Problem 6
An international society has its members from six different countries. The list of members contains names, numbered . 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 6 of 6: Conclude
In plain words
The pigeonhole descent forces a monochromatic additive relation.
Detailed analysis
Thus some member's number is the sum of two numbers of members from his own country, or twice one such number, as required.