RMO–1994
1.A leaf is torn from a paperback novel.The sum of the numbers on the remaining pages
   is 15000. What are the page numbers on the torn leaf.

2. In the triangle ABC, the incircle touches the sides BC, CA and AB respectively  at D,
    E and F.  If  the radius of the incircle is 4 units and if BD, CE and AF are consecutive
    integers, find the sides of the triangle ABC.

3. Find all 6-digit natural numbers a₁a₂a₃a₄a₅a₆ formed by using the digits   1, 2, 3, 4, 5,
    6, once each such that the number a₁a₂ . . . a_k is divisible by k, for 1<= k<= 6.

4. Solve the system of equations for real x and y :
5x{1 +1/(x² + y²)}= 12
5y{1 −1/(x² + y²)}= 4.

5. Let A be a set of 16 positive integers with the property that the product of any two
 distinct numbers of A will not exceed 1994. Show that there are two numbers a and
 b in A which are not relatively prime.

6. Let AC and BD be two chords of a circle with center O such that they intersect at
  right angles inside the circle at the point M. Suppose K and L are the mid-points of
  the chord AB and CD respectively. Prove that OKML is a parallelogram.

7. Find the number of all rational numbers m/n such that
(a) 0 < m/n < 1
(b) m and n are relatively prime
(c) mn = 25!

8. If a, b and c are positive real numbers such that a + b + c = 1, prove that
(1 + a)(1 + b)(1 + c)>= 8(1 − a)(1 − b)(1 − c).