5. Tina has been depositing 500 euros at the beginning of each year for five consecutive years into a bank that capitalizes annually at an interest rate of $5 \%$. How many years after the last deposit will she have 4500 euros?

## You have 120 minutes to solve the problems.

Solve each problem independently on the provided paper, with each problem on a separate page. Do not sign the sheets, write only your code. Write your answers in ink, clearly and neatly. Draw graphs of functions with a pencil. If you solve a problem in multiple ways, clearly indicate which solution the evaluator should grade. If you make a mistake, strike it through.

Each problem will be graded on a scale of 0 to 6 points.

The National Competition Committee wishes you great success.

## 7th Mathematics Knowledge Competition for Students of Secondary Technical and Vocational Schools

National Competition, April 21, 2007

## Solutions to Problems and Scoring Guide

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

A correct method is one that

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

If an intermediate or final result can be recognized, guessed, read from a diagram, or calculated mentally, the contestant generally receives all the points provided. However, if the solution is guessed (and cannot be arrived at without calculation), such a solution is scored with 0 points.

A contestant who has only partially solved a problem, with correct but incomplete steps, cannot receive more than half of the possible points.

## First Year
5. We use the formula $G=\frac{a\left(1.05^{5}-1\right)}{1.05-1} \cdot 1.05^{n}$. We substitute the data, rearrange the equation $1.05^{n}=1.63$ and calculate $n=10$ years.

![](https://cdn.mathpix.com/cropped/2024_06_07_cd02ddf507985fa73630g-11.jpg?height=68&width=1636&top_left_y=2533&top_left_x=210)

Calculation of the principal after five years 2765.81 euros..........................................................................................

Using the formula for compound interest $G=G \cdot r^{n} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$

![](https://cdn.mathpix.com/cropped/2024_06_07_cd02ddf507985fa73630g-12.jpg?height=54&width=1642&top_left_y=224&top_left_x=207)

![](https://cdn.mathpix.com/cropped/2024_06_07_cd02ddf507985fa73630g-12.jpg?height=52&width=1636&top_left_y=271&top_left_x=210)

Answer: $n=10$ years....................................................................................................