国际数学奥林匹克IMO试题(官方版)1982_eng.pdf

Twenty-third International Olympiad, 1982 1982/1. The function f(n) is defi ned for all positive integers n and takes on non-negative integer values. Also, for all m,n f(m + n) f(m) f(n) = 0 or 1 f(2) = 0,f(3) 0, and f(9999) = 3333. Determine f(1982). 1982/2. A non-isosceles triangle A1A2A3is given with sides a1,a2,a3(aiis the side opposite Ai). For all i = 1,2,3,Miis the midpoint of side ai, and Ti. is the point where the incircle touches side ai. Denote by Si the refl ection of Tiin the interior bisector of angle Ai. Prove that the lines M1,S1,M2S2, and M3S3are concurrent. 1982/3. Consider the infi nite sequences xn of positive real numbers with the following properties: x0= 1,and for all i 0,xi+1 xi. (a) Prove that for every such sequence, there is an n 1 such that x2 0 x1 + x2 1 x2 + + x2 n1 xn 3.999. (b) Find such a sequence for which x2 0 x1 + x2 1 x2 + + x2 n1 xn 4. 1982/4. Prove that if n is a positive integer such that the equation x3 3xy2+ y3= n has a solution in integers (x,y), then it has at least three such solutions. Show that the equation has no solutions in integers when n = 2891. 1982/5. The diagonals AC and CE of the regular hexagon ABCDEF are divided by the inner points M and N, respectively, so that AM AC = CN CE = r. Determine r if B,M, and N are collinear. 1982/6. Let S be a square with sides of length 100, and let L be a path within S which does not meet itself and which is composed of line segments A0A1,A1A2,An1Anwith A06= An. Suppose that for every point P of the boundary of S there is a point of L at a distance from P not greater than 1/2. Prove that there are two points X and Y in L such that the distance between X and Y is not greater than 1, and the length of that part of L which lies between X and Y is not smaller than 198. 。