<s>[INST] Constant such that $\max\left(\frac{5}{5-3c},\frac{5b}{5-3d}\right)\geq k\cdot\frac{2+3b}{5-c-2d}$ What is the greatest  constant $k>0$ such that
$$\max\left(\frac{5}{5-3c},\frac{5b}{5-3d}\right)\geq k\cdot\frac{2+3b}{5-c-2d}$$
for all $0\leq b\leq 1$ and $0\leq c\leq d\leq 1$?
The right-hand side looks like a weighted sum of the two terms on the left-hand side, but not quite. If we plug in $b=1$ and $c=d$, then all three terms are equal, so $k\leq 1$. 
On the other hand, we have $k\geq 3/5$. Indeed, we will show that $$\frac{5}{5-3c}\geq\frac35\cdot\frac{2+3b}{5-c-2d}.$$
 Note that $d\leq 1$ and $2+3b\leq 5$, so it suffices to show $$\frac{1}{5-3c}\geq\frac35\cdot\frac{1}{3-c},$$
or $$5(3-c)\geq 3(5-3c)$$
or $$15-5c\geq 15-9c$$
which is true. But the bound is not tight here, since we must have $b=d=1$ and $c=0$, and we have $\max(5, 5/2)\geq 5/3$. (The term $\frac{5b}{5-3d}$, which we did not use at all, is large.)
Update: By dividing the cases into whether $b\leq 3/5$ (and compare with the first term in the $\max$ if so) or $b\geq 3/5$ (and compare with the second term in the $\max$ if so), we can show that $k\geq 15/19$. Moreover, as Aravind pointed out in the comments, we have $k\leq 15/16$. So the gap is now between $15/19$ and $15/16$.
Update 2: WolframAlpha confirms that $k=15/16$ is the right answer. The question is now how to prove it: 
http://www.wolframalpha.com/input/?i=find+minimum+of+max(5%2F(5-3c),(5b)%2F(5-3d))*(5-c-2d)%2F(2%2B3b)+for+0%3C%3Db%3C%3D1+and+0%3C%3Dc%3C%3Dd%3C%3D1
 [/INST] The original problem is equivalent to finding the minimum value of
$$
\max (\frac{5}{5-3c}, \frac{5b}{5-3d}) \cdot \frac{5-c-2d}{2+3b}
$$
in the region $\{(b, c, d) \mid 0 \leq b \leq 1 \wedge 0 \leq c \leq d \leq 1\}$. We consider two cases below, namely,


*

*Case 1: $\frac{5}{5-3c} \leq \frac{5b}{5-3d}$;

*Case 2: $\frac{5}{5-3c} \geq \frac{5b}{5-3d}$.

Case 1: The problem can be stated as
\begin{align}
\text{minimize}\quad & f(b, c, d) = \frac{5-c-2d}{5-3d}\cdot \frac{5b}{2+3b} \\
\text{subject to}\quad & b \geq \frac{5-3d}{5-3c} \quad\quad (\text{by the condition of Case 1}) \\
& 0 \leq b \leq 1 \\
& 0 \leq c \leq d \leq 1
\end{align}
Note that the term $\frac{5b}{2+3b} = \frac{5}{3} - \frac{10}{6+9b}$ in $f(b, c, d)$ is a strictly increasing function of $b$. To minimize $f(b, c, d)$, $b$ should be minimized, for fixed $c$ and $d$. Therefore, 
$$b = \frac{5-3d}{5-3c} \tag{1}$$
Substituting $b$ with (1) in $f$, we obtain that
$$
f(b, c, d) = \frac{25-5c-10d}{25-6c-9d} = \frac{5}{6} + \frac{\frac{25}{6} - \frac{15}{6}d}{25 - 6c - 9d}
$$
For fixed $d$, $f(b, c, d)$ is minimized when $c$ is minimized, i.e., $c = 0$. We further obtain that
$$
f(b, c, d) = \frac{5}{6} + \frac{\frac{25}{6} - \frac{15}{6}d}{25 - 9d} = \frac{5}{6} + \frac{5}{18} - \frac{\frac{50}{18}}{25-9d}
$$
Setting $d = 1$, we obtain the minimum value $\frac{5}{6} + \frac{5}{18} - \frac{50}{18\cdot16} = \frac{15}{16}$.

Case 2: The problem can be stated as:
\begin{align}
\text{minimize}\quad & g(b, c, d) = \frac{5-c-2d}{5-3c}\cdot \frac{5}{2+3b} \\
\text{subject to}\quad & b \leq \frac{5-3d}{5-3c} \quad\quad (\text{by the condition of Case 2}) \\
& 0 \leq b \leq 1 \\
& 0 \leq c \leq d \leq 1
\end{align}
and the solution is similar to that of Case 1. First, the term $\frac{5}{2+3b}$ is a strictly decreasing function of $b$. Thus, we set $b = \frac{5-3d}{5-3c}$ and obtain
$$
g(b, c, d) = \frac{25-5c-10d}{25-6c-9d}
$$
which is the same function as that in Case 1. So the minimum value is still $\frac{15}{16}$.
</s>