Found problems: 2265
2012 National Olympiad First Round, 33
Let $ABCDA'B'C'D'$ be a rectangular prism with $|AB|=2|BC|$. $E$ is a point on the edge $[BB']$ satisfying $|EB'|=6|EB|$. Let $F$ and $F'$ be the feet of the perpendiculars from $E$ at $\triangle AEC$ and $\triangle A'EC'$, respectively. If $m(\widehat{FEF'})=60^{\circ}$, then $|BC|/|BE| = ? $
$ \textbf{(A)}\ \sqrt\frac53 \qquad \textbf{(B)}\ \sqrt\frac{15}2 \qquad \textbf{(C)}\ \frac32\sqrt{15} \qquad \textbf{(D)}\ 5\sqrt\frac53 \qquad \textbf{(E)}\ \text{None}$
2014 District Olympiad, 3
Let $ABCDEF$ be a regular hexagon with side length $a$. At point $A$, the perpendicular $AS$, with length $2a\sqrt{3}$, is erected on the hexagon's plane. The points $M, N, P, Q,$ and $R$ are the projections of point $A$ on the lines $SB, SC, SD, SE,$ and $SF$, respectively.
[list=a]
[*]Prove that the points $M, N, P, Q, R$ lie on the same plane.
[*]Find the measure of the angle between the planes $(MNP)$ and $(ABC)$.[/list]
2012 NIMO Problems, 6
A square is called [i]proper[/i] if its sides are parallel to the coordinate axes. Point $P$ is randomly selected inside a proper square $S$ with side length 2012. Denote by $T$ the largest proper square that lies within $S$ and has $P$ on its perimeter, and denote by $a$ the expected value of the side length of $T$. Compute $\lfloor a \rfloor$, the greatest integer less than or equal to $a$.
[i]Proposed by Lewis Chen[/i]
2017 Harvard-MIT Mathematics Tournament, 8
[b]U[/b]ndecillion years ago in a galaxy far, far away, there were four space stations in the three-dimensional space, each pair spaced 1 light year away from each other. Admiral Ackbar wanted to establish a base somewhere in space such that the sum of squares of the distances from the base to each of the stations does not exceed 15 square light years. (The sizes of the space stations and the base are negligible.) Determine the volume, in cubic light years, of the set of all possible locations for the Admiral’s base.
2005 District Olympiad, 3
Prove that if the circumcircles of the faces of a tetrahedron $ABCD$ have equal radii, then $AB=CD$, $AC=BD$ and $AD=BC$.
2016-2017 SDML (Middle School), 8
An ice cream cone has radius $1$ and height $4$ inches. What is the number of inches in the radius of a sphere of ice cream which has the same volume of the cone?
$\text{(A) }\frac{1}{2}\qquad\text{(B) }1\qquad\text{(C) }\frac{3}{2}\qquad\text{(D) }2\qquad\text{(E) }\frac{5}{2}$
1997 All-Russian Olympiad Regional Round, 10.4
Given a cube with a side of $4$. Is it possible to completely cover $3$ of its faces, which have a common vertex, with sixteen rectangular paper strips measuring $1 \times3$?
2007 iTest Tournament of Champions, 5
Let $c$ be the number of ways to choose three vertices of an $6$-dimensional cube that form an equilateral triangle. Find the remainder when $c$ is divided by $2007$.
Kyiv City MO 1984-93 - geometry, 1991.10.5
Diagonal sections of a regular 8-gon pyramid, which are drawn through the smallest and largest diagonals of the base, are equal. At what angle is the plane passing through the vertex, the pyramids and the smallest diagonal of the base inclined to the base?
[hide=original wording]Діагональні перерізи правильної 8-кутної піраміди, які Проведені через найменшу і найбільшу діагоналі основи, рівновеликі. Під яким кутом до основи нахилена площина, що проходить через вершину, піраміди і найменшу діагональ основи?[/hide]
2019 PUMaC Geometry B, 2
A right cone in $xyz$-space has its apex at $(0,0,0)$, and the endpoints of a diameter on its base are $(12,13,-9)$ and $(12,-5,15)$. The volume of the cone can be expressed as $a\pi$. What is $a$?
2006 JHMT, 5
An ant is on the bottom edge of a right circular cone with base area $\pi$ and slant length $6$. What is the shortest distance that the ant has to travel to loop around the cone and come back to its starting position?
May Olympiad L2 - geometry, 2003.5
An ant, which is on an edge of a cube of side $8$, must travel on the surface and return to the starting point. It's path must contain interior points of the six faces of the cube and should visit only once each face of the cube. Find the length of the path that the ant can carry out and justify why it is the shortest path.
1988 National High School Mathematics League, 4
Given three planes $\alpha,\beta,\gamma$. Intersection angle between any two planes are all $\theta$.$\alpha\cap\beta=a,\beta\cap\gamma=b,\gamma\cap\alpha=c$.
Given two conditions:
A: $\theta>\frac{\pi}{3}$
B: $a,b,c$ share one point.
$(\text{A})$A is sufficient but unnecessary condition of B.
$(\text{B})$A is necessary but insufficient condition of B.
$(\text{C})$A is sufficient and necessary condition of B.
$(\text{D})$None above
2005 QEDMO 1st, 4 (Z1)
Solve the equation $x^3+2y^3+5z^3=0$ in integers.
1957 Polish MO Finals, 6
A cube is given with base $ ABCD $, where $ AB = a $ cm. Calculate the distance of the line $ BC $ from the line passing through the point $ A $ and the center $ S $ of the face opposite the base.
III Soros Olympiad 1996 - 97 (Russia), 11.5
All faces of the parallelepiped $ABCDA_1B_1C_1D_1$ are equal rhombuses. Plane angles at vertex $A$ are equal. Points $K$ and $M$ are taken on the edges $A_1B_1$ and $A_1D_1$. It is known that $A_1K = a$, $A_1M = b$, and$ a + b$ is an edge of the parallelepiped. Prove that the plane $AKM$ touches the sphere inscribed in the parallelepiped. Let us denote by $Q$ the touchpoint of this sphere with the plane $AKM $. In what ratio does the straight line $AQ$ divide the segment $KM$?
1991 Arnold's Trivium, 77
Find the eigenvalues and their multiplicities of the Laplace operator $\Delta = \text{div grad}$ on a sphere of radius $R$ in Euclidean space of dimension $n$.
1995 AIME Problems, 12
Pyramid $OABCD$ has square base $ABCD,$ congruent edges $\overline{OA}, \overline{OB}, \overline{OC},$ and $\overline{OD},$ and $\angle AOB=45^\circ.$ Let $\theta$ be the measure of the dihedral angle formed by faces $OAB$ and $OBC.$ Given that $\cos \theta=m+\sqrt{n},$ where $m$ and $n$ are integers, find $m+n.$
1986 IMO Longlists, 11
Prove that the sum of the face angles at each vertex of a tetrahedron is a straight angle if and only if the faces are congruent triangles.
1969 IMO Longlists, 26
$(GBR 3)$ A smooth solid consists of a right circular cylinder of height $h$ and base-radius $r$, surmounted by a hemisphere of radius $r$ and center $O.$ The solid stands on a horizontal table. One end of a string is attached to a point on the base. The string is stretched (initially being kept in the vertical plane) over the highest point of the solid and held down at the point $P$ on the hemisphere such that $OP$ makes an angle $\alpha$ with the horizontal. Show that if $\alpha$ is small enough, the string will slacken if slightly displaced and no longer remain in a vertical plane. If then pulled tight through $P$, show that it will cross the common circular section of the hemisphere and cylinder at a point $Q$ such that $\angle SOQ = \phi$, $S$ being where it initially crossed this section, and $\sin \phi = \frac{r \tan \alpha}{h}$.
1983 National High School Mathematics League, 11
For a regular hexahedron and a regular octahedron, all their faces are regular triangles, whose lengths of each side are $a$. Their inradius are $r_1,r_2$. $\frac{r_1}{r_2}=\frac{m}{n}, \gcd(m,n)=1$. Then $mn=$________.
2007 Estonia National Olympiad, 2
A 3-dimensional chess board consists of $ 4 \times 4 \times 4$ unit cubes. A rook can step from any unit cube K to any other unit cube that has a common face with K. A bishop can step from any unit cube K to any other unit cube that has a common edge with K, but does not have a common face. One move of both a rook and a bishop consists of an arbitrary positive number of consecutive steps in the same direction. Find the average number of possible moves for either piece, where the average is taken over all possible starting cubes K.
2017 China Team Selection Test, 6
A plane has no vertex of a regular dodecahedron on it,try to find out how many edges at most may the plane intersect the regular dodecahedron?
1936 Moscow Mathematical Olympiad, 031
Given three planes and a ball in space. In space, find the number of different ways of placing another ball so that it would be tangent the three given planes and the given ball. It is assumed that the balls can only touch externally.
1989 IMO Longlists, 67
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}.$