Found problems: 2265
2001 Moldova Team Selection Test, 2
Let $A_i$ and $A_i^{'}$ $(i=1,2,3,4)$ be diametrically opposite vertexes of a rectangular cuboid and $M{}$ a point inside it. Prove that $S\leq\sum_{i=1}^{4}MA_i\cdot MA_i^{'}$, where $S{}$ is the total surface area of the rectangular cuboid.
2007 Estonia National Olympiad, 1
Consider a cylinder and a cone with a common base such that the volume of the
part of the cylinder enclosed in the cone equals the volume of the part of the cylinder outside the cone. Find the ratio of the height of the cone to the height of the cylinder.
2012 Tournament of Towns, 6
(a) A point $A$ is marked inside a sphere. Three perpendicular lines drawn through $A$ intersect the sphere at six points. Prove that the centre of gravity of these six points does not depend on the choice of such three lines.
(b) An icosahedron with the centre $A$ is placed inside a sphere (its centre does not necessarily coincide with the centre of the sphere). The rays going from $A$ to the vertices of the icosahedron mark $12$ points on the sphere. Then the icosahedron is rotated about its centre. New rays mark new $12$ points on the sphere. Let $O$ and $N$ be the centres of mass of old and new points respectively. Prove that $O = N$.
2013 ELMO Shortlist, 2
For what polynomials $P(n)$ with integer coefficients can a positive integer be assigned to every lattice point in $\mathbb{R}^3$ so that for every integer $n \ge 1$, the sum of the $n^3$ integers assigned to any $n \times n \times n$ grid of lattice points is divisible by $P(n)$?
[i]Proposed by Andre Arslan[/i]
2023 AMC 12/AHSME, 13
A rectangular box $\mathcal{P}$ has distinct edge lengths $a, b,$ and $c$. The sum of the lengths of all $12$ edges of $\mathcal{P}$ is $13$, the sum of the areas of all $6$ faces of $\mathcal{P}$ is $\frac{11}{2}$, and the volume of $\mathcal{P}$ is $\frac{1}{2}$. What is the length of the longest interior diagonal connecting two vertices of $\mathcal{P}$?
$\textbf{(A)}~2\qquad\textbf{(B)}~\frac{3}{8}\qquad\textbf{(C)}~\frac{9}{8}\qquad\textbf{(D)}~\frac{9}{4}\qquad\textbf{(E)}~\frac{3}{2}$
2005 Tuymaada Olympiad, 4
In a triangle $ABC$, let $A_{1}$, $B_{1}$, $C_{1}$ be the points where the excircles touch the sides $BC$, $CA$ and $AB$ respectively. Prove that $A A_{1}$, $B B_{1}$ and $C C_{1}$ are the sidelenghts of a triangle.
[i]Proposed by L. Emelyanov[/i]
1985 IMO Longlists, 9
A polyhedron has $12$ faces and is such that:
[b][i](i)[/i][/b] all faces are isosceles triangles,
[b][i](ii)[/i][/b] all edges have length either $x$ or $y$,
[b][i](iii)[/i][/b] at each vertex either $3$ or $6$ edges meet, and
[b][i](iv)[/i][/b] all dihedral angles are equal.
Find the ratio $x/y.$
2019 District Olympiad, 2
Let $ABCDA'B'C'D'$ be a rectangular parallelepiped and $M,N, P$ projections of points $A, C$ and $B'$ respectively on the diagonal $BD'$.
a) Prove that $BM + BN + BP = BD'$.
b) Prove that $3 (AM^2 + B'P^2 + CN^2)\ge 2D'B^2$ if and only if $ABCDA'B'C'D'$ is a cube.
2001 Tournament Of Towns, 5
Nine points are drawn on the surface of a regular tetrahedron with an edge of $1$ cm. Prove that among these points there are two located at a distance (in space) no greater than $0.5$ cm.
1972 USAMO, 2
A given tetrahedron $ ABCD$ is isoceles, that is, $ AB\equal{}CD$, $ AC\equal{}BD$, $ AD\equal{}BC$. Show that the faces of the tetrahedron are acute-angled triangles.
2019 Flanders Math Olympiad, 1
Two touching balls with radii $a$ and $b$ are enclosed in a cylindrical tin of diameter $d$ . Both balls hit the top surface and the shell of the cylinder. The largest ball also hits the bottom surface. Show that $\sqrt{d} =\sqrt{a} +\sqrt{b}$
[img]https://1.bp.blogspot.com/-O4B3P3bghFs/Xy1fDv9zGkI/AAAAAAAAMSQ/ePLVnsXsRi0mz3SWBpIzfGdsizWoLmGVACLcBGAsYHQ/s0/flanders%2B2019%2Bp1.png[/img]
2010 Saint Petersburg Mathematical Olympiad, 5
$SABCD$ is quadrangular pyramid. Lateral faces are acute triangles with orthocenters lying in one plane. $ABCD$ is base of pyramid and $AC$ and $BD$ intersects at $P$, where $SP$ is height of pyramid. Prove that $AC \perp BD$
2020 Jozsef Wildt International Math Competition, W25
In the Crelle $[ABCD]$ tetrahedron, we note with $A',B',C',A'',B'',C''$ the tangent points of the hexatangent sphere $\varphi(J,\rho)$, associated with the tetrahedron, with the edges $|BC|,|CA|,|AB|,|DA|,|DB|,|DC|$. Show that these inequalities occur:
a)
$$2\sqrt3R\ge6\rho\ge A'A''+B'B''+C'C''\ge6\sqrt3r$$
b)
$$4R^2\ge12\rho^2\ge(A'A'')^2+(B'B'')^2+(C'C'')^2\ge36r^2$$
c)
$$\frac{8R^3}{3\sqrt3}\ge8\rho^3\ge A'A''\cdot B'B''\cdot C'C''\ge24\sqrt3r^3$$
where $r,R$ is the length of the radius of the sphere inscribed and respectively circumscribed to the tetrahedron.
[i]Proposed by Marius Olteanu[/i]
2023 CCA Math Bonanza, TB2
How many ways are there to color a tetrahedron’s faces, edges, and vertices in red, green, and blue so that no face shares a color with any of its edges, and no edge shares a color with any of its endpoints? (Rotations and reflections are considered distinct.)
[i]Tiebreaker #2[/i]
1992 Romania Team Selection Test, 3
Let $ABCD$ be a tetrahedron; $B', C', D'$ be the midpoints of the edges $AB, AC, AD$; $G_A, G_B, G_C, G_D$ be the barycentres of the triangles $BCD, ACD, ABD, ABC$, and $G$ be the barycentre of the tetrahedron. Show that $A, G, G_B, G_C, G_D$ are all on a sphere if and only if $A, G, B', C', D'$ are also on a sphere.
[i]Dan Brânzei[/i]
2017 Swedish Mathematical Competition, 3
Given the segments $AB$ and $CD$ not necessarily on the same plane. Point $X$ is the midpoint of the segment $AB$, and the point $Y$ is the midpoint of $CD$. Given that point $X$ is not on line $CD$, and that point $Y$ is not on line $AB$, prove that $2 | XY | \le | AD | + | BC |$. When is equality achieved?
2001 AMC 12/AHSME, 8
Which of the cones listed below can be formed from a $ 252^\circ$ sector of a circle of radius $ 10$ by aligning the two straight sides?
[asy]import graph;unitsize(1.5cm);defaultpen(fontsize(8pt));draw(Arc((0,0),1,-72,180),linewidth(.8pt));draw(dir(288)--(0,0)--(-1,0),linewidth(.8pt));label("$10$",(-0.5,0),S);draw(Arc((0,0),0.1,-72,180));label("$252^{\circ}$",(0.05,0.05),NE);[/asy]
[asy]
import three;
picture mainframe;
defaultpen(fontsize(11pt));
picture conePic(picture pic, real r, real h, real sh)
{
size(pic, 3cm);
triple eye = (11, 0, 5);
currentprojection = perspective(eye);
real R = 1, y = 2;
triple center = (0, 0, 0);
triple radPt = (0, R, 0);
triple negRadPt = (0, -R, 0);
triple heightPt = (0, 0, y);
draw(pic, arc(center, radPt, negRadPt, heightPt, CW));
draw(pic, arc(center, radPt, negRadPt, heightPt, CCW), linetype("8 8"));
draw(pic, center--radPt, linetype("8 8"));
draw(pic, center--heightPt, linetype("8 8"));
draw(pic, negRadPt--heightPt--radPt);
label(pic, (string) r, center--radPt, dir(270));
if (h != 0)
{
label(pic, (string) h, heightPt--center, dir(0));
}
if (sh != 0)
{
label(pic, (string) sh, heightPt--radPt, dir(0));
}
return pic;
}
picture pic1;
pic1 = conePic(pic1, 6, 0, 10);
picture pic2;
pic2 = conePic(pic2, 6, 10, 0);
picture pic3;
pic3 = conePic(pic3, 7, 0, 10);
picture pic4;
pic4 = conePic(pic4, 7, 10, 0);
picture pic5;
pic5 = conePic(pic5, 8, 0, 10);
picture aux1; picture aux2; picture aux3;
add(aux1, pic1.fit(), (0,0), W);
label(aux1, "$\textbf{(A)}$", (0,0), 22W, linewidth(4));
label(aux1, "$\textbf{(B)}$", (0,0), 3E);
add(aux1, pic2.fit(), (0,0), 35E);
add(aux2, aux1.fit(), (0,0), W);
label(aux2, "$\textbf{(C)}$", (0,0), 3E);
add(aux2, pic3.fit(), (0,0), 35E);
add(aux3, aux2.fit(), (0,0), W);
label(aux3, "$\textbf{(D)}$", (0,0), 3E);
add(aux3, pic4.fit(), (0,0), 35E);
add(mainframe, aux3.fit(), (0,0), W);
label(mainframe, "$\textbf{(E)}$", (0,0), 3E);
add(mainframe, pic5.fit(), (0,0), 35E);
add(mainframe.fit(), (0,0), N);
[/asy]
2004 Indonesia MO, 1
Determine the number of positive odd and even factor of $ 5^6\minus{}1$.
1968 German National Olympiad, 2
Which of all planes, the one and the same body diagonal of a cube with the edge length $a$, cuts out a cut figure with the smallest area from the cube? Calculate the area of such a cut figure.
[hide=original wording]Welche von allen Ebenen, die eine und dieselbe Korperdiagonale eines Wurfels mit der Kantenlange a enthalten, schneiden aus den W¨urfel eine Schnittfigur kleinsten Flacheninhaltes heraus? Berechnen
Sie den Fl¨acheninhalt solch einer Schnittfigur![/hide]
1972 IMO Shortlist, 7
Given four distinct parallel planes, prove that there exists a regular tetrahedron with a vertex on each plane.
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. )
2012 Today's Calculation Of Integral, 780
Let $n\geq 3$ be integer. Given a regular $n$-polygon $P$ with side length 4 on the plane $z=0$ in the $xyz$-space.Llet $G$ be a circumcenter of $P$. When the center of the sphere $B$ with radius 1 travels round along the sides of $P$, denote by $K_n$ the solid swept by $B$.
Answer the following questions.
(1) Take two adjacent vertices $P_1,\ P_2$ of $P$. Let $Q$ be the intersection point between the perpendicular dawn from $G$ to $P_1P_2$,
prove that $GQ>1$.
(2) (i) Express the area of cross section $S(t)$ in terms of $t,\ n$ when $K_n$ is cut by the plane $z=t\ (-1\leq t\leq 1)$.
(ii) Express the volume $V(n)$ of $K_n$ in terms of $n$.
(3) Denote by $l$ the line which passes through $G$ and perpendicular to the plane $z=0$. Express the volume $W(n)$ of the solid by generated by a rotation of $K_n$ around $l$ in terms of $n$.
(4) Find $\lim_{n\to\infty} \frac{V(n)}{W(n)} .$
1996 Romania National Olympiad, 4
a) Let $AB CD$ be a regular tetrahedron. On the sides $AB$, $AC$ and $AD$, the points $M$, $N$ and $P$, are considered. Determine the volume of the tetrahedron $AMNP$ in terms of $x, y, z$, where $x=AM$, $y=AN$, $z=AP$.
b) Show that for any real numbers $x, y, z, t, u, v \in (0, 1)$ :
$$xyz + uv(1- x) + (1- y)(1- v)t + (1- z)(1- w)(1- t) < 1.$$
1992 IMO Longlists, 40
The colonizers of a spherical planet have decided to build $N$ towns, each having area $1/1000$ of the total area of the planet. They also decided that any two points belonging to different towns will have different latitude and different longitude. What is the maximal value of $N$?
2006 Sharygin Geometry Olympiad, 25
In the tetrahedron $ABCD$ , the dihedral angles at the $BC, CD$, and $DA$ edges are equal to $\alpha$, and for the remaining edges equal to $\beta$. Find the ratio $AB / CD$.