Found problems: 2265
PEN H Problems, 33
Does there exist an integer such that its cube is equal to $3n^2 +3n+7$, where $n$ is integer?
2006 Tournament of Towns, 5
Can a regular octahedron be inscribed in a cube in such a way that all vertices of the octahedron are on cube's edges? (4)
2022 HMIC, 2
Does there exist a regular pentagon whose vertices lie on the edges of a cube?
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]
1990 Bulgaria National Olympiad, Problem 6
The base $ABC$ of a tetrahedron $MABC$ is an equilateral triangle, and the lateral edges $MA,MB,MC$ are sides of a triangle of the area $S$. If $R$ is the circumradius and $V$ the volume of the tetrahedron, prove that $RS\ge2V$. When does equality hold?
1985 AMC 12/AHSME, 20
A wooden cube with edge length $ n$ units (where $ n$ is an integer $ >2$) is painted black all over. By slices parallel to its faces, the cube is cut into $ n^3$ smaller cubes each of unit length. If the number of smaller cubes with just one face painted black is equal to the number of smaller cubes completely free of paint, what is $ n$?
$ \textbf{(A)}\ 5 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ \text{none of these}$
2012 239 Open Mathematical Olympiad, 8
We call a tetrahedron divisor of a parallelepiped if the parallelepiped can be divided into $6$ copies of that tetrahedron. Does there exist a parallelepiped that it has at least two different divisor tetrahedrons?
1983 USAMO, 4
Six segments $S_1, S_2, S_3, S_4, S_5,$ and $S_6$ are given in a plane. These are congruent to the edges $AB, AC, AD, BC, BD,$ and $CD$, respectively, of a tetrahedron $ABCD$. Show how to construct a segment congruent to the altitude of the tetrahedron from vertex $A$ with straight-edge and compasses.
2001 All-Russian Olympiad, 4
A sphere with center on the plane of the face $ABC$ of a tetrahedron $SABC$ passes through $A$, $B$ and $C$, and meets the edges $SA$, $SB$, $SC$ again at $A_1$, $B_1$, $C_1$, respectively. The planes through $A_1$, $B_1$, $C_1$ tangent to the sphere meet at $O$. Prove that $O$ is the circumcenter of the tetrahedron $SA_1B_1C_1$.
1992 Tournament Of Towns, (357) 6
Consider a polyhedron having $100$ edges.
(a) Find the maximal possible number of its edges which can be intersected by a plane (not containing any vertices of the polyhedron) if the polyhedron is convex.
(b) Prove that for a non-convex polyhedron this number
i. can be as great as $96$,
ii. cannot be as great as $100$.
(A Andjans, Riga
1976 USAMO, 4
If the sum of the lengths of the six edges of a trirectangular tetrahedron $ PABC$ (i.e., $ \angle APB \equal{} \angle BPC \equal{} \angle CPA \equal{} 90^\circ$) is $ S$, determine its maximum volume.
1966 IMO Longlists, 57
Is it possible to choose a set of $100$ (or $200$) points on the boundary of a cube such that this set is fixed under each isometry of the cube into itself? Justify your answer.
2014 NIMO Summer Contest, 3
A square and equilateral triangle have the same perimeter. If the triangle has area $16\sqrt3$, what is the area of the square?
[i]Proposed by Evan Chen[/i]
2008 AMC 10, 21
A cube with side length $ 1$ is sliced by a plane that passes through two diagonally opposite vertices $ A$ and $ C$ and the midpoints $ B$ and $ D$ of two opposite edges not containing $ A$ and $ C$, ac shown. What is the area of quadrilateral $ ABCD$?
[asy]import three;
size(200);
defaultpen(fontsize(8)+linewidth(0.7));
currentprojection=obliqueX;
dotfactor=4;
draw((0.5,0,0)--(0,0,0)--(0,0,1)--(0,0,0)--(0,1,0),linetype("4 4"));
draw((0.5,0,1)--(0,0,1)--(0,1,1)--(0.5,1,1)--(0.5,0,1)--(0.5,0,0)--(0.5,1,0)--(0.5,1,1));
draw((0.5,1,0)--(0,1,0)--(0,1,1));
dot((0.5,0,0));
label("$A$",(0.5,0,0),WSW);
dot((0,1,1));
label("$C$",(0,1,1),NE);
dot((0.5,1,0.5));
label("$D$",(0.5,1,0.5),ESE);
dot((0,0,0.5));
label("$B$",(0,0,0.5),NW);[/asy]$ \textbf{(A)}\ \frac {\sqrt6}{2} \qquad \textbf{(B)}\ \frac {5}{4} \qquad \textbf{(C)}\ \sqrt2 \qquad \textbf{(D)}\ \frac {3}{2} \qquad \textbf{(E)}\ \sqrt3$
2008 Harvard-MIT Mathematics Tournament, 1
A $ 3\times3\times3$ cube composed of $ 27$ unit cubes rests on a horizontal plane. Determine the number of ways of selecting two distinct unit cubes from a $ 3\times3\times1$ block (the order is irrelevant) with the property that the line joining the centers of the two cubes makes a $ 45^\circ$ angle with the horizontal plane.
1997 Turkey MO (2nd round), 3
Let $D_{1}, D_{2}, . . . , D_{n}$ be rectangular parallelepipeds in space, with edges parallel to the $x, y, z$ axes. For each $D_{i}$, let $x_{i} , y_{i} , z_{i}$ be the lengths of its projections onto the $x, y, z$ axes, respectively. Suppose that for all pairs $D_{i}$ , $D_{j}$, if at least one of $x_{i} < x_{j}$ , $y_{i} < y_{j}$, $z_{i} < z_{j}$ holds, then $x_{i} \leq x_{j}$ , $y_{i} \leq y_{j}$, and $z_{i} < z_{j}$ . If the volume of the region $\bigcup^{n}_{i=1}{D_{i}}$ equals 1997, prove that there is a subset $\{D_{i_{1}}, D_{i_{2}}, . . . , D_{i_{m}}\}$ of the set $\{D_{1}, . . . , D_{n}\}$ such that
$(i)$ if $k \not= l $ then $D_{i_{k}} \cap D_{i_{l}} = \emptyset $, and
$(ii)$ the volume of $\bigcup^{m}_{k=1}{D_{i_{k}}}$ is at least 73.
2006 Tournament of Towns, 4
Is it possible to split a prism into disjoint set of pyramids so that each pyramid has its base on one base of the prism, while its vertex on another base of the prism ? (6)
2013 IPhOO, 8
A right-triangulated prism made of benzene sits on a table. The hypotenuse makes an angle of $30^\circ$ with the horizontal table. An incoming ray of light hits the hypotenuse horizontally, and leaves the prism from the vertical leg at an acute angle of $ \gamma $ with respect to the vertical leg. Find $\gamma$, in degrees, to the nearest integer. The index of refraction of benzene is $1.50$.
[i](Proposed by Ahaan Rungta)[/i]
2013 Stanford Mathematics Tournament, 15
Suppose we climb a mountain that is a cone with radius $100$ and height $4$. We start at the bottom of the mountain (on the perimeter of the base of the cone), and our destination is the opposite side of the mountain, halfway up (height $z = 2$). Our climbing speed starts at $v_0=2$ but gets slower at a rate inversely proportional to the distance to the mountain top (so at height $z$ the speed $v$ is $(h-z)v_0/h$). Find the minimum time needed to get to the destination.
1998 All-Russian Olympiad, 7
A tetrahedron $ABCD$ has all edges of length less than $100$, and contains two nonintersecting spheres of diameter $1$. Prove that it contains a sphere of diameter $1.01$.
1986 ITAMO, 5
Given an acute triangle $T$ with sides $a,b,c$, find the tetrahedra with base $T$ whose all faces are acute triangles of the same area.
2012 Online Math Open Problems, 18
The sum of the squares of three positive numbers is $160$. One of the numbers is equal to the sum of the other two. The difference between the smaller two numbers is $4.$ What is the difference between the cubes of the smaller two numbers?
[i]Author: Ray Li[/i]
[hide="Clarification"]The problem should ask for the positive difference.[/hide]
2021 Abels Math Contest (Norwegian MO) Final, 4a
A tetrahedron $ABCD$ satisfies $\angle BAC=\angle CAD=\angle DAB=90^o$. Show that the areas of its faces satisfy the equation $area(BAC)^2 + area(CAD)^2 + area(DAB)^2 = area(BCD)^2$.
.
1986 Austrian-Polish Competition, 6
Let $M$ be the set of all tetrahedra whose inscribed and circumscribed spheres are concentric. If the radii of these spheres are denoted by $r$ and $R$ respectively, find the possible values of $R/r$ over all tetrahedra from $M$ .
2004 AMC 12/AHSME, 22
Three mutually tangent spheres of radius $ 1$ rest on a horizontal plane. A sphere of radius $ 2$ rests on them. What is the distance from the plane to the top of the larger sphere?
$ \textbf{(A)}\ 3 \plus{} \frac {\sqrt {30}}{2} \qquad \textbf{(B)}\ 3 \plus{} \frac {\sqrt {69}}{3} \qquad \textbf{(C)}\ 3 \plus{} \frac {\sqrt {123}}{4}\qquad \textbf{(D)}\ \frac {52}{9}\qquad \textbf{(E)}\ 3 \plus{} 2\sqrt2$