3rd Austrian-Polish 1980


1. A, B, C are infinite arithmetic progressions of integers. {1, 2, 3, 4, 5, 6, 7, 8} is a subset of their union. Show that 1980 also belongs to their union.
2. 1 = a₁ < a₂ < a₃ < ... is an infinite sequence of integers such that a_n < 2n-1. Show that every positive integer is the difference of two members of the sequence.
3. P is an interior point of a tetrahedron. Show that the sum of the six angles subtended by the sides at P is greater than 540^o.
4. If S is a non-empty set of positive integers, let p(S) be the reciprocal of the product of the members of S. Show that ∑ p(S) = n, where there sum is taken over all non-empty subsets of {1, 2, ... , n}.
5. ABC is a triangle. A' is any point on the segment BC other than B and C. P_A is the perpendicular bisector of AA'. The lines P_B and P_C are defined similarly. Show that no point can lie on all of P_A, P_B and P_C.
6. The real numbers x₁, x₂, x₃, ... satisfy \|x_m+n - x_m - x_n\| ≤ 1 for all m, n. Show that \|x_m/m - x_n/n\| < 1/m + 1/n for all m, n.
7. Find the largest n for which we can find n-1 distinct positive integers a_i such that a_i = b_i + 1980/b_i for some integer b_i, where b₁b₂ ... b_n-1 divides 1980.
8. Given 1980 points in the plane, no two a distance < 1 apart, show that we can find a subset of 220 points, no two a distance < √3 apart.
9. C is an arbitrary point on the tangent to the circle K at A. D₁, D₂, E₁, E₂ are any points on the circle such that C, D₁, E₁ are collinear in that order, and C, D₂, E₂ are collinear in that order. AB is a diameter of the circle and the tangent at B meets the lines AD₁, AD₂, AE₁, AE₂ at M₁, M₂, N₁, N₂ respectively. Show that M₁M₂ = N₁N₂.

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.