H4. The largest of four different real numbers is $d$. When the numbers are summed in pairs, the four largest sums are $9,10,12$ and 13 .
What are the possible values of $d$ ?
Solution
Let the other three different numbers be $a, b$ and $c$, in increasing order. Then each of them is less than $d$, so that $c+d$ is the largest sum of a pair. The next largest is $b+d$, because it is larger than any other sum of a pair. But we do not know whether $b+c$ or $a+d$ is next (though each of these is larger than $a+c$, which in turn is larger than $a+b$ ). There are thus two cases to deal with, depending on whether $b+c \leqslant a+d$ or $a+d<b+c$.
$$
b+c \leqslant a+d
$$

We have
$$
\begin{aligned}
b+c & =9, \\
a+d & =10, \\
b+d & =12 \\
\text { and } \quad c+d & =13 .
\end{aligned}
$$

From equations (1), (3) and (4), we find that $2 d=16$, so that $d=8$.
$$
a+d<b+c
$$

We have
$$
\begin{aligned}
a+d & =9, \\
b+c & =10, \\
b+d & =12 \\
\text { and } \quad c+d & =13 .
\end{aligned}
$$

From equations (6) to (8), we find that $2 d=15$, so that $d=7.5$.

In each case, it is possible to find the values of $a, b$ and $c$ from the equations, and to check that these fit the conditions in the question.
Therefore the possible values of $d$ are 7.5 and 8 .