Found problems: 2265
1974 Chisinau City MO, 78
Each point of the sphere of radius $r\ge 1$ is colored in one of $n$ colors ($n \ge 2$), and for each color there is a point on the sphere colored in this color. Prove that there are points $A_i$, $B_i$, $i= 1, ..., n$ on the sphere such that the colors of the points $A_1, ..., A_n$ are pairwise different and the color of the point $B_i$ at a distance of $1$ from $A_i$ is different from the color of the point $A_1, i= 1, ..., n$
1993 Vietnam National Olympiad, 1
The tetrahedron $ABCD$ has its vertices on the fixed sphere $S$. Prove that $AB^{2}+AC^{2}+AD^{2}-BC^{2}-BD^{2}-CD^{2}$ is minimum iff $AB\perp AC,AC\perp AD,AD\perp AB$.
1997 All-Russian Olympiad, 4
An $n\times n\times n$ cube is divided into unit cubes. We are given a closed non-self-intersecting polygon (in space), each of whose sides joins the centers of two unit cubes sharing a common face. The faces of unit cubes which intersect the polygon are said to be distinguished. Prove that the edges of the unit cubes may be colored in two colors so that each distinguished face has an odd number of edges of each color, while each nondistinguished face has an even number of edges of each color.
[i]M. Smurov[/i]
2009 Spain Mathematical Olympiad, 3
Some edges are painted in red. We say that a coloring of this kind is [i]good[/i], if for each vertex of the polyhedron, there exists an edge which concurs in that vertex and is not painted red. Moreover, we say that a coloring where some of the edges of a regular polyhedron is [i]completely good[/i], if in addition to being [i]good[/i], no face of the polyhedron has all its edges painted red. What regular polyhedrons is equal the maximum number of edges that can be painted in a [i]good[/i] color and a [i]completely good[/i]? Explain your answer.
2017 CCA Math Bonanza, I11
$480$ $1$ cm unit cubes are used to build a block measuring $6$ cm by $8$ cm by $10$ cm. A tiny ant then chews his way in a straight line from one vertex of the block to the furthest vertex. How many cubes does the ant pass through? The ant is so tiny that he does not "pass through" cubes if he is merely passing through where their edges or vertices meet.
[i]2017 CCA Math Bonanza Individual Round #11[/i]
2007 Princeton University Math Competition, 6
Joe has $1729$ randomly oriented and randomly arranged unit cubes, which are initially unpainted. He makes two cubes of sidelengths $9$ and $10$ or of sidelengths $1$ and $12$ (randomly chosen). These cubes are dipped into white paint. Then two more cubes of sidelengths $1$ and $12$ or $9$ and $10$ are formed from the same unit cubes, again randomly oriented and randomly arranged, and dipped into paint. Joe continues this process until every side of every unit cube is painted. After how many times of doing this is the expected number of painted faces closest to half of the total?
1953 Moscow Mathematical Olympiad, 240
Let $AB$ and $A_1B_1$ be two skew segments, $O$ and $O_1$ their respective midpoints. Prove that $OO_1$ is shorter than a half sum of $AA_1$ and $BB_1$.
2018 Stanford Mathematics Tournament, 7
Two equilateral triangles $ABC$ and $DEF$, each with side length $1$, are drawn in $2$ parallel planes such that when one plane is projected onto the other, the vertices of the triangles form a regular hexagon $AF BDCE$. Line segments $AE$, $AF$, $BF$, $BD$, $CD,$ and $CE$ are drawn, and suppose that each of these segments also has length $1$. Compute the volume of the resulting solid that is formed.
1999 Harvard-MIT Mathematics Tournament, 10
Pyramid $EARLY$ is placed in $(x,y,z)$ coordinates so that $E=(10,10,0),A=(10,-10,0)$, $R=(-10,-10,0)$, $L=(-10,10,0)$, and $Y=(0,0,10)$. Tunnels are drilled through the pyramid in such a way that one can move from $(x,y,z)$ to any of the $9$ points $(x,y,z-1)$, $(x\pm 1,y,z-1)$, $(x,y\pm 1, z-1)$, $(x\pm 1, y\pm 1, z-1)$. Sean starts at $Y$ and moves randomly down to the base of the pyramid, choosing each of the possible paths with probability $\dfrac{1}{9}$. What is the probability that he ends up at the point $(8,9,0)$?
1982 Kurschak Competition, 1
A cube of integral dimensions is given in space so that all four vertices of one of the faces are lattice points. Prove that the other four vertices are also lattice points.
1966 IMO Longlists, 3
A regular triangular prism has the altitude $h,$ and the two bases of the prism are equilateral triangles with side length $a.$ Dream-holes are made in the centers of both bases, and the three lateral faces are mirrors. Assume that a ray of light, entering the prism through the dream-hole in the upper base, then being reflected once by any of the three mirrors, quits the prism through the dream-hole in the lower base. Find the angle between the upper base and the light ray at the moment when the light ray entered the prism, and the length of the way of the light ray in the interior of the prism.
2015 AMC 10, 21
Tetrahedron $ABCD$ has $AB=5$, $AC=3$, $BC=4$, $BD=4$, $AD=3$, and $CD=\tfrac{12}5\sqrt2$. What is the volume of the tetrahedron?
$\textbf{(A) }3\sqrt2\qquad\textbf{(B) }2\sqrt5\qquad\textbf{(C) }\dfrac{24}5\qquad\textbf{(D) }3\sqrt3\qquad\textbf{(E) }\dfrac{24}5\sqrt2$
2010 All-Russian Olympiad Regional Round, 11.6
At the base of the quadrangular pyramid $SABCD$ lies the parallelogram $ABCD$. Prove that for any point $O$ inside the pyramid, the sum of the volumes of the tetrahedra $OSAB$ and $OSCD$ is equal to the sum of the volumes of the tetrahedra $OSBC$ and $OSDA$ .
1996 Denmark MO - Mohr Contest, 3
This year's gift idea from BabyMath consists of a series of nine colored plastic containers of decreasing size, alternating in shape like a cube and a sphere. All containers can open and close with a convenient hinge, and each container can hold just about anything next in line. The largest and smallest container are both cubes. Determine the relationship between the edge lengths of these cubes.
2003 National Olympiad First Round, 10
Which of the followings is congruent (in $\bmod{25}$) to the sum in of integers $0\leq x < 25$ such that $x^3+3x^2-2x+4 \equiv 0 \pmod{25}$?
$
\textbf{(A)}\ 3
\qquad\textbf{(B)}\ 4
\qquad\textbf{(C)}\ 17
\qquad\textbf{(D)}\ 22
\qquad\textbf{(E)}\ \text{None of the preceding}
$
1983 IMO Longlists, 41
Let $E$ be the set of $1983^3$ points of the space $\mathbb R^3$ all three of whose coordinates are integers between $0$ and $1982$ (including $0$ and $1982$). A coloring of $E$ is a map from $E$ to the set {red, blue}. How many colorings of $E$ are there satisfying the following property: The number of red vertices among the $8$ vertices of any right-angled parallelepiped is a multiple of $4$ ?
2014 AMC 10, 19
Where is AMC 10a No.19? Thanks
2004 Italy TST, 1
At the vertices $A, B, C, D, E, F, G, H$ of a cube, $2001, 2002, 2003, 2004, 2005, 2008, 2007$ and $2006$ stones respectively are placed. It is allowed to move a stone from a vertex to each of its three neighbours, or to move a stone to a vertex from each of its three neighbours. Which of the following arrangements of stones at $A, B, \ldots , H$ can be obtained?
$(\text{a})\quad 2001, 2002, 2003, 2004, 2006, 2007, 2008, 2005;$
$(\text{b})\quad 2002, 2003, 2004, 2001, 2006, 2005, 2008, 2007;$
$(\text{c})\quad 2004, 2002, 2003, 2001, 2005, 2008, 2007, 2006.$
2011 District Round (Round II), 4
Let $M$ be a set of six distinct positive integers whose sum is $60$. These numbers are written on the faces of a cube, one number to each face. A [i]move[/i] consists of choosing three faces of the cube that share a common vertex and adding $1$ to the numbers on those faces. Determine the number of sets $M$ for which it’s possible, after a finite number of moves, to produce a cube all of whose sides have the same number.
1956 Czech and Slovak Olympiad III A, 2
In a given plane $\varrho$ consider a convex quadrilateral $ABCD$ and denote $E=AC\cap BD.$ Moreover, consider a point $V\notin\varrho$. On rays $VA,VB,VC,VD$ find points $A',B',C',D'$ respectively such that $E,A',B',C',D'$ are coplanar and $A'B'C'D'$ is a parallelogram. Discuss conditions of solvability.
1987 AIME Problems, 2
What is the largest possible distance between two points, one on the sphere of radius 19 with center $(-2, -10, 5)$ and the other on the sphere of radius 87 with center $(12, 8, -16)$?
2006 Sharygin Geometry Olympiad, 25
In the tetrahedron $ABCD$ , the dihedral angles at the $BC, CD$, and $DA$ edges are equal to $\alpha$, and for the remaining edges equal to $\beta$. Find the ratio $AB / CD$.
1992 Romania Team Selection Test, 10
In a tetrahedron $VABC$, let $I$ be the incenter and $A',B',C'$ be arbitrary points on the edges $AV,BV,CV$, and let $S_a,S_b,S_c,S_v$ be the areas of triangles $VBC,VAC,VAB,ABC$, respectively. Show that points $A',B',C',I$ are coplanar if and only if $\frac{AA'}{A'V}S_a +\frac{BB'}{B'V}S_b +\frac{CC'}{C'V}S_c = S_v$
1973 IMO Shortlist, 5
A circle of radius 1 is located in a right-angled trihedron and touches all its faces. Find the locus of centers of such circles.
PEN F Problems, 9
Prove that every positive rational number can be represented in the form \[\frac{a^{3}+b^{3}}{c^{3}+d^{3}}\] for some positive integers $a, b, c$, and $d$.