36th Polish Mathematical Olympiad Problems 1985

36th Polish Mathematical Olympiad Problems 1985

A1.  Find the largest k such that for every positive integer n we can find at least k numbers in the set {n+1, n+2, ... , n+16} which are coprime with n(n+17).

A2.  Given a square side 1 and 2n positive reals a₁, b₁, ... , a_n, b_n each ≤ 1 and satisfying ∑ a_ib_i ≥ 100. Show that the square can be covered with rectangles R_i with sides length (a_i, b_i) parallel to the square sides.

A3.  The function f : R → R satisfies f(3x) = 3f(x) - 4f(x)³ for all real x and is continuous at x = 0. Show that |f(x)| ≤ 1 for all x.

B1.  P is a point inside the triangle ABC is a triangle. The distance of P from the lines BC, CA, AB is d_a, d_b, d_c respectively. Show that 2/(1/d_a + 1/d_b + 1/d_c) < r < (d_a + d_b + d_c)/2, where r is the inradius.

B2.  p(x,y) is a polynomial such that p(cos t, sin t) = 0 for all real t. Show that there is a polynomial q(x,y) such that p(x,y) = (x² + y² - 1) q(x,y).

B3.  There is a convex polyhedron with k faces. Show that if k/2 of the faces are such that no two have a common edge, then the polyhedron cannot have an inscribed sphere.