5. With a stick of unknown length, we also want to determine the unknown width and height of a door. If we lay the stick horizontally across the door, it is 2 cubits longer than the width of the door. If the stick is placed vertically, it is 1 cubit longer than the height of the door. The stick fits exactly into the opening of the door if placed diagonally. Calculate the width and height of the door and the length of the stick.

## You have 120 minutes to solve.

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 pen, 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 out.

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

The National Competition Committee wishes you great success.

## 8th Competition in Mathematics Knowledge for Students of Secondary Technical and Vocational Schools  National Competition, April 19, 2008

## Problems for 4th Year Students
5. Let's denote the length of the pole as $d$, the width of the door as $x$, and the height of the door as $y$. The relationships are given by $x=d-2$, $y=d-1$, and $x^{2}+y^{2}=d^{2}$. Solving the system of three equations with three unknowns leads to the equation $d^{2}-6 d+5=0$ with solutions $d_{1}=1$ and $d_{2}=5$. The solution $d=1$ is not valid. From $d=5$, the solutions $x=3$ and $y=4$ also follow.

$x=$ width of the door

$y=$ height of the door

$d=$ length of the pole

Individual equations:

$d=x+2$

1 point

$d=y+1$

1 point

$d^{2}=x^{2}+y^{2}$ 1 point

Organized equation $d^{2}-6 d+5=0$ 1 point

Solutions $d_{1}=1$ and $d_{2}=5$....

1 point

Answer: Width of the door $x=3$, height of the door $y=4$, length of the pole $d=5$....

1 point

## Fourth Year