2006 AMC 10A Problem 19

Below is the professionally curated solution for Problem 19 of the 2006 AMC 10A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2006 AMC 10A solutions, or check the answer key.

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Concepts:angle sum arithmetic sequence counting integers in a range

Difficulty rating: 1630

19.

How many non-similar triangles have angles whose degree measures are distinct positive integers in arithmetic progression?

$0$

$1$

$59$

$89$

$178$

Solution:

Let the angles be $n - d,$ $n,$ $n + d.$ Their sum is $3n = 180,$ so $n = 60.$

The measures are distinct positive integers, so $d \ge 1,$ and $n - d \gt 0$ forces $d \lt 60.$ Thus $d \in \{1, 2, \ldots, 59\},$ giving $59$ non-similar triangles.

Thus, the correct answer is C.

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