Rubanov I.S.

For four different integers, all their pairwise sums and pairwise products were calculated.

The obtained sums and products were written on the board. What is the smallest number of different numbers that could end up on the board?

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If we take the numbers $-1,0,1,2$, then, as is easy to check, each of the numbers written on the board will be equal to $-2,-1,0$, 1,2 or 3 - a total of 6 different values.

We will show that fewer than six different numbers on the board could not have occurred. Let the numbers taken be $ac+d$. If $ac+d$.

The second method. Let $u$ and $v$ be the two largest in absolute value among the numbers $a, b, c, d$, with $|u| \leq|v|$. If $|u| \geq 2$, then $|u v| \geq 2|v|$, which is greater than any sum. If, however, $|u| \leq 1$, then among the original numbers there must be $-1,0,1$. If $v>0$, at least 6 different numbers are written on the board: $-1,0,1, v,-v, v+1$.

The case $v<0$ is handled similarly.

## Answer

6 numbers.

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