Bërn's Constant

Taking logorithms we get

$\displaystyle \ln s_n = \frac{1}{n}\ln\prod_{i=1}^n\sqrt{\sin\frac{i}{n}}=\frac{1}{2n}\sum_{i=1}^n\ln\sin\frac{i}{n}$

we know from the definition of definite integrals that

$\displaystyle \int_0^1f(x)\mathrm dx=\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^n f\left(\frac{i}{n}\right)$

so

$\displaystyle \lim_{n\rightarrow\infty} \ln s_n = \frac{1}{2}\int_0^1\ln\sin x\mathrm dx$

This integral is difficult, so I decided to solve it numerically

Mathematica outputs that $\displaystyle \int_0^1\ln\sin x\mathrm dx \approx -1.056720206$

and finally,

$\displaystyle \lim_{n\rightarrow\infty} \frac{1}{s_n} = e^{ - \lim \ln s_n} = e^{ \frac{1}{2} 1.056720206} \approx 1.69614$

we obtain $\left\lfloor 1.69614 \right\rfloor = \boxed{1}$