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 an = 20 + n2, and dn = gcd(an, an+1). Find the set of all values that are taken by dn and show by examples that each of these values are attained.

3. Solve for real x: \frac{1}{x} + \frac{1}{2x} = 9x) +\frac{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 is parallel to 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 three real numbers
|\frac{x}{y} - \frac{y}{x}|, |\frac{y}{z} - \frac{z}{y}|, |\frac{z}{x} - \frac{z}{a}|
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 ≠ B;
(b) A ∪ B = X
[e.g., if X = {a, b, c, d}, then {{a, b}, {b, c, d}}, {{a}, {b, c, d}}, {Φ, {a, b, c, d}} are some of the admissible pairs.]
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