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

Note: This is the pretty much the same solution as that of NP-hard.
Let $x=5-3c$, $y=5-3d$, $t=\dfrac{5}{3k}$.
Then we are looking for the minimum $t$ for which:
$min(x,\dfrac{y}{b}) \leq t\dfrac{(x+2y)}{(3b+2)}$.
with $2 \leq y \leq x \leq 5$ and $0 \leq b \leq 1$.
Consider the case $x \leq \dfrac{y}{b}$.
Then sufficient is: $t \geq \dfrac{(3b+2)x}{(x+2y)}$ and since $b \leq \dfrac{y}{x}$, it is sufficient that
$t \geq \dfrac{2x+3y}{x+2y}$ and the max value of the expression on the right is $\dfrac{16}{9}$.
Thus in this case, $t =\dfrac{16}{9}$ suffices which corresponds to $k=\dfrac{15}{16}$.
In the case when $\dfrac{y}{b} \leq x$, it is sufficient to have:
$t \geq \dfrac{(3b+2)}{(x+2y)}y/b=(3+\dfrac{2}{b})\dfrac{y}{x+2y}$ and using $b \geq \dfrac{y}{x}$, we once again get the same
calculation as above.
