RMO–1999

1. Prove that the inradius of a right-angled triangle with integer sides is an integer.

2. Find the number of positive integers which divide 10⁹⁹⁹ but not 10⁹⁹⁸.

3. Let ABCD be a square and M,N points on sides AB,BC, respectably, such that angle
MDN =45 ^o. If R is the midpoint of MN show that RP = RQ where P,Q are the points of
intersection of AC with the lines MD,ND.

4. If p, q, r are the roots of the cubic equation x³ − 3px² + 3q²x − r³= 0, show that
 p = q = r.

5. If a, b, c are the sides of a triangle prove the following inequality:
a/(c+ a − b)+ b/(a + b − c)+ c/(b + c − a) >= 3.

6. Find all solutions in integers m, n of the equation
(m − n)² =4mn/(m + n − 1).

7. Find the number of quadratic polynomials, ax2+bx+c, which satisfy the following
 conditions:
(a) a, b, c are distinct;
(b) a, b, c 2 {1, 2, 3, . . . 1999} and
(c) x + 1 divides ax² + bx + c.