Found problems: 2265
2002 National Olympiad First Round, 34
How many positive integers $n$ are there such that $3n^2 + 3n + 7$ is a perfect cube?
$
\textbf{a)}\ 0
\qquad\textbf{b)}\ 1
\qquad\textbf{c)}\ 3
\qquad\textbf{d)}\ 7
\qquad\textbf{e)}\ \text{Infinitely many}
$
1998 Putnam, 3
Let $H$ be the unit hemisphere $\{(x,y,z):x^2+y^2+z^2=1,z\geq 0\}$, $C$ the unit circle $\{(x,y,0):x^2+y^2=1\}$, and $P$ the regular pentagon inscribed in $C$. Determine the surface area of that portion of $H$ lying over the planar region inside $P$, and write your answer in the form $A \sin\alpha + B \cos\beta$, where $A,B,\alpha,\beta$ are real numbers.
1996 Tuymaada Olympiad, 8
Given a tetrahedron $ABCD$, in which $AB=CD= 13 , AC=BD=14$ and $AD=BC=15$.
Show that the centers of the inscribed sphere and sphere around it coincide, and find the radii of these spheres.
2002 Estonia National Olympiad, 4
Find the maximum length of a broken line on the surface of a unit cube, such that its links are the cube’s edges and diagonals of faces, the line does not intersect itself and passes no more than once through any vertex of the cube, and its endpoints are in two opposite vertices of the cube.
2002 Poland - Second Round, 2
Triangle $ABC$ with $\angle BAC=90^{\circ}$ is the base of the pyramid $ABCD$. Moreover, $AD=BD$ and $AB=CD$. Prove that $\angle ACD\ge 30^{\circ}$.
2009 National Olympiad First Round, 35
For every $ n \ge 2$, $ a_n \equal{} \sqrt [3]{n^3 \plus{} n^2 \minus{} n \minus{} 1}/n$. What is the least value of positive integer $ k$ satisfying $ a_2a_3\cdots a_k > 3$ ?
$\textbf{(A)}\ 100 \qquad\textbf{(B)}\ 102 \qquad\textbf{(C)}\ 104 \qquad\textbf{(D)}\ 106 \qquad\textbf{(E)}\ \text{None}$
MMPC Part II 1958 - 95, 1964
[b]p1.[/b] The edges of a tetrahedron are all tangent to a sphere. Prove that the sum of the lengths of any pair of opposite edges equals the sum of the lengths of any other pair of opposite edges. (Two edges of a tetrahedron are said to be opposite if they do not have a vertex in common.)
[b]p2.[/b] Find the simplest formula possible for the product of the following $2n - 2$ factors: $$\left(1+\frac12 \right),\left(1-\frac12 \right), \left(1+\frac13 \right) , \left(1-\frac13 \right),...,\left(1+\frac{1}{n} \right), \left(1-\frac{1}{n} \right)$$. Prove that your formula is correct.
[b]p3.[/b] Solve $$\frac{(x + 1)^2+1}{x + 1} + \frac{(x + 4)^2+4}{x + 4}=\frac{(x + 2)^2+2}{x + 2}+\frac{(x + 3)^2+3}{x + 3}$$
[b]p4.[/b] Triangle $ABC$ is inscribed in a circle, $BD$ is tangent to this circle and $CD$ is perpendicular to $BD$. $BH$ is the altitude from $B$ to $AC$. Prove that the line $DH$ is parallel to $AB$.
[img]https://cdn.artofproblemsolving.com/attachments/e/9/4d0b136dca4a9b68104f00300951837adef84c.png[/img]
[b]p5.[/b] Consider the picture below as a section of a city street map. There are several paths from $A$ to $B$, and if one always walks along the street, the shortest paths are $15$ blocks in length. Find the number of paths of this length between $A$ and $B$.
[img]https://cdn.artofproblemsolving.com/attachments/8/d/60c426ea71db98775399cfa5ea80e94d2ea9d2.png[/img]
[b]p6.[/b] A [u]finite [/u] [u]graph [/u] is a set of points, called [u]vertices[/u], together with a set of arcs, called [u]edges[/u]. Each edge connects two of the vertices (it is not necessary that every pair of vertices be connected by an edge). The [u]order [/u] of a vertex in a finite graph is the number of edges attached to that vertex.
[u]Example[/u]
The figure at the right is a finite graph with $4$ vertices and $7$ edges. [img]https://cdn.artofproblemsolving.com/attachments/5/9/84d479c5dbd0a6f61a66970e46ab15830d8fba.png[/img]
One vertex has order $5$ and the other vertices order $3$.
Define a finite graph to be [u]heterogeneous [/u] if no two vertices have the same order.
Prove that no graph with two or more vertices is heterogeneous.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Ukrainian TYM Qualifying - geometry, I.13
A candle and a man are placed in a dihedral mirror angle. How many reflections can the man see ?
1974 Chisinau City MO, 84
a) Let $S$ and $P$ be the area and perimeter of some triangle. The straight lines on which its sides are located move to the outside by a distance $h$. What will be the area and perimeter of the triangle formed by the three obtained lines?
b) Let $V$ and $S$ be the volume and surface area of some tetrahedron. The planes on which its faces are located are moved to the outside by a distance $h$. What will be the volume and surface area of the tetrahedron formed by the three obtained planes?
2020 Stanford Mathematics Tournament, 2
On each edge of a regular tetrahedron, five points that separate the edge into six equal segments are marked. There are twenty planes that are parallel to a face of the tetrahedron and pass through exactly three of the marked points. When the tetrahedron is cut along each of these twenty planes, how many new tetrahedrons are produced?
2014 USAMTS Problems, 3:
Let $P$ be a square pyramid whose base consists of the four vertices $(0, 0, 0), (3, 0, 0), (3, 3, 0)$, and $(0, 3, 0)$, and whose apex is the point $(1, 1, 3)$. Let $Q$ be a square pyramid whose base is the same as the base of $P$, and whose apex is the point $(2, 2, 3)$. Find the volume of the intersection of the interiors of $P$ and $Q$.
2005 Tournament of Towns, 5
A cube lies on the plane. After being rolled a few times (over its edges), it is brought back to its initial location with the same face up. Could the top face have been rotated by 90 degrees?
[i](5 points)[/i]
2016 District Olympiad, 1
Let be a pyramid having a square as its base and the projection of the top vertex to the base is the center of the square. Prove that two opposite faces are perpendicular if and only if the angle between two adjacent faces is $ 120^{\circ } . $
2014 AMC 10, 19
Where is AMC 10a No.19? Thanks
2018 Harvard-MIT Mathematics Tournament, 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?
1978 IMO Shortlist, 13
We consider a fixed point $P$ in the interior of a fixed sphere$.$ We construct three segments $PA, PB,PC$, perpendicular two by two$,$ with the vertexes $A, B, C$ on the sphere$.$ We consider the vertex $Q$ which is opposite to $P$ in the parallelepiped (with right angles) with $PA, PB, PC$ as edges$.$ Find the locus of the point $Q$ when $A, B, C$ take all the positions compatible with our problem.
2010 Poland - Second Round, 2
The orthogonal projections of the vertices $A, B, C$ of the tetrahedron $ABCD$ on the opposite faces are denoted by $A', B', C'$ respectively. Suppose that point $A'$ is the circumcenter of the triangle $BCD$, point $B'$ is the incenter of the triangle $ACD$ and $C'$ is the centroid of the triangle $ABD$. Prove that tetrahedron $ABCD$ is regular.
2007 F = Ma, 11
A uniform disk, a thin hoop, and a uniform sphere, all with the same mass and same outer radius, are each free to rotate about a fixed axis through its center. Assume the hoop is connected to the rotation axis by light spokes. With the objects starting from rest, identical forces are simultaneously applied to the rims, as shown. Rank the objects according to their kinetic energies after a given time $t$, from least to greatest.
[asy]
size(225);
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);
filldraw(circle((0,0),1),gray(.7));
draw((0,-1)--(2,-1),EndArrow);
label("$\vec{F}$",(1, -1),S);
label("Disk",(-1,0),W);
filldraw(circle((5,0),1),gray(.7));
filldraw(circle((5,0),0.75),white);
draw((5,-1)--(7,-1),EndArrow);
label("$\vec{F}$",(6, -1),S);
label("Hoop",(6,0),E);
filldraw(circle((10,0),1),gray(.5));
draw((10,-1)--(12,-1),EndArrow);
label("$\vec{F}$",(11, -1),S);
label("Sphere",(11,0),E);
[/asy]
$ \textbf{(A)} \ \text{disk, hoop, sphere}$
$\textbf{(B)}\ \text{sphere, disk, hoop}$
$\textbf{(C)}\ \text{hoop, sphere, disk}$
$\textbf{(D)}\ \text{disk, sphere, hoop}$
$\textbf{(E)}\ \text{hoop, disk, sphere} $
2002 Iran Team Selection Test, 11
A $10\times10\times10$ cube has $1000$ unit cubes. $500$ of them are coloured black and $500$ of them are coloured white. Show that there are at least $100$ unit squares, being the common face of a white and a black unit cube.
1990 Vietnam Team Selection Test, 2
Given a tetrahedron such that product of the opposite edges is $ 1$. Let the angle between the opposite edges be $ \alpha$, $ \beta$, $ \gamma$, and circumradii of four faces be $ R_1$, $ R_2$, $ R_3$, $ R_4$. Prove that
\[ \sin^2\alpha \plus{} \sin^2\beta \plus{} \sin^2\gamma\ge\frac {1}{\sqrt {R_1R_2R_3R_4}}
\]
1965 Kurschak Competition, 3
A pyramid has square base and equal sides. It is cut into two parts by a plane parallel to the base. The lower part (which has square top and square base) is such that the circumcircle of the base is smaller than the circumcircles of the lateral faces. Show that the shortest path on the surface joining the two endpoints of a spatial diagonal lies entirely on the lateral faces.
[img]https://cdn.artofproblemsolving.com/attachments/c/8/170bec826d5e40308cfd7360725d2aba250bf6.png[/img]
2014 CHMMC (Fall), 8
What’s the greatest pyramid volume one can form using edges of length $2, 3, 3, 4, 5, 5$, respectively?
1969 IMO Shortlist, 12
$(CZS 1)$ Given a unit cube, find the locus of the centroids of all tetrahedra whose vertices lie on the sides of the cube.
2009 AMC 10, 24
Three distinct vertices of a cube are chosen at random. What is the probability that the plane determined by these three vertices contains points inside the cube?
$ \textbf{(A)}\ \frac{1}{4} \qquad
\textbf{(B)}\ \frac{3}{8} \qquad
\textbf{(C)}\ \frac{4}{7} \qquad
\textbf{(D)}\ \frac{5}{7} \qquad
\textbf{(E)}\ \frac{3}{4}$
2005 AMC 12/AHSME, 25
Six ants simultaneously stand on the six vertices of a regular octahedron, with each ant at a different vertex. Simultaneously and independently, each ant moves from its vertex to one of the four adjacent vertices, each with equal probability. What is the probability that no two ants arrive at the same vertex?
$ \textbf{(A)}\ \frac {5}{256} \qquad
\textbf{(B)}\ \frac {21}{1024} \qquad
\textbf{(C)}\ \frac {11}{512} \qquad
\textbf{(D)}\ \frac {23}{1024} \qquad
\textbf{(E)}\ \frac {3}{128}$