Problem 22 · 2015 Math Kangaroo
Stretch
Algebra & Patterns
sum-constraintcasework
Five positive whole numbers, which are not necessarily all different, are written on five cards. Peter calculates the sum of each pair of cards. He obtains only three different results, namely 57, 70 and 83. What is the biggest number that is written on one of the cards?
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Answer: C — 48
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Hint 1 of 2
With only three distinct pairwise sums, the five numbers take very few distinct values.
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Hint 2 of 2
The smallest sum is the two smallest cards and the largest sum the two largest; use 57, 70, 83 to pin them.
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Approach: deduce the card values from the three pair-sums
- Only three sums (57, 70, 83) appear, so the cards take just a couple of distinct values plus an odd one out.
- The biggest sum 83 pairs the two largest cards; combined with 57 and 70 this forces a largest card of 48.
- Indeed 35 + 48 = 83, while repeated smaller values give 57 and 70.
- So the biggest number on a card is 48 (C).
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