18. Five positive integers (not necessarily all different) are written on five cards. Boris calculates the sum of the numbers on every pair of cards. He obtains only three different totals: 57, 70 and 83 . What is the largest integer on any card?
A 35
B 42
C 48
D 53
E 82
18. C Let the five integers be $a, b, c, d$ and $e$ with $a \leqslant b \leqslant c \leqslant d \leqslant e$. The smallest total is 57 , which is an odd number so $b \neq a$. Similarly, the largest total is 83 , which is also an odd number so $d \neq e$. Hence we now have $a<b \leqslant c \leqslant d<e$ and $a+b=57$ and $d+e=83$. Only one possible total remains and so $b=c=d$ with $c+d=70$. This gives $c=d(=b)=35$ and therefore $e$, the largest integer, is $83-35=48$ (whilst $a=22$ ).