44th Polish Mathematical Olympiad Problems 1993

44th Polish Mathematical Olympiad Problems 1993

A1.  Find all rational solutions to:
t² - w² + z² = 2xy
t² - y² + w² = 2xz
t² - w² + x² = 2yz.

A2.  A circle center O is inscribed in the quadrilateral ABCD. AB is parallel to and longer than CD and has midpoint M. The line OM meets CD at F. CD touches the circle at E. Show that DE = CF iff AB = 2CD.

A3.  g(k) is the greatest odd divisor of k. Put f(k) = k/2 + k/g(k) for k even, and 2^(k+1)/2 for k odd. Define the sequence x₁, x₂, x₃, ... by x₁ = 1, x_n+1 = f(x_n). Find n such that x_n = 800.

B1.  P is a convex polyhedron with all faces triangular. The vertices of P are each colored with one of three colors. Show that the number of faces with three vertices of different colors is even.

B2.  Find all real-valued functions f on the reals such that f(-x) = -f(x), f(x+1) = f(x) + 1 for all x, and f(1/x) = f(x)/x² for x ≠ 0.

B3.  Is the volume of a tetrahedron determined by the areas of its faces and its circumradius?