1991 AMC 12 Problem 12

Question

The measures (in degrees) of the interior angles of a convex hexagon form an arithmetic sequence of positive integers. Let

m^\circ

be the measure of the largest interior angle of the hexagon. The largest possible value of

m^\circ

is

165^\circ

167^\circ

170^\circ

175^\circ

179^\circ

Write the angles as

a,a+d,\ldots,a+5d

and use their sum

The equation forces

d

to be even, and convexity bounds the largest angle below

180^\circ

The angle sum is

720^\circ,

so

\begin{aligned} 6a+15d&=720,\\ 2a+5d&=240. \end{aligned}

Hence

d

is even. The largest angle is

a+5d=120+\frac52d,

which must be less than

180.

Thus

d\lt24,

and the largest allowable even value is

d=22.

It gives

a=65

and

m=65+5(22)=175.

Thus the correct answer is D.

Problem 12 in other years