Problem 10 · 2017 Math Kangaroo
Medium
Logic & Word Problems
sum-constraintcasework
Only four players scored goals in a handball game, and each scored a different number of goals. Michael scored the fewest. If the other three players scored 20 goals in total, what is the greatest number of goals Michael could have scored?
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Answer: C — 4
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Hint 1 of 2
Michael scored the fewest, and all four totals differ; the other three add to 20.
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Hint 2 of 2
To make Michael's count as big as possible, keep the other three just barely above him and distinct.
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Approach: push the other three players as close to Michael as possible
- Michael scored the fewest, so the other three each scored more than him, and all four totals are different.
- To let Michael score a lot, the other three should be just barely bigger: the three smallest different scores above Michael are Michael+1, Michael+2 and Michael+3.
- If Michael scored 4, the others would be at least 5, 6, 7 = 18, which fits inside 20 (for example 5, 6, 9 add to 20).
- If Michael scored 5, the others would be at least 6, 7, 8 = 21, which is already more than 20 — too big.
- So Michael could score at most 4 (C).
Same idea with algebra
If Michael scores \(m\), the smallest the other three can total is \((m+1)+(m+2)+(m+3)=3m+6\). We need \(3m+6\le 20\), so \(m\le 4\).
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