The exercise I am trying to solve is the following:
"Let $X$ be a $\delta$-hyperbolic space and $A \subseteq X$ a bounded subset with diameter $R$. Show that there exists $p \in X$ such that $A \subseteq B_{R/2 + 10 \delta}(p)$ and that for any other $q \in X$ with the same property we have $d(p,q) \leq 100 \delta$."
I would especially appreciate hints regarding where to look for $p$ and also regarding the strange values of the constants. I think that the problem might be solved in a way similar to Lemma ${III.\Gamma.3.3}$ of Bridson and Haefliger's "Metric spaces of non-positive curvature", but as I said I am a little taken aback from the "weird" values of the constants.
When one works with Gromov-hyperbolic spaces, the constants generally do not matter, so large upper bounds are introduced to get simple constants. Below is an argument which should help you:
Let $x,y \in A$ be two points satisfying $d(x,y) \geq R-\epsilon$ for some small $\epsilon$ (maybe take $\epsilon=\delta$) and let $p$ denote a midpoint of $x,y$. If $z \in A$, consider the triangle $(x,y,z)$, and use the hyperbolicity, combined with the fact that $d(x,z),d(y,z) \leq R$, to bound $d(p,z)$ with respect to $R$.
If $q$ is your other point, notice that $d(x,q)+d(q,y)$ is almost $R$, so $q$ is near to a geodesic between $x$ and $y$. Since geodesics fellow travel in hyperbolic spaces, you should be able to deduce that $d(p,q)$ is small.
However, I do not know whether the constants produced by the argument are those you are looking for.