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

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+4h²_a.
(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 ≠ 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.

Back