2010 AMC 12B Problems/Problem 21
Let , and let be a polynomial with integer coefficients such that
What is the smallest possible value of ?
There must be some polynomial such that
Then, plugging in values of we get
Thus, must be a multiple of .
Now we show that there exists such that We have Thus, for some From here it is clear that exists, since we can take
Therefore, our answer is
The evenly-spaced data suggests using discrete derivatives to tackle this problem. First, note that any polynomial of degree
can also be written as
Moreover, the coefficients are integers for iff the coefficients are integers for . This latter form is convenient for calculating discrete derivatives of .
The discrete derivative of a function is the related function defined as
With this definition, it's easy to see that for any positive integer we have
This in turn allows us to use successive discrete derivatives evaluated at to calculate all of the coefficients using
We can also calculate the following table of discrete derivatives based on the data points given in the problem statement:
Thus we can read down the column for to find that for . Interestingly, even if we choose to have degree greater than , the coefficients of lowest order always satisfy these conditions. Moreover, it's straightforward to show that the of degree with satisfying these conditions will fit the data given in the problem statement. Thus, to find minimal necessary and sufficient conditions on the value of , we need only consider these equations. As a result, with integer coefficients fitting the given data exists iff divides for . In other words, it's necessary and sufficient that
The last condition holds iff divides evenly into . Since such will also satisfy the first conditions, our answer is .
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