Solution:

We shall solve the problem for $A=\{1,2, \ldots, t\}$. Let $F_{0}=F_{1}=1$, $F_{n+1}=F_{n}+F_{n-1}$ for $n \geq 1$ be the Fibonacci sequence. We shall prove by induction that if $F_{n-1}<t \leq F_{n}, n \geq 2$, then the desired sum equals $n$.

Since for $t=2$ and $t=3$ Peter needs 2 or 3 leva, respectively, the assertion is true for $n=2$ and $n=3$.

Suppose that it is true for $n=k$.

Choose $t \in\left(F_{k}, F_{k+1}\right]$ and let Peter ask a question set having $s$ elements. If $s \in\left(F_{k-1}, F_{k}\right]$ and the answer is "yes" then Peter gives Ivan 2 leva and by the induction hypothesis he needs additional $k$ leva, i.e. in total $k+2$ leva.
If $s \leq F_{k-1}$, then $t-s \geq F_{k}+1-F_{k-1}=F_{k-2}+1$. If Peter receives answer "yes" then he pays 2 leva and he needs additional $k-1$ leva, i.e. in total $k+1$ leva. It remains to notice that if Peter asks a question set with $F_{k-1}$ elements and the answer is "yes" then he needs $2+k-1=k+1$ leva and for answer "no" he needs $1+k-1=k$ leva.
Since $F_{10}=89$ and $F_{11}>89$, the desired number equals 11 .