## Task A-3.7.

Marko places coins on some fields of a $3 \times 3$ board, and then records how many coins are in each individual row and column. How many coins at least must Marko place on the board if he wants these six numbers to be mutually distinct?
## Solution.

Let $Z$ be the sum of all six numbers (coins in rows and columns). 1 point

If all six numbers are different, the sum of these numbers $Z$ is at least $0+1+2+3+4+5=15. \quad 3$ points

Notice that $Z$ is an even number because it is equal to twice the number of coins on the board.

Due to its evenness, $Z \geqslant 16$, which means Marko must use at least eight coins.

An example shows that Marko can succeed with exactly eight coins:

| 0 | 0 | 0 |
| :--- | :--- | :--- |
| 0 | 0 | 2 |
| 1 | 3 | 2 |

In this example, the numbers in the rows are 0, 2, and 6, and the sums in the columns are 1, 3, and 4.

## SCHOOL/CITY COMPETITION IN MATHEMATICS

4th grade - high school - A variant

## January 28, 2019.

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