Problem 1.Suppose that $\xi>0$ and $\sin(2\xi)<0$. Let $$b_\nu=(N-v+1)\tfrac{\pi}{\xi}\quad\mbox{for}\quad\nu=1,\dots,N:=\big[\tfrac{\xi}{\pi}\big].$$ Prove that $$\mathrm{sgn}(\sin \xi)\int_{-1}^1t\prod_{\nu=1}^n(t^2-b_\nu^2)\sin(\xi t)\,dt>0$$ for every $n=1,2,\dots, N$.

Problem 2.Suppose that $\xi>0$ and $\sin(2\xi)>0$. Let $$a_\nu=(N-v+\tfrac12)\tfrac{\pi}{\xi}\quad\mbox{for}\quad\nu=1,\dots,N:=\big[\tfrac{\xi}{\pi}\big].$$ Prove that $$\mathrm{sgn}(\sin \xi)\int_{-1}^1\prod_{\nu=1}^n(t^2-a_\nu^2)\cos(\xi t)\,dt>0$$ for every $n=1,2,\dots, N$.

(The problem is posed 29.10.2016 by Marija Stanić on page 24 of Volume 1 of the Lviv Scottish Book).

More information on these problems can be found in this paper of G.Milovanovic, A.Cvetkovic, and M.Stanic.