2013 AIME I Problems/Problem 8
The domain of the function is a closed interval of length , where and are positive integers and . Find the remainder when the smallest possible sum is divided by 1000.
We know that the domain of is , so . Now we can apply the definition of logarithms: Since the domain of has length , we have that
A larger value of will also result in a larger value of since meaning and increase about linearly for large and . So we want to find the smallest value of that also results in an integer value of . The problem states that . Thus, first we try : Now, we try : Since is the smallest value of that results in an integral value, we have minimized , which is .
We start with the same method as above. The domain of the arcsin function is , so .
For to be an integer, must divide , and . To minimize , should be as small as possible because increasing will decrease , the amount you are subtracting, and increase , the amount you are adding; this also leads to a small which clearly minimizes .
We let equal , the smallest factor of that isn't . Then we have
, so the answer is .
Solution 3 (Operation Quadratics)
Note that we need , and this eventually gets to . From there, break out the quadratic formula and note that Then we realize that the square root, call it , must be an integer. Then
Observe carefully that ! It is not difficult to see that to minimize the sum, we want to minimize as much as possible. Seeing that is even, we note that a belongs in each factor. Now, since we want to minimize to minimize , we want to distribute the factors so that their ratio is as small as possible (sum is thus minimum). The smallest allocation of and fails; the next best is and , in which and . That is our best solution, upon which we see that , thus .
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