54th Polish Mathematical Olympiad Problems 2003

54th Polish Mathematical Olympiad Problems 2003

A1.  ABC is acute-angled. M is the midpoint of AB. A line through M meets the lines CA, CB at K, L with CK = CL. O is the circumcenter of CKL and CD is an altitude of ABC. Show that OD = OM.

A2.  0 ≤ k₁ < k₂ < ... < k_n are integers. 0 < a < 1 is a real. Show that (1-a)(a^k₁ + a^k₂ + ... + a^k_n)² < (1+a)(a^2k₁ + a^2k₂ + ... + a^2k_n).

A3.  Find all polynomials p(x) with integer coefficients such that p(n) divides 2ⁿ - 1 for n = 1, 2, 3, ... .

B1.  p is a prime and a, b, c, are distinct positive integers less than p such that a³ = b³ = c³ mod p. Show that a² + b² + c² is divisible by a + b + c.

B2.  ABCD is a tetrahedron. The insphere touches the face ABC at H. The exsphere opposite D (which also touches the face ABC and the three planes containing the other faces) touches the face ABC at O. If O is the circumcenter of ABC, show that H is the orthocenter of ABC.

B3.  n is even. Show that there is a permutation a₁a₂...a_n of 12...n such that a_i+1 ∈ {2a_i, 2a_i-1, 2a_i-n, 2a_i-n-1} for i = 1, 2, ... , n (and we use the cyclic subscript convention, so that a_n+1 means a₁).