Solving System of Equations
Let $X$ be a random variable distributed uniformly over $[0,20]$. Your constraints imply
$$ \mathbb{E}[f(X)] \geq \frac{K}{20}, \qquad \mathbb{E}[g(X)] \geq \frac{Q}{20}. $$
We conclude that
$$ \mathbb{E}[f(X)+g(X)] \geq \frac{K+Q}{20}, $$
and so there is some point $x \in [0,20]$ such that $f(x) + g(x) \geq (K+Q)/20$. In particular, $R \geq (K+Q)/20$. This bound is tight, as shown by the functions
$$ f(x) = \frac{K}{20} \mathbf{1}_{x \in [0,20]}, \qquad g(x) = \frac{Q}{20} \mathbf{1}_{x \in [0,20]}. $$