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IMO预选题1977
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.
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