British Mathematical Olympiad
The British Mathematical Olympiad is a national math competition held in the United Kingdom. Solvers who score over a certain threshold in the Senior Mathematical Challenge are automatically entered to the first round, but others can register for the first round.
Contents
[hide]Structure
The British Mathematical Olympiad is divided into two rounds. In the first round (BMO 1), solvers have 3.5 hours to solve 6 problems. High scorers can move on into the second round (BMO 2), where solvers have 3.5 hours to solve 4 problems.
For both rounds, each problem is worth 10 points. Like most Olympiads, complete solutions are required in order to get full credit.
Participants who submit a solution with the highest quality in BMO 2 can earn the Christopher Bradley elegance prize.
Resources
21-th Mathematical Olympiad 1985 Problem 5
A circular hoop of radius 4 cm is held fixed in a horizontal plane. A cylinder with radius 4 cm and length 6 cm rests on the hoop with its axis horizontal, and with each of its two circular ends touching the hoop at two points. The cylinder is free to move subject to the condition that each of its circular ends always touches the hoop at two points. Find, with proof, the locus of the centre of one of the cylinder’s circular ends.
Solution
Let the centroid of the cylinder be the point The side surface of the cylinder is shown by dark blue.
Let the center of one of the circular ends be the point This end is shown by green. Its edge is a purple circle
Let the center of the hoop be
The hoop is shown by red.
Let cross
at point
Therefore
cross
at second point symmetrical to
with respect to the plane
Let
be the sphere of radius
centered at
Part of this sphere is shown in the diagram by yellow. Let the cylinder is glued to the sphere and point
is fixed.
In this case and
both lie on
and point
lies on the sphere centered at
with radius
The clime “The cylinder is free to move subject to the condition that each of its circular ends always touches the hoop at two points” has the equivalent form “The sphere is free to move with fixed center subject to the condition that cross
”
If the sphere rotates around an axis then point
moves along circle with axis
Let the sphere rotates around an axis perpendicular and
Axis view is shown on the diagram. We rotate
together with
in counterclockwise direction. Point
moves along
till point
in the plane
where
tought
” The point
moves to extreme position
If one rotate
in clockwise direction, point
moves to position
dsymmetric to
with respect
The locus of the point A is the belt with a width of 1.68 cm located on a sphere with a radius of 3 cm symmetrically to the circumference of the great circle located parallel hoop.
vladimir.shelomovskii@gmail.com, vvsss
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