1991 AHSME Problems/Problem 12
Problem
The measures (in degrees) of the interior angles of a convex hexagon form an arithmetic sequence of integers. Let be the measure of the largest interior angle of the hexagon. The largest possible value of , in degrees, is
(A) 165 (B) 167 (C) 170 (D) 175 (E) 179
Solution
The angles must add to . Now, let the arithmetic progression have first term and difference , so by the sum formula, we get Now, and are both divisible by (as is an integer), so is divisible by , and thus is divisible by 5, so is divisible by . As the hexagon is convex, must be less than , so as it is a multiple of , it can be at most , and indeed this is possible with
See also
1991 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.