Found problems: 2265
2025 Korea Winter Program Practice Test, P3
$n$ assistants start simultaneously from one vertex of a cube-shaped planet with edge length $1$. Each assistant moves along the edges of the cube at a constant speed of $2, 4, 8, \cdots, 2^n$, and can only change their direction at the vertices of the cube. The assistants can pass through each other at the vertices, but if they collide at any point that is not a vertex, they will explode. Determine the maximum possible value of $n$ such that the assistants can move infinitely without any collisions.
2000 Harvard-MIT Mathematics Tournament, 7
A regular tetrahedron of volume $1$ is filled with water of total volume $\frac{7}{16}$. Is it possible that the center of the tetrahedron lies on the surface of the water? How about in a cube of volume $1$?
1996 German National Olympiad, 3
Let be given an arbitrary tetrahedron $ABCD$ with volume $V$. Consider all lines which pass through the barycenter $S$ of the tetrahedron and intersect the edges $AD,BD,CD$ at points $A',B',C$ respectively. It is known that among the obtained tetrahedra there exists one with the minimal volume. Express this minimal volume in terms of $V$
2000 Belarusian National Olympiad, 6
A vertex of a tetrahedron is called perfect if the three edges at this vertex are sides of a certain triangle. How many perfect vertices can a tetrahedron have?
PEN H Problems, 41
Suppose that $A=1,2,$ or $3$. Let $a$ and $b$ be relatively prime integers such that $a^{2}+Ab^2 =s^3$ for some integer $s$. Then, there are integers $u$ and $v$ such that $s=u^2 +Av^2$, $a =u^3 - 3Avu^2$, and $b=3u^{2}v -Av^3$.
2002 Romania National Olympiad, 3
Let $[ABCDEF]$ be a frustum of a regular pyramid. Let $G$ and $G'$ be the centroids of bases $ABC$ and $DEF$ respectively. It is known that $AB=36,DE=12$ and $GG'=35$.
$a)$ Prove that the planes $(ABF),(BCD),(CAE)$ have a common point $P$, and the planes $(DEC),(EFA),(FDB)$ have a common point $P'$, both situated on $GG'$.
$b)$ Find the length of the segment $[PP']$.
2012 Tournament of Towns, 1
Each vertex of a convex polyhedron lies on exactly three edges, at least two of which have the same length. Prove that the polyhedron has three edges of the same length.
1997 National High School Mathematics League, 10
Bottom surface of triangular pyramid $S-ABC$ is an isosceles right triangle (hypotenuse is $AB$). $SA=SB=SC=AB=2$, and $S,A,B,C$ are on a sphere with center of $O$. The distance of $O$ to plane $ABC$ is________.
2003 District Olympiad, 1
Let $ABC$ be an equilateral triangle. On the plane $(ABC)$ rise the perpendiculars $AA'$ and $BB'$ on the same side of the plane, so that $AA' = AB$ and $BB' =\frac12 AB$. Determine the measure the angle between the planes $(ABC)$ and $(A'B'C')$.
1968 Czech and Slovak Olympiad III A, 4
Four different points $A,B,C,D$ are given in space such that $AC\perp BD,AD\perp BC.$ Show there is a sphere containing midpoits of all 7 segments $AB,AC,AD,BC,BD,CD.$
1961 IMO Shortlist, 6
Consider a plane $\epsilon$ and three non-collinear points $A,B,C$ on the same side of $\epsilon$; suppose the plane determined by these three points is not parallel to $\epsilon$. In plane $\epsilon$ take three arbitrary points $A',B',C'$. Let $L,M,N$ be the midpoints of segments $AA', BB', CC'$; Let $G$ be the centroid of the triangle $LMN$. (We will not consider positions of the points $A', B', C'$ such that the points $L,M,N$ do not form a triangle.) What is the locus of point $G$ as $A', B', C'$ range independently over the plane $\epsilon$?
1982 Austrian-Polish Competition, 8
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 $XY$.
2021/2022 Tournament of Towns, P7
A starship is located in a halfspace at the distance $a$ from its boundary. The crew knows this but does not know which direction to move to reach the boundary plane. The starship may travel through the space by any path, may measure the way it has already travelled and has a sensor that signals when the boundary is reached. Is it possible to reach the boundary for sure, having passed no more than:
$a)14a$
$b)13a$?
1999 Federal Competition For Advanced Students, Part 2, 2
Let $\epsilon$ be a plane and $k_1, k_2, k_3$ be spheres on the same side of $\epsilon$. The spheres $k_1, k_2, k_3$ touch the plane at points $T_1, T_2, T_3$, respectively, and $k_2$ touches $k_1$ at $S_1$ and $k_3$ at $S_3$. Prove that the lines $S_1T_1$ and $S_3T_3$ intersect on the sphere $k_2$. Describe the locus of the intersection point.
2020 Purple Comet Problems, 30
Four small spheres each with radius $6$ are each internally tangent to a large sphere with radius $17$. The four small spheres form a ring with each of the four spheres externally tangent to its two neighboring small spheres. A sixth intermediately sized sphere is internally tangent to the large sphere and externally tangent to each of the four small spheres. Its radius is $\frac{m}{n}$ , where m and n are relatively prime positive integers. Find $m + n$.
[img]https://cdn.artofproblemsolving.com/attachments/7/2/25955cd6f22bc85f2f3c5ba8cd1ee0821c9d50.png[/img]
2011 Oral Moscow Geometry Olympiad, 2
Line $\ell $ intersects the plane $a$. It is known that in this plane there are $2011$ straight lines equidistant from $\ell$ and not intersecting $\ell$. Is it true that $\ell$ is perpendicular to $a$?
2014 Online Math Open Problems, 29
Let $ABCD$ be a tetrahedron whose six side lengths are all integers, and let $N$ denote the sum of these side lengths. There exists a point $P$ inside $ABCD$ such that the feet from $P$ onto the faces of the tetrahedron are the orthocenter of $\triangle ABC$, centroid of $\triangle BCD$, circumcenter of $\triangle CDA$, and orthocenter of $\triangle DAB$. If $CD = 3$ and $N < 100{,}000$, determine the maximum possible value of $N$.
[i]Proposed by Sammy Luo and Evan Chen[/i]
1975 Miklós Schweitzer, 12
Assume that a face of a convex polyhedron $ P$ has a common edge with every other face. Show that there exists a simple closed polygon that consists of edges of $ P$ and passes through all vertices.
[i]L .Lovasz[/i]
2024 AIME, 15
Let $\mathcal{B}$ be the set of rectangular boxes that have volume $23$ and surface area $54$. Suppose $r$ is the least possible radius of a sphere that can fit any element of $\mathcal{B}$ inside it. Then $r^{2}$ can be expressed as $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
Champions Tournament Seniors - geometry, 2008.4
Given a quadrangular pyramid $SABCD$, the basis of which is a convex quadrilateral $ABCD$. It is known that the pyramid can be tangent to a sphere. Let $P$ be the point of contact of this sphere with the base $ABCD$. Prove that $\angle APB + \angle CPD = 180^o$.
2004 Iran MO (3rd Round), 26
Finitely many points are given on the surface of a sphere, such that every four of them lie on the surface of open hemisphere. Prove that all points lie on the surface of an open hemisphere.
2021 CCA Math Bonanza, T6
Three spheres have radii $144$, $225$, and $400$, are pairwise externally tangent to each other, and are all tangent to the same plane at $A$, $B$, and $C$. Compute the area of triangle $ABC$.
[i]2021 CCA Math Bonanza Team Round #6[/i]
2018 District Olympiad, 3
Let $ABCDA'B'C'D'$ be the rectangular parallelepiped.
Let $M, N, P$ be midpoints of the edges $[AB], [BC],[BB']$ respectively . Let $\{O\} = A'N \cap C'M$.
a) Prove that the points $D, O, P$ are collinear.
b) Prove that $MC' \perp (A'PN)$ if and only if $ABCDA'B'C'D'$ is a cube.
1990 Poland - Second Round, 2
In space, a point $O$ and a finite set of vectors $ \overrightarrow{v_1},\ldots,\overrightarrow{v_n} $ are given . We consider the set of points $ P $ for which the vector $ \overrightarrow{OP} $can be represented as a sum $ a_1 \overrightarrow{v_1} + \ldots + a_n\overrightarrow{v_n} $with coefficients satisfying the inequalities $ 0 \leq a_i \leq 1 $ $( i = 1, 2, \ldots, n $). Decide whether this set can be a tetrahedron.
1997 AMC 12/AHSME, 23
In the figure, polygons $ A$, $ E$, and $ F$ are isosceles right triangles; $ B$, $ C$, and $ D$ are squares with sides of length $ 1$; and $ G$ is an equilateral triangle. The figure can be folded along its edges to form a polyhedron having the polygons as faces. The volume of this polyhedron is
$ \textbf{(A)}\ 1/2\qquad \textbf{(B)}\ 2/3\qquad \textbf{(C)}\ 3/4\qquad \textbf{(D)}\ 5/6\qquad \textbf{(E)}\ 4/3$
[asy]
size(180);
defaultpen(linewidth(.7pt)+fontsize(10pt));
draw((-1,1)--(2,1));
draw((-1,0)--(1,0));
draw((-1,1)--(-1,0));
draw((0,-1)--(0,3));
draw((1,2)--(1,0));
draw((-1,1)--(1,1));
draw((0,2)--(1,2));
draw((0,3)--(1,2));
draw((0,-1)--(2,1));
draw((0,-1)--((0,-1) + sqrt(2)*dir(-15)));
draw(((0,-1) + sqrt(2)*dir(-15))--(1,0));
label("$\textbf{A}$",foot((0,2),(0,3),(1,2)),SW);
label("$\textbf{B}$",midpoint((0,1)--(1,2)));
label("$\textbf{C}$",midpoint((-1,0)--(0,1)));
label("$\textbf{D}$",midpoint((0,0)--(1,1)));
label("$\textbf{E}$",midpoint((1,0)--(2,1)),NW);
label("$\textbf{F}$",midpoint((0,-1)--(1,0)),NW);
label("$\textbf{G}$",midpoint((0,-1)--(1,0)),2SE);[/asy]