C. 945. On 1 March, 2008, the Hungarian National Bank withdrew 1 and 2-forint (HUF) coins from circulation. At the cash desk, prices of individual items are not rounded, but the total is rounded to the nearest five. An information leaflet formulates the rule of rounding as follows: Totals ending in 1 or 2 are rounded down to 0; totals ending in 3 or 4 are rounded up to 5; totals ending in 6 or 7 are rounded down to 5; totals ending in 8 or 9 are rounded up to 0. Martin buys two croissants in the little shop on the corner every morning. In a few days, he saves exactly the price of a croissant. Given that a croissant costs more than 10 forints, what may be the price of a croissant so that he can save that amount in the shortest possible time?
C. 947. A rabbit is sitting at a point A, at a distance of 160 m from a straight railway track. The perpendicular projection of A onto the track is T. A train is approaching T at a speed of 30 m/s. The distance of the front of the train from point T is 300 m initially. The rabbit can run at 15 m/s. Can he cross the track in any direction before the train comes?
C. 949. The midpoint of the base AB of an isosceles triangle ABC is F, and its orthocentre is M. Given that the centroid of the triangle lies on the inscribed circle and , what may be the lengths of the sides of the triangle?
B. 4095. Given infinitely many cuboids with edges parallel to the axes of the coordinate space, one vertex at the origin and all vertices at points of non-negative integer coordinates. Is it always possible to select two, one of which contains the other?
B. 4096. The points of tangency of the inscribed circle on the sides of a triangle ABC are E, F and G. The distances of an arbitrary point P of the inscribed circle from the sides are a, b and c, and its distances from the lines FG, EG and EF are e, f and g. Prove that abc=efg.
A. 455. Let H be a set with n elements and let each of the families and consist of p subsets of H such that these 2p subsets are pairwise distinct. Suppose that for every and , the sets A and B have at least one common element. Show that .
A. 456. The point D lies in the triangle ABC such that the circles inscribed in triangles ABD, BCD, and CAD pairwise touch each other. On lines BC, CA, AB, AD, BD, CD, denote the touching points by A1, B1, C1, A2, B2, C2, respectively. Let lines B1C2 and B2C1 meet at E, and let lines A1C2 and A2C1 meet at F. Show that the lines AF, BE, and C1D are concurrent.