Found problems: 2265
2013 Online Math Open Problems, 44
Suppose tetrahedron $PABC$ has volume $420$ and satisfies $AB = 13$, $BC = 14$, and $CA = 15$. The minimum possible surface area of $PABC$ can be written as $m+n\sqrt{k}$, where $m,n,k$ are positive integers and $k$ is not divisible by the square of any prime. Compute $m+n+k$.
[i]Ray Li[/i]
1985 ITAMO, 15
Three 12 cm $\times$ 12 cm squares are each cut into two pieces $A$ and $B$, as shown in the first figure below, by joining the midpoints of two adjacent sides. These six pieces are then attached to a regular hexagon, as shown in the second figure, so as to fold into a polyhedron. What is the volume (in $\text{cm}^3$) of this polyhedron?
[asy]
defaultpen(fontsize(10));
size(250);
draw(shift(0, sqrt(3)+1)*scale(2)*rotate(45)*polygon(4));
draw(shift(-sqrt(3)*(sqrt(3)+1)/2, -(sqrt(3)+1)/2)*scale(2)*rotate(165)*polygon(4));
draw(shift(sqrt(3)*(sqrt(3)+1)/2, -(sqrt(3)+1)/2)*scale(2)*rotate(285)*polygon(4));
filldraw(scale(2)*polygon(6), white, black);
pair X=(2,0)+sqrt(2)*dir(75), Y=(-2,0)+sqrt(2)*dir(105), Z=(2*dir(300))+sqrt(2)*dir(225);
pair[] roots={2*dir(0), 2*dir(60), 2*dir(120), 2*dir(180), 2*dir(240), 2*dir(300)};
draw(roots[0]--X--roots[1]);
label("$B$", centroid(roots[0],X,roots[1]));
draw(roots[2]--Y--roots[3]);
label("$B$", centroid(roots[2],Y,roots[3]));
draw(roots[4]--Z--roots[5]);
label("$B$", centroid(roots[4],Z,roots[5]));
label("$A$", (1+sqrt(3))*dir(90));
label("$A$", (1+sqrt(3))*dir(210));
label("$A$", (1+sqrt(3))*dir(330));
draw(shift(-10,0)*scale(2)*polygon(4));
draw((sqrt(2)-10,0)--(-10,sqrt(2)));
label("$A$", (-10,0));
label("$B$", centroid((sqrt(2)-10,0),(-10,sqrt(2)),(sqrt(2)-10, sqrt(2))));[/asy]
I Soros Olympiad 1994-95 (Rus + Ukr), 11.4
The wire is bent in the form of a square with side $2$. Find the volume of the body consisting of all points in space located at a distance not exceeding $1$ from at least one point of the wire.
2019 HMNT, 10
For dessert, Melinda eats a spherical scoop of ice cream with diameter $2$ inches. She prefers to eat her ice cream in cube-like shapes, however. She has a special machine which, given a sphere placed in space, cuts it through the planes $x = n$, $y = n$, and $z = n$ for every integer $n$ (not necessarily positive). Melinda centers the scoop of ice cream uniformly at random inside the cube $0 \le x, y,z \le 1$, and then cuts it into pieces using her machine. What is the expected number of pieces she cuts the ice cream into?
2006 Stanford Mathematics Tournament, 1
Given $ \triangle{ABC}$, where $ A$ is at $ (0,0)$, $ B$ is at $ (20,0)$, and $ C$ is on the positive $ y$-axis. Cone $ M$ is formed when $ \triangle{ABC}$ is rotated about the $ x$-axis, and cone $ N$ is formed when $ \triangle{ABC}$ is rotated about the $ y$-axis. If the volume of cone $ M$ minus the volume of cone $ N$ is $ 140\pi$, find the length of $ \overline{BC}$.
1965 Poland - Second Round, 6
Prove that there is no polyhedron whose every plane section is a triangle.
2000 Mongolian Mathematical Olympiad, Problem 3
A cube of side $n$ is cut into $n^3$ unit cubes, and m of these cubes are marked so that the centers of any three marked cubes do not form a right-angled triangle with legs parallel to sides of the cube. Find the maximum possible value of $m$.
1967 Czech and Slovak Olympiad III A, 2
Let $ABCD$ be a tetrahedron such that \[AB^2+CD^2=AC^2+BD^2=AD^2+BC^2.\] Show that at least one of its faces is an acute triangle.
2013 District Olympiad, 3
Let be the regular hexagonal prism $ABCDEFA'B C'D'E'F'$ with the base edge of $12$ and the height of $12 \sqrt{3}$. We denote by $N$ the middle of the edge $CC'$.
a) Prove that the lines $BF'$ and $ND$ are perpendicular
b) Calculate the distance between the lines $BF'$ and $ND$.
2014 NIMO Problems, 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]
2017 Sharygin Geometry Olympiad, P24
Two tetrahedrons are given. Each two faces of the same tetrahedron are not similar, but each face of the first tetrahedron is similar to some face of the second one. Does this yield that these tetrahedrons are similar?
1968 All Soviet Union Mathematical Olympiad, 104
Three spheres are constructed so that the edges $[AB], [BC], [AD]$ of the tetrahedron $ABCD$ are their respective diameters. Prove that the spheres cover all the tetrahedron.
2003 All-Russian Olympiad, 4
The inscribed sphere of a tetrahedron $ABCD$ touches $ABC,ABD,ACD$ and $BCD$ at $D_1,C_1,B_1$ and $A_1$ respectively. Consider the plane equidistant from $A$ and plane $B_1C_1D_1$ (parallel to $B_1C_1D_1$) and the three planes defined analogously for the vertices $B,C,D$. Prove that the circumcenter of the tetrahedron formed by these four planes coincides with the circumcenter of tetrahedron of $ABCD$.
1967 IMO Shortlist, 5
Faces of a convex polyhedron are six squares and 8 equilateral triangles and each edge is a common side for one triangle and one square. All dihedral angles obtained from the triangle and square with a common edge, are equal. Prove that it is possible to circumscribe a sphere around the polyhedron, and compute the ratio of the squares of volumes of that polyhedron and of the ball whose boundary is the circumscribed sphere.
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]
2010 Princeton University Math Competition, 8
There is a point source of light in an empty universe. What is the minimum number of solid balls (of any size) one must place in space so that any light ray emanating from the light source intersects at least one ball?
2011 India IMO Training Camp, 1
Find all positive integer $n$ satisfying the conditions
$a)n^2=(a+1)^3-a^3$
$b)2n+119$ is a perfect square.
1962 Kurschak Competition, 3
$P$ is any point of the tetrahedron $ABCD$ except $D$. Show that at least one of the three distances $DA$, $DB$, $DC$ exceeds at least one of the distances $PA$, $PB$ and $PC$.
2014 AMC 12/AHSME, 19
A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is twice that of the sphere. What is the ratio of the radius of the bottom base of the truncated cone to the radius of the top base of the truncated cone?
[asy]
real r=(3+sqrt(5))/2;
real s=sqrt(r);
real Brad=r;
real brad=1;
real Fht = 2*s;
import graph3;
import solids;
currentprojection=orthographic(1,0,.2);
currentlight=(10,10,5);
revolution sph=sphere((0,0,Fht/2),Fht/2);
//draw(surface(sph),green+white+opacity(0.5));
//triple f(pair t) {return (t.x*cos(t.y),t.x*sin(t.y),t.x^(1/n)*sin(t.y/n));}
triple f(pair t) {
triple v0 = Brad*(cos(t.x),sin(t.x),0);
triple v1 = brad*(cos(t.x),sin(t.x),0)+(0,0,Fht);
return (v0 + t.y*(v1-v0));
}
triple g(pair t) {
return (t.y*cos(t.x),t.y*sin(t.x),0);
}
surface sback=surface(f,(3pi/4,0),(7pi/4,1),80,2);
surface sfront=surface(f,(7pi/4,0),(11pi/4,1),80,2);
surface base = surface(g,(0,0),(2pi,Brad),80,2);
draw(sback,rgb(0,1,0));
draw(sfront,rgb(.3,1,.3));
draw(base,rgb(.4,1,.4));
draw(surface(sph),rgb(.3,1,.3));
[/asy]
$ \textbf {(A) } \dfrac {3}{2} \qquad \textbf {(B) } \dfrac {1+\sqrt{5}}{2} \qquad \textbf {(C) } \sqrt{3} \qquad \textbf {(D) } 2 \qquad \textbf {(E) } \dfrac {3+\sqrt{5}}{2} $
1999 AIME Problems, 15
Consider the paper triangle whose vertices are $(0,0), (34,0),$ and $(16,24).$ The vertices of its midpoint triangle are the midpoints of its sides. A triangular pyramid is formed by folding the triangle along the sides of its midpoint triangle. What is the volume of this pyramid?
2007 Tournament Of Towns, 7
There are $100$ boxes, each containing either a red cube or a blue cube. Alex has a sum of money initially, and places bets on the colour of the cube in each box in turn. The bet can be anywhere from $0$ up to everything he has at the time. After the bet has been placed, the box is opened. If Alex loses, his bet will be taken away. If he wins, he will get his bet back, plus a sum equal to the bet. Then he moves onto the next box, until he has bet on the last one, or until he runs out of money. What is the maximum factor by which he can guarantee to increase his amount of money, if he knows that the exact number of blue cubes is
[list][b](a)[/b] $1$;
[b](b)[/b] some integer $k$, $1 < k \leq 100$.[/list]
1953 Czech and Slovak Olympiad III A, 4
Consider skew lines $a,b$ and a plane $\rho$ that intersect both of the lines (but does not contain any of them). Choose such points $X\in a,Y\in b$ that $XY\parallel\rho.$ Find the locus of midpoints $M$ of all segments $XY,$ when $X$ moves along line $a$.
1992 Baltic Way, 3
Find an infinite non-constant arithmetic progression of natural numbers such that each term is neither a sum of two squares, nor a sum of two cubes (of natural numbers).
1990 Romania Team Selection Test, 4
Let $M$ be a point on the edge $CD$ of a tetrahedron $ABCD$ such that the tetrahedra $ABCM$ and $ABDM$ have the same total areas. We denote by $\pi_{AB}$ the plane $ABM$. Planes $\pi_{AC},...,\pi_{CD}$ are analogously defined. Prove that the six planes $\pi_{AB},...,\pi_{CD}$ are concurrent in a certain point $N$, and show that $N$ is symmetric to the incenter $I$ with respect to the barycenter $G$.
1991 Arnold's Trivium, 92
Find the orders of the subgroups of the group of rotations of the cube, and find its normal subgroups.