## Task A-1.5.

Azra thought of four real numbers and on the board wrote down the sums of all possible pairs of the thought-of numbers, and then erased one of those sums. The numbers left on the board are $-2,1,2,3$ and 6. What numbers did Azra think of?
## Solution.

Let $a, b, c$ and $d$ be the imagined numbers. Without loss of generality, assume that the deleted sum is $c+d$. Then the numbers written on the board are $a+b, a+c, a+d, b+c$, and $b+d$, i.e.,

$$
\{-2,1,2,3,6\}=\{a+b, a+c, a+d, b+c, b+d\}
$$

Since

$$
(a+c)+(b+d)=(a+d)+(b+c)
$$

among the numbers on the board, we can choose two pairs of numbers with the same sum.

By checking

$$
\begin{aligned}
& -2+1=-1, \quad-2+2=0, \quad-2+3=1, \quad-2+6=4, \quad 1+2=3 \\
& 1+3=4, \quad 1+6=7, \quad 2+3=5, \quad 2+6=8, \quad 3+6=9
\end{aligned}
$$

we determine that $-2+6=1+3=4$.

We conclude that the sum of all the numbers Azra imagined is 4.

Among the numbers on the board, the number 2 does not appear in the previous equality (so $a+b=2$) and therefore the deleted number $c+d$ is equal to $4-2=2$.

Now let's change the labels.

Let the imagined numbers be $a, b, c, d$, but such that $a \leqslant b \leqslant c \leqslant d$.

We know that $\{-2,1,2,2,3,6\}=\{a+b, a+c, a+d, b+c, b+d, c+d\}$.

Since $a \leqslant b \leqslant c \leqslant d$, it is clear that $a+b$ is the smallest sum, and $c+d$ is the largest, so $a+b=-2$, $c+d=6$.

Now we can easily conclude that $a+c$ is smaller than all sums except $a+b$, and analogously $b+d$ is larger than all sums except $c+d$. This means that $a+c=1$ and $b+d=3$. Finally, $a+d=b+c=2$.

We have $2a = (a+b) + (a+c) - (b+c) = -2 + 1 - 2 = -3$, so $a = -\frac{3}{2}$.

Further, $b = -2 - a = -\frac{1}{2}$, $c = 1 - a = \frac{5}{2}$, and $d = 2 - a = \frac{7}{2}$.

Let's check that the other equalities are also satisfied:

$b+c = -\frac{1}{2} + \frac{5}{2} = 2$, \quad $b+d = -\frac{1}{2} + \frac{7}{2} = 3$, \quad $c+d = \frac{5}{2} + \frac{7}{2} = 6$.

Azra imagined the numbers $-\frac{3}{2}, -\frac{1}{2}, \frac{5}{2}$, and $\frac{7}{2}$.

2nd grade - high school - A variant
April 3, 2012.