Pigeonhole principle applications

Question

Let ${\{a_1, a_2,\ldots, a_9\}\subset \mathbb{N}}$ with $\displaystyle\sum_{i=1}^9 a_i = 30.$ Then, show that there exist $ i,j, k\in \{1, 2, 3, 4, 5, 6, 7, 8, 9\},\ i,j,k$ pairwise different, such that $a_i+a_j+a_k \geq 12$. You may use pigeonhole principle

Answer 1

Say $a_1\ge a_2\ge\cdots$. If $a_1+a_2+a_3<12 $ then $a_4+\cdots+a_9\ge19$ and so $a_4\ge\lceil 19/6\rceil=4$. Then $a_1,a_2,a_3\ge4$ and $a_1+a_2+a_3\ge12$, a contradiction.