Found problems: 2265
1990 French Mathematical Olympiad, Problem 2
A game consists of pieces of the shape of a regular tetrahedron of side $1$. Each face of each piece is painted in one of $n$ colors, and by this, the faces of one piece are not necessarily painted in different colors. Determine the maximum possible number of pieces, no two of which are identical.
1997 Vietnam National Olympiad, 3
In the unit cube, given 75 points, no three of which are collinear. Prove that there exits a triangle whose vertices are among the given points and whose area is not greater than 7/72.
1996 Czech And Slovak Olympiad IIIA, 2
Let $AP,BQ$ and $CR$ be altitudes of an acute-angled triangle $ABC$. Show that for any point $X$ inside the triangle $PQR$ there exists a tetrahedron $ABCD$ such that $X$ is the point on the face $ABC$ at the greatest distance from $D$ (measured along the surface of the tetrahedron).
1971 IMO, 1
All faces of the tetrahedron $ABCD$ are acute-angled. Take a point $X$ in the interior of the segment $AB$, and similarly $Y$ in $BC, Z$ in $CD$ and $T$ in $AD$.
[b]a.)[/b] If $\angle DAB+\angle BCD\ne\angle CDA+\angle ABC$, then prove none of the closed paths $XYZTX$ has minimal length;
[b]b.)[/b] If $\angle DAB+\angle BCD=\angle CDA+\angle ABC$, then there are infinitely many shortest paths $XYZTX$, each with length $2AC\sin k$, where $2k=\angle BAC+\angle CAD+\angle DAB$.
1999 Croatia National Olympiad, Problem 2
The base of a pyramid $ABCDV$ is a rectangle $ABCD$ with the sides $AB=a$ and $BC=b$, and all lateral edges of the pyramid have length $c$. Find the area of the intersection of the pyramid with a plane that contains the diagonal $BD$ and is parallel to $VA$.
2011 Iran MO (3rd Round), 1
A regular dodecahedron is a convex polyhedra that its faces are regular pentagons. The regular dodecahedron has twenty vertices and there are three edges connected to each vertex. Suppose that we have marked ten vertices of the regular dodecahedron.
[b]a)[/b] prove that we can rotate the dodecahedron in such a way that at most four marked vertices go to a place that there was a marked vertex before.
[b]b)[/b] prove that the number four in previous part can't be replaced with three.
[i]proposed by Kasra Alishahi[/i]
1984 IMO Longlists, 10
Assume that the bisecting plane of the dihedral angle at edge $AB$ of the tetrahedron $ABCD$ meets the edge $CD$ at point $E$. Denote by $S_1, S_2, S_3$, respectively the areas of the triangles $ABC, ABE$, and $ABD$. Prove that no tetrahedron exists for which $S_1, S_2, S_3$ (in this order) form an arithmetic or geometric progression.
2004 Tournament Of Towns, 3
The perpendicular projection of a triangular pyramid on some plane has the largest possible area. Prove that this plane is parallel to either a face or two opposite edges of the pyramid.
1998 Harvard-MIT Mathematics Tournament, 10
G. H. Hardy once went to visit Srinivasa Ramanujan in the hospital, and he started the conversation with: “I came here in taxi-cab number $1729$. That number seems dull to me, which I hope isn’t a bad omen.” “Nonsense,” said Ramanujan. “The number isn’t dull at all. It’s quite interesting. It’s the smallest number that can be expressed as the sum of two cubes in two different ways.” Ramujan had immediately seen that $1729=12^3+1^3=10^3+9^3$. What is the smallest positive integer representable as the sum of the cubes of [i]three[/i] positive integers in two different ways?
2000 IMC, 4
Let $OABC$ be a tetrahedon with $\angle BOC=\alpha,\angle COA =\beta$ and $\angle AOB =\gamma$. The angle between the faces $OAB$ and $OAC$ is $\sigma$ and the angle between the faces $OAB$ and $OBC$ is $\rho$.
Show that $\gamma > \beta \cos\sigma + \alpha \cos\rho$.
2018 Stanford Mathematics Tournament, 2
What is the largest possible height of a right cylinder with radius $3$ that can fit in a cube with side length $12$?
2018 Bundeswettbewerb Mathematik, 4
We are given six points in space with distinct distances, no three of them collinear. Consider all triangles with vertices among these points.
Show that among these triangles there is one such that its longest side is the shortest side in one of the other triangles.
2016 Saint Petersburg Mathematical Olympiad, 3
In a tetrahedron, the midpoints of all the edges lie on the same sphere. Prove that it's altitudes intersect at one point.
1973 Bulgaria National Olympiad, Problem 6
In the tetrahedron $ABCD$, $E$ and $F$ are the midpoints of $BC$ and $AD$, $G$ is the midpoint of the segment $EF$. Construct a plane through $G$ intersecting the segments $AB$, $AC$, $AD$ in the points $M,N,P$ respectively in such a way that the sum of the volumes of the tetrahedrons $BMNP$, $CMNP$ and $DMNP$ to be minimal.
[i]H. Lesov[/i]
2023 LMT Fall, 7
Isabella is making sushi. She slices a piece of salmon into the shape of a solid triangular prism. The prism is $2$ cm thick, and its triangular faces have side lengths $7$ cm, $ 24$cm, and $25$ cm. Find the volume of this piece of salmon in cm$^3$.
[i]Proposed by Isabella Li[/i]
2023 All-Russian Olympiad, 6
The plane $\alpha$ intersects the edges $AB$, $BC$, $CD$ and $DA$ of the tetrahedron $ABCD$ at points $X, Y, Z$ and $T$, respectively. It turned out, that points $Y$ and $T$ lie on a circle $\omega$ constructed with segment $XZ$ as the diameter. Point $P$ is marked in the plane $\alpha$ so that the lines $P Y$ and $P T$ are tangent to the circle $\omega$.Prove that the midpoints of the edges are $AB$, $BC$, $CD,$ $DA$ and the point $P$ lie in the same plane.
2010 CHMMC Fall, 4
Dagan has a wooden cube. He paints each of the six faces a different color. He then cuts up the cube to get eight identically-sized smaller cubes, each of which now has three painted faces and three unpainted faces. He then puts the smaller cubes back together into one larger cube such that no unpainted face is visible. Compute the number of different cubes that Dagan can make this way. Two cubes are considered the same if one can be rotated to obtain the other. You may express your answer either as an integer or as a product of prime numbers.
2009 Today's Calculation Of Integral, 402
Consider a right circular cylinder with radius $ r$ of the base, hight $ h$. Find the volume of the solid by revolving the cylinder about a diameter of the base.
2000 District Olympiad (Hunedoara), 4
Consider the pyramid $ VABCD, $ where $ V $ is the top and $ ABCD $ is a rectangular base. If $ \angle BVD = \angle AVC, $ then prove that the triangles $ VAC $ and $ VBD $ share the same perimeter and area.
2008 USAPhO, 1
A charged particle with charge $q$ and mass $m$ is given an initial kinetic energy $K_0$ at the middle of a uniformly charged spherical region of total charge $Q$ and radius $R$. $q$ and $Q$ have opposite signs. The spherically charged region is not free to move. Throughout this problem consider electrostatic forces only.
[asy]
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);
size(100);
filldraw(circle((0,0),1),gray(.8));
draw((0,0)--(0.5,sqrt(3)/2),EndArrow);
label("$R$",(0.25,sqrt(3)/4),SE);
[/asy]
(a) Find the value of $K_0$ such that the particle will just reach the boundary of the spherically charged region.
(b) How much time does it take for the particle to reach the boundary of the region if it starts with the kinetic energy $K_0$ found in part (a)?
1956 Czech and Slovak Olympiad III A, 2
In a given plane $\varrho$ consider a convex quadrilateral $ABCD$ and denote $E=AC\cap BD.$ Moreover, consider a point $V\notin\varrho$. On rays $VA,VB,VC,VD$ find points $A',B',C',D'$ respectively such that $E,A',B',C',D'$ are coplanar and $A'B'C'D'$ is a parallelogram. Discuss conditions of solvability.
2014 Purple Comet Problems, 30
Three mutually tangent spheres each with radius $5$ sit on a horizontal plane. A triangular pyramid has a base that is an equilateral triangle with side length $6$, has three congruent isosceles triangles for vertical faces, and has height $12$. The base of the pyramid is parallel to the plane, and the vertex of the pyramid is pointing downward so that it is between the base and the plane. Each of the three vertical faces of the pyramid is tangent to one of the spheres at a point on the triangular face along its altitude from the vertex of the pyramid to the side of length $6$. The distance that these points of tangency are from the base of the pyramid is $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[asy]
size(200);
defaultpen(linewidth(0.8));
pair X=(-.6,.4),A=(-.4,2),B=(-.7,1.85),C=(-1.1,2.05);
picture spherex;
filldraw(spherex,unitcircle,white);
draw(spherex,(-1,0)..(-.2,-.2)..(1,0)^^(0,1)..(-.2,-.2)..(0,-1));
add(shift(-0.5,0.6)*spherex);
filldraw(X--A--C--cycle,gray);
draw(A--B--C^^X--B);
add(shift(-1.5,0.2)*spherex);
add(spherex);
[/asy]
1994 Poland - First Round, 8
In a regular pyramid with a regular $n$-gon as a base, the dihedral angle between a lateral face and the base is equal to $\alpha$, and the angle between a lateral edge and the base is equal to $\beta$. Prove that
$sin^2 \alpha - sin^2 \beta \leq tg^2 \frac{\pi}{2n}$.
1999 Gauss, 22
Forty-two cubes with 1 cm edges are glued together to form a solid rectangular block. If the perimeter of the base of the block is 18 cm, then the height, in cm, is
$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ \dfrac{7}{3} \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$
2005 Iran MO (2nd round), 3
In one galaxy, there exist more than one million stars. Let $M$ be the set of the distances between any $2$ of them. Prove that, in every moment, $M$ has at least $79$ members. (Suppose each star as a point.)