Problem from Putnam 2023
At the time of posting this article, Putnam 2023 was sat just 2 months ago, in December 2023. In this article, we tackle one of the first problems of the competition. Do you have what it takes to figure it out?
Suppose we differentiate this first. Then we yield a product of sines and cosines as follows:
Where we just differentiated each term of the product separately using product rule. Now, if we were to differentiate again, this would be quite long and tedious. Luckily, it only asks for |f’’_n(0)| though, not f’’_n(x) itself. Notice that, when we differentiate these, either we have two sine terms in a single product or none at all. If we have a sine term within a product, we know that sin(0) = 0 and thus we don’t really care about this product. Additionally, cos(0) = 1. So really the only thing we care about is the terms when we differentiate the sin terms again, and so we have:
And so, we just need to find when the sum of squares first takes a value greater than 2023. We note that the formula for this is:
For n = 17 we have |f_n’’(0)| = 1785. For n = 18 we have |f_n’’(0)| = 2109.
Therefore, the smallest n for which this occurs is 18.
