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Combinatorics Difficulty 2.2 Junior Find the answer Canada

In the diagram, the target shown has three scoring areas. An
arrow that hits the centre circle is worth $10$ points, an arrow that hits the shaded middle ring is worth $5$ points, and an arrow that hits the outer ring is worth $1$ point.

Three arrows are shot and each hits the target. Which of the
following cannot be the total score for the three arrows?

Pick one

Solution

The correct answer is $13$ . Why? Since the scoring areas $10$ and $5$ are each multiples of $5$ , then any combination of these two scores is also a multiple of $5$ . The closest multiple of $5$ less than $13$ is $10$ , and so to obtain a total score of $13$ , a minimum of three arrows must each score $1$ point (since $13-10=3$ ). However, at least one arrow is needed to score $10$ points, and so at least four arrows are needed to score $13$ points. We confirm that each of the remaining answers is possible since $16=10+5+1$ , $11=5+5+1$ , $7=5+1+1$ , and $20=10+5+5$ .

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Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.