Found problems: 2265
IV Soros Olympiad 1997 - 98 (Russia), 10.11
A plane intersecting a unit cube divides it into two polyhedra. It is known that for each polyhedron the distance between any two points of it does not exceeds $\frac32$ m. What can be the cross-sectional area of a cube drawn by a plane?
2011 Mediterranean Mathematics Olympiad, 3
A regular tetrahedron of height $h$ has a tetrahedron of height $xh$ cut off by a plane parallel to the base. When the remaining frustrum is placed on one of its slant faces on a horizontal plane, it is just on the point of falling over. (In other words, when the remaining frustrum is placed on one of its slant faces on a horizontal plane, the projection of the center of gravity G of the frustrum is a point of the minor base of this slant face.)
Show that $x$ is a root of the equation $x^3 + x^2 + x = 2$.
1988 Austrian-Polish Competition, 6
Three rays $h_1,h_2,h_3$ emanating from a point $O$ are given, not all in the same plane. Show that if for any three points $A_1,A_2,A_3$ on $h_1,h_2,h_3$ respectively, distinct from $O$, the triangle $A_1A_2A_3$ is acute-angled, then the rays $h_1,h_2,h_3$ are pairwise orthogonal.
1990 Tournament Of Towns, (255) 3
(a) Some vertices of a dodecahedron are to be marked so that each face contains a marked vertex. What is the smallest number of marked vertices for which this is possible?
(b) Answer the same question, but for an icosahedron.
(G. Galperin, Moscow)
(Recall that a dodecahedron has $12$ pentagonal faces which meet in threes at each vertex, while an icosahedron has $20$ triangular faces which meet in fives at each vertex.)
1902 Eotvos Mathematical Competition, 2
Let $S$ be a given sphere with center $O$ and radius $r$. Let $P$ be any point outside then sphere $S$, and let $S'$ be the sphere with center $P$ and radius $PO$. Denote by $F$ the area of the surface of the part of $S'$ that lies inside $S$. Prove that $F$ is independent of the particular point $P$ chosen.
IV Soros Olympiad 1997 - 98 (Russia), 11.9
Cut pyramid $ABCD$ into $8$ equal and similar pyramids, if:
a) $AB = BC = CD$, $\angle ABC =\angle BCD = 90^o$, dihedral angle at edge $BC$ is right
b) all plane angles at vertex $B$ are right and $AB = BC = BD\sqrt2$.
Note. Whether there are other types of triangular pyramids that can be cut into any number similar to the original pyramids (their number is not necessarily $8$ and the pyramids are not necessarily equal to each other) is currently unknown
2020 USOMO, 2
An empty $2020 \times 2020 \times 2020$ cube is given, and a $2020 \times 2020$ grid of square unit cells is drawn on each of its six faces. A [i]beam[/i] is a $1 \times 1 \times 2020$ rectangular prism. Several beams are placed inside the cube subject to the following conditions:
[list=]
[*]The two $1 \times 1$ faces of each beam coincide with unit cells lying on opposite faces of the cube. (Hence, there are $3 \cdot {2020}^2$ possible positions for a beam.)
[*]No two beams have intersecting interiors.
[*]The interiors of each of the four $1 \times 2020$ faces of each beam touch either a face of the cube or the interior of the face of another beam.
[/list]
What is the smallest positive number of beams that can be placed to satisfy these conditions?
[i]Proposed by Alex Zhai[/i]
1977 IMO Longlists, 56
The four circumcircles of the four faces of a tetrahedron have equal radii. Prove that the four faces of the tetrahedron are congruent triangles.
1967 IMO, 2
Prove that a tetrahedron with just one edge length greater than $1$ has volume at most $ \frac{1}{8}.$
2002 Flanders Math Olympiad, 4
A lamp is situated at point $A$ and shines inside the cube. A (massive) square is hung on the midpoints of the 4 vertical faces. What's the area of its shadow?
[img]http://www.mathlinks.ro/Forum/album_pic.php?pic_id=285[/img]
2010 Iran MO (3rd Round), 2
[b]rolling cube[/b]
$a$,$b$ and $c$ are natural numbers. we have a $(2a+1)\times (2b+1)\times (2c+1)$ cube. this cube is on an infinite plane with unit squares. you call roll the cube to every side you want. faces of the cube are divided to unit squares and the square in the middle of each face is coloured (it means that if this square goes on a square of the plane, then that square will be coloured.)
prove that if any two of lengths of sides of the cube are relatively prime, then we can colour every square in plane.
time allowed for this question was 1 hour.
2006 Austrian-Polish Competition, 3
$ABCD$ is a tetrahedron.
Let $K$ be the center of the incircle of $CBD$.
Let $M$ be the center of the incircle of $ABD$.
Let $L$ be the gravycenter of $DAC$.
Let $N$ be the gravycenter of $BAC$.
Suppose $AK$, $BL$, $CM$, $DN$ have one common point.
Is $ABCD$ necessarily regular?
2007 Bundeswettbewerb Mathematik, 3
A set $ E$ of points in the 3D space let $ L(E)$ denote the set of all those points which lie on lines composed of two distinct points of $ E.$ Let $ T$ denote the set of all vertices of a regular tetrahedron. Which points are in the set $ L(L(T))?$
2013 Putnam, 2
Let $C=\bigcup_{N=1}^{\infty}C_N,$ where $C_N$ denotes the set of 'cosine polynomials' of the form \[f(x)=1+\sum_{n=1}^Na_n\cos(2\pi nx)\] for which:
(i) $f(x)\ge 0$ for all real $x,$ and
(ii) $a_n=0$ whenever $n$ is a multiple of $3.$
Determine the maximum value of $f(0)$ as $f$ ranges through $C,$ and prove that this maximum is attained.
1997 Tournament Of Towns, (536) 1
A cube is cut into 99 smaller cubes, exactly 98 of which are unit cubes. Find the volume of the original cube.
(V Proizvolov)
1987 IMO Shortlist, 4
Let $ABCDEFGH$ be a parallelepiped with $AE \parallel BF \parallel CG \parallel DH$. Prove the inequality
\[AF + AH + AC \leq AB + AD + AE + AG.\]
In what cases does equality hold?
[i]Proposed by France.[/i]
Denmark (Mohr) - geometry, 2005.1
This figure is cut out from a sheet of paper. Folding the sides upwards along the dashed lines, one gets a (non-equilateral) pyramid with a square base. Calculate the area of the base.
[img]https://1.bp.blogspot.com/-lPpfHqfMMRY/XzcBIiF-n2I/AAAAAAAAMW8/nPs_mLe5C8srcxNz45Wg-_SqHlRAsAmigCLcBGAsYHQ/s0/2005%2BMohr%2Bp1.png[/img]
1996 National High School Mathematics League, 10
Give two congruent regular triangular pyramids, stick their bottom surfaces together. Then ,it becomes a hexahedron with all dihedral angles equal. The length of the shortest edge of the hexahedron is $2$. Then, the furthest distance between two vertexes is________.
1996 Czech and Slovak Match, 3
The base of a regular quadrilateral pyramid $\pi$ is a square with side length $2a$ and its lateral edge has length a$\sqrt{17}$. Let $M$ be a point inside the pyramid. Consider the five pyramids which are similar to $\pi$ , whose top vertex is at $M$ and whose bases lie in the planes of the faces of $\pi$ . Show that the sum of the surface areas of these five pyramids is greater or equal to one fifth the surface of $\pi$ , and find for which $M$ equality holds.
1979 Austrian-Polish Competition, 8
Let $A,B,C,D$ be four points in space, and $M$ and $N$ be the midpoints of $AC$ and $BD$, respectively. Show that $$AB^2+BC^2+CD^2+DA^2 = AC^2+BD^2+4MN^2$$
2008 China Western Mathematical Olympiad, 2
Given $ x,y,z\in (0,1)$ satisfying that
$ \sqrt{\frac{1 \minus{} x}{yz}} \plus{} \sqrt{\frac{1 \minus{} y}{xz}} \plus{} \sqrt{\frac{1 \minus{} z}{xy}} \equal{} 2$.
Find the maximum value of $ xyz$.
2015 CCA Math Bonanza, I13
Let $ABCD$ be a tetrahedron such that $AD \perp BD$, $BD \perp CD$, $CD \perp AD$ and $AD=10$, $BD=15$, $CD=20$. Let $E$ and $F$ be points such that $E$ lies on $BC$, $DE \perp BC$, and $ADEF$ is a rectangle. If $S$ is the solid consisting of all points inside $ABCD$ but outside $FBCD$, compute the volume of $S$.
[i]2015 CCA Math Bonanza Individual Round #13[/i]
2016 Fall CHMMC, 12
For a positive real number $a$, let $C$ be the cube with vertices at $(\pm a, \pm a, \pm a)$ and let $T$ be the tetrahedron with vertices at $(2a,2a,2a),(2a, -2a, -2a),(-2a, 2a, -2a),(-2a, -2a, -2a)$. If the intersection of $T$ and $C$ has volume $ka^3$ for some $k$, find $k$.
2018 HMNT, 8
Tessa has a unit cube, on which each vertex is labeled by a distinct integer between 1 and 8 inclusive. She also has a deck of 8 cards, 4 of which are black and 4 of which are white. At each step she draws a card from the deck, and[list][*]if the card is black, she simultaneously replaces the number on each vertex by the sum of the three numbers on vertices that are distance 1 away from the vertex;[*]if the card is white, she simultaneously replaces the number on each vertex by the sum of the three numbers on vertices that are distance $\sqrt2$ away from the vertex.[/list]When Tessa finishes drawing all cards of the deck, what is the maximum possible value of a number that is on the cube?
2007 Harvard-MIT Mathematics Tournament, 1
A cube of edge length $s>0$ has the property that its surface area is equal to the sum of its volume and five times its edge length. Compute all possible values of $s$.