2011 UNCO Math Contest II Problems/Problem 9

Problem

Let $T(n)$ be the number of ways of selecting three distinct numbers from $\left\{1, 2, 3,\cdots ,n\right\}$ so that they are the lengths of the sides of a triangle. As an example, $T(5) = 3$ ; the only possibilities are $\{2-3-4\},\{ 2-4-5\}$ , and $\{3-4-5\}$ .

(a) Determine a recursion for T(n).

(b) Determine a closed formula for T(n).

Solution

(a) $T(n+1)+T(n)=\binom{n}{3}$

(b) $T(N) = \binom{N-1}{3}-\binom{N-2}{3}+\binom{N-3}{3}-\binom{N-4}{3}+\cdots$

I found (a) $T(n) = T(n - 1) + 2(n-4)$ and (b) $T(n) = 13 + n (n - 7).$

2011 UNCO Math Contest II (Problems • Answer Key • Resources)
Preceded by Problem 8	Followed by Problem 10
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2011 UNCO Math Contest II Problems/Problem 9

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