AoPS Academy
be a positive integer. Consider necklaces of length n whose beads are colored in one of three colors, say red, green, or blue, with exactly two beads of each color (so 
). A rotation of the necklace or a reflection (flipping) is considered the same necklace. But now impose the extra condition that no two beads of the same color are adjacent around the circle. Determine the number 
of such distinct necklaces (up to rotation and reflection).
and 
meet at two distinct points 
and 
. Choose any line 
through , and let meet again at 
(other than ) and meet again at 
. On , let 
be the midpoint of the minor arc 
(i.e., the point on such that 
subtends the arc ), and on let 
be the midpoint of the minor arc 
. Prove that![\[
MN \parallel \text{(radical axis of } \omega_1, \omega_2).
\]](http://latex.artofproblemsolving.com/6/6/4/6647f0f7182bdf5096ec5f2956a272803082f4f7.png)
be the foot of the altitude from the vertex 
in the acute-angled 
. Let 
be the midpoint of side 
, and 
be the circumcentre of . Line 
meets line 
at 
. Prove that the points 
, and lie on a circle.
. What is the smallest 
such that the tetrahedron can be passed through a loop of twine of length ?
.
so that it can loop around one edge.
. Doesn't the problem ask equivalently for the parallel projection (i.e. shadow) of the tetrahedron on a plane with the smallest circumference? So take the plane perpendicular to one edge.
is of course sufficient. Start with one edge and strip it through till the opposite edge. At each moment, the string forms a rectangle of circumference .
![$f:\mathbb (0;1] \to\mathbb R$](http://latex.artofproblemsolving.com/7/9/9/799657d25f079cf2f55cd420ee41aa57bebc68a9.png)
such that![$$f(x)+f(1-x)=0\; \forall x \in (0;1] $$](http://latex.artofproblemsolving.com/7/d/e/7de74760f30216f280a168cd1e8c5737cb6bff0c.png)
number, note that the problem asks for a contour of any shape, not just a circular one.