# Sums in an Arithmetic Progression

Let A be any set of 20 distinct integers chosen from the arithmetic progression 1, 4, 7,..., 100. Prove that there must be two distinct integers in A whose sum is 104.

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Copyright © 1996-2018 Alexander Bogomolny
Let A be any set of 19 distinct integers chosen from the arithmetic progression 1, 4, 7,..., 100. Prove that there must be two distinct integers in A whose sum is 104.

### Solution

The general term of the progression is in the form 3n + 1. 1 = 3×0 + 1, 100 = 3×33 + 1. There are 34 terms in all. We may split all the terms of the arithmetic progression into 18 subsets: two singletons {1} and {52} and 16 pairs

### Reference

- 18.S34 Problem Solving Seminar, Fall 2007, MIT Opencourseware

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Copyright © 1996-2018 Alexander Bogomolny