## Task 4 - 330524

Rita calculates the three numbers

$$
1+9-9+3=a, \quad 1 \cdot 9+9-3=b, \quad 1 \cdot 9 \cdot 9 \cdot 3=c
$$

She considers further possibilities for filling in the boxes in the row

$$
1 \square 9 \square 9 \square 3=
$$

with signs that are either + or - or $\cdot$. She is looking for all those insertions where the number to be calculated is greater than 30 but less than 100.

Find all such insertions; show that you have found all of them!

Add the numbers that result from these insertions!

To the sum thus found, add the product of the two smallest numbers found between 30 and 100! Finally, add the numbers calculated above as $a, b$, and $c$!
The number to be calculated can only be greater than 30 if the symbol $\cdot$ stands in at least one of the two boxes between 9 and 9 or between 9 and 3.

If it stands between 9 and 3, then there cannot be a - before it (it would subtract a number that is too large), nor can there be a $\cdot$ (because 1 combined with $9 \cdot 9 \cdot 3=243$ through an operation +, - or $\cdot$ does not yield a result between 30 and 100). Therefore, there must be a + before it.

Furthermore, there cannot be a - between 1 and 9 (again, it would subtract a number that is too large). If the symbol stands between 9 and 9, then only + or - can follow (since $\cdot$ has already been disproven), and there cannot be a - before it (it would subtract a number that is too large).

Thus, the following substitutions result in numbers to be calculated between 30 and 100:

$$
\begin{array}{r}
1+9+9 \cdot 3=37 \\
1 \cdot 9+9 \cdot 3=36 \\
1+9 \cdot 9+3=85 \\
1 \cdot 9 \cdot 9+3=84 \\
1+9 \cdot 9-3=79 \\
1 \cdot 9 \cdot 9-3=78
\end{array}
$$

With the sum $s$ of these numbers, the further required additions are $s=37+36+85+84+79+78=399$; $36 \cdot 37=1332, a+b+c=4+15+243=262$, in total 1993.

Solutions of the 2nd round 1993 taken from [5]

### 2.35 XXXIV. Olympiad 1994

### 2.35.1 1st Round 1994, Class 5