# Problem 4.

Given a square with a side length of 5 cm. On its four sides, the vertices of a second square are located; on the four sides of the second square, the vertices of a third square, and so on. For what smallest natural $n$ will the sum of the areas of the first $n$ squares be guaranteed to be greater than 49 cm $^{2}$?
Answer: 6

Solution. A square inscribed in the given square has the smallest area if its vertices are the midpoints of the sides of the original square (this follows, for example, from the fact that the area of a square is half the square of its diagonal). In this case, the area of the inscribed square is half the area of the original. Thus, the areas of the squares of minimum area form a geometric progression with the first term 5 and the common ratio 0.5. The sum of the first $n$ terms of this progression will first exceed 49 when $n=6$ (this can be obtained by enumeration or from the formula for the sum of the terms of a geometric progression).

## B-2

A square with a side of 7 cm is given. On the four sides of this square, the vertices of a second square are located; on the four sides of the second square, the vertices of a third square, and so on. For what smallest natural $n$ will the sum of the areas of the first $n$ squares be guaranteed to be greater than 97 cm $^{2}$?

Answer: 7

## B-3

A square with a side of 3 cm is given. On the four sides of this square, the vertices of a second square are located; on the four sides of the second square, the vertices of a third square, and so on. For what smallest natural $n$ will the sum of the areas of the first $n$ squares be guaranteed to be greater than 17 cm $^{2}$?

Answer: 5

## B-4

A square with a side of 9 cm is given. On the four sides of this square, the vertices of a second square are located; on the four sides of the second square, the vertices of a third square, and so on. For what smallest natural $n$ will the sum of the areas of the first $n$ squares be guaranteed to be greater than 160 cm $^{2}$?

Answer: 7

## Lomonosov Olympiad for Schoolchildren in Mathematics

Preliminary stage $2020 / 21$ academic year for $7-8$ grades

## B-1

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