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 1 of 6: Choose a large country
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In plain words

Pigeonhole creates the first reservoir of differences.

⌈19786⌉=330\left\lceil\frac{1978}{6}\right\rceil=330
Detailed analysis

Among 19781978 members in 66 countries, one country contains at least 330330 members. List their numbers as a1<⋯<a330a_1<\cdots<a_{330}.