Found problems: 2265
1981 All Soviet Union Mathematical Olympiad, 321
A number is written in the each vertex of a cube. It is allowed to add one to two numbers written in the ends of one edge. Is it possible to obtain the cube with all equal numbers if the numbers were initially as on the pictures:
2007 Princeton University Math Competition, 2
A black witch's hat is in the classic shape of a cone on top of a circular brim. The cone has a slant height of $18$ inches and a base radius of $3$ inches. The brim has a radius of $5$ inches. What is the total surface area of the hat?
1972 IMO Longlists, 5
Given a pyramid whose base is an $n$-gon inscribable in a circle, let $H$ be the projection of the top vertex of the pyramid to its base. Prove that the projections of $H$ to the lateral edges of the pyramid lie on a circle.
2009 Germany Team Selection Test, 1
Consider cubes of edge length 5 composed of 125 cubes of edge length 1 where each of the 125 cubes is either coloured black or white. A cube of edge length 5 is called "big", a cube od edge length is called "small". A posititve integer $ n$ is called "representable" if there is a big cube with exactly $ n$ small cubes where each row of five small cubes has an even number of black cubes whose centres lie on a line with distances $ 1,2,3,4$ (zero counts as even number).
(a) What is the smallest and biggest representable number?
(b) Construct 45 representable numbers.
2012 District Olympiad, 4
Consider a tetrahedron $ABCD$ in which $AD \perp BC$ and $AC \perp BD$. We denote by $E$ and $F$ the projections of point $B$ on the lines $AD$ and $AC$, respectively. If $M$ and $N$ are the midpoints of the segments $[AB]$ and $[CD]$, respectively, show that $MN \perp EF$
1986 All Soviet Union Mathematical Olympiad, 440
Consider all the tetrahedrons $AXBY$, circumscribed around the sphere. Let $A$ and $B$ points be fixed. Prove that the sum of angles in the non-plane quadrangle $AXBY$ doesn't depend on points $X$ and $Y$ .
1985 Bundeswettbewerb Mathematik, 3
From a point in space, $n$ rays are issuing, whereas the angle among any two of these rays is at least $30^{\circ}$. Prove that $n < 59$.
2003 AMC 10, 3
A solid box is $ 15$ cm by $ 10$ cm by $ 8$ cm. A new solid is formed by removing a cube $ 3$ cm on a side from each corner of this box. What percent of the original volume is removed?
$ \textbf{(A)}\ 4.5 \qquad
\textbf{(B)}\ 9 \qquad
\textbf{(C)}\ 12 \qquad
\textbf{(D)}\ 18 \qquad
\textbf{(E)}\ 24$
1998 Harvard-MIT Mathematics Tournament, 2
A cube with sides 1m in length is filled with water, and has a tiny hole through which the water drains into a cylinder of radius $1\text{ m}$. If the water level in the cube is falling at a rate of $1 \text{ cm/s}$, at what rate is the water level in the cylinder rising?
2004 AMC 12/AHSME, 22
Three mutually tangent spheres of radius $ 1$ rest on a horizontal plane. A sphere of radius $ 2$ rests on them. What is the distance from the plane to the top of the larger sphere?
$ \textbf{(A)}\ 3 \plus{} \frac {\sqrt {30}}{2} \qquad \textbf{(B)}\ 3 \plus{} \frac {\sqrt {69}}{3} \qquad \textbf{(C)}\ 3 \plus{} \frac {\sqrt {123}}{4}\qquad \textbf{(D)}\ \frac {52}{9}\qquad \textbf{(E)}\ 3 \plus{} 2\sqrt2$
2012 Kyoto University Entry Examination, 2
Given a regular tetrahedron $OABC$. Take points $P,\ Q,\ R$ on the sides $OA,\ OB,\ OC$ respectively. Note that $P,\ Q,\ R$ are different from the vertices of the tetrahedron $OABC$. If $\triangle{PQR}$ is an equilateral triangle, then prove that three sides $PQ,\ QR,\ RP$ are pararell to three sides $AB,\ BC,\ CA$ respectively.
30 points
1973 Polish MO Finals, 3
A polyhedron $W$ has the following properties:
(i) It possesses a center of symmetry.
(ii) The section of $W$ by a plane passing through the center of symmetry and one of its edges is always a parallelogram.
(iii) There is a vertex of $W$ at which exactly three edges meet.
Prove that $W$ is a parallelepiped.
2010 Princeton University Math Competition, 8
There is a point source of light in an empty universe. What is the minimum number of solid balls (of any size) one must place in space so that any light ray emanating from the light source intersects at least one ball?
1963 Poland - Second Round, 6
From the point $ S $ of space arise $ 3 $ half-lines: $ SA $, $ SB $ and $ SC $, none of which is perpendicular to both others. Through each of these rays, a plane is drawn perpendicular to the plane containing the other two rays. Prove that the drawn planes intersect along one line $ d $.
1979 Bulgaria National Olympiad, Problem 2
Points $P,Q,R,S$ are taken on respective edges $AC$, $AB$, $BD$, and $CD$ of a tetrahedron $ABCD$ so that $PR$ and $QS$ intersect at point $N$ and $PS$ and $QR$ intersect at point $M$. The line $MN$ meets the plane $ABC$ at point $L$. Prove that the lines $AL$, $BP$, and $CQ$ are concurrent.
1986 IMO Shortlist, 21
Let $ABCD$ be a tetrahedron having each sum of opposite sides equal to $1$. Prove that
\[r_A + r_B + r_C + r_D \leq \frac{\sqrt 3}{3}\]
where $r_A, r_B, r_C, r_D$ are the inradii of the faces, equality holding only if $ABCD$ is regular.
2020 Jozsef Wildt International Math Competition, W25
In the Crelle $[ABCD]$ tetrahedron, we note with $A',B',C',A'',B'',C''$ the tangent points of the hexatangent sphere $\varphi(J,\rho)$, associated with the tetrahedron, with the edges $|BC|,|CA|,|AB|,|DA|,|DB|,|DC|$. Show that these inequalities occur:
a)
$$2\sqrt3R\ge6\rho\ge A'A''+B'B''+C'C''\ge6\sqrt3r$$
b)
$$4R^2\ge12\rho^2\ge(A'A'')^2+(B'B'')^2+(C'C'')^2\ge36r^2$$
c)
$$\frac{8R^3}{3\sqrt3}\ge8\rho^3\ge A'A''\cdot B'B''\cdot C'C''\ge24\sqrt3r^3$$
where $r,R$ is the length of the radius of the sphere inscribed and respectively circumscribed to the tetrahedron.
[i]Proposed by Marius Olteanu[/i]
1990 Brazil National Olympiad, 3
Each face of a tetrahedron is a triangle with sides $a, b,$c and the tetrahedon has circumradius 1. Find $a^2 + b^2 + c^2$.
Denmark (Mohr) - geometry, 2000.2
Three identical spheres fit into a glass with rectangular sides and bottom and top in the form of regular hexagons such that every sphere touches every side of the glass. The glass has volume $108$ cm$^3$. What is the sidelength of the bottom?
[img]https://1.bp.blogspot.com/-hBkYrORoBHk/XzcDt7B83AI/AAAAAAAAMXs/P5PGKTlNA7AvxkxMqG-qxqDVc9v9cU0VACLcBGAsYHQ/s0/2000%2BMohr%2Bp2.png[/img]
2011 Sharygin Geometry Olympiad, 25
Three equal regular tetrahedrons have the common center. Is it possible that all faces of the polyhedron that forms their intersection are equal?
2012 AIME Problems, 5
In the accompanying figure, the outer square has side length 40. A second square S' of side length 15 is constructed inside S with the same center as S and with sides parallel to those of S. From each midpoint of a side of S, segments are drawn to the two closest vertices of S'. The result is a four-pointed starlike figure inscribed in S. The star figure is cut out and then folded to form a pyramid with base S'. Find the volume of this pyramid.
[asy]
draw((0,0)--(8,0)--(8,8)--(0,8)--(0,0));
draw((2.5,2.5)--(4,0)--(5.5,2.5)--(8,4)--(5.5,5.5)--(4,8)--(2.5,5.5)--(0,4)--(2.5,2.5)--(5.5,2.5)--(5.5,5.5)--(2.5,5.5)--(2.5,2.5));
[/asy]
1995 Rioplatense Mathematical Olympiad, Level 3, 3
Given a regular tetrahedron with edge $a$, its edges are divided into $n$ equal segments, thus obtaining $n + 1$ points: $2$ at the ends and $n - 1$ inside. The following set of planes is considered:
$\bullet$ those that contain the faces of the tetrahedron, and
$\bullet$ each of the planes parallel to a face of the tetrahedron and containing at least one of the points determined above.
Now all those points $P$ that belong (simultaneously) to four planes of that set are considered. Determine the smallest positive natural $n$ so that among those points $P$ the eight vertices of a square-based rectangular parallelepiped can be chosen.
1956 Moscow Mathematical Olympiad, 331
Given a closed broken line $A_1A_2A_3...A_n$ in space and a plane intersecting all its segments, $A_1A_2$ at $B_1, A_2A_3$ at $B_2$ ,$... $, $A_nA_1$ at $B_n$, prove that
$$\frac{A_1B_1}{B_1A_2}\cdot \frac{A_2B_2}{B_2A_3}\cdot \frac{A_3B_3}{B_3A_4}\cdot ...\cdot \frac{A_nB_n}{B_nA_1}= 1$$.
2024 Euler Olympiad, Round 1, 9
Ants, named Anna and Bob, are located at vertices \(A\) and \(B\) respectively of a cube \(ABCD A_1 B_1 C_1 D_1\), with a sugar cube placed at vertex \(C_1\). It is known that Bob can move at a speed of $20$ meters per minute. Determine the minimum speed in integer meters per minute that Anna must be able to travel in order to reach the sugar cube at \(C_1\) before Bob.
[i]Proposed by Tamar Turashvili, Georgia [/i]
2004 Germany Team Selection Test, 1
Let $a_{ij}$ $i=1,2,3$; $j=1,2,3$ be real numbers such that $a_{ij}$ is positive for $i=j$ and negative for $i\neq j$.
Prove the existence of positive real numbers $c_{1}$, $c_{2}$, $c_{3}$ such that the numbers \[a_{11}c_{1}+a_{12}c_{2}+a_{13}c_{3},\qquad a_{21}c_{1}+a_{22}c_{2}+a_{23}c_{3},\qquad a_{31}c_{1}+a_{32}c_{2}+a_{33}c_{3}\] are either all negative, all positive, or all zero.
[i]Proposed by Kiran Kedlaya, USA[/i]