LR3

Part of 2019 Costa Rica - Final Round

Problems(1)

shaded area of n-th square A_n < 2/3 wanted, sequence of squares

Source: OLCOMA Costa Rica National Olympiad, Final Round, 2019 Shortlist LC3 day1 (logical reasoning)

Consider the following sequence of squares (side

1

), in each step the central square is divided into equal parts and colored as shown in the figure: https://cdn.artofproblemsolving.com/attachments/9/0/6874ab5aecadf2112fbe4a196ab3091ab8b31a.png Square 1 Square 2 Square 3
Let

A_n

n \in N

n> 1

be the shaded area of square

n

A_n <\frac23

geometrySquaressquareareas