Found problems: 2265
1987 IMO Longlists, 38
Let $S_1$ and $S_2$ be two spheres with distinct radii that touch externally. The spheres lie inside a cone $C$, and each sphere touches the cone in a full circle. Inside the cone there are $n$ additional solid spheres arranged in a ring in such a way that each solid sphere touches the cone $C$, both of the spheres $S_1$ and $S_2$ externally, as well as the two neighboring solid spheres. What are the possible values of $n$?
[i]Proposed by Iceland.[/i]
2000 AMC 8, 22
A cube has edge length $2$. Suppose that we glue a cube of edge length $1$ on top of the big cube so that one of its faces rests entirely on the top face of the larger cube. The percent increase in the surface area (sides, top, and bottom) from the original cube to the new solid formed is closest to
[asy]
draw((0,0)--(2,0)--(3,1)--(3,3)--(2,2)--(0,2)--cycle);
draw((2,0)--(2,2));
draw((0,2)--(1,3));
draw((1,7/3)--(1,10/3)--(2,10/3)--(2,7/3)--cycle);
draw((2,7/3)--(5/2,17/6)--(5/2,23/6)--(3/2,23/6)--(1,10/3));
draw((2,10/3)--(5/2,23/6));
draw((3,3)--(5/2,3));
[/asy]
$\text{(A)}\ 10 \qquad \text{(B)}\ 15 \qquad \text{(C)}\ 17 \qquad \text{(D)}\ 21 \qquad \text{(E)}\ 25$
2019 CMIMC, 2
How many ways are there to color the vertices of a cube red, blue, or green such that no edge connects two vertices of the same color? Rotations and reflections are considered distinct colorings.
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.
2005 Romania National Olympiad, 3
Let the $ABCA'B'C'$ be a regular prism. The points $M$ and $N$ are the midpoints of the sides $BB'$, respectively $BC$, and the angle between the lines $AB'$ and $BC'$ is of $60^\circ$. Let $O$ and $P$ be the intersection of the lines $A'C$ and $AC'$, with respectively $B'C$ and $C'N$.
a) Prove that $AC' \perp (OPM)$;
b) Find the measure of the angle between the line $AP$ and the plane $(OPM)$.
[i]Mircea Fianu[/i]
2007 Middle European Mathematical Olympiad, 3
A tetrahedron is called a [i]MEMO-tetrahedron[/i] if all six sidelengths are different positive integers where one of them is $ 2$ and one of them is $ 3$. Let $ l(T)$ be the sum of the sidelengths of the tetrahedron $ T$.
(a) Find all positive integers $ n$ so that there exists a MEMO-Tetrahedron $ T$ with $ l(T)\equal{}n$.
(b) How many pairwise non-congruent MEMO-tetrahedrons $ T$ satisfying $ l(T)\equal{}2007$ exist? Two tetrahedrons are said to be non-congruent if one cannot be obtained from the other by a composition of reflections in planes, translations and rotations. (It is not neccessary to prove that the tetrahedrons are not degenerate, i.e. that they have a positive volume).
1977 AMC 12/AHSME, 23
If the solutions of the equation $x^2+px+q=0$ are the cubes of the solutions of the equation $x^2+mx+n=0$, then
$\textbf{(A) }p=m^3+3mn\qquad\textbf{(B) }p=m^3-3mn\qquad$
$\textbf{(C) }p+q=m^3\qquad\textbf{(D) }\left(\frac{m}{n}\right)^2=\frac{p}{q}\qquad \textbf{(E) }\text{none of these}$
1982 IMO Longlists, 13
A regular $n$-gonal truncated pyramid is circumscribed around a sphere. Denote the areas of the base and the lateral surfaces of the pyramid by $S_1, S_2$, and $S$, respectively. Let $\sigma$ be the area of the polygon whose vertices are the tangential points of the sphere and the lateral faces of the pyramid. Prove that
\[\sigma S = 4S_1S_2 \cos^2 \frac{\pi}{n}.\]
1967 Miklós Schweitzer, 5
Let $ f$ be a continuous function on the unit interval $ [0,1]$. Show that \[ \lim_{n \rightarrow \infty} \int_0^1... \int_0^1f(\frac{x_1+...+x_n}{n})dx_1...dx_n=f(\frac12)\] and \[ \lim_{n \rightarrow \infty} \int_0^1... \int_0^1f (\sqrt[n]{x_1...x_n})dx_1...dx_n=f(\frac1e).\]
1994 Abels Math Contest (Norwegian MO), 1a
In a half-ball of radius $3$ is inscribed a cylinder with base lying on the base plane of the half-ball, and another such cylinder with equal volume. If the base-radius of the first cylinder is $\sqrt3$, what is the base-radius of the other one?
1997 Vietnam Team Selection Test, 1
Let $ ABCD$ be a given tetrahedron, with $ BC \equal{} a$, $ CA \equal{} b$, $ AB \equal{} c$, $ DA \equal{} a_1$, $ DB \equal{} b_1$, $ DC \equal{} c_1$. Prove that there is a unique point $ P$ satisfying
\[ PA^2 \plus{} a_1^2 \plus{} b^2 \plus{} c^2 \equal{} PB^2 \plus{} b_1^2 \plus{} c^2 \plus{} a^2 \equal{} PC^2 \plus{} c_1^2 \plus{} a^2 \plus{} b^2 \equal{} PD^2 \plus{} a_1^2 \plus{} b_1^2 \plus{} c_1^2
\]
and for this point $ P$ we have $ PA^2 \plus{} PB^2 \plus{} PC^2 \plus{} PD^2 \ge 4R^2$, where $ R$ is the circumradius of the tetrahedron $ ABCD$. Find the necessary and sufficient condition so that this inequality is an equality.
1979 Chisinau City MO, 182
Prove that a section of a cube by a plane cannot be a regular pentagon.
2020 AMC 10, 2
Carl has $5$ cubes each having side length $1$, and Kate has $5$ cubes each having side length $2$. What is the total volume of the $10$ cubes?
$\textbf{(A) }24 \qquad \textbf{(B) }25 \qquad \textbf{(C) } 28\qquad \textbf{(D) } 40\qquad \textbf{(E) } 45$
1990 National High School Mathematics League, 15
In pyramid $M-ABCD$, bottom surface $ABCD$ is a square. $MA=MC,MA\perp AB$. If the area of $\triangle AMD$ is $1$, find the maximum value of radius of sphere that can be put inside the pyramid.
2023 IMC, 9
We say that a real number $V$ is [i]good[/i] if there exist two closed convex subsets $X$, $Y$ of the unit cube in $\mathbb{R}^3$, with volume $V$ each, such that for each of the three coordinate planes (that is, the planes spanned by any two of the three coordinate axes), the projections of $X$ and $Y$ onto that plane are disjoint.
Find $\sup \{V\mid V\ \text{is good}\}$.
2005 Sharygin Geometry Olympiad, 20
Let $I$ be the center of the sphere inscribed in the tetrahedron $ABCD, A ', B', C ', D'$ be the centers of the spheres circumscribed around the tetrahedra $IBCD, ICDA, IDAB, IABC$, respectively. Prove that the sphere circumscribed around $ABCD$ lies entirely inside the circumscribed around $A'B'C'D '$.
1985 Iran MO (2nd round), 3
Find the angle between two common sections of the page $2x+y-z=0$ and the cone $4x^2-y^2+3z^2=0.$
2003 Romania National Olympiad, 1
Let be a tetahedron $ OABC $ with $ OA\perp OB\perp OC\perp OA. $ Show that
$$ OH\le r\left( 1+\sqrt 3 \right) , $$
where $ H $ is the orthocenter of $ ABC $ and $ r $ is radius of the inscribed spere of $ OABC. $
[i]Valentin Vornicu[/i]
2006 Harvard-MIT Mathematics Tournament, 9
Four spheres, each of radius $r$, lie inside a regular tetrahedron with side length $1$ such that each sphere is tangent to three faces of the tetrahedron and to the other three spheres. Find $r$.
2010 Tournament Of Towns, 3
A $1\times 1\times 1$ cube is placed on an $8\times 8$ chessboard so that its bottom face coincides with a square of the chessboard. The cube rolls over a bottom edge so that the adjacent face now lands on the chessboard. In this way, the cube rolls around the chessboard, landing on each square at least once. Is it possible that a particular face of the cube never lands on the chessboard?
2006 Romania National Olympiad, 3
Let $ABCDA_1B_1C_1D_1$ be a cube and $P$ a variable point on the side $[AB]$. The perpendicular plane on $AB$ which passes through $P$ intersects the line $AC'$ in $Q$. Let $M$ and $N$ be the midpoints of the segments $A'P$ and $BQ$ respectively.
a) Prove that the lines $MN$ and $BC'$ are perpendicular if and only if $P$ is the midpoint of $AB$.
b) Find the minimal value of the angle between the lines $MN$ and $BC'$.
2008 Harvard-MIT Mathematics Tournament, 3
A $ 3\times3\times3$ cube composed of $ 27$ unit cubes rests on a horizontal plane. Determine the number of ways of selecting two distinct unit cubes from a $ 3\times3\times1$ block (the order is irrelevant) with the property that the line joining the centers of the two cubes makes a $ 45^\circ$ angle with the horizontal plane.
1935 Moscow Mathematical Olympiad, 015
Triangles $\vartriangle ABC$ and $\vartriangle A_1B_1C_1$ lie on different planes. Line $AB$ intersects line $A_1B_1$, line $BC$ intersects line $B_1C_1$ and line $CA$ intersects line $C_1A_1$. Prove that either the three lines $AA_1, BB_1, CC_1$ meet at one point or that they are all parallel.
1998 Romania National Olympiad, 4
Let $ABCD$ be an arbitrary tetrahedron. The bisectors of the angles $\angle BDC$, $\angle CDA$ and $\angle ADB$ intersect $BC$, $CA$ and $AB$, in the points $M$, $N$, $P$, respectively.
a) Show that the planes $(ADM)$, $(BDN)$ and $(CDP)$ have a common line $d$.
b) Let the points $A' \in (AD)$, $B' \in (BD)$ and $C' \in (CD)$ be such that $(AA') = (BB') = (CC')$ ; show that if $G$ and $G'$ are the centroids of $ABC$ and $A'B'C'$ then the lines $GG'$ and $d$ are either parallel or identical.
2007 ITest, 38
Find the largest positive integer that is equal to the cube of the sum of its digits.