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Problem 54

Combinatorics Difficulty 1.1 Multiple choice CEMC Cayley · Canada · 2022

The pattern of shapes $\$\triangle, \square, \heartsuit, \diamondsuit, \bigcirc\$$ is repeated to form
the sequence $\triangle, \square, \heartsuit, \diamondsuit, \bigcirc, \triangle, \square, \heartsuit, \diamondsuit, \bigcirc, \ldots$ The 22nd shape in the sequence
is

Pick one

Official solution

The sequence consists of a pattern of 5 shapes that are
repeated.

The first repetitions of this pattern end on the 5th, 10th, 15th, 20th,
and 25th shapes.

This means that the 22nd shape is the 2nd shape after the 20th shape,
and so is the 2nd shape in the pattern.

Thus, the 22nd shape is $\square$ .

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