Olympiad problems

2006 AMC 8, 18

Tags: geometry , 3D geometry

A cube with 3-inch edges is made using 27 cubes with 1-inch edges. Nineteen of the smaller cubes are white and eight are black. If the eight black cubes are placed at the corners of the larger cube, what fraction of the surface area of the larger cube is white? $ \textbf{(A)}\ \dfrac{1}{9} \qquad \textbf{(B)}\ \dfrac{1}{4} \qquad \textbf{(C)}\ \dfrac{4}{9} \qquad \textbf{(D)}\ \dfrac{5}{9} \qquad \textbf{(E)}\ \dfrac{19}{27}$

1999 Brazil Team Selection Test, Problem 4

Tags: geometry , 3D geometry

Assume that it is possible to color more than half of the surfaces of a given polyhedron so that no two colored surfaces have a common edge. (a) Describe one polyhedron with the above property. (b) Prove that one cannot inscribe a sphere touching all the surfaces of a polyhedron with the above property.

2005 USAMTS Problems, 5

Tags: USAMTS , geometry , 3D geometry , sphere

Sphere $S$ is inscribed in cone $C$. The height of $C$ equals its radius, and both equal $12+12\sqrt2$. Let the vertex of the cone be $A$ and the center of the sphere be $B$. Plane $P$ is tangent to $S$ and intersects $\overline{AB}$. $X$ is the point on the intersection of $P$ and $C$ closest to $A$. Given that $AX=6$, find the area of the region of $P$ enclosed by the intersection of $C$ and $P$.

2017 CMI B.Sc. Entrance Exam, 2

Tags: CMI , Chennai Mathematical Institute , B.Sc , math , CS , 2017 , 3D geometry

Let $L$ be the line of intersection of the planes $~x+y=0~$ and $~y+z=0$. [b](a)[/b] Write the vector equation of $L$, i.e. find $(a,b,c)$ and $(p,q,r)$ such that $$L=\{(a,b,c)+\lambda(p,q,r)~~\vert~\lambda\in\mathbb{R}\}$$ [b](b)[/b] Find the equation of a plane obtained by $x+y=0$ about $L$ by $45^{\circ}$.

2004 District Olympiad, 3

Tags: geometry , 3D geometry , tetrahedron , geometric transformation , rotation

On the tetrahedron $ ABCD $ make the notation $ M,N,P,Q, $ for the midpoints of $ AB,CD,AC, $ respectively, $ BD. $ Additionally, we know that $ MN $ is the common perpendicular of $ AB,CD, $ and $ PQ $ is the common perpendicular of $ AC,BD. $ Show that $ AB=CD, BC=DA, AC=BD. $

2009 Putnam, A1

Tags: Putnam , function , linear algebra , matrix , geometry , 3D geometry , geometric transformation

Let $ f$ be a real-valued function on the plane such that for every square $ ABCD$ in the plane, $ f(A)\plus{}f(B)\plus{}f(C)\plus{}f(D)\equal{}0.$ Does it follow that $ f(P)\equal{}0$ for all points $ P$ in the plane?

2004 Harvard-MIT Mathematics Tournament, 7

Tags: geometry , 3D geometry , tetrahedron , graph theory

We have a polyhedron such that an ant can walk from one vertex to another, traveling only along edges, and traversing every edge exactly once. What is the smallest possible total number of vertices, edges, and faces of this polyhedron?

1980 IMO, 5

Tags: geometry , 3D geometry , sphere , function , combinatorics proposed , combinatorics

In the Euclidean three-dimensional space, we call [i]folding[/i] of a sphere $S$ every partition of $S \setminus \{x,y\}$ into disjoint circles, where $x$ and $y$ are two points of $S$. A folding of $S$ is called [b]linear[/b] if the circles of the [i]folding[/i] are obtained by the intersection of $S$ with a family of parallel planes or with a family of planes containing a straight line $D$ exterior to $S$. Is every [i]folding[/i] of a sphere $S$ [b]linear[/b]?

1994 Brazil National Olympiad, 1

Tags: geometry , 3D geometry , combinatorics proposed , combinatorics

The edges of a cube are labeled from 1 to 12 in an arbitrary manner. Show that it is not possible to get the sum of the edges at each vertex the same. Show that we can get eight vertices with the same sum if one of the labels is changed to 13.

2005 National High School Mathematics League, 2

Tags: geometry , 3D geometry

Four points in space $A,B,C,D$ satisfy that $|AB|=3,|BC|=7,|CD|=11,|DA|=9$, then the number of values of $\overrightarrow{AC}\cdot\overrightarrow{BD}$ is $\text{(A)}$ Only one. $\text{(B)}$ Two. $\text{(C)}$ Three. $\text{(D)}$ Infinitely many.

2016 PUMaC Combinatorics B, 3

Tags: geometry , 3D geometry , rotation

Chitoge is painting a cube; she can paint each face either black or white, but she wants no vertex of the cube to be touching three faces of the same color. In how many ways can Chitoge paint the cube? Two paintings of a cube are considered to be the same if you can rotate one cube so that it looks like the other cube.

2016 Postal Coaching, 5

Tags: geometry , incircle , 3D geometry , prism

Two triangles $ABC$ and $DEF$ have the same incircle. If a circle passes through $A,B,C,D,E$ prove that it also passes through $F$.

1980 IMO Longlists, 15

Tags: geometry , 3D geometry , tetrahedron , sphere , triangle inequality , angles , IMO Shortlist

Prove that the sum of the six angles subtended at an interior point of a tetrahedron by its six edges is greater than 540°.

1938 Moscow Mathematical Olympiad, 039

Tags: geometry , reflection , 3D geometry

The following operation is performed over points $O_1, O_2, O_3$ and $A$ in space. The point $A$ is reflected with respect to $O_1$, the resultant point $A_1$ is reflected through $O_2$, and the resultant point $A_2$ through $O_3$. We get some point $A_3$ that we will also consecutively reflect through $O_1, O_2, O_3$. Prove that the point obtained last coincides with $A$..

2021 AIME Problems, 10

Tags: geometry , 3D geometry , sphere , AMC , AIME , AIME II

Two spheres with radii $36$ and one sphere with radius $13$ are each externally tangent to the other two spheres and to two different planes $\mathcal{P}$ and $\mathcal{Q}$. The intersection of planes $\mathcal{P}$ and $\mathcal{Q}$ is the line $\ell$. The distance from line $\ell$ to the point where the sphere with radius $13$ is tangent to plane $\mathcal{P}$ is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [img]https://imgur.com/1mfBNNL.png[/img]

1989 Tournament Of Towns, (206) 4

Tags: combinatorics , combinatorial geometry , geometry , 3D geometry

Can one draw , on the surface of a Rubik's cube , a closed path which crosses each little square exactly once and does not pass through any vertex of a square? (S . Fomin, Leningrad)

2008 Sharygin Geometry Olympiad, 23

Tags: geometry , 3D geometry , sphere , geometry proposed

(V.Protasov, 10--11) In the space, given two intersecting spheres of different radii and a point $ A$ belonging to both spheres. Prove that there is a point $ B$ in the space with the following property: if an arbitrary circle passes through points $ A$ and $ B$ then the second points of its meet with the given spheres are equidistant from $ B$.

2010 All-Russian Olympiad, 2

Tags: geometry , 3D geometry , tetrahedron , incenter , analytic geometry , geometry proposed

Could the four centers of the circles inscribed into the faces of a tetrahedron be coplanar? (vertexes of tetrahedron not coplanar)

1992 Polish MO Finals, 2

Tags: geometry , 3D geometry , pyramid , sphere , vector , geometry unsolved

The base of a regular pyramid is a regular $2n$-gon $A_1A_2...A_{2n}$. A sphere passing through the top vertex $S$ of the pyramid cuts the edge $SA_i$ at $B_i$ (for $i = 1, 2, ... , 2n$). Show that $\sum\limits_{i=1}^n SB_{2i-1} = \sum\limits_{i=1}^n SB_{2i}$.

1992 Putnam, A6

Tags: geometry , 3D geometry , sphere , probability , tetrahedron , symmetry , geometric transformation

Four points are chosen at random on the surface of a sphere. What is the probability that the center of the sphere lies inside the tetrahedron whose vertices are at the four points?

2002 Iran Team Selection Test, 3

Tags: geometry , 3D geometry , sphere , inequalities , geometry unsolved

A "[i]2-line[/i]" is the area between two parallel lines. Length of "2-line" is distance of two parallel lines. We have covered unit circle with some "2-lines". Prove sum of lengths of "2-lines" is at least 2.

Denmark (Mohr) - geometry, 2000.2

Tags: geometry , 3D geometry , sphere , prism , hexagon

Three identical spheres fit into a glass with rectangular sides and bottom and top in the form of regular hexagons such that every sphere touches every side of the glass. The glass has volume $108$ cm$^3$. What is the sidelength of the bottom? [img]https://1.bp.blogspot.com/-hBkYrORoBHk/XzcDt7B83AI/AAAAAAAAMXs/P5PGKTlNA7AvxkxMqG-qxqDVc9v9cU0VACLcBGAsYHQ/s0/2000%2BMohr%2Bp2.png[/img]

1955 Poland - Second Round, 6

Tags: 3D geometry , geometry , fixed

Inside the trihedral angle $ OABC $, whose plane angles $ AOB $, $ BOC $, $ COA $ are equal, a point $ S $ is chosen equidistant from the faces of this angle. Through point $ S $ a plane is drawn that intersects the edges $ OA $, $ OB $, $ OC $ at points $ M $, $ N $, $ P $, respectively. Prove that the sum $$ \frac{1}{OM} + \frac{1}{ON} + \frac{1}{OP}$$ has a constant value, i.e. independent of the position of the plane $ MNP $.

1969 IMO Shortlist, 58

Tags: pigeonhole principle , geometry , 3D geometry , tetrahedron , IMO Shortlist , IMO Longlist

$(SWE 1)$ Six points $P_1, . . . , P_6$ are given in $3-$dimensional space such that no four of them lie in the same plane. Each of the line segments $P_jP_k$ is colored black or white. Prove that there exists one triangle $P_jP_kP_l$ whose edges are of the same color.

2008 Iran MO (3rd Round), 6

Tags: geometry , 3D geometry , sphere

There are five research labs on Mars. Is it always possible to divide Mars to five connected congruent regions such that each region contains exactly on research lab. [img]http://i37.tinypic.com/f2iq8g.png[/img]