2008-I-16

$\begin{array}{rcl}
b & = & \dfrac{10+24}{2} \\
& = & 17
\end{array}$

Also, we have

$\begin{array}{rcl}
10 & = & \dfrac{a+b}{2} \\
10 & = & \dfrac{a+17}{2} \\
20 & = & a + 17 \\
a & = & 3
\end{array}$

1. The salaries tax charged for the citizen

   $\begin{array}{cl}
   = & P \times 20\% \\
   = & \$ 0.2 P
   \end{array}$

2. For the salaries tax of a citizen being charged at the standard rate,

   $\begin{array}{rcl}
   30~000\times 3\% + 30~000\times 10\% + 30~000\times 17\% & & \\
   + (P-172~000-90~000)\times 24\% & \ge & 0.2P \\
   0.24P – 53~880 & \ge & 0.2P \\
   0.04P & \ge & 53~880 \\
   P & \ge & 1~347~000
   \end{array}$

   Therefore, the least net total income of the citizen is $\$1~347~000$.

$\begin{array}{cl}
= & 1~400~000 \times 0.2 \\
= & \$280~000
\end{array}$

The amount after the 1st month

$\begin{array}{cl}
= & 23~000 \times (1+\dfrac{3\%}{12}) \\
= & \$23~000(1.002~5)
\end{array}$

The amount after the 2nd month

$\begin{array}{cl}
= & [23~000 + 23~000(1.002~5)] (1+ \dfrac{3\%}{12}) \\
= & \$[23~000(1.002~5) + 23~000(1.002~5)^2]
\end{array}$

Hence, the amount after the 12th month

$\begin{array}{cl}
= & 23~000(1.002~5) + 23~000(1.002~5)^2 + \cdots + 23~000(1.002~5)^{12} \\
= & \dfrac{23~000(1.002~5)(1-(1.002~5)^{12})}{1-1.002~5} \\
= & \$280~526.370~6 \\
> & \$280~000
\end{array}$

Therefore, Peter has enough money to pay his salaries tax on the due day.