Mathematics Olympiad proplem 1995

Here are the IMO 1995 problems as posted by Abhilash thakur

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.