Maths Olympiad Prep

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Problem 275

Combinatorics Difficulty 2.3 Find the answer CEMC Cayley · Canada · 2020

Shaded and unshaded squares are arranged in rows so that:

the first row consists of one unshaded square,
each row begins with an unshaded square,
the squares in each row alternate between unshaded and shaded, and
each row after the first has two more squares than the previous row.

The first 4 rows are shown.

The number of shaded squares in the 2020th row is

20222022
20212021
20202020
20192019
20182018

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

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Official solution

The 1st row has 0 shaded squares and 1 unshaded square.

The 2nd row has 1 shaded square and 2 unshaded squares.

The 3rd row has 2 shaded squares and 3 unshaded squares.

The 4th row has 3 shaded squares and 4 unshaded squares.

Because each row has 2 more squares than the previous row and the squares in each row alternate betweeen unshaded and shaded, then each row has exactly 1 more shaded square than the previous row.

This means that, moving from the 4th row to the 2020th row, a total of 20204=20162020 - 4 = 2016 additional shaded squares are added.

Thus, the 2020th row has 3+2016=20193 + 2016 = 2019 shaded squares.

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