Found problems: 2265
Ukrainian TYM Qualifying - geometry, XI.13
On the plane there are two cylindrical towers with radii of bases $r$ and $R$. Find the set of all those points of the plane from which these towers are visible at the same angle. Consider the case of more towers.
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.
ICMC 5, 3
A set of points has [i]point symmetry[/i] if a reflection in some point maps the set to itself. Let $\cal P$ be a solid convex polyhedron whose orthogonal projections onto any plane have point symmetry. Prove that $\cal P$ has point symmetry.
[i]Proposed by Ethan Tan[/i]
1957 AMC 12/AHSME, 17
A cube is made by soldering twelve $ 3$-inch lengths of wire properly at the vertices of the cube. If a fly alights at one of the vertices and then walks along the edges, the greatest distance it could travel before coming to any vertex a second time, without retracing any distance, is:
$ \textbf{(A)}\ 24\text{ in.}\qquad
\textbf{(B)}\ 12\text{ in.}\qquad
\textbf{(C)}\ 30\text{ in.}\qquad
\textbf{(D)}\ 18\text{ in.}\qquad
\textbf{(E)}\ 36\text{ in.}$
1986 Balkan MO, 2
Let $ABCD$ be a tetrahedron and let $E,F,G,H,K,L$ be points lying on the edges $AB,BC,CD$ $,DA,DB,DC$ respectively, in such a way that
\[AE \cdot BE = BF \cdot CF = CG \cdot AG= DH \cdot AH=DK \cdot BK=DL \cdot CL.\]
Prove that the points $E,F,G,H,K,L$ all lie on a sphere.
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
$
2010 All-Russian Olympiad, 1
Let $a \neq b a,b \in \mathbb{R}$ such that $(x^2+20ax+10b)(x^2+20bx+10a)=0$ has no roots for $x$. Prove that $20(b-a)$ is not an integer.
2009 Flanders Math Olympiad, 4
The maximum number of solid regular tetrahedrons can be placed against each other so that one of their edges coincides with a given line segment in space?
[hide=original wording]Hoeveel massieve regelmatige viervlakken kan men maximaal tegen mekaar plaatsen
zodat ´e´en van hun ribben samenvalt met een gegeven lijnstuk in de ruimte?[/hide]
2009 Romanian Master of Mathematics, 3
Given four points $ A_1, A_2, A_3, A_4$ in the plane, no three collinear, such that
\[ A_1A_2 \cdot A_3 A_4 \equal{} A_1 A_3 \cdot A_2 A_4 \equal{} A_1 A_4 \cdot A_2 A_3,
\]
denote by $ O_i$ the circumcenter of $ \triangle A_j A_k A_l$ with $ \{i,j,k,l\} \equal{} \{1,2,3,4\}.$ Assuming $ \forall i A_i \neq O_i ,$ prove that the four lines $ A_iO_i$ are concurrent or parallel.
[i]Nikolai Ivanov Beluhov, Bulgaria[/i]
2022 JHMT HS, 2
Four mutually externally tangent spherical apples of radius $4$ are placed on a horizontal flat table. Then, a spherical orange of radius $3$ is placed such that it rests on all the apples. Find the distance from the center of the orange to the table.
2011 Benelux, 3
If $k$ is an integer, let $\mathrm{c}(k)$ denote the largest cube that is less than or equal to $k$. Find all positive integers $p$ for which the following sequence is bounded:
$a_0 = p$ and $a_{n+1} = 3a_n-2\mathrm{c}(a_n)$ for $n \geqslant 0$.
2010 Romania National Olympiad, 3
Let $VABCD$ be a regular pyramid, having the square base $ABCD$. Suppose that on the line $AC$ lies a point $M$ such that $VM=MB$ and $(VMB)\perp (VAB)$. Prove that $4AM=3AC$.
[i]Mircea Fianu[/i]
1978 All Soviet Union Mathematical Olympiad, 266
Prove that for every tetrahedron there exist two planes such that the projection areas on those planes ratio is not less than $\sqrt 2$.
1982 All Soviet Union Mathematical Olympiad, 348
The $KLMN$ tetrahedron (triangle pyramid) vertices are situated inside or on the faces or on the edges of the $ABCD$ tetrahedron. Prove that perimeter of $KLMN$ is less than $4/3$ perimeter of $ABCD$.
1986 Czech And Slovak Olympiad IIIA, 3
Prove that the entire space can be partitioned into “crosses” made of seven unit cubes as shown in the picture.
[img]https://cdn.artofproblemsolving.com/attachments/2/b/77c4a4309170e8303af321daceccc4010da334.png[/img]
1992 IMO Longlists, 76
Given any triangle $ABC$ and any positive integer $n$, we say that $n$ is a [i]decomposable[/i] number for triangle $ABC$ if there exists a decomposition of the triangle $ABC$ into $n$ subtriangles with each subtriangle similar to $\triangle ABC$. Determine the positive integers that are decomposable numbers for every triangle.
1987 IMO Shortlist, 10
Let $S_1$ and $S_2$ be two spheres with distinct radii that touch externally. The spheres lie inside a cone $C$, and each sphere touches the cone in a full circle. Inside the cone there are $n$ additional solid spheres arranged in a ring in such a way that each solid sphere touches the cone $C$, both of the spheres $S_1$ and $S_2$ externally, as well as the two neighboring solid spheres. What are the possible values of $n$?
[i]Proposed by Iceland.[/i]
2012 Online Math Open Problems, 8
An $8 \times 8 \times 8$ cube is painted red on $3$ faces and blue on $3$ faces such that no corner is surrounded by three faces of the same color. The cube is then cut into $512$ unit cubes. How many of these cubes contain both red and blue paint on at least one of their faces?
[i]Author: Ray Li[/i]
[hide="Clarification"]The problem asks for the number of cubes that contain red paint on at least one face and blue paint on at least one other face, not for the number of cubes that have both colors of paint on at least one face (which can't even happen.)[/hide]
2005 National High School Mathematics League, 10
In tetrahedron $ABCD$, the volume of tetrahedron $ABCD$ is $\frac{1}{6}$, and $\angle ACB=45^{\circ},AD+BC+\frac{AC}{\sqrt2}=3$, then $CD=$________.
1991 Arnold's Trivium, 70
Calculate the mean value of the solid angle by which the disc $x^2 + y^2 \le 1$ lying in the plane $z = 0$ is seen from points of the sphere $x^2 + y^2 + (z-2)^2 = 1$.
1969 IMO Longlists, 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 $1$.
2018 Singapore MO Open, 1
Consider a regular cube with side length $2$. Let $A$ and $B$ be $2$ vertices that are furthest apart. Construct a sequence of points on the surface of the cube $A_1$, $A_2$, $\ldots$, $A_k$ so that $A_1=A$, $A_k=B$ and for any $i = 1,\ldots, k-1$, the distance from $A_i$ to $A_{i+1}$ is $3$. Find the minimum value of $k$.
1959 IMO Shortlist, 6
Two planes, $P$ and $Q$, intersect along the line $p$. The point $A$ is given in the plane $P$, and the point $C$ in the plane $Q$; neither of these points lies on the straight line $p$. Construct an isosceles trapezoid $ABCD$ (with $AB \parallel CD$) in which a circle can be inscribed, and with vertices $B$ and $D$ lying in planes $P$ and $Q$ respectively.
PEN H Problems, 33
Does there exist an integer such that its cube is equal to $3n^2 +3n+7$, where $n$ is integer?
1996 Polish MO Finals, 1
$ABCD$ is a tetrahedron with $\angle BAC = \angle ACD$ and $\angle ABD = \angle BDC$. Show that $AB = CD$.