51st Polish Mathematical Olympiad Problems 2000

51st Polish Mathematical Olympiad Problems 2000

A1.  How many solutions in non-negative reals are there to the equations:
x₁ + x_n² = 4x_n
x₂ + x₁² = 4x₁
...
x_n + x_n-1² = 4x_n-1?

A2.  The triangle ABC has AC = BC. P is a point inside the triangle such that ∠PAB = ∠PBC. M is the midpoint of AB. Show that ∠APM + ∠BPC = 180^o.

A3.  The sequence a₁, a₂, a₃, ... is defined as follows. a₁ and a₂ are primes. a_n is the greatest prime divisor of a_n-1 + a_n-2 + 2000. Show that the sequence is bounded.

B1.  PA₁A₂...A_n is a pyramid. The base A₁A₂...A_n is a regular n-gon. The apex P is placed so that the lines PA_i all make an angle 60^o with the plane of the base. For which n is it possible to find B_i on PA_i for i = 2, 3, ... , n such that A₁B₂ + B₂B₃ + B₃B₄ + ... + B_n-1B_n + B_nA₁ < 2A₁P?

B2.  For each n ≥ 2 find the smallest k such that given any subset S of k squares on an n x n chessboard we can find a subset T of S such that every row and column of the board has an even number of squares in T.

B3.  p(x) is a polynomial of odd degree which satisfies p(x²-1) = p(x)² - 1 for all x. Show that p(x) = x.