Found problems: 2265
2007 Iran MO (2nd Round), 2
Two vertices of a cube are $A,O$ such that $AO$ is the diagonal of one its faces. A $n-$run is a sequence of $n+1$ vertices of the cube such that each $2$ consecutive vertices in the sequence are $2$ ends of one side of the cube. Is the $1386-$runs from $O$ to itself less than $1386-$runs from $O$ to $A$ or more than it?
1967 German National Olympiad, 6
Prove the following theorem:
If there are $n$ pairs of different points $P_i$, $i = 1, 2, ..., n$, $n > 2$ in three dimensions space, such that each of them is at a smaller distance from one and the same point $Q$ than any other $P_i$, then $n < 15$.
2023 AIME, 14
A cube-shaped container has vertices $A$, $B$, $C$, and $D$ where $\overline{AB}$ and $\overline{CD}$ are parallel edges of the cube, and $\overline{AC}$ and $\overline{BD}$ are diagonals of the faces of the cube. Vertex $A$ of the cube is set on a horizontal plane $\mathcal P$ so that the plane of the rectangle $ABCD$ is perpendicular to $\mathcal P$, vertex $B$ is $2$ meters above $\mathcal P$, vertex $C$ is $8$ meters above $\mathcal P$, and vertex $D$ is $10$ meters above $\mathcal P$. The cube contains water whose surface is $7$ meters above $\mathcal P$. The volume of the water is $\tfrac mn$ cubic meters, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[asy]
size(250);
defaultpen(linewidth(0.6));
pair A = origin, B = (6,3), X = rotate(40)*B, Y = rotate(70)*X, C = X+Y, Z = X+B, D = B+C, W = B+Y;
pair P1 = 0.8*C+0.2*Y, P2 = 2/3*C+1/3*X, P3 = 0.2*D+0.8*Z, P4 = 0.63*D+0.37*W;
pair E = (-20,6), F = (-6,-5), G = (18,-2), H = (9,8);
filldraw(E--F--G--H--cycle,rgb(0.98,0.98,0.2));
fill(A--Y--P1--P4--P3--Z--B--cycle,rgb(0.35,0.7,0.9));
draw(A--B--Z--X--A--Y--C--X^^C--D--Z);
draw(P1--P2--P3--P4--cycle^^D--P4);
dot("$A$",A,S);
dot("$B$",B,S);
dot("$C$",C,N);
dot("$D$",D,N);
label("$\mathcal P$",(-13,4.5));
[/asy]
2024 AMC 12/AHSME, 23
A right pyramid has regular octagon $ABCDEFGH$ with side length $1$ as its base and apex $V.$ Segments $\overline{AV}$ and $\overline{DV}$ are perpendicular. What is the square of the height of the pyramid?
$
\textbf{(A) }1 \qquad
\textbf{(B) }\frac{1+\sqrt2}{2} \qquad
\textbf{(C) }\sqrt2 \qquad
\textbf{(D) }\frac32 \qquad
\textbf{(E) }\frac{2+\sqrt2}{3} \qquad
$
2021 Sharygin Geometry Olympiad, 24
A truncated trigonal pyramid is circumscribed around a sphere touching its bases at points $T_1, T_2$. Let $h$ be the altitude of the pyramid, $R_1, R_2$ be the circumradii of its bases, and $O_1, O_2$ be the circumcenters of the bases. Prove that $$R_1R_2h^2 = (R_1^2-O_1T_1^2)(R_2^2-O_2T_2^2).$$
2014 BMT Spring, 10
A plane intersects a sphere of radius $10$ such that the distance from the center of the sphere to the plane is $9$. The plane moves toward the center of the bubble at such a rate that the increase in the area of the intersection of the plane and sphere is constant, and it stops once it reaches the center of the circle. Determine the distance from the center of the sphere to the plane after two-thirds of the time has passed.
1968 Yugoslav Team Selection Test, Problem 6
Prove that the incenter coincides with the circumcenter of a tetrahedron if and only if each pair of opposite edges are of equal length.
1995 Yugoslav Team Selection Test, Problem 3
Let $SABCD$ be a pyramid with the vertex $S$ whose all edges are equal. Points $M$ and $N$ on the edges $SA$ and $BC$ respectively are such that $MN$ is perpendicular to both $SA$ and $BC$. Find the ratios $SM:MA$ and $BN:NC$.
1960 Czech and Slovak Olympiad III A, 2
Consider a cube $ABCDA'B'C'D'$ (where $ABCD$ is a square and $AA' \parallel BB' \parallel CC' \parallel DD'$) and a point $P$ on the line $AA'$. Construct center $S$ of a sphere which has plane $ABB'$ as a plane of symmetry, $P$ lies on the sphere and $p = AB$, $q = A'D'$ are its tangent lines. Discuss conditions of solvability with respect to different position of the point $P$ (on line $AA'$).
1949-56 Chisinau City MO, 60
Show that the sum of the distances from any point of a regular tetrahedron to its faces is equal to the height of this tetrahedron.
1969 IMO Longlists, 32
$(GDR 4)$ Find the maximal number of regions into which a sphere can be partitioned by $n$ circles.
1922 Eotvos Mathematical Competition, 1
Given four points $A,B,C,D$ in space, find a plane, $S$, equidistant from all four points and having $A$ and $C$ on one side, $B$ and $D$ on the other.
1950 Moscow Mathematical Olympiad, 186
A spatial quadrilateral is circumscribed around a sphere. Prove that all the tangent points lie in one plane.
2014 Math Prize For Girls Problems, 10
An ant is on one face of a cube. At every step, the ant walks to one of its four neighboring faces with equal probability. What is the expected (average) number of steps for it to reach the face opposite its starting face?
1971 IMO Shortlist, 16
Let $P_1$ be a convex polyhedron with vertices $A_1,A_2,\ldots,A_9$. Let $P_i$ be the polyhedron obtained from $P_1$ by a translation that moves $A_1$ to $A_i$. Prove that at least two of the polyhedra $P_1,P_2,\ldots,P_9$ have an interior point in common.
2013 Poland - Second Round, 6
Decide, whether exist tetrahedrons $T$, $T'$ with walls $S_1$, $S_2$, $S_3$, $S_4$ and $S_1'$, $S_2'$, $S_3'$, $S_4'$, respectively, such that for $i = 1, 2, 3, 4$ triangle $S_i$ is similar to triangle $S_i'$, but despite this, tetrahedron $T$ is not similar to tetrahedron $T'$.
2019 Puerto Rico Team Selection Test, 6
Starting from a pyramid $T_0$ whose edges are all of length $2019$, we construct the Figure $T_1$ when considering the triangles formed by the midpoints of the edges of each face of $T_0$, building in each of these new pyramid triangles with faces identical to base. Then the bases of these new pyramids are removed. Figure $T_2$ is constructed by applying the same process from $T_1$ on each triangular face resulting from $T_1$, and so on for $T_3, T_4, ...$
Let $D_0= \max \{d(x,y)\}$, where $x$ and $y$ are vertices of $T_0$ and $d(x,y)$ is the distance between $x$ and $y$. Then we define $D_{n + 1} = \max \{d (x, y) |d (x, y) \notin \{D_0, D_1,...,D_n\}$, where $x, y$ are vertices of $T_{n+1}$.
Find the value of $D_n$ for all $n$.
2014 Sharygin Geometry Olympiad, 7
Prove that the smallest dihedral angle between faces of an arbitrary tetrahedron is not greater than the dihedral angle between faces of a regular tetrahedron.
(S. Shosman, O. Ogievetsky)
2002 Moldova National Olympiad, 3
Let $ P$ be a polyhedron whose all edges are congruent and tangent to a sphere. Suppose that one of the facesof $ P$ has an odd number of sides. Prove that all vertices of $ P$ lie on a single sphere.
2002 Croatia National Olympiad, Problem 2
Consider the cube with the vertices $A(1,1,1)$, $B(-1,1,1)$, $C(-1,-1,1)$, $D(1,-1,1)$ and $A',B',C',D'$ symmetric to $A,B,C,D$ respectively with respect to the origin $O$. Let $T$ be a point not on the circumsphere of the cube and let $OT=d$. Denote $\alpha=\angle ATA'$, $\beta=\angle BTB'$, $\gamma=\angle CTC'$, $\delta=\angle DTD'$. Prove that
$$\tan^2\alpha+\tan^2\beta+\tan^2\gamma+\tan^2\delta=\frac{32d^2}{\left(d^2-3\right)^2}.$$
1988 Tournament Of Towns, (187) 4
Each face of a cube has been divided into four equal quarters and each quarter is painted with one of three available colours. Quarters with common sides are painted with different colours . Prove that each of the available colours was used in painting $8$ quarters.
2012-2013 SDML (Middle School), 2
A regular tetrahedron with $5$-inch edges weighs $2.5$ pounds. What is the weight in pounds of a similarly constructed regular tetrahedron that has $6$-inch edges? Express your answer as a decimal rounded to the nearest hundredth.
1984 Bundeswettbewerb Mathematik, 4
A sphere is touched by all the four sides of a (space) quadrilateral. Prove that all the four touching points are in the same plane.
1986 National High School Mathematics League, 4
None face of a tetrahedron is isosceles triangle. How many kinds of lengths of edges do the tetrahedron have at least?
$\text{(A)}3\qquad\text{(B)}4\qquad\text{(C)}5\qquad\text{(D)}6$
I Soros Olympiad 1994-95 (Rus + Ukr), 11.4
A tetrahedron $ABCD$ is given, in which each pair of adjacent edges are equal segments. Let $O$ be the center of the sphere inscribed in this tetrahedron . $X$ is an arbitrary point inside the tetrahedron, $X \ne O$. The line $OX$ intersects the planes of the faces of the tetrahedron at the points marked by $A_1$, $B_1$, $C_1$, $D_1$. Prove that
$$\frac{A_1X}{A_1O} +\frac{B_1X}{B_1O} +\frac{C_1X}{C_1O}+\frac{D_1X}{D_1O}=4$$