Found problems: 2265
Let $AA',BB',CC'$ be parallel lines not lying in the same plane. Denote $U$ the intersection of the planes $A'BC,AB'C,ABC'$ and $V$ the intersection of the planes $AB'C',A'BC',A'B'C$. Show that the line $UV$ is parallel with $AA'$.
Which of the following can be used to distinguish a solid ball from a hollow sphere of the same radius and mass?
$\textbf{(A)}$ Measurements of the orbit of a test mass around the object.
$\textbf{(B)}$ Measurements of the time it takes the object to roll down an inclined plane.
$\textbf{(C)}$ Measurements of the tidal forces applied by the object to a liquid body.
$\textbf{(D)}$ Measurements of the behavior of the object as it oats in water.
$\textbf{(E)}$ Measurements of the force applied to the object by a uniform gravitational field.
Suppose you have a sphere tangent to the $xy$-plane with its center having positive $z$-coordinate. If it is projected from a point $P=(0,b,a)$ to the $xy$-plane, it gives the conic section $y=x^2$. If we write $a=\tfrac pq$ where $p,q$ are integers, find $p+q$.
Two regular pentagons $ABCDE$ and $AEKPL$ are given in space, such that $\angle DAK = 60^{\circ}$. Let $M$, $N$ and $S$ be the midpoints of $AE$, $CD$ and $EK$. Prove that:
[list=a]
[*] $\triangle NMS$ is a right triangle;
[*] planes $(ACK)$ and $(BAL)$ are perpendicular.
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[i]Ukraine Olympiad[/i]
Given a point $M$ inside a right tetrahedron. Prove that at least one tetrahedron edge is seen from the $M$ in an angle, that has a cosine not greater than $-1/3$. (e.g. if $A$ and $B$ are the vertices, corresponding to that edge, $cos(\widehat{AMB}) \le -1/3$)
Ten adults enter a room, remove their shoes, and toss their shoes into a pile. Later, a child randomly pairs each left shoe with a right shoe without regard to which shoes belong together. The probability that for every positive integer $k<5,$ no collection of $k$ pairs made by the child contains the shoes from exactly $k$ of the adults is $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Given a tetrahedron $ABCD$, let $x = AB \cdot CD$, $y = AC \cdot BD$, and $z = AD \cdot BC$. Prove that there exists a triangle with edges $x, y, z.$
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 $m$.
A tetrahedron is a polyhedron composed of four triangular faces. Faces $ABC$ and $BCD$ of a tetrahedron $ABCD$ meet at an angle of $\pi/6$. The area of triangle $\triangle ABC$ is $120$. The area of triangle $\triangle BCD$ is $80$, and $BC = 10$. What is the volume of the tetrahedron? We call the volume of a tetrahedron as one-third the area of it's base times it's height.
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 $1 \times3$?
A lampshade is part of the surface of a right circular cone whose axis is vertical. Its upper and lower edges are two horizontal circles. Two points are selected on the upper smaller circle and four points on the lower larger circle. Each of these six points has three of the others that are its nearest neighbors at a distance $d$ from it. By distance is meant the shortest distance measured over the curved survace of the lampshade. Prove that the area of the lampshade is $d^2(2\theta + \sqrt 3)$ where $\cot \frac {\theta}{2} = \frac{3}{\theta}.$
A sphere is inscribed in an $n$-angled pyramid. Prove that if we align all side faces of the pyramid with the base plane, flipping them around the corresponding edges of the base, then
(1) all tangent points of these faces to the sphere would coincide with one point, $H$, and
(2) the vertices of the faces would lie on a circle centered at $H$.
Three spheres are all externally tangent to a plane and to each other. Suppose that the radii of these spheres are $6$, $8$, and, $10$. The tangency points of these spheres with the plane form the vertices of a triangle. Determine the largest integer that is smaller than the perimeter of this triangle.
Prove that for every point $M$ on the surface of a regular tetrahedron there exists a point $M'$ such that there are at least three different curves on the surface joining $M$ to $M'$ with the smallest possible length among all curves on the surface joining $M$ to $M'$.
A paper equilateral triangle of side length $2$ on a table has vertices labeled $A,B,C.$ Let $M$ be the point on the sheet of paper halfway between $A$ and $C.$ Over time, point $M$ is lifted upwards, folding the triangle along segment $BM,$ while $A,B,$ and $C$ on the table. This continues until $A$ and $C$ touch. Find the maximum volume of tetrahedron $ABCM$ at any time during this process.
Place the integers $1,2 , \ldots, n^{3}$ in the cells of a $n\times n \times n$ cube such that every number appears once. For any possible enumeration, write down the maximal difference between any two adjacent cells (adjacent means having a common vertex). What is the minimal number noted down?
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]
Find out the maximum value of the numbers of edges of a solid regular octahedron that we can see from a point out of the regular octahedron.(We define we can see an edge $AB$ of the regular octahedron from point $P$ outside if and only if the intersection of non degenerate triangle $PAB$ and the solid regular octahedron is exactly edge $AB$.
Find the least positive integers $m,k$ such that
a) There exist $2m+1$ consecutive natural numbers whose sum of cubes is also a cube.
b) There exist $2k+1$ consecutive natural numbers whose sum of squares is also a square.
The author is Vasile Suceveanu
Let $n$ be a positive integer. A child builds a wall along a line with $n$ identical cubes. He lays the first cube on the line and at each subsequent step, he lays the next cube either on the ground or on the top of another cube, so that it has a common face with the previous one. How many such distinct walls exist?
Let $Q$ be a point inside a cube. Prove that there are infinitely many lines $l$ so that $AQ=BQ$ where $A$ and $B$ are the two points of intersection of $l$ and the surface of the cube.
Is there a six-link broken line in space that passes through all the vertices of a given cube?
A convex polyhedron has no diagonals (every pair of vertices are connected by an edge). Prove that it is a tetrahedron.
A triangular pyramid $ABCD$ is given. Prove that $\frac Rr > \frac ah$, where $R$ is the radius of the circumscribed sphere, $r$ is the radius of the inscribed sphere, $a$ is the length of the longest edge, $h$ is the length of the shortest altitude (from a vertex to the opposite face).
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 $G$.