Found problems: 2265
1996 Vietnam National Olympiad, 2
Given a trihedral angle Sxyz. A plane (P) not through S cuts Sx,Sy,Sz respectively at A,B,C. On the plane (P), outside triangle ABC, construct triangles DAB,EBC,FCA which are confruent to the triangles SAB,SBC,SCA respectively. Let (T) be the sphere lying inside Sxyz, but not inside the tetrahedron SABC, toucheing the planes containing the faces of SABC. Prove that (T) touches the plane (P) at the circumcenter of triangle DEF.
2021 Iran MO (3rd Round), 1
Is it possible to arrange natural numbers 1 to 8 on vertices of a cube such that each number divides sum of the three numbers sharing an edge with it?
1969 IMO Shortlist, 55
For each of $k=1,2,3,4,5$ find necessary and sufficient conditions on $a>0$ such that there exists a tetrahedron with $k$ edges length $a$ and the remainder length $1$.
2002 AMC 10, 16
Two walls and the ceiling of a room meet at right angles at point $P$. A fly is in the air one meter from one wall, eight meters from the other wall, and $9$ meters from point $P$. How many meters is the fly from the ceiling?
$\textbf{(A) }\sqrt{13}\qquad\textbf{(B) }\sqrt{14}\qquad\textbf{(C) }\sqrt{15}\qquad\textbf{(D) }4\qquad\textbf{(E) }\sqrt{17}$
1985 Iran MO (2nd round), 3
Find the angle between two common sections of the page $2x+y-z=0$ and the cone $4x^2-y^2+3z^2=0.$
1989 ITAMO, 3
Prove that, for every tetrahedron $ABCD$, there exists a unique point $P$ in the interior of the tetrahedron such that the tetrahedra $PABC,PABD,PACD,PBCD$ have equal volumes.
2017 District Olympiad, 4
An ant can move from a vertex of a cube to the opposite side of its diagonal only on the edges and the diagonals of the faces such that it doesn’t trespass a second time through the path made. Find the distance of the maximum journey this ant can make.
2008 Pre-Preparation Course Examination, 3
Prove that we can put $ \Omega(\frac1{\epsilon})$ points on surface of a sphere with radius 1 such that distance of each of these points and the plane passing through center and two of other points is at least $ \epsilon$.
1975 Putnam, B2
A [i]slab[/i] is the set of points strictly between two parallel planes. Prove that a countable sequence of slabs, the sum of whose thicknesses converges, cannot fill space.
1956 Poland - Second Round, 6
Prove that if in a tetrahedron $ ABCD $ the segments connecting the vertices of the tetrahedron with the centers of circles inscribed in opposite faces intersect at one point, then
$$AB \cdot CD = AC \cdot BD = AD \cdot BC$$
and that the converse also holds.
2002 Putnam, 2
Given any five points on a sphere, show that some four of them must lie on a closed hemisphere.
2004 AMC 10, 7
A grocer stacks oranges in a pyramid-like stack whose rectangular base is $ 5$ oranges by $ 8$ oranges. Each orange above the first level rests in a pocket formed by four oranges in the level below. The stack is completed by a single row of oranges. How many oranges are in the stack?
$ \textbf{(A)}\ 96 \qquad
\textbf{(B)}\ 98 \qquad
\textbf{(C)}\ 100 \qquad
\textbf{(D)}\ 101 \qquad
\textbf{(E)}\ 134$
1982 IMO Longlists, 13
A regular $n$-gonal truncated pyramid is circumscribed around a sphere. Denote the areas of the base and the lateral surfaces of the pyramid by $S_1, S_2$, and $S$, respectively. Let $\sigma$ be the area of the polygon whose vertices are the tangential points of the sphere and the lateral faces of the pyramid. Prove that
\[\sigma S = 4S_1S_2 \cos^2 \frac{\pi}{n}.\]
1991 National High School Mathematics League, 1
The number of regular triangles that three apexes are among eight vertex of a cube is
$\text{(A)}4\qquad\text{(B)}8\qquad\text{(C)}12\qquad\text{(D)}24$
1957 Polish MO Finals, 6
A cube is given with base $ ABCD $, where $ AB = a $ cm. Calculate the distance of the line $ BC $ from the line passing through the point $ A $ and the center $ S $ of the face opposite the base.
MIPT Undergraduate Contest 2019, 1.1 & 2.1
In $\mathbb{R}^3$, let there be a cube $Q$ and a sequence of other cubes, all of which are homothetic to $Q$ with coefficients of homothety that are each smaller than $1$. Prove that if this sequence of homothetic cubes completely fills $Q$, the sum of their coefficients of homothety is not less than $4$.
2021 AIME Problems, 6
Segments $\overline{AB}, \overline{AC},$ and $\overline{AD}$ are edges of a cube and $\overline{AG}$ is a diagonal through the center of the cube. Point $P$ satisfies $BP=60\sqrt{10}$, $CP=60\sqrt{5}$, $DP=120\sqrt{2}$, and $GP=36\sqrt{7}$. Find $AP.$
2011 Bundeswettbewerb Mathematik, 4
Let $ABCD$ be a tetrahedron that is not degenerate and not necessarily regular, where sides $AD$ and $BC$ have the same length $a$, sides $BD$ and $AC$ have the same length $b$, side $AB$ has length $c_1$ and the side $CD$ has length $c_2$. There is a point $P$ for which the sum of the distances to the vertices of the tetrahedron is minimal. Determine this sum depending on the quantities $a, b, c_1$ and $c_2$.
2012 China Second Round Olympiad, 5
Suppose two regular pyramids with the same base $ABC$: $P-ABC$ and $Q-ABC$ are circumscribed by the same sphere. If the angle formed by one of the lateral face and the base of pyramid $P-ABC$ is $\frac{\pi}{4}$, find the tangent value of the angle formed by one of the lateral face and the base of the pyramid $Q-ABC$.
2016 Postal Coaching, 5
Two triangles $ABC$ and $DEF$ have the same incircle. If a circle passes through $A,B,C,D,E$ prove that it also passes through $F$.
1982 Czech and Slovak Olympiad III A, 1
Given a tetrahedron $ABCD$ and inside the tetrahedron points $K, L, M, N$ that do not lie on a plane. Denote also the centroids of $P$, $Q$, $R$, $S$ of the tetrahedrons $KBCD$, $ALCD$, $ABMD$, $ABCN$ do not lie on a plane. Let $T$ be the centroid of the tetrahedron ABCD, $T_o$ be the centroid of the tetrahedron $PQRS$ and $T_1$ be the centroid of the tetrahedron $KLMN$.
a) Prove that the points $T, T_0, T_1$ lie in one straight line.
b) Determine the ratio $|T_0T| : |T_0 T_1|$.
1998 Hong kong National Olympiad, 2
The underside of a pyramid is a convex nonagon , paint all the diagonals of the nonagon and all the ridges of the pyramid into white and black , prove : there exists a triangle ,the colour of its three sides are the same . ( PS:the sides of the nonagon is not painted. )
2005 AMC 12/AHSME, 10
A wooden cube $ n$ units on a side is painted red on all six faces and then cut into $ n^3$ unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is $ n$?
$ \textbf{(A)}\ 3\qquad
\textbf{(B)}\ 4\qquad
\textbf{(C)}\ 5\qquad
\textbf{(D)}\ 6\qquad
\textbf{(E)}\ 7$
2023 Sharygin Geometry Olympiad, 24
A tetrahedron $ABCD$ is give. A line $\ell$ meets the planes $ABC,BCD,CDA,DAB$ at points $D_0,A_0,B_0,C_0$ respectively. Let $P$ be an arbitrary point not lying on $\ell$ and the planes of the faces, and $A_1,B_1,C_1,D_1$ be the second common points of lines $PA_0,PB_0,PC_0,PD_0$ with the spheres $PBCD,PCDA,PDAB,PABC$ respectively. Prove $P,A_1,B_1,C_1,D_1$ lie on a circle.
1966 IMO Shortlist, 44
What is the greatest number of balls of radius $1/2$ that can be placed within a rectangular box of size $10 \times 10 \times 1 \ ?$