MathLabs

Problem 6

An international society has its members from six different countries. The list of members contains 19781978 names, numbered 1,2,…,19781,2,\ldots,1978. Prove that there is at least one member whose number is the sum of the numbers of two members from his own country, or twice as large as the number of one member from his own country.
Step 2 of 6: Form differences
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In plain words

A same-color difference immediately closes the proof.

ai−a1(2≤i≤330)a_i-a_1\quad(2\le i\le330)
Detailed analysis

If one difference ai−a1a_i-a_1 has the same country as a1a_1, then ai=a1+(ai−a1)a_i=a_1+(a_i-a_1) is the required sum. Therefore assume all 329329 differences lie in the other 55 countries.