Hamilton 2017 Q4
4. 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 \(d\)?
Hamilton Olympiad 2017, Question 4. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: 7.5 and 8
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 \(a + d < b + c\).
\(\boldsymbol{b + c \leqslant a + d}\)
\(\boldsymbol{a + d < b + c}\)
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.
Solution © UK Mathematics Trust, from the Hamilton Olympiad 2017 solutions.