Good afternoon. I have a couple of questions regarding a problem on the 2010 AMC 12A/AHSME. it's more on the reasoning of something I was using to try and solve the problem (turned out not to be fruitful). Here's the question.
Let $f(x)=\log_{10}\left(\sin{\pi x}\cdot\sin{2\pi x}\cdot ...\cdot \sin{8\pi x}\right)$. The intersection of the domain of $f$ with the interval $[0,1]$ is a union of disjoint open intervals. What is $n$?
I already answered the question ($n=12$). But the strategy for accomplishing this task on my end was very tedious and got me thinking that there was a better. I literally graphed and highlighted the domains of the individual sine functions above. Then, since the argument of the logarithm is a product, the intervals in the domain generated would mean that the product would be positive in a particular open interval $(a,b)\in[0,1]$ when there exists an even number of indexed sine functions where, for a particular $c\in[a,b]$, $\sin{ck\pi}<0$. This made it easy to generate the sets, albeit as i mentioned, quite tedious. But also, what happens if $k$ is large? It becomes a massive problem.
One of the solutions I was thinking of was to look at the derivative of $f$ and see how many maxes or mins occur. Since there are swathes of sine functions and each of those functions is composite, and that huge product is in itself the inner function of the broader logarithm, this seemed impossible without a CAS like Mathematica (which I haven't tried yet but will). So then I thought to transform the function based upon rules of logarithms. So
$$f(x)=\log_{10}\left(\sin{\pi x}\cdot\sin{2\pi x}\cdot ...\cdot \sin{8\pi x}\right)=\sum_{k=1}^{8}\log_{10}\sin{k\pi x}$$
But this approach doesn't seem to work (and it seems that the rules of logarithms doesn't take into account for the product since the domains of both functions appear to be different). So outside of a CAS, this again seems like a black hole of problems.
My questions are this: 1. Are there any mathematical papers that can handle the analysis of this type of problem or any generalized function similar to this one? 2. Are there patterns that emerge for increasing $k$? The number of intervals for the problem in question has the following sequence for $k=1...20$
$$1,1,2,4,6,6,9,12,16,16,21,26,30,32,36,44,54,51,60,68,...$$
I checked the OEIS and there was nothing. I basically plotted the function in desmos and counted intervals where the function existed and went with that. I am not 100% convinced that this is the sequence, but if it's not, I don't know where i made my error.
Any insight, papers, references, etc are welcome and super helpful. Thank you in advance.
Here is a simple PARI/GP program to calculate the number of intervals in $[0, 1]$ where $\prod_{j=2}^n \sin(j \pi x)$ is positive. It collects the zeros of all factors in a list, sorts them in increasing order and then counts the number of non-degenerate intervals on which the product is positive.
P(n) = {
my (zeros = List());
for (j=2, n, for (k=1,j-1, listput(zeros, k/j)));
listput(zeros, 1);
my (xcoords = vecsort(Vec(zeros)), x = 0, sign = -1, count = 0);
for(i=1, #xcoords,
sign = -sign;
if (xcoords[i] > x && sign == 1, count += 1);
x = xcoords[i];
);
count
};
vector(30, n, P(n))
(You can run it in the PARI/GP online calculator). Output:
%2 = [1, 1, 2, 4, 6, 6, 9, 12, 16, 16, 21, 26, 30, 32, 36, 44, 54, 51, 60, 68, 74, 75, 86, 98, 104, 106, 115, 128, 144, 139]
