37th Spanish 2001


A1. Show that the graph of the polynomial p(x) is symmetric about the point (a,b) iff there is a polynomial q(x) such that p(x) = b + (x-1) q((x-a)²).
A2. P is a point inside the triangle ABC equidistant from A and B. Exterior triangles BQC and CRA are constructed similar to APB. Show that P, Q, C, R are collinear or form a parallelogram.
A3. Five segments are such that any three of them can be used to form a triangle. Show that at least one of these triangles is acute-angled.
B1. Can we arrange the digits 0 to 9 into a 3 x 3 array so that the six numbers (the three rows left to right, and the three columns top to bottom) add up to 2001?
B2. ABCD is a quadrilateral inscribed in a circle radius 1 with AB a diameter. It has an inscribed circle. Show that CD ≤ 2√5 - 4.
B3. Let N be the set of positive integers. Find a function f : N → N such that f(1) = f(2ⁿ) = 1 for all n ∈ N, and f(2ⁿ + m) = f(n) + 1 for m < 2ⁿ. Find the maximum value of f(n) for n ≤ 2001. Find the smallest n such that f(n) = 2001.

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.