RMO–1997

1. Let P be an interior point of a triangle ABC and let BP and CP meet AC and AB in E
    and F respectively. If [BPF] = 4, [BPC] = 8 and [CPE] = 13, find [AFPE]. (Here [·]
    denotes the area of a triangle or a quadrilateral, as the case may be.)

2. For each positive integer n, define a_n = 20 + n², and d_n = gcd(a_n, a_n+1). Find the set
    of all values that are taken by d_n and show by examples that each of these values are
    attained.

3. Solve for real x:
    1/[x]+1/[2x]= (9x) +1/3,
    where [x] is the greatest integer less than or equal to x and (x) = x − [x],[e.g.[3.4] = 3
    and (3.4) = 0.4].

4. In a quadrilateral ABCD, it is given that AB||CD and the diagonals AC and BD are
    perpendicular to each other.
    Show that:
    (a) AD.BC >= AB.CD;
    (b) AD + BC >= AB + CD.

5. Let x, y and z be three distinct real positive numbers. Determine with proof whether
    or not the 3 real numbers |x/y−y/x|, |y/z−z/y|, |z/x−x/z| can be the lengths of the sides
    of a triangle.

6. Find the number of unordered pairs {A,B} (i.e., the pairs {A,B}and{B,A} are
    considered to be the same) of subsets of an n-element set X which satisfy the conditions:
    (a) A not equal to B;
    (b) AUB = X
   [e.g., if X = {a, b, c, d}, then {{a, b}, {b, c, d}}, {{a}, {b, c, d}}, { , {a, b, c, d}} are
   some of theadmissible pairs.]