2002 AIME II Problems/Problem 8
Find the least positive integer for which the equation has no integer solutions for . (The notation means the greatest integer less than or equal to .)
Note that if , then either , or . Either way, we won't skip any natural numbers.
The greatest such that is . (The inequality simplifies to , which is easy to solve by trial, as the solution is obviously .)
We can now compute:
From the observation above (and the fact that ) we know that all integers between and will be achieved for some values of . Similarly, for we obviously have .
Hence the least positive integer for which the equation has no integer solutions for is .
After getting that , for ease of computation above, we can use the fact that varies solely based on and checking these gives us that the pattern fails at giving us as the answer.
Rewriting the given information and simplifying it a bit, we have
Now note that in order for there to be no integer solutions to we must have We seek the smallest such A bit of experimentation yields that is the smallest solution, as for it is true that Furthermore, is the smallest such case. (If unsure, we could check if the result holds for and as it turns out, it doesn't.) Therefore, the answer is
In this solution we use inductive reasoning and a lot of trial and error. Depending on how accurately you can estimate, the solution will come quicker or slower.
Using values of as and we can find the corresponding values of relatively easily. For , is in the range ; for , is the the range , etc: . For any positive integer is in a range of .
Now we try testing to get a better understanding of what our solution will look like. Obviously, there will be no solution for , but we are more interested in how the range will compute to. Using the formula we got above, the range will be . Testing any integer from will result in the same range. Also, notice that each and every one of them have no solution for . Testing gives a range of , and gives . They each have a solution for , and their range is only one value. Therefore, we can assume with relative safety that the integer we want is the lowest integer that follows this equation
Now we can easily guess and check starting from . After a few tests it's not difficult to estimate a few jumps, and it took me only a few minutes to realize the answer was somewhere in the forties (You could also use the fact that ). Then it's just a matter of checking them until we get . Alternatively, you could use the equation above and proceed with one of the other two solutions listed.
Edited and ed by PhunsukhWangdu
Here is an intuitive way to approximate the answer is around : For the function its derivative is , which should be close to because we need to find the smallest skipped integer. The rest of the steps are the same as Solution 1.
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