1997 AIME Problems/Problem 9
Problem
Given a nonnegative real number , let denote the fractional part of ; that is, , where denotes the greatest integer less than or equal to . Suppose that is positive, , and . Find the value of .
Solution 1
Looking at the properties of the number, it is immediately guess-able that (the golden ratio) is the answer. The following is the way to derive that:
Since , . Thus , and it follows that . Noting that is a root, this factors to , so (we discard the negative root).
Our answer is . Complex conjugates reduce the second term to . The first term we can expand by the binomial theorem to get . The answer is .
Note that to determine our answer, we could have also used other properties of like .
Solution 2
Find as shown above. Note that, since is a root of the equation , , and . Also note that, since is a root of , . The expression we wish to calculate then becomes . Plugging in , we plug in to get an answer of .
Solution 3
Find as shown above. Note that satisfies the equation (this is the equation we solved to get it). Then, we can simplify as follows using the fibonacci numbers:
So we want since is equivalent to .
Solution 4
As Solution 1 stated, . . So, , , , , .
Therefore,
Another way to factor :
See also
1997 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 8 |
Followed by Problem 10 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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