<s>[INST] Ceiling inequality? I know $\lceil$$\frac{a}{3}$$\rceil$ + $\lceil$$\frac{b}{3}$$\rceil$ $\le$ $\lceil$$\frac{a + b}{3}$$\rceil$ + 1
what do we know about $\lceil$$\frac{a}{3}$$\rceil$ + $\lceil$$\frac{b}{3}$$\rceil$ + $\lceil$$\frac{c}{3}$$\rceil$ + $\lceil$$\frac{d}{3}$$\rceil$?
 [/INST] For $x_i=3q_i+r_i$ we have $\lceil \frac {x_i}3\rceil=q_i+\lceil \frac {r_i}3\rceil$.
For a sum you'll get $\sum\limits_{i=1}^4 \lceil \frac{x_i}{3}\rceil=\sum\limits_{i=1}^4 q_i+\sum\limits_{i=1}^4 \lceil \frac{r_i}{3}\rceil$
While $\bigg\lceil\sum\limits_{i=1}^4  \frac{x_i}{3}\bigg\rceil=\sum\limits_{i=1}^4 q_i+\bigg\lceil\sum\limits_{i=1}^4 \frac{r_i}{3}\bigg\rceil$
So we have to compare $A=\sum\limits_{i=1}^4 \lceil \frac{r_i}{3}\rceil$ with $B=\bigg\lceil\sum\limits_{i=1}^4 \frac{r_i}{3}\bigg\rceil$ for all possible values of the $r_i\in\{0,1,2\}$.
$\begin{array}{|l|cc|c|}
\hline r_i & A & B & A-B\\ \hline
0000 & 0 & 0 & 0 \\
0001 & 1 & 1 & 0 \\
0002 & 1 & 1 & 0 \\
0011 & 2 & 1 & 1 \\
0012 & 2 & 1 & 1 \\
0022 & 2 & 2 & 0 \\
0111 & 3 & 1 & 2 \\
0112 & 3 & 2 & 1 \\
0122 & 3 & 2 & 1 \\
0222 & 3 & 2 & 1 \\
1111 & 4 & 2 & 2 \\
1112 & 4 & 2 & 2 \\
1122 & 4 & 2 & 2 \\
1222 & 4 & 3 & 1 \\
2222 & 4 & 3 & 1 \\ \hline
\end{array}$
Thus the maximum is $2$ and finally we get:

$\displaystyle \big\lceil\frac a3\big\rceil+\big\lceil\frac b3\big\rceil+\big\lceil\frac c3\big\rceil+\big\lceil\frac d3\big\rceil\le\big\lceil\frac {a+b+c+d}3\big\rceil+2$

</s>