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]
1998 Czech And Slovak Olympiad IIIA, 3
A sphere is inscribed in a tetrahedron $ABCD$. The tangent planes to the sphere parallel to the faces of the tetrahedron cut off four smaller tetrahedra. Prove that sum of all the $24$ edges of the smaller tetrahedra equals twice the sum of edges of the tetrahedron $ABCD$.
1979 IMO Longlists, 41
Prove the following statement: There does not exist a pyramid with square base and congruent lateral faces for which the measures of all edges, total area, and volume are integers.
1992 AIME Problems, 7
Faces $ABC$ and $BCD$ of tetrahedron $ABCD$ meet at an angle of $30^\circ$. The area of face $ABC$ is $120$, the area of face $BCD$ is $80$, and $BC=10$. Find the volume of the tetrahedron.
2009 Princeton University Math Competition, 2
Tetrahedron $ABCD$ has sides of lengths, in increasing order, $7, 13, 18, 27, 36, 41$. If $AB=41$, then what is the length of $CD$?
1956 Moscow Mathematical Olympiad, 329
Consider positive numbers $h, s_1, s_2$, and a spatial triangle $\vartriangle ABC$. How many ways are there to select a point $D$ so that the height of tetrahedron $ABCD$ drawn from $D$ equals $h$, and the areas of faces $ACD$ and $BCD$ equal $s_1$ and $s_2$, respectively?
1976 IMO Longlists, 2
Let $P$ be a set of $n$ points and $S$ a set of $l$ segments. It is known that:
$(i)$ No four points of $P$ are coplanar.
$(ii)$ Any segment from $S$ has its endpoints at $P$.
$(iii)$ There is a point, say $g$, in $P$ that is the endpoint of a maximal number of segments from $S$ and that is not a vertex of a tetrahedron having all its edges in $S$.
Prove that $l \leq \frac{n^2}{3}$
1965 German National Olympiad, 3
Two parallelograms $ABCD$ and $A'B'C'D'$ are given in space. Points $A'',B'',C'',D''$ divide the segments $AA',BB',CC',DD'$ in the same ratio. What can be said about the quadrilateral $A''B''C''D''$?
1989 Bulgaria National Olympiad, Problem 5
Prove that the perpendiculars, drawn from the midpoints of the edges of the base of a given tetrahedron to the opposite lateral edges, have a common point if and only if the circumcenter of the tetrahedron, the centroid of the base, and the top vertex of the tetrahedron are collinear.
1999 Brazil Team Selection Test, Problem 4
Assume that it is possible to color more than half of the surfaces of a given polyhedron so that no two colored surfaces have a common edge.
(a) Describe one polyhedron with the above property.
(b) Prove that one cannot inscribe a sphere touching all the surfaces of a polyhedron with the above property.
1984 Austrian-Polish Competition, 1
Prove that if the feet of the altitudes of a tetrahedron are the incenters of the corresponding faces, then the tetrahedron is regular.
1984 Putnam, A1
Let $A$ be a solid $a\times b\times c$ rectangular brick, where $a,b,c>0$. Let $B$ be the set of all points which are a distance of at most one from some point of $A$. Express the volume of $B$ as a polynomial in $a,b,$ and $c$.