3. From five ones, five twos, five threes, five fours, and five fives, we form five five-digit numbers that read the same forwards and backwards (e.g., 32 223), and then we add these numbers. What is the smallest and the largest value that the resulting sum can have?

The school - written part of the 1st round, category C takes place

on Tuesday, January 28, 2003

so that it starts in the morning and the contestants have 4 hours of pure time to solve the problems. For each problem, a contestant can earn 6 points, and a successful solver is a student who scores 10 points or more. These details will be communicated to the students before the start of the competition.
3. Let's denote and write in the decimal system five five-digit numbers that read the same forwards as backwards and are composed of the given digits:

$$
\begin{aligned}
& \overline{a_{1} b_{1} c_{1} b_{1} a_{1}}=a_{1} \cdot 10^{4}+b_{1} \cdot 10^{3}+c_{1} \cdot 10^{2}+b_{1} \cdot 10+a_{1}, \\
& \overline{a_{2} b_{2} c_{2} b_{2} a_{2}}=a_{2} \cdot 10^{4}+b_{2} \cdot 10^{3}+c_{2} \cdot 10^{2}+b_{2} \cdot 10+a_{2}, \\
& \overline{a_{3} b_{3} c_{3} b_{3} a_{3}}=a_{3} \cdot 10^{4}+b_{3} \cdot 10^{3}+c_{3} \cdot 10^{2}+b_{3} \cdot 10+a_{3}, \\
& \overline{a_{4} b_{4} c_{4} b_{4} a_{4}}=a_{4} \cdot 10^{4}+b_{4} \cdot 10^{3}+c_{4} \cdot 10^{2}+b_{4} \cdot 10+a_{4}, \\
& \overline{a_{5} b_{5} c_{5} b_{5} a_{5}}=a_{5} \cdot 10^{4}+b_{5} \cdot 10^{3}+c_{5} \cdot 10^{2}+b_{5} \cdot 10+a_{5} .
\end{aligned}
$$

Among the digits $c_{1}, c_{2}, c_{3}, c_{4}, c_{5}$, there is exactly one 1, one 2, one 3, one 4, and one 5. If, for example, the digit 1 were missing in the hundreds place of the five numbers, it would have to appear an odd number of times (five times) in the other places, which is not possible due to the symmetry of the numbers.

For the sum $S$ of the considered numbers, we have

$$
\begin{aligned}
S= & \overline{a_{1} b_{1} c_{1} b_{1} a_{1}}+\overline{a_{2} b_{2} c_{2} b_{2} a_{2}}+\overline{a_{3} b_{3} c_{3} b_{3} a_{3}}+\overline{a_{4} b_{4} c_{4} b_{4} a_{4}}+\overline{a_{5} b_{5} c_{5} b_{5} a_{5}}= \\
= & \left(a_{1}+a_{2}+a_{3}+a_{4}+a_{5}\right) \cdot\left(10^{4}+1\right)+\left(b_{1}+b_{2}+b_{3}+b_{4}+b_{5}\right) \cdot\left(10^{3}+10\right)+ \\
& +\left(c_{1}+c_{2}+c_{3}+c_{4}+c_{5}\right) \cdot 10^{2}= \\
= & 10001 \cdot\left(a_{1}+a_{2}+a_{3}+a_{4}+a_{5}\right)+1010 \cdot\left(b_{1}+b_{2}+b_{3}+b_{4}+b_{5}\right)+ \\
& +100 \cdot(1+2+3+4+5)= \\
= & 10001 \cdot\left(a_{1}+a_{2}+a_{3}+a_{4}+a_{5}\right)+1010 \cdot\left(b_{1}+b_{2}+b_{3}+b_{4}+b_{5}\right)+1500 .
\end{aligned}
$$

Considering the digits available, the sum $S$ will be the smallest if

$$
\begin{aligned}
& a_{1}+a_{2}+a_{3}+a_{4}+a_{5}=1+1+2+2+3=9, \\
& b_{1}+b_{2}+b_{3}+b_{4}+b_{5}=5+5+4+4+3=21.
\end{aligned}
$$

The smallest possible sum is thus

$$
S_{\min }=10001 \cdot 9+1010 \cdot 21+1500=112719
$$

and can be obtained, for example, as the sum

$$
S_{\min }=13131+14241+24342+25452+35553.
$$

Similarly, the sum $S$ will be the largest if

$$
\begin{aligned}
& a_{1}+a_{2}+a_{3}+a_{4}+a_{5}=5+5+4+4+3=21, \\
& b_{1}+b_{2}+b_{3}+b_{4}+b_{5}=1+1+2+2+3=9.
\end{aligned}
$$

The largest possible sum is thus

$$
S_{\max }=10001 \cdot 21+1010 \cdot 9+1500=220611
$$

and can be obtained, for example, as the sum

$$
S_{\max }=53535+52425+42324+41214+31113.
$$

For a complete solution, 6 points are awarded, of which 2 points for determining the smallest possible value, 1 point for constructing specific numbers, 2 points for determining the largest possible value, and 1 point for constructing specific numbers. If the solver constructs both correct examples but does not explain at least briefly why a larger or smaller sum cannot be obtained, award 4 points.