Problem 10 · 2016 Math Kangaroo
Medium
Counting & Probability
complementary-counting
Sven writes five different single-digit positive whole numbers on a board. He realises that no sum of two of these numbers is equal to 10. Which of the following numbers has Sven definitely written on the board?
Show answer
Answer: E — 5
Show hints
Hint 1 of 2
Group the digits 1–9 into pairs that add to 10.
Still stuck? Show hint 2 →
Hint 2 of 2
You can use at most one number from each pair; count how many that allows.
Show solution
Approach: pigeonhole on pairs
- The pairs summing to 10 are {1,9},{2,8},{3,7},{4,6}; the digit 5 has no partner.
- From four pairs you may pick only one each — at most 4 numbers.
- To reach five numbers you must also use 5, so 5 is definitely written.
Mark:
· log in to save