## Task A-2.7.

Two teams are playing handball. Neither team has scored 30 or more goals. The recorder at the beginning of the game and after each goal scored records the score and calculates the sum of all digits in the score. For example, with a score of 15 : 6, the sum of the digits is 12. How many times during the game could the recorder have recorded a score where the total sum of the digits is 10?
## Solution.

At each change of the result, exactly one of the numbers $\overline{a b}$ and $\overline{c d}$ increases by one, while the other remains unchanged.

Consider some (fixed) $a$ and $c$. Let $b$ and $d$ be such that $a+b+c+d=10$. With each subsequent change of the result, one of them will increase, and the other will not decrease. Therefore, the sum $a+b+c+d$ increases as long as $a$ and $c$ remain unchanged. In other words, for each given pair of numbers $a$ and $c$, there can be at most one result in the same match that satisfies the conditions of the problem.

From the text of the problem, we see that the digits $a$ and $c$ can only be 0, 1, and 2.

The number $a$ initially has the value 0, and later it can only increase from 0 to 1 and then from 1 to 2. The same applies to the number $c$.

Without loss of generality, assume that $a$ is the first of these two numbers to increase from 0 to 1. Then in that match, we can have at most five different pairs $(a, c)$:

$$
(0,0) \rightarrow(1,0) \rightarrow(1,1) \rightarrow(2,1) \rightarrow(2,2)
$$

or

$$
(0,0) \rightarrow(1,0) \rightarrow(1,1) \rightarrow(1,2) \rightarrow(2,2)
$$

or

$$
(0,0) \rightarrow(1,0) \rightarrow(2,0) \rightarrow(2,1) \rightarrow(2,2)
$$

Therefore, the sum can be equal to 10 at most five times.

It remains to show that these results are indeed possible. By solving the equation $a+b+c+d=10$ for each pair $(a, c)$ and considering that the result of a team cannot decrease during the match, we see that one possible sequence of results is, for example:

$$
06: 04, \quad 13: 06, \quad 21: 07, \quad 22: 15, \quad 25: 21
$$

Note: The result alone, i.e., the answer without justification, is worth 0 points. 8 points are awarded for proving that there can be at most five different results, and the remaining 2 points are awarded for an example of such a sequence.

## SCHOOL/CITY COMPETITION IN MATHEMATICS

3rd grade - high school - A variant
January 4, 2020.

## IF A STUDENT HAS A DIFFERENT APPROACH TO SOLVING THE PROBLEM, THE COMMITTEE IS OBLIGED TO GRADE AND EVALUATE THAT APPROACH APPROPRIATELY.