Found problems: 2265
PEN R Problems, 7
Show that the number $r(n)$ of representations of $n$ as a sum of two squares has $\pi$ as arithmetic mean, that is \[\lim_{n \to \infty}\frac{1}{n}\sum^{n}_{m=1}r(m) = \pi.\]
1990 Romania Team Selection Test, 4
The six faces of a hexahedron are quadrilaterals. Prove that if seven its vertices lie on a sphere, then the eighth vertex also lies on the sphere.
1967 IMO Longlists, 27
Which regular polygon can be obtained (and how) by cutting a cube with a plane ?
2016 Purple Comet Problems, 30
Some identically sized spheres are piled in $n$ layers in the shape of a square pyramid with one sphere in the top layer, 4 spheres in the second layer, 9 spheres in the third layer, and so forth so that the bottom layer has a square array of $n^2$ spheres. In each layer the centers of the spheres form a square grid so that each sphere is tangent to any sphere adjacent to it on the grid. Each sphere in an upper level is tangent to the four spheres directly below it. The diagram shows how the first three layers of spheres are stacked. A square pyramid is built around the pile of spheres so that the sides of the pyramid are tangent to the spheres on the outside of the pile. There is a positive integer $m$ such that as $n$ gets large, the ratio of the volume of the pyramid to the total volume inside all of the spheres approaches $\frac{\sqrt{m}}{\pi}$. Find $m$.
[center][img]https://snag.gy/bIwyl6.jpg[/img][/center]
1987 Traian Lălescu, 1.4
Let $ ABCD $ be a regular tetahedron and $ M,N $ be middlepoints for $ AD, $ respectively, $ BC. $ Through a point $ P $ that is on segment $ MN, $ passes a plane perpendicular on $ MN, $ and meets the sides $ AB,AC,CD,BD $ of the tetahedron at $ E,F,G, $ respectively, $ H. $
[b]a)[/b] Prove that the perimeter of the quadrilateral $ EFGH $ doesn't depend on $ P. $
[b]b)[/b] Determine the maximum area of $ EFGH $ (depending on a side of the tetahedron).
1972 Vietnam National Olympiad, 4
Let $ABCD$ be a regular tetrahedron with side $a$. Take $E,E'$ on the edge $AB, F, F'$ on the edge $AC$ and $G,G'$ on the edge AD so that $AE =a/6,AE' = 5a/6,AF= a/4,AF'= 3a/4,AG = a/3,AG'= 2a/3$. Compute the volume of $EFGE'F'G'$ in term of $a$ and find the angles between the lines $AB,AC,AD$ and the plane $EFG$.
1938 Moscow Mathematical Olympiad, 040
What is the largest number of parts into which $n$ planes can divide space?
We assume that the set of planes is non-degenerate in the sense that any three planes intersect in one point and no four planes have a common point (and for n=2 it is necessary to require that the planes are not parallel).
1957 Poland - Second Round, 3
Given a cube with edge $ AB = a $ cm. Point $ M $ of segment $ AB $ is distant from the diagonal of the cube, which is oblique to $ AB $, by $ k $ cm. Find the distance of point $ M $ from the midpoint $ S $ of segment $ AB $.
1995 AMC 12/AHSME, 6
The figure shown can be folded into the shape of a cube. In the resulting cube, which of the lettered faces is opposite the face marked $x$?
[asy]
defaultpen(linewidth(0.7));
path p=origin--(0,1)--(1,1)--(1,2)--(2,2)--(2,3);
draw(p^^(2,3)--(4,3)^^shift(2,0)*p^^(2,0)--origin);
draw(shift(1,0)*p, dashed);
label("$x$", (0.3,0.5), E);
label("$A$", (1.3,0.5), E);
label("$B$", (1.3,1.5), E);
label("$C$", (2.3,1.5), E);
label("$D$", (2.3,2.5), E);
label("$E$", (3.3,2.5), E);[/asy]
$
\mathbf{(A)}\; A\qquad
\mathbf{(B)}\; B\qquad
\mathbf{(C)}\; C\qquad
\mathbf{(D)}\; D\qquad
\mathbf{(E)}\; E$
1982 Miklós Schweitzer, 7
Let $ V$ be a bounded, closed, convex set in $ \mathbb{R}^n$, and denote by $ r$ the radius of its circumscribed sphere (that is, the radius of the smallest sphere that contains $ V$). Show that $ r$ is the only real number with the following property: for any finite number of points in $ V$, there exists a point in $ V$ such that the arithmetic mean of its distances from the other points is equal to $ r$.
[i]Gy. Szekeres[/i]
1966 IMO Longlists, 6
Let $m$ be a convex polygon in a plane, $l$ its perimeter and $S$ its area. Let $M\left( R\right) $ be the locus of all points in the space whose distance to $m$ is $\leq R,$ and $V\left(R\right) $ is the volume of the solid $M\left( R\right) .$
[i]a.)[/i] Prove that \[V (R) = \frac 43 \pi R^3 +\frac{\pi}{2} lR^2 +2SR.\]
Hereby, we say that the distance of a point $C$ to a figure $m$ is $\leq R$ if there exists a point $D$ of the figure $m$ such that the distance $CD$ is $\leq R.$ (This point $D$ may lie on the boundary of the figure $m$ and inside the figure.)
additional question:
[i]b.)[/i] Find the area of the planar $R$-neighborhood of a convex or non-convex polygon $m.$
[i]c.)[/i] Find the volume of the $R$-neighborhood of a convex polyhedron, e. g. of a cube or of a tetrahedron.
[b]Note by Darij:[/b] I guess that the ''$R$-neighborhood'' of a figure is defined as the locus of all points whose distance to the figure is $\leq R.$
2006 Purple Comet Problems, 15
A concrete sewer pipe fitting is shaped like a cylinder with diameter $48$ with a cone on top. A cylindrical hole of diameter $30$ is bored all the way through the center of the fitting as shown. The cylindrical portion has height $60$ while the conical top portion has height $20$. Find $N$ such that the volume of the concrete is $N \pi$.
[asy]
import three;
size(250);
defaultpen(linewidth(0.7)+fontsize(10)); pen dashes = linewidth(0.7) + linetype("2 2");
currentprojection = orthographic(0,-15,5);
draw(circle((0,0,0), 15),dashes);
draw(circle((0,0,80), 15));
draw(scale3(24)*((-1,0,0)..(0,-1,0)..(1,0,0)));
draw(shift((0,0,60))*scale3(24)*((-1,0,0)..(0,-1,0)..(1,0,0)));
draw((-24,0,0)--(-24,0,60)--(-15,0,80)); draw((24,0,0)--(24,0,60)--(15,0,80));
draw((-15,0,0)--(-15,0,80),dashes); draw((15,0,0)--(15,0,80),dashes);
draw("48", (-24,0,-20)--(24,0,-20));
draw((-15,0,-20)--(-15,0,-17)); draw((15,0,-20)--(15,0,-17));
label("30", (0,0,-15));
draw("60", (50,0,0)--(50,0,60));
draw("20", (50,0,60)--(50,0,80));
draw((50,0,60)--(47,0,60));[/asy]
2006 Spain Mathematical Olympiad, 2
The dimensions of a wooden octahedron are natural numbers. We painted all its surface (the six faces), cut it by planes parallel to the cubed faces of an edge unit and observed that exactly half of the cubes did not have any painted faces. Prove that the number of octahedra with such property is finite.
(It may be useful to keep in mind that $\sqrt[3]{\frac{1}{2}}=1,79 ... <1,8$).
[hide=original wording] Las dimensiones de un ortoedro de madera son enteras. Pintamos toda su superficie (las seis caras), lo cortamos mediante planos paralelos a las caras en cubos de una unidad de arista y observamos que exactamente la mitad de los cubos no tienen ninguna cara pintada. Probar que el número de ortoedros con tal propiedad es finito[/hide]
2011 Polish MO Finals, 2
In a tetrahedron $ABCD$, the four altitudes are concurrent at $H$. The line $DH$ intersects the plane $ABC$ at $P$ and the circumsphere of $ABCD$ at $Q\neq D$. Prove that $PQ=2HP$.
1992 Vietnam National Olympiad, 1
Let $ABCD$ be a tetrahedron satisfying
i)$\widehat{ACD}+\widehat{BCD}=180^{0}$, and
ii)$\widehat{BAC}+\widehat{CAD}+\widehat{DAB}=\widehat{ABC}+\widehat{CBD}+\widehat{DBA}=180^{0}$.
Find value of $[ABC]+[BCD]+[CDA]+[DAB]$ if we know $AC+CB=k$ and $\widehat{ACB}=\alpha$.
2023 CCA Math Bonanza, T2
How many ways are there to fill an $8\times8\times8$ cube with $1\times1\times8$ sticks? Rotations and reflections are considered distinct.
[i]Team #2[/i]
1967 IMO Shortlist, 2
Is it possible to find a set of $100$ (or $200$) points on the boundary of a cube such that this set remains fixed under all rotations which leave the cube fixed ?
1976 Bulgaria National Olympiad, Problem 3
In the space is given a tetrahedron with length of the edge $2$. Prove that distances from some point $M$ to all of the vertices of the tetrahedron are integer numbers if and only if $M$ is a vertex of tetrahedron.
[i]J. Tabov[/i]
2006 Romania National Olympiad, 1
We consider a prism with 6 faces, 5 of which are circumscriptible quadrilaterals. Prove that all the faces of the prism are circumscriptible quadrilaterals.
1985 Traian Lălescu, 1.3
We have a parallelepiped $ ABCDA'B'C'D' $ in which the top ($ A'B'C'D' $) and the ground ($ ABCD $) are connected by four vertical edges, and $ \angle DAB=30^{\circ} . $ Through $ AB, $ a plane inersects the parallelepiped at an angle of $ 30 $ with respect to the ground, delimiting two interior sections. Find the area of these interior sections in function of the length of $ AA'. $
1979 Canada National Olympiad, 2
It is known in Euclidean geometry that the sum of the angles of a triangle is constant. Prove, however, that the sum of the dihedral angles of a tetrahedron is not constant.
1970 Vietnam National Olympiad, 5
A plane $p$ passes through a vertex of a cube so that the three edges at the vertex make equal angles with $p$. Find the cosine of this angle. Find the positions of the feet of the perpendiculars from the vertices of the cube onto $p$. There are 28 lines through two vertices of the cube and 20 planes through three vertices of the cube. Find some relationship between these lines and planes and the plane $p$.
1990 AIME Problems, 14
The rectangle $ABCD$ below has dimensions $AB = 12 \sqrt{3}$ and $BC = 13 \sqrt{3}$. Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at $P$. If triangle $ABP$ is cut out and removed, edges $\overline{AP}$ and $\overline{BP}$ are joined, and the figure is then creased along segments $\overline{CP}$ and $\overline{DP}$, we obtain a triangular pyramid, all four of whose faces are isosceles triangles. Find the volume of this pyramid.
[asy]
pair D=origin, A=(13,0), B=(13,12), C=(0,12), P=(6.5, 6);
draw(B--C--P--D--C^^D--A);
filldraw(A--P--B--cycle, gray, black);
label("$A$", A, SE);
label("$B$", B, NE);
label("$C$", C, NW);
label("$D$", D, SW);
label("$P$", P, N);
label("$13\sqrt{3}$", A--D, S);
label("$12\sqrt{3}$", A--B, E);[/asy]
2002 Romania National Olympiad, 4
The right prism $[A_1A_2A_3\ldots A_nA_1'A_2'A_3'\ldots A_n'],n\in\mathbb{N},n\ge 3$, has a convex polygon as its base. It is known that $A_1A_2'\perp A_2A_3',A_2A_3'\perp A_3A_4',$$\ldots A_{n-1}A_n'\perp A_nA_1', A_nA_1'\perp A_1A_2'$. Show that:
$a)$ $n=3$;
$b)$ the prism is regular.
1972 Spain Mathematical Olympiad, 3
Given a regular hexagonal prism. Find a polygonal line that, starting from a vertex of the base, runs through all the lateral faces and ends at the vertex of the face top, located on the same edge as the starting vertex, and has a minimum length.