Question 15
In this paper, let $\lfloor x\rfloor$ denote the greatest integer not exceeding $x$. For examples, $\lfloor5\rfloor=5$, $\lfloor2.8\rfloor=2$, and $\lfloor-2.3\rfloor=-3$.
Assume that $x\ge y\ge z\ge\dfrac\pi{12}$ and $x+y+z=\dfrac\pi2$. Let $M$ and $m$ be the largest possible value and the smallest possible value of $\cos x\sin y\cos z$ respectively. Determine the value of $\left\lfloor\dfrac Mm\right\rfloor$.
Assume that $x\ge y\ge z\ge\dfrac\pi{12}$ and $x+y+z=\dfrac\pi2$. Let $M$ and $m$ be the largest possible value and the smallest possible value of $\cos x\sin y\cos z$ respectively. Determine the value of $\left\lfloor\dfrac Mm\right\rfloor$.