43rd Polish Mathematical Olympiad Problems 1992

43rd Polish Mathematical Olympiad Problems 1992

A1.  Segments AC and BD meet at P, and |PA| = |PD|, |PB| = |PC|. O is the circumcenter of the triangle PAB. Show that OP and CD are perpendicular.

A2.  Find all functions f : Q⁺ → Q⁺, where Q⁺ is the positive rationals, such that f(x+1) = f(x) + 1 and f(x³) = f(x)³ for all x.

A3.  Show that for real numbers x₁, x₂, ... , x_n we have ∑_i=1^m (∑_j=1ⁿ x_ix_j/(i+j) ) ≥ 0. When do we have equality?

B1.  The functions f₀, f₁, f₂, ... are defined on the reals by f₀(x) = 8 for all x, f_n+1(x) = √(x² + 6f_n(x)). For all n solve the equation f_n(x) = 2x.

B2.  The base of a regular pyramid is a regular 2n-gon A₁A₂...A_2n. A sphere passes through the apex S of the pyramid and cuts the edge SA_i at B_i (for i = 1, 2, ... , 2n). Show that ∑ SB_2i-1 = ∑ SB_2i.

B3.  Show that k³! is divisible by (k!)^k2+k+1.