Problem 6

An international society has its members from six different countries. The list of members contains

1978

names, numbered

1,2,\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 3 of 6: Repeat the difference descent

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In plain words

Each descent either proves the claim or reduces the number of available countries.

\left\lceil\frac{329}{5}\right\rceil=66,\quad\left\lceil\frac{65}{4}\right\rceil=17

Detailed analysis

At least $66$ of the differences share one country. Take their $65$ differences from the smallest; if a difference lands in any already used country, the nested difference representation gives the required same-country sum. Otherwise at least $17$ of these differences share a new country.