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}. Ai is a randomly chosen subset of Ai-1 (with each subset having equal probability). Show that the expected number of elements of Ak is n/2k.

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


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