Regional Mathematical Olympiad – 2003

1. Let ABC be a triangle in which AB = AC and ∠CAB = 90⁰. Suppose M and N are points on the hypotenuse BC such that BM²+CN² = MN². Prove that ∠MAN = 45⁰.

3. Let a, b, c be three positive real numbers such that a + b + c = 1. Prove that among the three numbers a − ab, b − bc, c − ca there is one which is at most 1/4 and there is one which is at least 2/9.

4. Find the number of ordered triples (x, y, z) of nonnegative integers satisfying the conditions:
(i) x ≤ y ≤ z;
(ii) x + y + z ≤ 100.

5. Suppose P is an interior point of a triangle ABC such that the ratios , , are all equal. Find the common value of these ratios.[Here d(X, Y Z) denotes the perpendicular distance from a point X to the line Y Z.]

6. Find all real numbers a for which the equation
x² + (a − 2)x + 1 = 3|x|
has exactly three distinct real solutions in x.

7. Consider the set X = {1, 2, 3, . . . , 9, 10}. Find two disjoint nonempty subsets A and B of X such that
(a) A ∪ B = X;
(b) prod(A) is divisible by prod(B), where for any finite set of numbers C, prod(C) denotes the product of all numbers in C ;
(c) the quotient prod(A)/prod(B) is as small as possible.

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