40th Polish Mathematical Olympiad Problems 1989

40th Polish Mathematical Olympiad Problems 1989

A1.  An even number of politicians are sitting at a round table. After a break, they come back and sit down again in arbitrary places. Show that there must be two people with the same number of people sitting between them as before the break.

A2.  k₁, k₂, k₃ are three circles. k₂ and k₃ touch externally at P, k₃ and k₁ touch externally at Q, and k₁ and k₂ touch externally at R. The line PQ meets k₁ again at S, the line PR meets k₁ again at T. The line RS meets k₂ again at U, and the line QT meets k₃ again at V. Show that P, U, V are collinear.

A3.  The edges of a cube are labeled from 1 to 12. Show that there must exist at least eight triples (i, j, k) with 1 ≤ i < j < k ≤ 12 so that the edges i, j, k are consecutive edges of a path. But show that the labeling can be done so that we cannot find nine such triples.

B1.  n, k are positive integers. A₀ is the set {1, 2, ... , n}. A_i is a randomly chosen subset of A_i-1 (with each subset having equal probability). Show that the expected number of elements of A_k is n/2^k.

B2.  Three circles of radius a are drawn on the surface of a sphere of radius r. Each pair of circles touches externally and the three circles all lie in one hemisphere. Find the radius of a circle on the surface of the sphere which touches all three circles.

B3.  Show that for positive reals a, b, c, d we have ((ab + ac + ad + bc + bd + cd)/6)^1/2 ≥ ((abc + abd + acd + bcd)/4)^1/3