A function $f: \mathbb{R} \rightarrow \mathbb{R}$ is essentially increasing if $f(s) \leq f(t)$ holds whenever $s \leq t$ are real numbers such that $f(s) \neq 0$ and $f(t) \neq 0$. Find the smallest integer $k$ such that for any 2022 real numbers $x_{1}, x_{2}, \ldots, x_{2022}$, there exist $k$ essentially increasing functions $f_{1}, \ldots, f_{k}$ such that 
$$ f_{1}(n)+f_{2}(n)+\cdots+f_{k}(n)=x_{n} \quad \text { for every } n=1,2, \ldots, 2022 $$
The answer is 11 and, more generally, if 2022 is replaced by $N$ then the answer is $\left\lfloor\log _{2} N\right\rfloor+1$.  『 Bound. Suppose for contradiction that $2^{k}-1>N$ and choose $x_{n}=-n$ for each $n=1, \ldots, N$. Now for each index $1 \leq n \leq N$, define  $$ S(n)=\left\{\text { indices } i \text { for which } f_{i}(n) \neq 0\right\} \subseteq\{1, \ldots, k\} $$  As each $S(n t)$ is nonempty, by pigeonhole, two $S(n)$ 's coincide, say $S(n)=S\left(n^{\prime}\right)$ for $nx_{n^{\prime}}$ in that case due to the essentially increasing condition.  【 Construction. It suffices to do $N=2^{k}-1$. Rather than drown the reader in notation, we'll just illustrate an example of the (inductive) construction for $k=4$. Empty cells are zero.  |  | $f_{1}$ | $f_{2}$ | $f_{3}$ | $f_{4}$ | | ---: | ---: | ---: | ---: | ---: | | $x_{1}=3$ | 3 |  |  |  | | $x_{2}=1$ | 10 | -9 |  |  | | $x_{3}=4$ |  | 4 |  |  | | $x_{4}=1$ | 100 | 200 | $-\mathbf{2 9 9}$ |  | | $x_{5}=5$ |  | 200 | -195 |  | | $x_{6}=9$ | 100 |  | -91 |  | | $x_{7}=2$ |  |  | 2 |  | | $x_{8}=6$ | 1000 | 2000 | 4000 | $-\mathbf{6 9 9 4}$ | | $x_{9}=5$ |  | 2000 | 4000 | -5995 | | $x_{10}=3$ | 1000 |  | 4000 | -4997 | | $x_{11}=5$ |  |  | 4000 | -3995 | | $x_{12}=8$ | 1000 | 2000 |  | -2992 | | $x_{13}=9$ |  | 2000 |  | -1991 | | $x_{14}=7$ | 1000 |  |  | -993 | | $x_{15}=9$ |  |  |  | 9 |  The general case is handled in the same way with powers of 10 replaced by powers of $B$, for a sufficiently large number $B$.

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