C. 850. An arbitrary interior point of a regular hexagon of unit side is reflected about the midpoint of each side, respectively. Calculate the area of the resulting hexagon.

C. 851. A fair coin is tossed 12 times in a row and the outcomes are listed. How many sequences of outcomes are possible in which there are no two consecutive heads?

B. 3904.ABC is an isosceles triangle. Drop a perpendicular from the midpoint D of the base BC onto the leg AC and denote the foot of the perpendicular by E. The midpoint of the line segment DE is F. Show that the lines BE and AF are perpendicular.

B. 3906. The skew lines e and f are perpendicular. The unit line segment AB has its endpoint A on e and endpoint B on f. What is the locus of the midpoint of AB as its endpoints are moving along the lines?

B. 3910.AB is a diameter in the circle k. Choose an arbitrary point E in the interior of the circle. The other intersections of the lines AE and BE with the circle are C and D, respectively. Prove that the value of the expression AC^{.}AE+BD^{.}BE is independent of the position of E.

B. 3911. Prove that it is possible to inscribe an ellipse in any acute triangle, such that the foci are the orthocentre and the circumcentre of the triangle, respectively.

A. 398. Given a circle k in the plane and a line lying outside k. Moreover, a point O is given on the circle. Define the binary operator + on the points of k as follows. For arbitrary points X and Y, denote by M_{XY} the intersection point of lines XY and . (If X=Y then take the tangent. If the lines are parallel then M_{XY} is the ideal point.) Construct the second intersection of OM_{XY} and k. (If the line is the tangent at O then also the second intersection is O.) Denote this point by X+Y (see the figure).

Show that operation + can be extended to the union of k, and the ideal point of such that the points with this operation form an Abelian group with the unity O; i.e. the operation has the following properties:

a) (X+Y)+Z=X+(Y+Z) for all points X, Y, Z;

b) X+Y=Y+X for all points X, Y;

c) X+O=X for all points X;

d) For an arbitrary point X there exists a point Y such that X+Y=O.

A. 399. Given n 3-element subset of a certain set S. Prove that if n=6 then the elements of S can be colored with two colors such that each subset contains both colors. Is the same true for n=7?