PK 2016 Q8
8. Sven wrote five different one-digit positive integers on a blackboard. He discovered that none of the sums of two different numbers on the board equalled 10. Which of the following numbers did Sven definitely write on the blackboard?
- A 1
- B 2
- C 3
- D 4
- E 5
Pink Kangaroo 2016, Question 8. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: E 5
Sven can choose from {1, 2, 3, 4, 5, 6, 7, 8, 9}, but can choose at most one from each of the pairs that add to 10: {1, 9}, {2, 8}, {3, 7}, {4, 6}. Since this gives him a maximum of 4 integers, he must always pick the digit 5.
Solution © UK Mathematics Trust, from the PK 2016 solutions.