Found problems: 2265
1967 IMO Longlists, 34
Faces of a convex polyhedron are six squares and 8 equilateral triangles and each edge is a common side for one triangle and one square. All dihedral angles obtained from the triangle and square with a common edge, are equal. Prove that it is possible to circumscribe a sphere around the polyhedron, and compute the ratio of the squares of volumes of that polyhedron and of the ball whose boundary is the circumscribed sphere.
VMEO III 2006 Shortlist, G3
The tetrahedron $OABC$ has all angles at vertex $O$ equal to $60^o$. Prove that $$AB \cdot BC + BC \cdot CA + CA \cdot AB \ge OA^2 + OB^2 + OC^2$$
1967 IMO Longlists, 32
Determine the volume of the body obtained by cutting the ball of radius $R$ by the trihedron with vertex in the center of that ball, it its dihedral angles are $\alpha, \beta, \gamma.$
2012 District Olympiad, 2
The pyramid $VABCD$ has base the rectangle ABCD, and the side edges are congruent. Prove that the plane $(VCD)$ forms congruent angles with the planes $(VAC)$ and $(BAC)$ if and only if $\angle VAC = \angle BAC $.
1992 AMC 12/AHSME, 19
For each vertex of a solid cube, consider the tetrahedron determined by the vertex and the midpoints of the three edges that meet at that vertex. The portion of the cube that remains when these eight tetrahedra are cut away is called a [i]cuboctahedron[/i]. The ratio of the volume of the cuboctahedron to the volume of the original cube is closest to which of these?
$ \textbf{(A)}\ 75\%\qquad\textbf{(B)}\ 78\%\qquad\textbf{(C)}\ 81\%\qquad\textbf{(D)}\ 84\%\qquad\textbf{(E)}\ 87\% $
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 $(12, 8, -16)$?
1972 Putnam, B5
Let $A,B,C$ and $D$ be non-coplanar points such that $\angle ABC=\angle ADC$ and $\angle BAD=\angle BCD$.
Show that $AB=CD$ and $AD=BC$.
Kyiv City MO Seniors 2003+ geometry, 2003.11.3
Let $x_1, x_2, x_3, x_4$ be the distances from an arbitrary point inside the tetrahedron to the planes of its faces, and let $h_1, h_2, h_3, h_4$ be the corresponding heights of the tetrahedron. Prove that $$\sqrt{h_1+h_2+h_3+h_4} \ge \sqrt{x_1}+\sqrt{x_2}+\sqrt{x_3}+\sqrt{x_4}$$
(Dmitry Nomirovsky)
1954 Kurschak Competition, 2
Every planar section of a three-dimensional body $B$ is a disk. Show that B must be a ball.
1987 National High School Mathematics League, 8
We have two triangles that lengths of its sides are $3,4,5$, one triangle that lengths of its sides are $4,5,\sqrt{41}$, one triangle that lengths of its sides are $\frac{5}{6}\sqrt2,4,5$. The number of tetrahedrons with such four surfaces is________.
1983 Spain Mathematical Olympiad, 7
A regular tetrahedron with an edge of $30$ cm rests on one of its faces. Assuming it is hollow, $2$ liters of water are poured into it. Find the height of the ''upper'' liquid and the area of the ''free'' surface of the water.
1979 IMO Longlists, 36
A regular tetrahedron $A_1B_1C_1D_1$ is inscribed in a regular tetrahedron $ABCD$, where $A_1$ lies in the plane $BCD$, $B_1$ in the plane $ACD$, etc. Prove that $A_1B_1 \ge\frac{ AB}{3}$.
2016 All-Russian Olympiad, 4
There is three-dimensional space. For every integer $n$ we build planes $ x \pm y\pm z = n$. All space is divided on octahedrons and tetrahedrons.
Point $(x_0,y_0,z_0)$ has rational coordinates but not lies on any plane. Prove, that there is such natural $k$ , that point $(kx_0,ky_0,kz_0)$ lies strictly inside the octahedron of partition.
2009 Harvard-MIT Mathematics Tournament, 2
The corner of a unit cube is chopped off such that the cut runs through the three vertices adjacent to the vertex of the chosen corner. What is the height of the cube when the freshly-cut face is placed on a table?
1998 French Mathematical Olympiad, Problem 1
A tetrahedron $ABCD$ satisfies the following conditions: the edges $AB,AC$ and $AD$ are pairwise orthogonal, $AB=3$ and $CD=\sqrt2$. Find the minimum possible value of
$$BC^6+BD^6-AC^6-AD^6.$$
2008 AMC 8, 16
A shape is created by joining seven unit cubes, as shown. What is the ratio of the volume in cubic units to the surface area in square units?
[asy]
import three;
defaultpen(linewidth(0.8));
real r=0.5;
currentprojection=orthographic(1,1/2,1/4);
draw(unitcube, white, thick(), nolight);
draw(shift(1,0,0)*unitcube, white, thick(), nolight);
draw(shift(1,-1,0)*unitcube, white, thick(), nolight);
draw(shift(1,0,-1)*unitcube, white, thick(), nolight);
draw(shift(2,0,0)*unitcube, white, thick(), nolight);
draw(shift(1,1,0)*unitcube, white, thick(), nolight);
draw(shift(1,0,1)*unitcube, white, thick(), nolight);[/asy]
$\textbf{(A)} \:1 : 6 \qquad\textbf{ (B)}\: 7 : 36 \qquad\textbf{(C)}\: 1 : 5 \qquad\textbf{(D)}\: 7 : 30\qquad\textbf{ (E)}\: 6 : 25$
1989 Brazil National Olympiad, 5
A tetrahedron is such that the center of the its circumscribed sphere is inside the tetrahedron.
Show that at least one of its edges has a size larger than or equal to the size of the edge of a regular tetrahedron inscribed in this same sphere.
2013 German National Olympiad, 4
Let $ABCDEFGH$ be a cube of sidelength $a$ and such that $AG$ is one of the space diagonals. Consider paths on the surface of this cube. Then determine the set of points $P$ on the surface for which the shortest path from $P$ to $A$ and from $P$ to $G$ have the same length $l.$ Also determine all possible values of $l$ depending on $a.$
1991 IberoAmerican, 1
Each vertex of a cube is assigned an 1 or a -1, and each face is assigned the product of the numbers assigned to its vertices. Determine the possible values the sum of these 14 numbers can attain.
2022 Chile National Olympiad, 4
In a right circular cone of wood, the radius of the circumference $T$ of the base circle measures $10$ cm, while every point on said circumference is $20$ cm away. from the apex of the cone. A red ant and a termite are located at antipodal points of $T$. A black ant is located at the midpoint of the segment that joins the vertex with the position of the termite. If the red ant moves to the black ant's position by the shortest possible path, how far does it travel?
2020 AMC 12/AHSME, 9
A three-quarter sector of a circle of radius $4$ inches together with its interior can be rolled up to form the lateral surface area of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches?
[asy]
draw(Arc((0,0), 4, 0, 270));
draw((0,-4)--(0,0)--(4,0));
label("$4$", (2,0), S);
[/asy]
$\textbf{(A)}\ 3\pi \sqrt5 \qquad\textbf{(B)}\ 4\pi \sqrt3 \qquad\textbf{(C)}\ 3 \pi \sqrt7 \qquad\textbf{(D)}\ 6\pi \sqrt3 \qquad\textbf{(E)}\ 6\pi \sqrt7$
2018 Iranian Geometry Olympiad, 4
We have a polyhedron all faces of which are triangle. Let $P$ be an arbitrary point on one of the edges of this polyhedron such that $P$ is not the midpoint or endpoint of this edge. Assume that $P_0 = P$. In each step, connect $P_i$ to the centroid of one of the faces containing it. This line meets the perimeter of this face again at point $P_{i+1}$. Continue this process with $P_{i+1}$ and the other face containing $P_{i+1}$. Prove that by continuing this process, we cannot pass through all the faces. (The centroid of a triangle is the point of intersection of its medians.)
Proposed by Mahdi Etesamifard - Morteza Saghafian
1999 AMC 8, 20
Figure 1 is called a "stack map." The numbers tell how many cubes are stacked in each position. Fig. 2 shows these cubes, and Fig. 3 shows the view of the stacked cubes as seen from the front.
Which of the following is the front view for the stack map in Fig. 4?
[asy]
unitsize(24);
draw((0,0)--(2,0)--(2,2)--(0,2)--cycle);
draw((1,0)--(1,2));
draw((0,1)--(2,1));
draw((5,0)--(7,0)--(7,1)--(20/3,4/3)--(20/3,13/3)--(19/3,14/3)--(16/3,14/3)--(16/3,11/3)--(13/3,11/3)--(13/3,2/3)--cycle);
draw((20/3,13/3)--(17/3,13/3)--(17/3,10/3)--(14/3,10/3)--(14/3,1/3));
draw((20/3,10/3)--(17/3,10/3)--(17/3,7/3)--(20/3,7/3));
draw((17/3,7/3)--(14/3,7/3));
draw((7,1)--(6,1)--(6,2)--(5,2)--(5,0));
draw((5,1)--(6,1)--(6,0));
draw((20/3,4/3)--(6,4/3));
draw((17/3,13/3)--(16/3,14/3));
draw((17/3,10/3)--(16/3,11/3));
draw((14/3,10/3)--(13/3,11/3));
draw((5,2)--(13/3,8/3));
draw((5,1)--(13/3,5/3));
draw((6,2)--(17/3,7/3));
draw((9,0)--(11,0)--(11,4)--(10,4)--(10,3)--(9,3)--cycle);
draw((11,3)--(10,3)--(10,0));
draw((11,2)--(9,2));
draw((11,1)--(9,1));
draw((13,0)--(16,0)--(16,2)--(13,2)--cycle);
draw((13,1)--(16,1));
draw((14,0)--(14,2));
draw((15,0)--(15,2));
label("Figure 1",(1,0),S);
label("Figure 2",(17/3,0),S);
label("Figure 3",(10,0),S);
label("Figure 4",(14.5,0),S);
label("$1$",(1.5,.2),N);
label("$2$",(.5,.2),N);
label("$3$",(.5,1.2),N);
label("$4$",(1.5,1.2),N);
label("$1$",(13.5,.2),N);
label("$3$",(14.5,.2),N);
label("$1$",(15.5,.2),N);
label("$2$",(13.5,1.2),N);
label("$2$",(14.5,1.2),N);
label("$4$",(15.5,1.2),N);[/asy]
[asy]
unitsize(18);
draw((0,0)--(3,0)--(3,2)--(1,2)--(1,4)--(0,4)--cycle);
draw((0,3)--(1,3));
draw((0,2)--(1,2)--(1,0));
draw((0,1)--(3,1));
draw((2,0)--(2,2));
draw((5,0)--(8,0)--(8,4)--(7,4)--(7,3)--(6,3)--(6,2)--(5,2)--cycle);
draw((8,3)--(7,3)--(7,0));
draw((8,2)--(6,2)--(6,0));
draw((8,1)--(5,1));
draw((10,0)--(12,0)--(12,4)--(11,4)--(11,3)--(10,3)--cycle);
draw((12,3)--(11,3)--(11,0));
draw((12,2)--(10,2));
draw((12,1)--(10,1));
draw((14,0)--(17,0)--(17,4)--(16,4)--(16,2)--(14,2)--cycle);
draw((17,3)--(16,3));
draw((17,2)--(16,2)--(16,0));
draw((17,1)--(14,1));
draw((15,0)--(15,2));
draw((19,0)--(22,0)--(22,4)--(20,4)--(20,1)--(19,1)--cycle);
draw((22,3)--(20,3));
draw((22,2)--(20,2));
draw((22,1)--(20,1)--(20,0));
draw((21,0)--(21,4));
label("(A)",(1.5,0),S);
label("(B)",(6.5,0),S);
label("(C)",(11,0),S);
label("(D)",(15.5,0),S);
label("(E)",(20.5,0),S);[/asy]
1967 IMO Longlists, 20
In the space $n \geq 3$ points are given. Every pair of points determines some distance. Suppose all distances are different. Connect every point with the nearest point. Prove that it is impossible to obtain (closed) polygonal line in such a way.
2003 AMC 8, 15
A fi gure is constructed from unit cubes. Each cube shares at least one face with another cube. What is the minimum number of cubes needed to build a fi gure with the front and side views shown?
[asy]
defaultpen(linewidth(0.8));
path p=unitsquare;
draw(p^^shift(0,1)*p^^shift(1,0)*p);
draw(shift(4,0)*p^^shift(5,0)*p^^shift(5,1)*p);
label("FRONT", (1,0), S);
label("SIDE", (5,0), S);[/asy]
$ \textbf{(A)}\ 3\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 7$