2006 AIME I Problems/Problem 8
Hexagon is divided into five rhombuses, and as shown. Rhombuses and are congruent, and each has area Let be the area of rhombus . Given that is a positive integer, find the number of possible values for .
Let denote the common side length of the rhombi. Let denote one of the smaller interior angles of rhombus . Then . We also see that . Thus can be any positive integer in the interval . and , so can be any integer between 1 and 89, inclusive. Thus the number of positive values for is .
Call the side of each rhombus w. w is the width of the rhombus. Call the height h, where . The height of rhombus T would be 2h, and the width would be . Substitute the first equation to get . Then the area of the rhombus would be . Combine like terms to get . This expression equals an integer K. specifically must be in the form . There is no restriction on h as long as it is a positive real number, so all we have to do is find all the positive possible values of for . Now, quick testing shows that and , but we must also test , because the product of two will make it an integer. is also less than , so we have numbers 1-44, times two because 0.5 can be added to each of time, plus 1, because 0.5 is also a valid value. (notice 0 is not valid because the height must be a positive number) That gives us
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