Problem

Source: India IMOTC Practice Test 1 Problem 3

Tags: combinatorics



In a conference, mathematicians from $11$ different countries participate and they have integer-valued ages between $27$ and $33$ years (including $27$ and $33$). There is at least one mathematician from each country, and there is at least one mathematician of each possible age between $27$ and $33$. Show that we can find at least five mathematicians $m_1, \ldots, m_5$ such that for any $i \in \{1, \ldots, 5 \}$ there are more mathematicians in the conference having the same age as $m_i$ than those having the same nationality as $m_i$. Proposed by S. Muralidharan