K. 355. Let F be the point lying closer to point A that divides side AB of a square ABCD in a 1:2 ratio. Let E be the point closer to D that divides side AD in a 1:2 ratio, and let G be the point closer to C that divides side CD in a 1:5 ratio. In what ratio do the diagonals of quadrilateral EFBG divide each other?
K. 358. In a warehouse, there are cubical boxes of the same mass and size stacked in the form of a large rectangular block. A worker is loading the boxes onto a truck. In each step, he transfers to the truck one vertical or horizontal layer of boxes, with a thickness of a single box. (He is able to approach the stack of boxes from any direction.) The mass of the boxes is a whole number of kilograms. The total mass of the boxes loaded in the first step is 60 kg, in the second step it is 84 kg, and in the third step it is 112 kg. What may be the total mass of all the boxes, in kilograms?
K. 360. The accompanying figure is drawn on a square lattice. The separation of consecutive parallel grid lines is 5 mm. The polygon consists of 999 line segments. Determine its length, rounded to the nearest metre.
C. 1146. Given one vertex and the midpoints of all the sides of a pentagon on the plane, prove that it is always possible to find a point on the plane, such that it forms a parallelogram with three of the six given points, and another parallelogram with the other three points.
C. 1149. The number of one-element subsets, the number of two-element subsets and the number of three-element subsets of a set of more than three elements are consecutive terms of an arithmetic progression. How many elements does the set have?
B. 4493. Let (n,k) denote the greatest common divisor of the positive integers n and k, and let [n,k] denote their least common multiple. Show that, for all positive integers a, b, c, the greatest common divisor of the numbers [a,b], [b,c], [c,a] equals the least common multiple of the numbers (a,b), (b,c), (c,a).
B. 4494.F is the midpoint of base BC of the isosceles triangle ABC. D is an interior point of the line segment BF. The perpendicular dropped from point D onto BC intersects side AB at M, and the line through point D, parallel to AB intersects side AC at P. Express the ratio of the areas of triangles AMP and ABC in terms of k=BD:BC.
B. 4495. Given the parallelogram ABCD and points F and G, such that AF=FC, BG=GD and given that triangles AFC and BGD are similar, prove that the line FG is perpendicular to a side of the parallelogram.
B. 4496. An absent minded professor is travelling in a train along a subway line having infinite length in both directions. He would like to get from Alpha station to Tau station. At every station, the probability that he looks up from his notes is the same p>0 (independently of each other). If he looks up and observes that he has reached his destination, then he gets off. If he can see that he has passed his destination, then he also gets off and gets on the train travelling in the opposite direction. If he looks up and sees that he has not reached his destination yet, he stays in his train. If he does not look up, he will stay on the train wherever he is. What is the probability that he will get off at Tau station after a while?
B. 4498. A vertex B of a convex quadrilateral ABCD is said to be ``special'' if the reflection of line BD in the bisector of angle ABC bisects the diagonal AC. Prove that if one vertex of a cyclic quadrilateral is special then all vertices are special.