3. The five integers $2,5,6,9,14$ are arranged into a different order. In the new arrangement, the sum of the first three integers is equal to the sum of the last three integers.
What is the middle number in the new arrangement?
A 2
B 5
C 6
D 9
E 14
Solution
E

Let the integers in the new arrangement be in the order $p, q, r, s, t$. Because the sum of the first three integers is the same as the sum of the last three,
$$
p+q+r=r+s+t,
$$
and hence
$$
p+q=s+t .
$$

We therefore see that the pair of integers $p, q$ has the same sum as the pair $s, t$. Also, the middle number, $r$, is the one that is not included in either of these pairs.
It is straightforward to check that $2+9=5+6$ and that 2,9 and 5,6 are the only two pairs of the given integers with the same sum.

Therefore the middle integer in the new arrangement is 14 , as this does not occur in either pair.