Found problems: 2265
2004 Alexandru Myller, 2
The medians from $ A $ to the faces $ ABC,ABD,ACD $ of a tetahedron $ ABCD $ are pairwise perpendicular.
Show that the edges from $ A $ have equal lengths.
[i]Dinu Șerbănescu[/i]
2005 Colombia Team Selection Test, 1
Let $a,b,c$ be integers such that $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}=3$ prove that $abc$ is a perfect cube!
1964 Poland - Second Round, 5
Given is a trihedral angle with edges $ SA $, $ SB $, $ SC $, all plane angles of which are acute, and the dihedral angle at edge $ SA $ is right. Prove that the section of this triangle with any plane perpendicular to any edge, at a point different from the vertex $ S $, is a right triangle.
2011 AIME Problems, 13
A cube with side length 10 is suspended above a plane. The vertex closest to the plane is labelled $A$. The three vertices adjacent to vertex $A$ are at heights 10, 11, and 12 above the plane. The distance from vertex $A$ to the plane can be expressed as $\tfrac{r-\sqrt{s}}{t}$, where $r$, $s$, and $t$ are positive integers, and $r+s+t<1000$. Find $r+s+t$.
1965 Spain Mathematical Olympiad, 6
We have an empty equilateral triangle with length of a side $l$. We put the triangle, horizontally, over a sphere of radius $r$. Clearly, if the triangle is small enough, the triangle is held by the sphere. Which is the distance between any vertex of the triangle and the centre of the sphere (as a function of $l$ and $r$)?
2006 National Olympiad First Round, 4
There are $27$ unit cubes. We are marking one point on each of the two opposing faces, two points on each of the other two opposing faces, and three points on each of the remaining two opposing faces of each cube. We are constructing a $3\times 3 \times 3$ cube with these $27$ cubes. What is the least number of marked points on the faces of the new cube?
$
\textbf{(A)}\ 54
\qquad\textbf{(B)}\ 60
\qquad\textbf{(C)}\ 72
\qquad\textbf{(D)}\ 90
\qquad\textbf{(E)}\ 96
$
1965 Bulgaria National Olympiad, Problem 4
In the space there are given crossed lines $s$ and $t$ such that $\angle(s,t)=60^\circ$ and a segment $AB$ perpendicular to them. On $AB$ it is chosen a point $C$ for which $AC:CB=2:1$ and the points $M$ and $N$ are moving on the lines $s$ and $t$ in such a way that $AM=2BN$. The angle between vectors $\overrightarrow{AM}$ and $\overrightarrow{BM}$ is $60^\circ$. Prove that:
(a) the segment $MN$ is perpendicular to $t$;
(b) the plane $\alpha$, perpendicular to $AB$ in point $C$, intersects the plane $CMN$ on fixed line $\ell$ with given direction in respect to $s$;
(c) all planes passing by $ell$ and perpendicular to $AB$ intersect the lines $s$ and $t$ respectively at points $M$ and $N$ for which $AM=2BN$ and $MN\perp t$.
2004 Purple Comet Problems, 17
We want to paint some identically-sized cubes so that each face of each cube is painted a solid color and each cube is painted with six different colors. If we have seven different colors to choose from, how many distinguishable cubes can we produce?
2014 Contests, 3
A tetrahedron $ABCD$ with acute-angled faces is inscribed in a sphere with center $O$. A line passing through $O$ perpendicular to plane $ABC$ crosses the sphere at point $D'$ that lies on the opposide side of plane $ABC$ than point $D$. Line $DD'$ crosses plane $ABC$ in point $P$ that lies inside the triangle $ABC$. Prove, that if $\angle APB=2\angle ACB$, then $\angle ADD'=\angle BDD'$.
1999 All-Russian Olympiad Regional Round, 11.4
A polyhedron is circumscribed around a sphere. Let's call its face [i]large [/i] if the projection of the sphere onto the plane of the face falls entirely within the face. Prove that there are no more than 6 large faces.
1966 IMO Shortlist, 23
Three faces of a tetrahedron are right triangles, while the fourth is not an obtuse triangle.
[i](a) [/i]Prove that a necessary and sufficient condition for the fourth face to be a right triangle is that at some vertex exactly two angles are right.
[i](b)[/i] Prove that if all the faces are right triangles, then the volume of the tetrahedron equals one -sixth the product of the three smallest edges not belonging to the same face.
1976 Bulgaria National Olympiad, Problem 5
It is given a tetrahedron $ABCD$ and a plane $\alpha$ intersecting the three edges passing through $D$. Prove that $\alpha$ divides the surface of the tetrahedron into two parts proportional to the volumes of the bodies formed if and only if $\alpha$ is passing through the center of the inscribed tetrahedron sphere.
2000 French Mathematical Olympiad, Exercise 2
Let $A,B,C$ be three distinct points in space, $(A)$ the sphere with center $A$ and radius $r$. Let $E$ be the set of numbers $R>0$ for which there is a sphere $(H)$ with center $H$ and radius $R$ such that $B$ and $C$ are outside the sphere, and the points of the sphere $(A)$ are strictly inside it.
(a) Suppose that $B$ and $C$ are on a line with $A$ and strictly outside $(A)$. Show that $E$ is nonempty and bounded, and determine its supremum in terms of the given data.
(b) Find a necessary and sufficient condition for $E$ to be nonempty and bounded
(c) Given $r$, compute the smallest possible supremum of $E$, if it exists.
2021 Purple Comet Problems, 8
Fiona had a solid rectangular block of cheese that measured $6$ centimeters from left to right, $5$ centimeters from front to back, and $4$ centimeters from top to bottom. Fiona took a sharp knife and sliced off a $1$ centimeter thick slice from the left side of the block and a $1$ centimeter slice from the right side of the block. After that, she sliced off a $1$ centimeter thick slice from the front side of the remaining block and a $1$ centimeter slice from the back side of the remaining block. Finally, Fiona sliced off a $1$ centimeter slice from the top of the remaining block and a $1$ centimeter slice from the bottom of the remaining block. Fiona now has $7$ blocks of cheese. Find the total surface area of those seven blocks of cheese measured in square centimeters.
VII Soros Olympiad 2000 - 01, 8.4
Paint the maximum number of vertices of the cube red so that you cannot select three of the red vertices that form an equilateral triangle.
2008 Harvard-MIT Mathematics Tournament, 12
Suppose we have an (infinite) cone $ \mathcal C$ with apex $ A$ and a plane $ \pi$. The intersection of $ \pi$ and $ \mathcal C$ is an ellipse $ \mathcal E$ with major axis $ BC$, such that $ B$ is closer to $ A$ than $ C$, and $ BC \equal{} 4$, $ AC \equal{} 5$, $ AB \equal{} 3$. Suppose we inscribe a sphere in each part of $ \mathcal C$ cut up by $ \mathcal E$ with both spheres tangent to $ \mathcal E$. What is the ratio of the radii of the spheres (smaller to larger)?
2020 HMIC, 3
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]
2012 Purple Comet Problems, 30
The diagram below shows four regular hexagons each with side length $1$ meter attached to the sides of a square. This figure is drawn onto a thin sheet of metal and cut out. The hexagons are then bent upward along the sides of the square so that $A_1$ meets $A_2$, $B_1$ meets $B_2$, $C_1$ meets $C_2$, and $D_1$ meets $D_2$. If the resulting dish is filled with water, the water will rise to the height of the corner where the $A_1$ and $A_2$ meet. there are relatively prime positive integers $m$ and $n$ so that the number of cubic meters of water the dish will hold is $\sqrt{\frac{m}{n}}$. Find $m+n$.
[asy]
/* File unicodetex not found. */
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */
import graph; size(7cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -4.3, xmax = 14.52, ymin = -8.3, ymax = 6.3; /* image dimensions */
draw((0,1)--(0,0)--(1,0)--(1,1)--cycle);
draw((1,1)--(1,0)--(1.87,-0.5)--(2.73,0)--(2.73,1)--(1.87,1.5)--cycle);
draw((0,1)--(1,1)--(1.5,1.87)--(1,2.73)--(0,2.73)--(-0.5,1.87)--cycle);
draw((0,0)--(1,0)--(1.5,-0.87)--(1,-1.73)--(0,-1.73)--(-0.5,-0.87)--cycle);
draw((0,1)--(0,0)--(-0.87,-0.5)--(-1.73,0)--(-1.73,1)--(-0.87,1.5)--cycle);
/* draw figures */
draw((0,1)--(0,0));
draw((0,0)--(1,0));
draw((1,0)--(1,1));
draw((1,1)--(0,1));
draw((1,1)--(1,0));
draw((1,0)--(1.87,-0.5));
draw((1.87,-0.5)--(2.73,0));
draw((2.73,0)--(2.73,1));
draw((2.73,1)--(1.87,1.5));
draw((1.87,1.5)--(1,1));
draw((0,1)--(1,1));
draw((1,1)--(1.5,1.87));
draw((1.5,1.87)--(1,2.73));
draw((1,2.73)--(0,2.73));
draw((0,2.73)--(-0.5,1.87));
draw((-0.5,1.87)--(0,1));
/* dots and labels */
dot((1.87,-0.5),dotstyle);
label("$C_1$", (1.72,-0.1), NE * labelscalefactor);
dot((1.87,1.5),dotstyle);
label("$B_2$", (1.76,1.04), NE * labelscalefactor);
dot((1.5,1.87),dotstyle);
label("$B_1$", (0.96,1.8), NE * labelscalefactor);
dot((-0.5,1.87),dotstyle);
label("$A_2$", (-0.26,1.78), NE * labelscalefactor);
dot((-0.87,1.5),dotstyle);
label("$A_1$", (-0.96,1.08), NE * labelscalefactor);
dot((-0.87,-0.5),dotstyle);
label("$D_2$", (-1.02,-0.18), NE * labelscalefactor);
dot((-0.5,-0.87),dotstyle);
label("$D_1$", (-0.22,-0.96), NE * labelscalefactor);
dot((1.5,-0.87),dotstyle);
label("$C_2$", (0.9,-0.94), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */
[/asy]
2003 AMC 12-AHSME, 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$
1966 AMC 12/AHSME, 10
If the sum of two numbers is 1 and their product is 1, then the sum of their cubes is:
$\text{(A)} \ 2 \qquad \text{(B)} \ -2-\frac{3i\sqrt{3}}{4} \qquad \text{(C)} \ 0 \qquad \text{(D)} \ -\frac{3i\sqrt{3}}{4} \qquad \text{(E)} \ -2$
1982 IMO Longlists, 10
Let $r_1, \ldots , r_n$ be the radii of $n$ spheres. Call $S_1, S_2, \ldots , S_n$ the areas of the set of points of each sphere from which one cannot see any point of any other sphere. Prove that
\[\frac{S_1}{r_1^2} + \frac{S_2}{r_2^2}+\cdots+\frac{S_n}{r_n^2} = 4 \pi.\]
1993 AMC 8, 17
Square corners, $5$ units on a side, are removed from a $20$ unit by $30$ unit rectangular sheet of cardboard. The sides are then folded to form an open box. The surface area, in square units, of the interior of the box is
[asy]
fill((0,0)--(20,0)--(20,5)--(0,5)--cycle,lightgray);
fill((20,0)--(20+5*sqrt(2),5*sqrt(2))--(20+5*sqrt(2),5+5*sqrt(2))--(20,5)--cycle,lightgray);
draw((0,0)--(20,0)--(20,5)--(0,5)--cycle);
draw((0,5)--(5*sqrt(2),5+5*sqrt(2))--(20+5*sqrt(2),5+5*sqrt(2))--(20,5));
draw((20+5*sqrt(2),5+5*sqrt(2))--(20+5*sqrt(2),5*sqrt(2))--(20,0));
draw((5*sqrt(2),5+5*sqrt(2))--(5*sqrt(2),5*sqrt(2))--(5,5),dashed);
draw((5*sqrt(2),5*sqrt(2))--(15+5*sqrt(2),5*sqrt(2)),dashed);
[/asy]
$\text{(A)}\ 300 \qquad \text{(B)}\ 500 \qquad \text{(C)}\ 550 \qquad \text{(D)}\ 600 \qquad \text{(E)}\ 1000$
2018 Hanoi Open Mathematics Competitions, 1
How many rectangles can be formed by the vertices of a cube? (Note: square is also a special rectangle).
A. $6$ B. $8$ C. $12$ D. $18$ E. $16$
2007 Pan African, 3
In a country, towns are connected by roads. Each town is directly connected to exactly three other towns. Show that there exists a town from which you can make a round-trip, without using the same road more than once, and for which the number of roads used is not divisible by $3$. (Not all towns need to be visited.)
1990 AIME Problems, 1
The increasing sequence $2,3,5,6,7,10,11,\ldots$ consists of all positive integers that are neither the square nor the cube of a positive integer. Find the 500th term of this sequence.