Soal Kompetisi Matematika : 4-th Mediterranean Mathematical Competition 2001

1. Let P and Q be points on a circle k. A chord AC of k passes through the midpoint M of PQ. Consider a trapezoid ABCD inscribed in k with AB ║ CD. Prove that the intersection point X of AD and BC depends only on k and P,Q.

2. Find al integers n for which the polynomial p(x) = x5 − nx − n − 2 can be represented as a product of two non-constant polynomials with integer coefficients.

3. Show that there exists a positive integer N such that the decimal repre- sentation of 2000N starts with the digits 200120012001.

4. Let S be the set of points inside a given equilateral triangle ABC with side 1 or on its boundary. For any M ∈ S, aM, bM, cM denote the distances from M to BC,CA,AB, respectively. Define

(a) Describe the set {M ∈ S | f(M) ≥ 0} geometrically.

(b) Find the minimum and maximum values of f(M) as well as the points in which these are attained.

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