Found problems: 2265
1993 AMC 12/AHSME, 29
Which of the following sets could NOT be the lengths of the external diagonals of a right rectangular prism [a "box"]? (An [i]external diagonal[/i] is a diagonal of one of the rectangular faces of the box.)
$ \textbf{(A)}\ \{4, 5, 6\} \qquad\textbf{(B)}\ \{4, 5, 7\} \qquad\textbf{(C)}\ \{4, 6, 7\} \qquad\textbf{(D)}\ \{5, 6, 7\} \qquad\textbf{(E)}\ \{5, 7, 8\} $
2022 USAMTS Problems, 2
Given a sphere, a great circle of the sphere is a circle on the sphere whose diameter is also a diameter of the sphere. For a given positive integer $n,$ the surface of a sphere is divided into several regions by $n$ great circles, and each region is colored black or white. We say that a coloring is good if any two adjacent regions (that share an arc as boundary, not just a finite number of points) have different colors. Find, with proof, all positive integers $n$ such that in every good coloring with $n$ great circles, the sum of the areas of the black regions is equal to the sum of the areas of the white regions.
1977 IMO Longlists, 9
Let $ABCD$ be a regular tetrahedron and $\mathbf{Z}$ an isometry mapping $A,B,C,D$ into $B,C,D,A$, respectively. Find the set $M$ of all points $X$ of the face $ABC$ whose distance from $\mathbf{Z}(X)$ is equal to a given number $t$. Find necessary and sufficient conditions for the set $M$ to be nonempty.
1985 All Soviet Union Mathematical Olympiad, 417
The $ABCDA_1B_1C_1D_1$ cube has unit length edges. Find the distance between two circumferences, one of those is inscribed into the $ABCD$ base, and another comes through points $A,C$ and $B_1$ .
2003 AIME Problems, 5
Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures 3 by 4 by 5 units. Given that the volume of this set is $(m + n \pi)/p$, where $m$, $n$, and $p$ are positive integers, and $n$ and $p$ are relatively prime, find $m + n + p$.
2001 Croatia National Olympiad, Problem 2
A piece of paper in the shape of a square $FBHD$ with side $a$ is given. Points $G,A$ on $FB$ and $E,C$ on $BH$ are marked so that $FG=GA=AB$ and $BE=EC=CH$. The paper is folded along $DG,DA,DC$ and $AC$ so that $G$ overlaps with $B$, and $F$ and $H$ overlap with $E$. Compute the volume of the obtained tetrahedron $ABCD$.
1981 Spain Mathematical Olympiad, 2
A cylindrical glass beaker is $8$ cm high and its circumference rim is $12$ cm wide . Inside, $3$ cm from the edge, there is a tiny drop of honey. In a point on its outer surface, belonging to the plane passing through the axis of the cylinder and for the drop of honey, and located $1$ cm from the base (or bottom) of the glass, there is a fly.
What is the shortest path that the fly must travel, walking on the surface from the glass, to the drop of honey, and how long is said path?
[hide=original wording]Un vaso de vidrio cil´ındrico tiene 8 cm de altura y su borde 12 cm de circunferencia. En su interior, a 3 cm del borde, hay una diminuta gota de miel. En un punto de su superficie exterior, perteneciente al plano que pasa por el eje del cilindro y por la gota de miel, y situado a 1 cm de la base (o fondo) del vaso, hay una mosca.
¿Cu´al es el camino m´as corto que la mosca debe recorrer, andando sobre la superficie del vaso, hasta la gota de miel, y qu´e longitud tiene dicho camino?[/hide]
2008 Pre-Preparation Course Examination, 3
Prove that we can put $ \Omega(\frac1{\epsilon})$ points on surface of a sphere with radius 1 such that distance of each of these points and the plane passing through center and two of other points is at least $ \epsilon$.
1986 Flanders Math Olympiad, 4
Given a cube in which you can put two massive spheres of radius 1.
What's the smallest possible value of the side - length of the cube?
Prove that your answer is the best possible.
1960 IMO, 6
Consider a cone of revolution with an inscribed sphere tangent to the base of the cone. A cylinder is circumscribed about this sphere so that one of its bases lies in the base of the cone. let $V_1$ be the volume of the cone and $V_2$ be the volume of the cylinder.
a) Prove that $V_1 \neq V_2$;
b) Find the smallest number $k$ for which $V_1=kV_2$; for this case, construct the angle subtended by a diamter of the base of the cone at the vertex of the cone.
2016 SDMO (Middle School), 4
There is an infinitely tall tetrahedral stack of spheres where each row of the tetrahedron consists of a triangular arrangement of spheres, as shown below. There is $1$ sphere in the top row (which we will call row $0$), $3$ spheres in row $1$, $6$ spheres in row $2$, $10$ spheres in row $3$, etc. The top-most sphere in row $0$ is assigned the number $1$. We then assign each other sphere the sum of the number(s) assigned to the sphere(s) which touch it in the row directly above it. Find a simplified expression in terms of $n$ for the sum of the numbers assigned to each sphere from row $0$ to row $n$.
[asy]
import three;
import solids;
size(8cm);
//currentprojection = perspective(1, 1, 10);
triple backright = (-2, 0, 0), backleft = (-1, -sqrt(3), 0), backup = (-1, -sqrt(3) / 3, 2 * sqrt(6) / 3);
draw(shift(2 * backleft) * surface(sphere(1,20)), white); //2
draw(shift(backleft + backright) * surface(sphere(1,20)), white); //2
draw(shift(2 * backright) * surface(sphere(1,20)), white); //3
draw(shift(backup + backleft) * surface(sphere(1,20)), white);
draw(shift(backup + backright) * surface(sphere(1,20)), white);
draw(shift(2 * backup) * surface(sphere(1,20)), white);
draw(shift(backleft) * surface(sphere(1,20)), white);
draw(shift(backright) * surface(sphere(1,20)), white);
draw(shift(backup) * surface(sphere(1,20)), white);
draw(surface(sphere(1,20)), white);
label("Row 0", 2 * backup, 15 * dir(20));
label("Row 1", backup, 25 * dir(20));
label("Row 2", O, 35 * dir(20));
dot(-backup);
dot(-7 * backup / 8);
dot(-6 * backup / 8);
dot((backleft - backup) + backleft * 2);
dot(5 * (backleft - backup) / 4 + backleft * 2);
dot(6 * (backleft - backup) / 4 + backleft * 2);
dot((backright - backup) + backright * 2);
dot(5 * (backright - backup) / 4 + backright * 2);
dot(6 * (backright - backup) / 4 + backright * 2);
[/asy]
1999 USAMTS Problems, 5
We say that a finite set of points is [i]well scattered[/i] on the surface of a sphere if every open hemisphere (half the surface of the sphere without its boundary) contains at least one of the points. The set $\{ (1,0,0), (0,1,0), (0,0,1) \}$ is not well scattered on the unit sphere (the sphere of radius $1$ centered at the origin), but if you add the correct point $P$ it becomes well scattered. Find, with proof, all possible points $P$ that would make the set well scattered.
2014 USAMTS Problems, 1:
The net of 20 triangles shown below can be folded to form a regular icosahedron. Inside each of the triangular faces, write a number from 1 to 20 with each number used exactly once. Any pair of numbers that are consecutive must be written on faces sharing an edge in the folded icosahedron, and additionally, 1 and 20 must also be on faces sharing an edge. Some numbers have been given to you. (No proof is necessary.)
[asy]
unitsize(1cm);
pair c(int a, int b){return (a-b/2,sqrt(3)*b/2);}
draw(c(0,0)--c(0,1)--c(-1,1)--c(1,3)--c(1,1)--c(2,2)--c(3,2)--c(4,3)--c(4,2)--c(3,1)--c(2,1)--c(2,-1)--c(1,-1)--c(1,-2)--c(0,-3)--c(0,-2)--c(-1,-2)--c(1,0)--cycle);
draw(c(0,0)--c(1,1)--c(0,1)--c(1,2)--c(0,2)--c(0,1),linetype("4 4"));
draw(c(4,2)--c(3,2)--c(3,1),linetype("4 4"));
draw(c(3,2)--c(1,0)--c(1,1)--c(2,1)--c(2,2),linetype("4 4"));
draw(c(1,-2)--c(0,-2)--c(0,-1)--c(1,-1)--c(1,0)--c(2,0)--c(0,-2),linetype("4 4"));
label("2",(c(0,2)+c(1,2))/2,S);
label("15",(c(1,1)+c(2,1))/2,S);
label("6",(c(0,1)+c(1,1))/2,N);
label("14",(c(0,0)+c(1,0))/2,N);[/asy]
2008 ITest, 91
Find the sum of all positive integers $n$ such that \[x^3+y^3+z^3=nx^2y^2z^2\] is satisfied by at least one ordered triplet of positive integers $(x,y,z)$.
1980 AMC 12/AHSME, 26
Four balls of radius 1 are mutually tangent, three resting on the floor and the fourth resting on the others. A tetrahedron, each of whose edges have length $s$, is circumscribed around the balls. Then $s$ equals
$\text{(A)} \ 4\sqrt 2 \qquad \text{(B)} \ 4\sqrt 3 \qquad \text{(C)} \ 2\sqrt 6 \qquad \text{(D)} \ 1+2\sqrt 6 \qquad \text{(E)} \ 2+2\sqrt 6$
1984 AIME Problems, 9
In tetrahedron $ABCD$, edge $AB$ has length 3 cm. The area of face $ABC$ is 15 $\text{cm}^2$ and the area of face $ABD$ is 12 $\text{cm}^2$. These two faces meet each other at a $30^\circ$ angle. Find the volume of the tetrahedron in $\text{cm}^3$.
2013 Sharygin Geometry Olympiad, 3
Let $X$ be a point inside triangle $ABC$ such that $XA.BC=XB.AC=XC.AC$. Let $I_1, I_2, I_3$ be the incenters of $XBC, XCA, XAB$. Prove that $AI_1, BI_2, CI_3$ are concurrent.
[hide]Of course, the most natural way to solve this is the Ceva sin theorem, but there is an another approach that may surprise you;), try not to use the Ceva theorem :))[/hide]
PEN P Problems, 5
Show that any positive rational number can be represented as the sum of three positive rational cubes.
1953 Polish MO Finals, 3
Through each vertex of a tetrahedron with a given volume $ V $, a plane is drawn parallel to the opposite face of the tetrahedron. Calculate the volume of the tetrahedron formed by these planes.
2014 Bulgaria National Olympiad, 3
A real number $f(X)\neq 0$ is assigned to each point $X$ in the space.
It is known that for any tetrahedron $ABCD$ with $O$ the center of the inscribed sphere, we have :
\[ f(O)=f(A)f(B)f(C)f(D). \]
Prove that $f(X)=1$ for all points $X$.
[i]Proposed by Aleksandar Ivanov[/i]
2013 Stanford Mathematics Tournament, 11
Sara has an ice cream cone with every meal. The cone has a height of $2\sqrt2$ inches and the base of the cone has a diameter of $2$ inches. Ice cream protrudes from the top of the cone in a perfect hempisphere. Find the surface area of the ice cream cone, ice cream included, in square inches.
2011 Cuba MO, 2
A cube of dimensions $20 \times 20 \times 20$ is constructed with blocks of $1 \times 2 \times 2$. Prove that there is a line that passes through the cube but not any block.
2003 Portugal MO, 1
The planet Caramelo is a cube with a $1$ km edge. This planet is going to be wrapped with foam anti-gluttons in order to prevent the presence of greedy ships less than $500$ meters from the planet. What the minimum volume of foam that must surround the planet?
2001 AIME Problems, 11
Club Truncator is in a soccer league with six other teams, each of which it plays once. In any of its 6 matches, the probabilities that Club Truncator will win, lose, or tie are each $\frac{1}{3}$. The probability that Club Truncator will finish the season with more wins than losses is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
1974 Spain Mathematical Olympiad, 1
It is known that a regular dodecahedron is a regular polyhedron with $12$ faces of equal pentagons and concurring $3$ edges in each vertex. It is requested to calculate, reasonably,
a) the number of vertices,
b) the number of edges,
c) the number of diagonals of all faces,
d) the number of line segments determined for every two vertices,
d) the number of diagonals of the dodecahedron.