49th Polish Mathematical Olympiad Problems 1998
A1. Find all solutions in positive integers to a + b + c = xyz, x + y + z = abc.
A2. Fn is the Fibonacci sequence F0 = F1 = 1, Fn+2 = Fn+1 + Fn. Find all pairs m > k ≥ 0 such that the sequence x0, x1, x2, ... defined by x0 = Fk/Fm and xn+1 = (2xn - 1)/(1 - xn) for xn ≠ 1, or 1 if xn = 1, contains the number 1.
A3. PABCDE is a pyramid with ABCDE a convex pentagon. A plane meets the edges PA, PB, PC, PD, PE in points A', B', C', D', E' distinct from A, B, C, D, E and P. For each of the quadrilaterals ABB'A', BCC'B, CDD'C', DEE'D', EAA'E' take the intersection of the diagonals. Show that the five intersections are coplanar.
B1. Define the sequence a1, a2, a3, ... by a1 = 1, an = an-1 + a[n/2]. Does the sequence contain infinitely many multiples of 7?
B2. The points D, E on the side AB of the triangle ABC are such that (AD/DB)(AE/EB) = (AC/CB)2. Show that ∠ACD = ∠BCE.
B3. S is a board containing all unit squares in the xy plane whose vertices have integer coordinates and which lie entirely inside the circle x2 + y2 = 19982. +1 is written in each square of S. An allowed move is to change the sign of every square in S in a given row, column or diagonal. Can we end up with all -1s by a sequence of allowed moves?
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