¶ 2001 AIME I Problem 13
Problem:
In a certain circle, the chord of a -degree arc is centimeters longer than the chord of a -degree arc, where . The length of the chord of a -degree arc is centimeters, where and are positive integers. Find .
Solution:
In the figure, points , and are concyclic, the degree sizes of arcs , and are all , and . Note that is the chord of a -degree arc. Let . Then , because is the chord of a -degree arc. In isosceles trapezoid , draw the altitude from to , and notice that divides into and . Because the right triangles and share the leg , it follows that
which simplifies to . Thus and .
Noting that is a cyclic isosceles trapezoid, apply Ptolemy's Theorem to obtain , or . Solve the equation to find that .
Query: If the restriction were removed, then the problem would have an additional solution. Can you find it?
The problems on this page are the property of the MAA's American Mathematics Competitions