Problem

For each integer $n\geq3$ , let $f(n)$ be the number of $3$ -element subsets of the vertices of the regular $n$ -gon that are the vertices of an isosceles triangle (including equilateral triangles). Find the sum of all values of $n$ such that $f(n+1)=f(n)+78$ .

Solution

Considering $n \pmod{6}$ , we have the following formulas:

Even and a multiple of 3: $\frac{n(n-4)}{2} + \frac{n}{3}$ Even and not a multiple of 3: $\frac{n(n-2)}{2}$ Odd and a multiple of 3: $\frac{n(n-3)}{2} + \frac{n}{3}$ Odd and not a multiple of 3: $\frac{n(n-1)}{2}$

Considering the six possibilities $n \equiv 0,1,2,3,4,5 \pmod{6}$ and solving, we find that the only valid solutions are $n = 36, 52, 157$ , from which the answer is $36 + 52 + 157 = \boxed{245}$ .

2017 AIME II (Problems • Answer Key • Resources)
Preceded by Problem 12	Followed by Problem 14
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2017 AIME II Problems/Problem 13

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