There are $n$ numbers $a_1,\ldots,a_n\in [0,1]$.

Their sum is $\sum_{i=1}^n a_i = s$, where $s$ is some integer.

We want to group them into sets so that the sum of each set is at least $t$, where $t$ is some integer.

Let $F(n,s,t)$ be the largest number of sets that we can always create (for any $a_i$).

What is $F(n,s,t)$?

**Example**. $F(n=8,s=7,t=1)=4$:

*Proof that $F(8,7,1)\geq 4$*: We can always create 4 sets by dividing the $8$ numbers arbitrarily into $4$ pairs. The sum of each pair is at most $2$, and the sum of all pairs is $7$, so the sum of each pair is at least $1$.*Proof that $F(8,7,1)\leq 4$*: We cannot always create 5 sets. Suppose for all $i$, $a_i=7/8$. In any $5$ sets, at least one set is a singleton so its sum is less than $1$.

Similarly, whenever $n$ is even, $F(n,n-1,1)=n/2$.

**What else is known on the function $F$?**

Currently I am particularly interested in the case $t=2$, but I will be happy for any more general references.

UPPER BOUND: $F(n,s,t)\leq \lfloor {s+1\over t+1}\rfloor$.
*Proof*. Suppose that $s+1$ numbers equal $s/(s+1)$ and the other $n-s-1$ numbers equal $0$. To create a set with sum at least $t$, we need $t+1$ nonzeros. So we can create at most $\lfloor {s+1\over t+1}\rfloor$ such sets.