2019 AIME II Problems/Problem 10
There is a unique angle between and such that for nonnegative integers , the value of is positive when is a multiple of , and negative otherwise. The degree measure of is , where and are relatively prime integers. Find .
Note that if is positive, then is in the first or third quadrant, so . Also notice that the only way can be positive for all that are multiples of is when , etc. are all the same value . This happens if , so . Therefore, the only possible values of theta between and are , , and . However does not work since is positive, and does not work because is positive. Thus, . .
As in the previous solution, we note that is positive when is in the first or third quadrant. In order for to be positive for all divisible by , we must have , , , etc to lie in the first or second quadrants. We already know that . We can keep track of the range of for each by considering the portion in the desired quadrants, which gives at which point we realize a pattern emerging. Specifically, the intervals repeat every after . We can use these repeating intervals to determine the desired value of since the upper and lower bounds will converge to such a value (since it is unique, as indicated in the problem). Let's keep track of the lower bound.
Initially, the lower bound is (at ), then increases to at . This then becomes at , at , at , at . Due to the observed pattern of the intervals, the lower bound follows a partial geometric series. Hence, as approaches infinity, the lower bound converges to -ktong
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