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 ∠MDN = 450. 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: \frac{a}{c+a-b}+\frac{b}{a+b-c}+\frac{c}{b+c-a} \ge 3

6. Find all solutions in integers m, n of the equation (m − n)2 = \frac{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 ∈ {1, 2, 3, . . . 1999} and
(c) x + 1 divides ax2 + bx + c.

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