Found problems: 2265
1999 National Olympiad First Round, 17
In a regular pyramid with top point $ T$ and equilateral base $ ABC$, let $ P$, $ Q$, $ R$, $ S$ be the midpoints of $ \left[AB\right]$, $ \left[BC\right]$, $ \left[CT\right]$ and $ \left[TA\right]$, respectively. If $ \left|AB\right| \equal{} 6$ and the altitude of pyramid is equal to $ 2\sqrt {15}$, then area of $ PQRS$ will be
$\textbf{(A)}\ 4\sqrt {15} \qquad\textbf{(B)}\ 8\sqrt {2} \qquad\textbf{(C)}\ 8\sqrt {3} \qquad\textbf{(D)}\ 6\sqrt {5} \qquad\textbf{(E)}\ 9\sqrt {2}$
2019 Jozsef Wildt International Math Competition, W. 41
For $n \in \mathbb{N}$, consider in $\mathbb{R}^3$ the regular tetrahedron with vertices $O(0, 0, 0)$, $A(n, 9n, 4n)$, $B(9n, 4n, n)$ and $C(4n, n, 9n)$. Show that the number $N$ of points $(x, y, z)$, $[x, y, z \in \mathbb{Z}]$ inside or on the boundary of the tetrahedron $OABC$ is given by$$N=\frac{343n^3}{3}+\frac{35n^2}{2}+\frac{7n}{6}+1$$
1966 IMO Shortlist, 60
Prove that the sum of the distances of the vertices of a regular tetrahedron from the center of its circumscribed sphere is less than the sum of the distances of these vertices from any other point in space.
1965 Czech and Slovak Olympiad III A, 4
Consider a container of a hollow cube $ABGCDEPF$ (where $ABGC$, $DEPF$ are squares and $AD\parallel BE\parallel GP\parallel CF$). The cube is placed on a table in a way that the space diagonal $AP=1$ is perpendicular to the table. Then, water is poured into the cube. Denote $x$ the length of part of $AP$ submerged in water. Determine the volume of water $y$ in terms of $x$ when
a) $0 < x \leq\frac13$,
b) $\frac13 < x \leq\frac12$.
1966 Bulgaria National Olympiad, Problem 4
It is given a tetrahedron with vertices $A,B,C,D$.
(a) Prove that there exists a vertex of the tetrahedron with the following property: the three edges of that tetrahedron through that vertex can form a triangle.
(b) On the edges $DA,DB$ and $DC$ there are given the points $M,N$ and $P$ for which:
$$DM=\frac{DA}n,\enspace DN=\frac{DB}{n+1}\enspace DP=\frac{DC}{n+2}$$where $n$ is a natural number. The plane defined by the points $M,N$ and $P$ is $\alpha_n$. Prove that all planes $\alpha_n$, $(n=1,2,3,\ldots)$ pass through a single straight line.
2011 Spain Mathematical Olympiad, 3
Let $A$, $B$, $C$, $D$ be four points in space not all lying on the same plane. The segments $AB$, $BC$, $CD$, and $DA$ are tangent to the same sphere. Prove that their four points of tangency are coplanar.
2002 Flanders Junior Olympiad, 3
Is it possible to number the $8$ vertices of a cube from $1$ to $8$ in such a way that the value of the sum on every edge is different?
1990 IMO Longlists, 27
A plane cuts a right circular cone of volume $ V$ into two parts. The plane is tangent to the circumference of the base of the cone and passes through the midpoint of the altitude. Find the volume of the smaller part.
[i]Original formulation:[/i]
A plane cuts a right circular cone into two parts. The plane is tangent to the circumference of the base of the cone and passes through the midpoint of the altitude. Find the ratio of the volume of the smaller part to the volume of the whole cone.
2012 JHMT, 7
What is the radius of the largest sphere that fits inside an octahedron of side length $1$?
1956 Moscow Mathematical Olympiad, 345
* Prove that if the trihedral angles at each of the vertices of a triangular pyramid are formed by the identical planar angles, then all faces of this pyramid are equal.
1994 Bundeswettbewerb Mathematik, 3
Let $A$ and $B$ be two spheres of different radii, both inscribed in a cone $K$. There are $m$ other, congruent spheres arranged in a ring such that each of them touches $A, B, K$ and two of the other spheres. Prove that this is possible for at most three values of $m.$
2017 Yasinsky Geometry Olympiad, 2
Prove that if all the edges of the tetrahedron are equal triangles (such a tetrahedron is called equilateral), then its projection on the plane of a face is a triangle.
1987 AMC 8, 7
The large cube shown is made up of $27$ identical sized smaller cubes. For each face of the large cube, the opposite face is shaded the same way. The total number of smaller cubes that must have at least one face shaded is
[asy]
unitsize(36);
draw((0,0)--(3,0)--(3,3)--(0,3)--cycle);
draw((3,0)--(5.2,1.4)--(5.2,4.4)--(3,3));
draw((0,3)--(2.2,4.4)--(5.2,4.4));
fill((0,0)--(0,1)--(1,1)--(1,0)--cycle,black);
fill((0,2)--(0,3)--(1,3)--(1,2)--cycle,black);
fill((1,1)--(1,2)--(2,2)--(2,1)--cycle,black);
fill((2,0)--(3,0)--(3,1)--(2,1)--cycle,black);
fill((2,2)--(3,2)--(3,3)--(2,3)--cycle,black);
draw((1,3)--(3.2,4.4));
draw((2,3)--(4.2,4.4));
draw((.733333333,3.4666666666)--(3.73333333333,3.466666666666));
draw((1.466666666,3.9333333333)--(4.466666666,3.9333333333));
fill((1.73333333,3.46666666666)--(2.7333333333,3.46666666666)--(3.46666666666,3.93333333333)--(2.46666666666,3.93333333333)--cycle,black);
fill((3,1)--(3.733333333333,1.466666666666)--(3.73333333333,2.46666666666)--(3,2)--cycle,black);
fill((3.73333333333,.466666666666)--(4.466666666666,.93333333333)--(4.46666666666,1.93333333333)--(3.733333333333,1.46666666666)--cycle,black);
fill((3.73333333333,2.466666666666)--(4.466666666666,2.93333333333)--(4.46666666666,3.93333333333)--(3.733333333333,3.46666666666)--cycle,black);
fill((4.466666666666,1.9333333333333)--(5.2,2.4)--(5.2,3.4)--(4.4666666666666,2.9333333333333)--cycle,black);
[/asy]
$\text{(A)}\ 10 \qquad \text{(B)}\ 16 \qquad \text{(C)}\ 20 \qquad \text{(D)}\ 22 \qquad \text{(E)}\ 24$
1993 Dutch Mathematical Olympiad, 4
Let $ C$ be a circle with center $ M$ in a plane $ V$, and $ P$ be a point not on the circle $ C$.
$ (a)$ If $ P$ is fixed, prove that $ AP^2\plus{}BP^2$ is a constant for every diameter $ AB$ of the circle $ C$.
$ (b)$ Let $ AB$ be a fixed diameter of $ C$ and $ P$ a point on a fixed sphere $ S$ not intersecting $ V$. Determine the points $ P$ on $ S$ that minimize $ AP^2\plus{}BP^2$.
1996 AIME Problems, 4
A wooden cube, whose edges are one centimeter long, rests on a horizontal surface. Illuminated by a point source of light that is $x$ centimeters directly above an upper vertex, the cube casts a shadow on the horizontal surface. The area of the shadow, which does not inclued the area beneath the cube is 48 square centimeters. Find the greatest integer that does not exceed $1000x.$
2007 Moldova National Olympiad, 11.7
Given a tetrahedron $VABC$ with edges $VA$, $VB$ and $VC$ perpendicular any two of them. The sum of the lengths of the tetrahedron's edges is $3p$. Find the maximal volume of $VABC$.
2009 Miklós Schweitzer, 11
Denote by $ H_n$ the linear space of $ n\times n$ self-adjoint complex matrices, and by $ P_n$ the cone of positive-semidefinite matrices in this space. Let us consider the usual inner product on $ H_n$
\[ \langle A,B\rangle \equal{} {\rm tr} AB\qquad (A,B\in H_n)\]
and its derived metric. Show that every $ \phi: P_n\to P_n$ isometry (that is a not necessarily surjective, distance preserving map with respect to the above metric) can be expressed as
\[ \phi(A) \equal{} UAU^* \plus{} X\qquad (A\in H_n)\]
or
\[ \phi(A) \equal{} UA^TU^* \plus{} X\qquad (A\in H_n)\]
where $ U$ is an $ n\times n$ unitary matrix, $ X$ is a positive-semidefinite matrix, and $ ^T$ and $ ^*$ denote taking the transpose and the adjoint, respectively.
2003 Austrian-Polish Competition, 6
$ABCD$ is a tetrahedron such that we can find a sphere $k(A,B,C)$ through $A, B, C$ which meets the plane $BCD$ in the circle diameter $BC$, meets the plane $ACD$ in the circle diameter $AC$, and meets the plane $ABD$ in the circle diameter $AB$. Show that there exist spheres $k(A,B,D)$, $k(B,C,D)$ and $k(C,A,D)$ with analogous properties.
1989 Tournament Of Towns, (216) 4
Is it possible to mark a diagonal on each little square on the surface of a Rubik 's cube so that one obtains a non-intersecting path?
(S . Fomin, Leningrad)
2006 District Olympiad, 3
We say that a prism is [i]binary[/i] if there exists a labelling of the vertices of the prism with integers from the set $\{-1,1\}$ such that the product of the numbers assigned to the vertices of each face (base or lateral face) is equal to $-1$.
a) Prove that any [i]binary[/i] prism has the number of total vertices divisible by 8;
b) Prove that any prism with 2000 vertices is [i]binary[/i].
2019 Olympic Revenge, 4
A regular icosahedron is a regular solid of $20$ faces, each in the form of an equilateral triangle, with $12$ vertices, so that each vertex is in $5$ edges.
Twelve indistinguishable candies are glued to the vertices of a regular icosahedron (one at each vertex), and four of these twelve candies are special. André and Lucas want to together create a strategy for the following game:
• First, André is told which are the four special sweets and he must remove exactly four sweets that are not special from the icosahedron and leave the solid on a table, leaving afterwards without communicating with Lucas.
• Later, Sponchi, who wants to prevent Lucas from discovering the special sweets, can pick up the icosahedron from the table and rotate it however he wants.
• After Sponchi makes his move, he leaves the room, Lucas enters and he must determine the four special candies out of the eight that remain in the icosahedron.
Determine if there is a strategy for which Lucas can always properly discover the four special sweets.
1963 Bulgaria National Olympiad, Problem 4
In the tetrahedron $ABCD$ three of the faces are right-angled triangles and the other is not an obtuse triangle. Prove that:
(a) the fourth wall of the tetrahedron is a right-angled triangle if and only if exactly two of the plane angles having common vertex with the some of vertices of the tetrahedron are equal.
(b) its volume is equal to $\frac16$ multiplied by the multiple of two shortest edges and an edge not lying on the same wall.
2023 Oral Moscow Geometry Olympiad, 4
Given isosceles tetrahedron $PABC$ (faces are equal triangles). Let $A_0$, $B_0$ and $C_0$ be the touchpoints of the circle inscribed in the triangle $ABC$ with sides $BC$, $AC$ and $AB$ respectively, $A_1$, $B_1$ and $C_1$ are the touchpoints of the excircles of triangles $PCA$, $PAB$ and $PBC$ with extensions of sides $PA$, $PB$ and $PC$, respectively (beyond points $A$, $B$, $C$). Prove that the lines $A_0A_1$, $B_0B_1$ and $C_0C_1$ intersect at one point.
2008 Princeton University Math Competition, A9
In tetrahedron $ABCD$ with circumradius $2$, $AB = 2$, $CD = \sqrt{7}$, and $\angle ABC = \angle BAD = \frac{\pi}{2}$. Find all possible angles between the planes containing $ABC$ and $ABD$.
2021 All-Russian Olympiad, 6
In tetrahedron $ABCS$ no two edges have equal length. Point $A'$ in plane $BCS$ is symmetric to $S$ with respect to the perpendicular bisector of $BC$. Points $B'$ and $C'$ are defined analagously. Prove that planes $ABC, AB'C', A'BC'$ abd $A'B'C$ share a common point.