Found problems: 2265
On a table are a cone, resting on the base, and six equal spheres tangent to the cone. Besides that, each sphere is tangent to the two adjacent spheres. Knowing that the radius $R$ of the base of the cone is half its height and determine the radius $r$ of the spheres.
Let $SABC$ be a pyramid with right angles at the vertex $S$. Points $A', B', C'$ lie on the edges $SA, SB, SC$ respectively in such a way that the triangles $ABC$ and $A'B'C'$ are similar. Does this yield that the planes $ABC$ and $A'B'C'$ are parallel?
$(CZS 1)$ Given a unit cube, find the locus of the centroids of all tetrahedra whose vertices lie on the sides of the cube.
The edges of a cube are labeled from 1 to 12 in an arbitrary manner. Show that it is not possible to get the sum of the edges at each vertex the same.
Show that we can get eight vertices with the same sum if one of the labels is changed to 13.
Let $P$ be a point inside a regular tetrahedron ABCD with edge length $1$. Show that $$d(P,AB)+d(P,AC)+d(P,AD)+d(P,BC)+d(P,BD)+d(P,CD) \ge \frac{3}{2} \sqrt2$$ , with equality only when $P$ is the centroid of $ABCD$.
Here $d(P,XY)$ denotes the distance from point $P$ to line $XY$.
We consider a fixed point $P$ in the interior of a fixed sphere$.$ We construct three segments $PA, PB,PC$, perpendicular two by two$,$ with the vertexes $A, B, C$ on the sphere$.$ We consider the vertex $Q$ which is opposite to $P$ in the parallelepiped (with right angles) with $PA, PB, PC$ as edges$.$ Find the locus of the point $Q$ when $A, B, C$ take all the positions compatible with our problem.
In one galaxy, there exist more than one million stars. Let $M$ be the set of the distances between any $2$ of them. Prove that, in every moment, $M$ has at least $79$ members. (Suppose each star as a point.)
Consider a container of a hollow cube $ABGCDEPF$ (where $ABGC$, $DEPF$ are squares and $AD\parallel BE\parallel GP\parallel CF$). The cube is placed on a table in a way that the space diagonal $AP=1$ is perpendicular to the table. Then, water is poured into the cube. Denote $x$ the length of part of $AP$ submerged in water. Determine the volume of water $y$ in terms of $x$ when
a) $0 < x \leq\frac13$,
b) $\frac13 < x \leq\frac12$.
Let $K$ and $M$ be points on the side $AB$ of a triangle $\triangle{ABC}$, and let $L$ and $N$ be points on the side $AC$. The point $K$ is between $M$ and $B$, and the point $L$ is between $N$ and $C$. If $\frac{BK}{KM}=\frac{CL}{LN}$, then prove that the orthocentres of the triangles $\triangle{ABC}$, $\triangle{AKL}$ and $\triangle{AMN}$ lie on one line.
Each vertex of a cube is to be labeled with an integer $1$ through $8$, with each integer being used once, in such a way that the sum of the four numbers on the vertices of a face is the same for each face. Arrangements that can be obtained from each other through rotations of the cube are considered to be the same. How many different arrangements are possible?
$\textbf{(A) } 1\qquad\textbf{(B) } 3\qquad\textbf{(C) }6 \qquad\textbf{(D) }12 \qquad\textbf{(E) }24$
Consider two solid spherical balls, one centered at $(0, 0, \frac{21}{2} )$ with radius $6$, and the other centered at $(0, 0, 1)$ with radius $\frac 92$ . How many points $(x, y, z)$ with only integer coordinates (lattice points) are there in the intersection of the
balls?
$\text{(A)}\ 7 \qquad \text{(B)}\ 9 \qquad \text{(C)}\ 11 \qquad \text{(D)}\ 13 \qquad \text{(E)}\ 15$
Consider the solid $Q$ obtained by attaching unit cubes $Q_1...Q_6$ at the six faces of a unit cube $Q$. Prove or disprove that the space can be filled up with such solids so that no two of them have a common interior point.
Consider the cube $ABCDA'B'C'D'$ ($ABCD$ and $A'B'C'D'$ are the upper and lower bases, repsectively, and edges $AA', BB', CC', DD'$ are parallel). The point $X$ moves at a constant speed along the perimeter of the square $ABCD$ in the direction $ABCDA$, and the point $Y$ moves at the same rate along the perimiter of the square $B'C'CB$ in the direction $B'C'CBB'$. Points $X$ and $Y$ begin their motion at the same instant from the starting positions $A$ and $B'$, respectively. Determine and draw the locus of the midpionts of the segments $XY$.
In the space is given a tetrahedron with length of the edge $2$. Prove that distances from some point $M$ to all of the vertices of the tetrahedron are integer numbers if and only if $M$ is a vertex of tetrahedron.
[i]J. Tabov[/i]
A three-quarter sector of a circle of radius $4$ inches together with its interior can be rolled up to form the lateral surface area of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches?
[asy]
draw(Arc((0,0), 4, 0, 270));
draw((0,-4)--(0,0)--(4,0));
label("$4$", (2,0), S);
[/asy]
$\textbf{(A)}\ 3\pi \sqrt5 \qquad\textbf{(B)}\ 4\pi \sqrt3 \qquad\textbf{(C)}\ 3 \pi \sqrt7 \qquad\textbf{(D)}\ 6\pi \sqrt3 \qquad\textbf{(E)}\ 6\pi \sqrt7$
A right pyramid has regular octagon $ABCDEFGH$ with side length $1$ as its base and apex $V.$ Segments $\overline{AV}$ and $\overline{DV}$ are perpendicular. What is the square of the height of the pyramid?
$
\textbf{(A) }1 \qquad
\textbf{(B) }\frac{1+\sqrt2}{2} \qquad
\textbf{(C) }\sqrt2 \qquad
\textbf{(D) }\frac32 \qquad
\textbf{(E) }\frac{2+\sqrt2}{3} \qquad
$
Let $SABCD$ be a pyramid with the vertex $S$ whose all edges are equal. Points $M$ and $N$ on the edges $SA$ and $BC$ respectively are such that $MN$ is perpendicular to both $SA$ and $BC$. Find the ratios $SM:MA$ and $BN:NC$.
Two satellites are orbiting the earth in the equatorial plane at an altitude $h$ above the surface. The distance between the satellites is always $d$, the diameter of the earth. For which $h$ is there always a point on the equator at which the two satellites subtend an angle of $90^\circ$?
We have eight light bulbs, placed on the eight lattice points (points with integer coordinates) in space that are $\sqrt{3}$ units away from the origin. Each light bulb can either be turned on or off. These lightbulbs are unstable, however. If two light bulbs that are at most 2 units apart are both on simultaneously, they both explode. Given that no explosions take place, how many possible configurations of on/off light bulbs exist?
[i]Proposed by Lewis Chen[/i]
How many $2 \times 2 \times 2$ cubes must be added to a $8 \times 8 \times 8$ cube to form a $12 \times 12 \times 12$ cube?
[i]Proposed by Evan Chen[/i]
Let $AB$ and $A_1B_1$ be two skew segments, $O$ and $O_1$ their respective midpoints. Prove that $OO_1$ is shorter than a half sum of $AA_1$ and $BB_1$.
Suppose $\mathcal{T}=A_0A_1A_2A_3$ is a tetrahedron with $\angle A_1A_0A_3 = \angle A_2A_0A_1 = \angle A_3A_0A_2 = 90^\circ$, $A_0A_1=5, A_0A_2=12$ and $A_0A_3=9$. A cube $A_0B_0C_0D_0E_0F_0G_0H_0$ with side length $s$ is inscribed inside $\mathcal{T}$ with $B_0\in \overline{A_0A_1}, D_0 \in \overline{A_0A_2}, E_0 \in \overline{A_0A_3}$, and $G_0\in \triangle A_1A_2A_3$; what is $s$?
Let $S$ be a great circle with pole $P$. On any great circle through $P$, two points $A$ and $B$ are chosen equidistant from $P$. For any [i] spherical triangle [/i] $ABC$ (the sides are great circles ares), where $C$ is on $S$, prove that the great circle are $CP$ is the angle bisector of angle $C$.
[b] Note. [/b] A great circle on a sphere is one whose center is the center of the sphere. A pole of the great circle $S$ is a point $P$ on the sphere such that the diameter through $P$ is perpendicular to the plane of $S$.
Let $P_1P_2P_3P_4$ be a tetrahedron in $\mathbb{R}^3$ and let $O$ be a point equidistant from each of its vertices. Suppose there exists a point $H$ such that for each $i$, the line $P_iH$ is perpendicular to the plane through the other three vertices. Line $P_1H$ intersects the plane through $P_2, P_3, P_4$ at $A$, and contains a point $B\neq P_1$ such that $OP_1=OB$. Show that $HB=3HA$.
[i]Michael Ren[/i]
$ABCD$ is a regular tetrahedron with side length $1$. Find the area of the cross section of $ABCD$ cut by the plane that passes through the midpoints of $AB$, $AC$, and $CD$.