2010/11 AHSMC Part 1 Problem 10

Problem 10 of 16

We are constructing isosceles but non-equilateral triangles with positive areas and integral side lengths between $1$ and $9$ inclusive. The number of such triangles which are non-congruent is

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Solution

We first count the triangles in which the equal sides of length $k$ are longer than the third side, which can be of length from $1$ to $k-1$ . Summing from $k = 1$ to $9$ , we have $0+1+2+\cdots+8 = 36$ such triangles. We now count the triangles in which the equal sides of length $k$ are shorter than the third side, which can be of length from $k + 1$ to $2k - 1$ . Summing from $k = 1$ to $9$ , we have $0 + 1 + 2 + 3 + 4 + 3 + 2 + 1 + 0 = 16$ . The total is $36 + 16 = 52$ . The answer is (C).

University of Alberta

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