Found problems: 2265
1978 Poland - Second Round, 5
Prove that there is no inclined plane such that any tetrahedron placed arbitrarily with a certain face on the plane will not fall over.
It means the following:
Given a plane $ \pi $ and a line $ l $ not perpendicular to it. Prove that there is a tetrahedron $ T $ such that for each of its faces $ S $ there is in the plane $ \pi $ a triangle $ ABC $ congruent to $ S $ and there is a point $ D $ such that the tetrahedron $ ABCD $ congruent to $ T $ and the line parallel to $ l $ passing through the center of gravity of the tetrahedron $ ABCD $ does not intersect the triangle $ ABC $.
Note. The center of gravity of a tetrahedron is the intersection point of the segments connecting the centers of gravity of the faces of this tetrahedron with the opposite vertices (it is known that such a point always exists).
2023 AMC 8, 17
A [i]regular octahedron[/i] has eight equilateral triangle faces with four faces meeting at each vertex. Jun will make the regular octahedron shown on the right by folding the piece of paper shown on the left. Which numbered face will end up to the right of $Q$?
[asy]
// Note: This diagram was not made by me.
import graph;
// The Solid
// To save processing time, do not use three (dimensions)
// Project (roughly) to two
size(15cm);
pair Fr, Lf, Rt, Tp, Bt, Bk;
Lf=(0,0);
Rt=(12,1);
Fr=(7,-1);
Bk=(5,2);
Tp=(6,6.7);
Bt=(6,-5.2);
draw(Lf--Fr--Rt);
draw(Lf--Tp--Rt);
draw(Lf--Bt--Rt);
draw(Tp--Fr--Bt);
draw(Lf--Bk--Rt,dashed);
draw(Tp--Bk--Bt,dashed);
label(rotate(-8.13010235)*slant(0.1)*"$Q$", (4.2,1.6));
label(rotate(21.8014095)*slant(-0.2)*"$?$", (8.5,2.05));
pair g = (-8,0); // Define Gap transform
real a = 8;
draw(g+(-a/2,1)--g+(a/2,1), Arrow()); // Make arrow
// Time for the NET
pair DA,DB,DC,CD,O;
DA = (6.92820323028,0);
DB = (3.46410161514,6);
DC = (DA+DB)/3;
CD = conj(DC);
O=(0,0);
transform trf=shift(3g+(0,3));
path NET = O--(-2*DA)--(-2DB)--(-DB)--(2DA-DB)--DB--O--DA--(DA-DB)--O--(-DB)--(-DA)--(-DA-DB)--(-DB);
draw(trf*NET);
label("$7$",trf*DC);
label("$Q$",trf*DC+DA-DB);
label("$5$",trf*DC-DB);
label("$3$",trf*DC-DA-DB);
label("$6$",trf*CD);
label("$4$",trf*CD-DA);
label("$2$",trf*CD-DA-DB);
label("$1$",trf*CD-2DA);
[/asy]
$\textbf{(A)}~1\qquad\textbf{(B)}~2\qquad\textbf{(C)}~3\qquad\textbf{(D)}~4\qquad\textbf{(E)}~5\qquad$
2003 District Olympiad, 1
In the interior of a cube we consider $\displaystyle 2003$ points. Prove that one can divide the cube in more than $\displaystyle 2003^3$ cubes such that any point lies in the interior of one of the small cubes and not on the faces.
1966 IMO Longlists, 44
What is the greatest number of balls of radius $1/2$ that can be placed within a rectangular box of size $10 \times 10 \times 1 \ ?$
2020 Polish Junior MO First Round, 7.
Consider the right prism with the rhombus with side $a$ and acute angle $60^{\circ}$ as a base. This prism was intersected by some plane intersecting its side edges, such that the cross-section of the prism and the plane is a square. Determine all possible lengths of the side of this square.
2016 PUMaC Combinatorics A, 1
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.
2005 QEDMO 1st, 8 (Z2)
Prove that if $n$ can be written as $n=a^2+ab+b^2$, then also $7n$ can be written that way.
2016 AIME Problems, 14
Equilateral $\triangle ABC$ has side length $600$. Points $P$ and $Q$ lie outside of the plane of $\triangle ABC$ and are on the opposite sides of the plane. Furthermore, $PA=PB=PC$, and $QA=QB=QC$, and the planes of $\triangle PAB$ and $\triangle QAB$ form a $120^{\circ}$ dihedral angle (The angle between the two planes). There is a point $O$ whose distance from each of $A,B,C,P$ and $Q$ is $d$. Find $d$.
2002 National High School Mathematics League, 9
Points $P_1,P_2,P_3,P_4$ are vertexes of a regular triangular pyramid, and $P_5,P_6,P_7,P_8,P_9,P_{10}$ midpoints of edges. The number of groups $(P_1,P_i,P_j,P_k)(1<i<j<k\leq10)$ that $P_1,P_i,P_j,P_k$ are coplane is________.
1984 All Soviet Union Mathematical Olympiad, 394
Prove that every cube's cross-section, containing its centre, has the area not less then its face's area.
2014 NIMO Problems, 1
Let $\eta(m)$ be the product of all positive integers that divide $m$, including $1$ and $m$. If $\eta(\eta(\eta(10))) = 10^n$, compute $n$.
[i]Proposed by Kevin Sun[/i]
2000 AIME Problems, 12
The points $A, B$ and $C$ lie on the surface of a sphere with center $O$ and radius 20. It is given that $AB=13, BC=14, CA=15,$ and that the distance from $O$ to triangle $ABC$ is $\frac{m\sqrt{n}}k,$ where $m, n,$ and $k$ are positive integers, $m$ and $k$ are relatively prime, and $n$ is not divisible by the square of any prime. Find $m+n+k.$
2015 AMC 12/AHSME, 16
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?
$\textbf{(A) }3\sqrt2\qquad\textbf{(B) }2\sqrt5\qquad\textbf{(C) }\dfrac{24}5\qquad\textbf{(D) }3\sqrt3\qquad\textbf{(E) }\dfrac{24}5\sqrt2$
1937 Eotvos Mathematical Competition, 2
Two circles in space are said to be tangent to each other if they have a corni-non tangent at the same point of tangency. Assume that there are three circles in space which are mutually tangent at three distinct points. Prove that they either alI lie in a plane or all lie on a sphere.
2001 Dutch Mathematical Olympiad, 3
A wooden beam $EFGH$ $ABCD$ is with three cuts in $8$ smaller ones sawn beams. Each cut is parallel to one of the three pair of opposit sides. Each pair of saw cuts is shown perpendicular to each other. The smaller bars at the corners $A, C, F$ and $H$ have a capacity of $9, 12, 8, 24$ respectively.(The proportions in the picture are not correct!!). Calculate content of the entire bar.
[asy]
unitsize (0.5 cm);
pair A, B, C, D, E, F, G, H;
pair x, y, z;
x = (1,0.5);
y = (-0.8,0.8);
z = (0,1);
B = (0,0);
C = 5*x;
A = 3*y;
F = 4*z;
E = A + F;
G = C + F;
H = A + C + F;
fill(y--3*y--(3*y + z)--(y + z)--cycle, gray(0.8));
fill(2*x--5*x--(5*x + z)--(2*x + z)--cycle, gray(0.8));
fill((y + z)--(y + 4*z)--(y + 4*z + 2*x)--(4*z + 2*x)--(2*x + z)--z--cycle, gray(0.8));
fill((2*x + y + 4*z)--(2*x + 3*y + 4*z)--(5*x + 3*y + 4*z)--(5*x + y + 4*z)--cycle, gray(0.8));
draw(B--C--G--H--E--A--cycle);
draw(B--F);
draw(E--F);
draw(G--F);
draw(y--(y + 4*z)--(y + 4*z + 5*x));
draw(2*x--(2*x + 4*z)--(2*x + 4*z + 3*y));
draw((3*y + z)--z--(5*x + z));
label("$A$", A, SW);
label("$B$", B, S);
label("$C$", C, SE);
label("$E$", E, NW);
label("$F$", F, SE);
label("$G$", G, NE);
label("$H$", H, N);
[/asy]
2005 Romania National Olympiad, 1
We consider a cube with sides of length 1. Prove that a tetrahedron with vertices in the set of the vertices of the cube has the volume $\dfrac 16$ if and only if 3 of the vertices of the tetrahedron are vertices on the same face of the cube.
[i]Dinu Serbanescu[/i]
2002 Baltic Way, 15
A spider and a fly are sitting on a cube. The fly wants to maximize the shortest path to the spider along the surface of the cube. Is it necessarily best for the fly to be at the point opposite to the spider?
(“Opposite” means “symmetric with respect to the centre of the cube”.)
2009 Purple Comet Problems, 15
We have twenty-seven $1$ by $1$ cubes. Each face of every cube is marked with a natural number so that two opposite faces (top and bottom, front and back, left and right) are always marked with an even number and an odd number where the even number is twice that of the odd number. The twenty-seven cubes are put together to form one $3$ by $3$ cube as shown. When two cubes are placed face-to-face, adjoining faces are always marked with an odd number and an even number where the even number is one greater than the odd number. Find the sum of all of the numbers on all of the faces of all the $1$ by $1$ cubes.
[asy]
import graph; size(7cm);
real labelscalefactor = 0.5;
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);
draw((-1,7)--(-1,4));
draw((-1,9.15)--(-3.42,8.21));
draw((-1,9.15)--(1.42,8.21));
draw((-1,7)--(1.42,8.21));
draw((1.42,7.21)--(-1,6));
draw((1.42,6.21)--(-1,5));
draw((1.42,5.21)--(-1,4));
draw((1.42,8.21)--(1.42,5.21));
draw((-3.42,8.21)--(-3.42,5.21));
draw((-3.42,7.21)--(-1,6));
draw((-3.42,8.21)--(-1,7));
draw((-1,4)--(-3.42,5.21));
draw((-3.42,6.21)--(-1,5));
draw((-2.61,7.8)--(-2.61,4.8));
draw((-1.8,4.4)--(-1.8,7.4));
draw((-0.2,7.4)--(-0.2,4.4));
draw((0.61,4.8)--(0.61,7.8));
label("2",(-1.07,9.01),SE*labelscalefactor);
label("9",(-1.88,8.65),SE*labelscalefactor);
label("1",(-2.68,8.33),SE*labelscalefactor);
label("3",(-0.38,8.72),SE*labelscalefactor);
draw((-1.8,7.4)--(0.63,8.52));
draw((-0.27,8.87)--(-2.61,7.8));
draw((-2.65,8.51)--(-0.2,7.4));
draw((-1.77,8.85)--(0.61,7.8));
label("7",(-1.12,8.33),SE*labelscalefactor);
label("5",(-1.9,7.91),SE*labelscalefactor);
label("1",(0.58,8.33),SE*labelscalefactor);
label("18",(-0.36,7.89),SE*labelscalefactor);
label("1",(-1.07,7.55),SE*labelscalefactor);
label("1",(-0.66,6.89),SE*labelscalefactor);
label("5",(-0.68,5.8),SE*labelscalefactor);
label("1",(-0.68,4.83),SE*labelscalefactor);
label("2",(0.09,7.27),SE*labelscalefactor);
label("1",(0.15,6.24),SE*labelscalefactor);
label("2",(0.11,5.26),SE*labelscalefactor);
label("1",(0.89,7.61),SE*labelscalefactor);
label("3",(0.89,6.63),SE*labelscalefactor);
label("9",(0.92,5.62),SE*labelscalefactor);
label("18",(-3.18,7.63),SE*labelscalefactor);
label("2",(-3.07,6.61),SE*labelscalefactor);
label("2",(-3.09,5.62),SE*labelscalefactor);
label("1",(-2.29,7.25),SE*labelscalefactor);
label("3",(-2.27,6.22),SE*labelscalefactor);
label("5",(-2.29,5.2),SE*labelscalefactor);
label("7",(-1.49,6.89),SE*labelscalefactor);
label("34",(-1.52,5.81),SE*labelscalefactor);
label("1",(-1.41,4.86),SE*labelscalefactor); [/asy]
2002 China Team Selection Test, 2
There are $ n$ points ($ n \geq 4$) on a sphere with radius $ R$, and not all of them lie on the same semi-sphere. Prove that among all the angles formed by any two of the $ n$ points and the sphere centre $ O$ ($ O$ is the vertex of the angle), there is at least one that is not less than $ \displaystyle 2 \arcsin{\frac{\sqrt{6}}{3}}$.
1976 IMO, 3
A box whose shape is a parallelepiped can be completely filled with cubes of side $1.$ If we put in it the maximum possible number of cubes, each of volume $2$, with the sides parallel to those of the box, then exactly $40$ percent of the volume of the box is occupied. Determine the possible dimensions of the box.
2006 German National Olympiad, 2
Five points are on the surface of of a sphere of radius $1$. Let $a_{\text{min}}$ denote the smallest distance (measured along a straight line in space) between any two of these points. What is the maximum value for $a_{\text{min}}$, taken over all arrangements of the five points?
2005 Harvard-MIT Mathematics Tournament, 1
The volume of a cube (in cubic inches) plus three times the total length of its edges (in inches) is equal to twice its surface area (in square inches). How many inches long is its long diagonal?
2005 Paraguay Mathematical Olympiad, 2
If you multiply the number of faces that a pyramid has with the number of edges of the pyramid, you get $5.100$. Determine the number of faces of the pyramid.
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.
2022 BMT, Tie 1
Let $ABCDEF GH$ be a unit cube such that $ABCD$ is one face of the cube and $\overline{AE}$, $\overline{BF}$, $\overline{CG}$, and $\overline{DH}$ are all edges of the cube. Points $I, J, K$, and $L$ are the respective midpoints of $\overline{AF}$, $\overline{BG}$, $\overline{CH}$, and $\overline{DE}$. The inscribed circle of $IJKL$ is the largest cross-section of some sphere. Compute the volume of this sphere.