Found problems: 2265
2016 HMNT, 3
Let $V$ be a rectangular prism with integer side lengths. The largest face has area $240$ and the smallest face has area $48$. A third face has area $x$, where $x$ is not equal to $48$ or $240$. What is the sum of all possible values of $x$?
2010 Oral Moscow Geometry Olympiad, 5
All edges of a regular right pyramid are equal to $1$, and all vertices lie on the side surface of a (infinite) right circular cylinder of radius $R$. Find all possible values of $R$.
1985 Bundeswettbewerb Mathematik, 2
The insphere of any tetrahedron has radius $r$. The four tangential planes parallel to the side faces of the tetrahedron cut from the tetrahedron four smaller tetrahedrons whose in-sphere radii are $r_1, r_2, r_3$ and $r_4$. Prove that $$r_1 + r_2 + r_3 + r_4 = 2r$$
2008 AMC 12/AHSME, 8
What is the volume of a cube whose surface area is twice that of a cube with volume $ 1$?
$ \textbf{(A)}\ \sqrt{2} \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ 2\sqrt{2} \qquad
\textbf{(D)}\ 4 \qquad
\textbf{(E)}\ 8$
2014 Polish MO Finals, 3
A tetrahedron $ABCD$ with acute-angled faces is inscribed in a sphere with center $O$. A line passing through $O$ perpendicular to plane $ABC$ crosses the sphere at point $D'$ that lies on the opposide side of plane $ABC$ than point $D$. Line $DD'$ crosses plane $ABC$ in point $P$ that lies inside the triangle $ABC$. Prove, that if $\angle APB=2\angle ACB$, then $\angle ADD'=\angle BDD'$.
2002 Portugal MO, 2
Consider five spheres with radius $10$ cm . Four of these spheres are arranged on a horizontal table so that its centers form a $20$ cm square and the fifth sphere is placed on them so that it touches the other four. What is the distance between center of this fifth sphere and the table?
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]
2023 Junior Balkan Team Selection Tests - Romania, P4
Given is a cube $3 \times 3 \times 3$ with $27$ unit cubes. In each such cube a positive integer is written. Call a $\textit {strip}$ a block $1 \times 1 \times 3$ of $3$ cubes. The numbers are written so that for each cube, its number is the sum of three other numbers, one from each of the three strips it is in. Prove that there are at least $16$ numbers that are at most $60$.
1985 ITAMO, 2
When a right triangle is rotated about one leg, the volume of the cone produced is $800 \pi$ $\text{cm}^3$. When the triangle is rotated about the other leg, the volume of the cone produced is $1920 \pi$ $\text{cm}^3$. What is the length (in cm) of the hypotenuse of the triangle?
1969 IMO Shortlist, 39
$(HUN 6)$ Find the positions of three points $A,B,C$ on the boundary of a unit cube such that $min\{AB,AC,BC\}$ is the greatest possible.
2009 AMC 10, 11
One dimension of a cube is increased by $ 1$, another is decreased by $ 1$, and the third is left unchanged. The volume of the new rectangular solid is $ 5$ less than that of the cube. What was the volume of the cube?
$ \textbf{(A)}\ 8 \qquad
\textbf{(B)}\ 27 \qquad
\textbf{(C)}\ 64 \qquad
\textbf{(D)}\ 125 \qquad
\textbf{(E)}\ 216$
1992 Romania Team Selection Test, 10
In a tetrahedron $VABC$, let $I$ be the incenter and $A',B',C'$ be arbitrary points on the edges $AV,BV,CV$, and let $S_a,S_b,S_c,S_v$ be the areas of triangles $VBC,VAC,VAB,ABC$, respectively. Show that points $A',B',C',I$ are coplanar if and only if $\frac{AA'}{A'V}S_a +\frac{BB'}{B'V}S_b +\frac{CC'}{C'V}S_c = S_v$
1983 AIME Problems, 11
The solid shown has a square base of side length $s$. The upper edge is parallel to the base and has length $2s$. All other edges have length $s$. Given that $s = 6 \sqrt{2}$, what is the volume of the solid?
[asy]
import three;
size(170);
pathpen = black+linewidth(0.65);
pointpen = black;
currentprojection = perspective(30,-20,10);
real s = 6 * 2^.5;
triple A=(0,0,0),B=(s,0,0),C=(s,s,0),D=(0,s,0),E=(-s/2,s/2,6),F=(3*s/2,s/2,6);
draw(F--B--C--F--E--A--B);
draw(A--D--E, dashed);
draw(D--C, dashed);
label("$2s$", (s/2, s/2, 6), N);
label("$s$", (s/2, 0, 0), SW);
[/asy]
Oliforum Contest IV 2013, 6
Let $P$ be a polyhedron whose faces are colored black and white so that there are more black faces and no two black faces are adjacent. Show that $P$ is not circumscribed about a sphere.
1994 Moldova Team Selection Test, 9
Let $O{}$ be the center of the circumscribed sphere of the tetrahedron $ABCD$. Let $L,M,N$ respectively be the midpoints of the segments $BC,CA,AB$. It is known that $AB+BC=AD+CD$, $BC+CA=BD+AD$, $CA+AB=CD+BD$. Prove that $\angle LOM=\angle MON=\angle NOL$. Find their value.
1989 IMO Shortlist, 21
Prove that the intersection of a plane and a regular tetrahedron can be an obtuse-angled triangle and that the obtuse angle in any such triangle is always smaller than $ 120^{\circ}.$
Indonesia Regional MO OSP SMA - geometry, 2003.3
The points $P$ and $Q$ are the midpoints of the edges $AE$ and $CG$ on the cube $ABCD.EFGH$ respectively. If the length of the cube edges is $1$ unit, determine the area of the quadrilateral $DPFQ$ .
2016 All-Russian Olympiad, 4
There is three-dimensional space. For every integer $n$ we build planes $ x \pm y\pm z = n$. All space is divided on octahedrons and tetrahedrons.
Point $(x_0,y_0,z_0)$ has rational coordinates but not lies on any plane. Prove, that there is such natural $k$ , that point $(kx_0,ky_0,kz_0)$ lies strictly inside the octahedron of partition.
1994 Denmark MO - Mohr Contest, 1
A wine glass with a cross section as shown has the property of an orange in shape as a sphere with a radius of $3$ cm just can be placed in the glass without protruding above glass. Determine the height $h$ of the glass.
[img]https://1.bp.blogspot.com/-IuLm_IPTvTs/XzcH4FAjq5I/AAAAAAAAMYY/qMi4ng91us8XsFUtnwS-hb6PqLwAON_jwCLcBGAsYHQ/s0/1994%2BMohr%2Bp1.png[/img]
1996 IberoAmerican, 1
Let $ n$ be a natural number. A cube of edge $ n$ may be divided in 1996 cubes whose edges length are also natural numbers. Find the minimum possible value for $ n$.
1974 IMO Longlists, 17
Show that there exists a set $S$ of $15$ distinct circles on the surface of a sphere, all having the same radius and such that $5$ touch exactly $5$ others, $5$ touch exactly $4$ others, and $5$ touch exactly $3$ others.
[i][General Problem: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=46&t=384764][/i]
2013 Tuymaada Olympiad, 5
Each face of a $7 \times 7 \times 7$ cube is divided into unit squares. What is the maximum number of squares that can be chosen so that no two chosen squares have a common point?
[i]A. Chukhnov[/i]
1987 IMO Longlists, 4
Let $a_1, a_2, a_3, b_1, b_2, b_3$ be positive real numbers. Prove that
\[(a_1b_2 + a_2b_1 + a_1b_3 + a_3b_1 + a_2b_3 + a_3b_2)^2 \geq 4(a_1a_2 + a_2a_3 + a_3a_1)(b_1b_2 + b_2b_3 + b_3b_1)\]
and show that the two sides of the inequality are equal if and only if $\frac{a_1}{b_1} = \frac{a_2}{b_2} = \frac{a_3}{b_3}.$
1999 VJIMC, Problem 1
Find the minimal $k$ such that every set of $k$ different lines in $\mathbb R^3$ contains either $3$ mutually parallel lines or $3$ mutually intersecting lines or $3$ mutually skew lines.
1966 IMO Shortlist, 7
For which arrangements of two infinite circular cylinders does their intersection lie in a plane?