GK 2015 Q18
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
Grey Kangaroo 2015, Question 18. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: C 48
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 \(a = 22\)).
Solution © UK Mathematics Trust, from the GK 2015 solutions.