47th Polish Mathematical Olympiad Problems 1996

47th Polish Mathematical Olympiad Problems 1996

A1.  Find all pairs (n,r) with n a positive integer and r a real such that 2x²+2x+1 divides (x+1)ⁿ - r.

A2.  P is a point inside the triangle ABC such that ∠PBC = ∠PCA < ∠PAB. The line PB meets the circumcircle of ABC again at E. The line CE meets the circumcircle of APE again at F. Show that area APEF/area ABP does not depend on P.

A3.  a_i, x_i are positive reals such that a₁ + a₂ + ... + a_n = x₁ + x₂ + ... + x_n = 1. Show that 2 ∑_i<j x_ix_j ≤ (n-2)/(n-1) + ∑ a_ix_i²/(1-a_i). When do we have equality?

B1.  ABCD is a tetrahedron with ∠BAC = ∠ACD and ∠ABD = ∠BDC. Show that AB = CD.

B2.  Let p(k) be the smallest prime not dividing k. Put q(k) = 1 if p(k) = 2, or the product of all primes < p(k) if p(k) > 2. Define the sequence x₀, x₁, x₂, ... by x₀ = 1, x_n+1 = xnp(x_n)/q(x_n). Find all n such that x_n = 111111.

B3.  Let S be the set of permutations a₁a₂...a_n of 123...n such that a_i ≥ i. An element of S is chosen at random. Find all n such that the probability that the chosen permutation satisfies a_i ≤ i+1 exceeds 1/3.