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Problem 173

Number theory Difficulty 1.4 Multiple choice CEMC Gauss (Grade 8) · Canada · 2023

When attempting to arrange a pile of peaches into groups of
threes, there are two peaches not in a group of three. Which of the
following choices could be the number of peaches in the original
pile?

Pick one

Official solution

Suppose we call the unknown vertex $V$ .

The side of the rectangle joining the points $(2,1)$ and $(2,5)$ is vertical, and so the opposite
side of the rectangle (the side joining $(4,1)$ to $V$ ) must also be vertical.

This means that $V$ has the same
$x$ -coordinate as $(4,1)$ , which is 4.

Similarly, the side of the rectangle joining the points $(2,1)$ and $(4,1)$ is horizontal, and so the opposite
side of the rectangle (the side joining $(2,5)$ to $V$ ) must also be horizontal.

This means that $V$ has the same
$y$ -coordinate as $(2,5)$ , which is 5.

Therefore, the coordinates of the fourth vertex of the rectangle are
$(4,5)$ .

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