5. The surface areas of five cubes are terms of a geometric sequence with a common ratio of 4. The edge of the smallest cube is $5 \mathrm{~cm}$ long. What is the sum of the surface areas of all five cubes? What is the surface area of the largest cube?[^1]

## You have 120 minutes to solve.

THE NATIONAL COMPETITION COMMITTEE WISHES YOU GREAT SUCCESS.

## Solutions to the problems and the scoring guide

A contestant who arrives at the solution by any correct method (even if not anticipated by the scoring guide) receives all possible points.

A correct method is one that

- sensibly takes into account the wording of the problem,
- leads to the solution of the problem,
- is mathematically correct and complete.

A contestant who has only partially solved the problem, with correct steps in the solution process but without a clear path to the final solution, cannot receive more than half of the possible points.

## First Year
5. We use the formula for the surface area of a cube $P=6 a^{2}$. We determine that the surface area of the smallest cube is the first term of a geometric sequence $a_{1}=6 \cdot 5^{2}$. We substitute the data into the formula for the sum of the first five surface areas of the cubes $S_{5}=\frac{6 \cdot 5^{2}\left(4^{5}-1\right)}{4-1}$. We calculate the sum $S_{5}=51150 \mathrm{~cm}^{2}$. Then we substitute the appropriate data into the formula for the fifth term of the sequence $a_{5}=6 \cdot 5^{2} \cdot 4^{4}$. We then calculate the surface areas $P_{5}=38400 \mathrm{~cm}^{2}$.

Using the formula for the surface area of a cube $P=6 a^{2} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$

![](https://cdn.mathpix.com/cropped/2024_06_07_59669f975316a35884bfg-14.jpg?height=55&width=1642&top_left_y=812&top_left_x=204)

![](https://cdn.mathpix.com/cropped/2024_06_07_59669f975316a35884bfg-14.jpg?height=69&width=1639&top_left_y=862&top_left_x=206)

![](https://cdn.mathpix.com/cropped/2024_06_07_59669f975316a35884bfg-14.jpg?height=57&width=1642&top_left_y=917&top_left_x=204)

![](https://cdn.mathpix.com/cropped/2024_06_07_59669f975316a35884bfg-14.jpg?height=60&width=1642&top_left_y=964&top_left_x=207)

![](https://cdn.mathpix.com/cropped/2024_06_07_59669f975316a35884bfg-14.jpg?height=57&width=1642&top_left_y=1017&top_left_x=207)


[^0]:    Solve the problems independently on the provided paper, and each problem on a separate page.

    Do not sign the sheets, write only your code.

    Write your answers with a pen clearly and neatly. The solution to each problem will be graded from 0 to 6 points.

[^1]:    Solve the problems independently on the provided paper, and each problem on a separate page.

    Do not sign the sheets, write only your code.

    Write your answers with a pen clearly and neatly. The solution to each problem will be graded from 0 to 6 points.