Here are the IMO 1995 problems as posted by Khanh Diep on sci.math: 1. Let A, B, C and D be four distinct points on a line, in that order. The circles with diameters AC and BD intersect at the points X and Y. The line XY meets BC at the point Z. Let P be a point on the line XY different from Z. The line CP intersects the circle with diameter AC at the points C and M, and the line BP intersects the circle with diameter BD at the points B and N. Prove that the lines AM, DN and XY are concurrent. 2. Let a, b and c be positive real numbers such that a*b*c=1. Prove that 1 1 1 3 ---------- + ---------- + ---------- >= - (a^3)(b+c) (b^3)(c+a) (c^3)(a+b) 2 3. Determine all integers n>3 for which there exist n points A1, A2, ..., An in the plane, and real numbers r1, r2, ..., rn satisfying the following two conditions: (i) no three of the points A1, A2, ..., An lie on a line; (ii) for each triple i, j, k (1 <= i < j < k <= n) the triangle AiAjAk has area equal to ri+rj+rk. 4. Find the maximum value of x[0] for which there exists a sequence of positive real numbers x[0], x[1], ..., x[1995] satisfying the two conditions: (i) x[0]=x[1995]; (ii) x[i-1] + 2/(x[i-1]) = 2x[i] + 1/x[i] for each i = 1, 2, ..., 1995. [Note: x[i] means x subscript i] 5. Let ABCDEF be a convex hexagon with AB=BC=CD and DE=EF=FA, and angle BCD=angle EFA=60 degrees. Let G and H be two points in the interior of the hexagon such that angle AGB=andgle DHE=120 degrees. Prove that AG+GB+GH+DH+HE >= CF 6. Let p be an odd prime number. Find the number of subsets of A of the set {1, 2, ..., 2p} such that (i) A has exactly p elements, and (ii) the sum of all the elements in A is divisible by p.