RMO–1996

1. The sides of a triangle are three consecutive integers and its inradius is four units.
    Determine the circumradius.

2. Find all triples (a, b, c) of positive integers such that (1 +1/a)(1 +1/b)(1 +1/c) = 3.

3. Solve for real number x and y:
     xy² = 15x² + 17xy + 15y²
     x²y = 20x² + 3y².

4. Suppose N is an n-digit positive integer such that
(a) all the n-digits are distinct; and
(b) the sum of any three consecutive digits is divisible by 5.
Prove that n is at most 6. Further, show that starting with any digit one can find a six-
digit number with these properties.

5. Let ABC be a triangle and h_a the altitude through A. Prove that(b + c)²  >= a² + 4h_a².
(As usual a, b, c denote the sides BC, CA, AB respectively.)

6.Given any positive integer n show that there are two positive rational numbers a and b,
 a npt equal to b,which are not  integers and which are such that a − b, a² − b², a³ − b³,
. . . ,aⁿ − bⁿ are all integers.

7. If A is a fifty-element subset of the set {1, 2, 3, . . . , 100} such that no two numbers
from A add up to 100 show that A contains a square.