52nd Polish Mathematical Olympiad Problems 2001

52nd Polish Mathematical Olympiad Problems 2001

A1.  Show that x₁ + 2x₂ + 3x₃ + ... + nx_n ≤ ½n(n-1) + x₁ + x₂² + x₃³ + ... + x_nⁿ for all non-negative reals x_i.

A2.  P is a point inside a regular tetrahedron with edge 1. Show that the sum of the distances from P to the vertices is at most 3.

A3.  The sequence x₁, x₂, x₃, ... is defined by x₁ = a, x₂ = b, x_n+2 = x_n+1 + x_n, where a and b are reals. A number c is a repeated value if it occurs in the sequence more than once. Show that we can choose a, b so that the sequence has more than 2000 repeated values, but not so that it has infinitely many repeated values.

B1.  a and b are integers such that 2ⁿa + b is a square for all non-negative integers n. Show that a = 0.

B2.  ABCD is a parallelogram. K is a point on the side BC and L is a point on the side CD such that BK·AD = DL·AB. DK and BL meet at P. Show that ∠DAP = ∠BAC.

B3.  Given a set of 2000 distinct positive integers under 10¹⁰⁰, show that one can find two non-empty disjoint subsets which have the same number of elements, the same sum and the same sum of squares.