AIME 2001 II · 第 5 题

Solution

Out of all ten-element subsets with distinct elements that do not possess the triangle property, we want to find the one with the smallest maximum element. Call this subset $\mathcal{S}$. Without loss of generality, consider any $a, b, c \,\in \mathcal{S}$ with $a < b < c$. $\,\mathcal{S}$ does not possess the triangle property, so $c \geq a + b$. We use this property to build up $\mathcal{S}$ from the smallest possible $a$ and $b$:

$$\mathcal{S} = \{\, 4,\, 5,\, 4+5, \,5+(4+5),\, \ldots\,\} = \{4, 5, 9, 14, 23, 37, 60, 97, 157, 254\}$$ $\mathcal{S}$ is the "smallest" ten-element subset without the triangle property, and since the set $\{4, 5, 6, \ldots, 253\}$ is the largest set of consecutive integers that does not contain this subset, it is also the largest set of consecutive integers in which all ten-element subsets possess the triangle property. Thus, our answer is $n = \fbox{253}$.

Note

If we wanted to find this for a much larger number (say 2001), we could have noted that this is a "quasi-Fibonacci" sequence with initial terms $4,5$ and built up an explicit function to find the $nth$ term. (The latter part is generally pretty annoying).

~Dhillonr25 ~Minor edit by Yiyj1