3rd Austrian-Polish Mathematics Competition 1980 Problems

3rd Austrian-Polish Mathematics Competition 1980 Problems

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.  [Missing]

Solutions

3rd APMC 1980 Problem 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.

Solution

If any two of 4, 6, 8 belong to the same set, then we are done. So assume (wlog) that 4 ∈ A, 6 ∈ B and 8 ∈ C. Then 5 cannot belong to A or B or we are done, so 5 ∈ C. Similarly 3 cannot belong to A or B or we are done, so 3 ∈ C. But if 3 and 5 are in C, then so is 7. Since 7 and 8 are in C, then so are all integers > 8 and hence 1980.

Thanks to Suat Namli