IMO预选题1977.pdfVIP

IMO LongList 1977 IMO ShortList/LongList Project Group June 19, 2004 1. (Bulgaria 1) Let N be the set of positive integers. Let f be a function defined on N, which satisfies the inequality f (n + 1) f (f (n)) for all n ∈ N. Prove that for any n we have f (n) = n. Remark: This question was chosen as sixth question in the IMO. 2. (Bulgaria 2) The pentagon ABCDE inscribed in a circle, for which BC CD and AB DE , is the basis of the pyramid with vertex S . If AS is the longest edge starting from S , prove that BS CS. 3. (Bulgaria 3) In a company of n persons each person has no more than d acquaintances and in that company there exist a group of k persons, k ≥ d, who are not acquainted to each other. Prove that the number of acquainted couples is not greater than n2 . 4 4. (Bulgaria 4) In general position n points are given in the space. Some pairs of these points are connected by line segments so that the number of segments equals n2 , and a triangle exists. 4 Prove that any point from which the maximal number of segments starts, is a vertex of a triangle. 5. (Federal Republic Of Germany 1) Let a, b be two natural numbers. When we divide a2 +b2 by a + b, we the the remainder r and the quotient q. Determine all pairs (a, b) for which q2 +r = 1977. Remark: This question was chosen as fifth question in the IMO. 6. (Federal Republic Of Germany 2) Describe all closed and bounded figures Φ in the plane, whose any two points are connectable by a semicircle lying in Φ. Remark: This problem was included in the IMO shortlist.