48th Polish Mathematical Olympiad Problems 1997

48th Polish Mathematical Olympiad Problems 1997

A1.  The positive integers x₁, x₂, ... , x₇ satisfy x₆ = 144, x_n+3 = x_n+2(x_n+1+x_n) for n = 1, 2, 3, 4. Find x₇.

A2.  Find all real solutions to 3(x² + y² + z²) = 1, x²y² + y²z² + z2x² = xyz(x + y + z)³.

A3.  ABCD is a tetrahedron. DE, DF, DG are medians of triangles DBC, DCA, DAB. The angles between DE and BC, between DF and CA, and between DG and AB are equal. Show that area DBC ≤ area DCA + area DAB.

B1.  The sequence a₁, a₂, a₃, ... is defined by a₁ = 0, a_n = a_[n/2] + (-1)^n(n+1)/2. Show that for any positive integer k we can find n in the range 2^k ≤ n < 2^k+1 such that a_n = 0.

B2.  ABCDE is a convex pentagon such that DC = DE and ∠C = ∠E = 90^o. F is a point on the side AB such that AF/BF = AE/BC. Show that ∠FCE = ∠FDE and ∠FEC = ∠BDC.

B3.  Given any n points on a unit circle show that at most n²/3 of the segments joining two points have length > √2.