Found problems: 2265
2024 Francophone Mathematical Olympiad, 2
Given a positive integer $n \ge 2$, let $\mathcal{P}$ and $\mathcal{Q}$ be two sets, each consisting of $n$ points in three-dimensional space. Suppose that these $2n$ points are distinct. Show that it is possible to label the points of $\mathcal{P}$ as $P_1,P_2,\dots,P_n$ and the points of $\mathcal{Q}$ as $Q_1,Q_2,\dots,Q_n$ such that for any indices $i$ and $j$, the balls of diameters $P_iQ_i$ and $P_jQ_j$ have at least one common point.
2000 Turkey Team Selection Test, 1
Show that any triangular prism of infinite length can be cut by a plane such that the resulting intersection is an equilateral triangle.
1997 AIME Problems, 1
How many of the integers between 1 and 1000, inclusive, can be expressed as the difference of the squares of two nonnegative integers?
1983 Bundeswettbewerb Mathematik, 1
The surface of a soccer ball is made up of black pentagons and white hexagons together. On the sides of each pentagon are nothing but hexagons, while on the sides of each border of hexagons alternately pentagons and hexagons. Determine from this information about the soccer ball , the number of its pentagons and its hexagons.
1980 Vietnam National Olympiad, 1
Prove that for any tetrahedron in space, it is possible to find two perpendicular planes such that ratio between the projections of the tetrahedron on the two planes lies in the interval $[\frac{1}{\sqrt{2}}, \sqrt{2}].$
1999 National High School Mathematics League, 12
The bottom surface of triangular pyramid $S-ABC$ is a regular triangle. Projection of $A$ on plane $SBC$ is $H$, which is the orthocenter of $\triangle SBC$. If $H-AB-C=30^{\circ},SA=2\sqrt3$, then the volume of $S-ABC$ is________.
2013 Polish MO Finals, 4
Given is a tetrahedron $ABCD$ in which $AB=CD$ and the sum of measures of the angles $BAD$ and $BCD$ equals $180$ degrees. Prove that the measure of the angle $BAD$ is larger than the measure of the angle $ADC$.
2000 Romania National Olympiad, 4
In the rectangular parallelepiped $ABCDA'B'C'D'$, the points $E$ and $F$ are the centers of the faces $ABCD$ and $ADD' A'$, respectively, and the planes $(BCF)$ and $(B'C'E)$ are perpendicular. Let $A'M \perp B'A$, $M \in B'A$ and $BN \perp B'C$, $N \in B'C$. Denote $n = \frac{C'D}{BN}$.
a) Show that $n \ge \sqrt2$. .
b) Express and in terms of $n$, the ratio between the volume of the tetrahedron $BB'M N$ and the volume of the parallelepiped $ABCDA'B'C'D'$.
1981 Spain Mathematical Olympiad, 3
Given the intersecting lines $ r$ and $s$, consider the lines $u$ and $v$ as such what:
a) $u$ is symmetric to $r$ with respect to $s$,
b) $v$ is symmetric to $s$ with respect to $r$ .
Determine the angle that the given lines must form such that $u$ and $v$ to be coplanar.
1981 Tournament Of Towns, (012) 1
We will say that two pyramids touch each other by faces if they have no common interior points and if the intersection of a face of one of them with a face of the other is either a triangle or a polygon. Is it possible to place $8$ tetrahedra in such a way that every two of them touch each other by faces?
(A Andjans, Riga)
1955 Czech and Slovak Olympiad III A, 2
Let $\mathsf{S}_1,\mathsf{S}_2$ be concentric spheres with radii $a,b$ respectively, where $a<b.$ Denote $ABCDA'B'C'D'$ a square cuboid ($ABCD,A'B'C'D$ are the squares and $AA'\parallel BB'\parallel CC'\parallel DD'$) such that $A,B,C,D\in\mathsf{S}_2$ and the plane $A'B'C'D'$ is tangent to $\mathsf{S}_1.$ Finally assume that \[\frac{AB}{AA'}=\frac ab.\] Compute the lengths $AB,AA'.$ How many of such cuboids exist (up to a congruence)?
1981 Bulgaria National Olympiad, Problem 3
A quadrilateral pyramid is cut by a plane parallel to the base. Suppose that a sphere $S$ is circumscribed and a sphere $\Sigma$ inscribed in the obtained solid, and moreover that the line through the centers of these two spheres is perpendicular to the base of the pyramid. Show that the pyramid is regular.
2006 China Team Selection Test, 1
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.
2015 AMC 10, 17
The centers of the faces of the right rectangular prism shown below are joined to create an octahedron, What is the volume of the octahedron?
[asy]
import three; size(2inch);
currentprojection=orthographic(4,2,2);
draw((0,0,0)--(0,0,3),dashed);
draw((0,0,0)--(0,4,0),dashed);
draw((0,0,0)--(5,0,0),dashed);
draw((5,4,3)--(5,0,3)--(5,0,0)--(5,4,0)--(0,4,0)--(0,4,3)--(0,0,3)--(5,0,3));
draw((0,4,3)--(5,4,3)--(5,4,0));
label("3",(5,0,3)--(5,0,0),W);
label("4",(5,0,0)--(5,4,0),S);
label("5",(5,4,0)--(0,4,0),SE);
[/asy]
$\textbf{(A) } \dfrac{75}{12}
\qquad\textbf{(B) } 10
\qquad\textbf{(C) } 12
\qquad\textbf{(D) } 10\sqrt2
\qquad\textbf{(E) } 15
$
2008 Sharygin Geometry Olympiad, 22
(A.Khachaturyan, 10--11) a) All vertices of a pyramid lie on the facets of a cube
but not on its edges, and each facet contains at least one vertex. What is the
maximum possible number of the vertices of the pyramid?
b) All vertices of a pyramid lie in the facet planes of a cube but not on the lines
including its edges, and each facet plane contains at least one vertex. What is the
maximum possible number of the vertices of the pyramid?
2013 Saint Petersburg Mathematical Olympiad, 3
Let $M$ and $N$ are midpoint of edges $AB$ and $CD$ of the tetrahedron $ABCD$, $AN=DM$ and $CM=BN$. Prove that $AC=BD$.
S. Berlov
1998 Poland - Second Round, 6
Prove that the edges $AB$ and $CD$ of a tetrahedron $ABCD$ are perpendicular if and only if there exists a parallelogram $CDPQ$ such that $PA = PB = PD$ and $QA = QB = QC$.
2022 JHMT HS, 2
Four mutually externally tangent spherical apples of radius $4$ are placed on a horizontal flat table. Then, a spherical orange of radius $3$ is placed such that it rests on all the apples. Find the distance from the center of the orange to the table.
Ukrainian TYM Qualifying - geometry, II.1
Inside a right cylinder with a radius of the base $R$ are placed $k$ ($k\ge 3$) of equal balls, each of which touches the side surface and the lower base of the cylinder and, in addition, exactly two other balls. After that, another equal ball is placed inside the cylinder so that it touches the upper base of the cylinder and all other balls. Find the volume $V (R, k)$ of the cylinder.
1995 IMO Shortlist, 6
Let $ A_1A_2A_3A_4$ be a tetrahedron, $ G$ its centroid, and $ A'_1, A'_2, A'_3,$ and $ A'_4$ the points where the circumsphere of $ A_1A_2A_3A_4$ intersects $ GA_1,GA_2,GA_3,$ and $ GA_4,$ respectively. Prove that
\[ GA_1 \cdot GA_2 \cdot GA_3 \cdot GA_ \cdot4 \leq GA'_1 \cdot GA'_2 \cdot GA'_3 \cdot GA'_4\]
and
\[ \frac{1}{GA'_1} \plus{} \frac{1}{GA'_2} \plus{} \frac{1}{GA'_3} \plus{} \frac{1}{GA'_4} \leq \frac{1}{GA_1} \plus{} \frac{1}{GA_2} \plus{} \frac{1}{GA_3} \plus{} \frac{1}{GA_4}.\]
2003 District Olympiad, 1
Find all functions $\displaystyle f : \mathbb N^\ast \to \mathbb N^\ast$ ($\displaystyle N^\ast = \{ 1,2,3,\ldots \}$) with the property that, for all $\displaystyle n \geq 1$, \[ f(1) + f(2) + \ldots + f(n) \] is a perfect cube $\leq n^3$.
[i]Dinu Teodorescu[/i]
1961 IMO, 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 $\epsilon$?
2000 Harvard-MIT Mathematics Tournament, 8
Let $\vec{v_1},\vec{v_2},\vec{v_3},\vec{v_4}$ and $\vec{v_5}$ be vectors in three dimensions. Show that for some $i,j$ in $1,2,3,4,5$, $\vec{v_i}\cdot \vec{v_j}\ge 0$.
1979 Czech And Slovak Olympiad IIIA, 2
Given a cuboid $Q$ with dimensions $a, b, c$, $a < b < c$. Find the length of the edge of a cube $K$ , which has parallel faces and a common center with the given cuboid so that the volume of the difference of the sets $Q \cup K$ and $Q \cap K$ is minimal.
1991 Arnold's Trivium, 16
What fraction of a $5$-dimensional cube is the volume of the inscribed sphere? What fraction is it of a $10$-dimensional cube?