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Problem 340

Combinatorics Difficulty 2.5 Multiple choice CEMC Gauss (Grade 8) · Canada · 2020

Francesca chooses an integer from the list $-4,-3,-2,-1,0,1,2,3,4,5,6$ and then a second integer that is larger than the first. How many such pairs of integers can she choose so that the sum of the pair is 3?

Pick one

Official solution

Solution 1

There are 60 minutes in 1 hour, and so there are $60+52=112$ minutes in 1 hour 52 minutes.

If Emil’s race time was 54 minutes, then Olivia’s race time was 4 minutes more, or 58 minutes.

In this case, their race times total $54+58=112$ minutes, as required.
Therefore, it took Olivia 58 minutes to run the race.

Solution 2

As in Solution 1, the total of their race times is 112 minutes.

If Emil’s race time was 4 minutes more, then his race time would be equal to Olivia’s, and the total of their race times would be $112+4=116$ minutes.

If their race times are equal and total 116 minutes, then they each finished the race in $116\div2=58$ minutes.
Therefore, it took Olivia 58 minutes to run the race.

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