Problem - 2795
Among nine randomly selected even numbers from $2$, $4$, $6$, $\cdots$, $28$, $30$, show that at least two of them whose sum is $34$.
Dividing these $15$ numbers into the following $8$ groups:
- $\{2\}$
- $\{4, 30\}$
- $\{6, 28\}$
- $\{8, 26\}$
- $\{10, 24\}$
- $\{12, 22\}$
- $\{14, 20\}$
- $\{16, 18\}$
Then, by the pigeonhole principle, two must be in the same group if nine numbers are chosen. The sum of these two numbers is $34$.