## Task 4 - 150524

Given a line $g$ and a point $A$ on it.

Construct four more points $B, C, D, E$ on this line $g$ such that they lie in this order on the same ray starting from $A$ and the following conditions are met:

(1) The segment $A B$ is $2.5 \mathrm{~cm}$ long.

(2) The segment $B C$ is $0.3 \mathrm{dm}$ longer than the segment $A B$.

(3) The segment $C E$ is as long as the sum of the segments $A B$ and $B C$.

(4) The segment $D E$ is $50 \mathrm{~mm}$ shorter than the segment $C E$.

Describe the construction, and determine the length of the segment $A D$ (in $\mathrm{cm}$)!
We first measure from $A$ along $g$ with the compass a segment of $2.5 \, \text{cm}$ in length. The other endpoint is 3.

Then, because $0.3 \, \text{dm} = 3 \, \text{cm}$ and $2.5 \, \text{cm} + 3 \, \text{cm} = 5.5 \, \text{cm}$, we measure from $B$ on the ray of $g$ that does not contain $A$, a segment of $5.5 \, \text{cm}$ in length and name its other endpoint $C$.

Since $2.5 \, \text{cm} + 5.5 \, \text{cm} = 8 \, \text{cm}$, we now measure from $C$ on the ray of $g$ that does not contain $B$, a segment of $8 \, \text{cm}$ in length and name its other endpoint $E$.

Since $D$ is between $C$ and $E$ according to the problem, we finally measure, because $50 \, \text{mm} = 5 \, \text{cm}$ and $8 \, \text{cm} - 5 \, \text{cm} = 3 \, \text{cm}$, from $E$ on the same ray of $g$ that contains $C$, a segment of $3 \, \text{cm}$ in length and name its other endpoint $D$.

Since $2.5 \, \text{cm} + 5.5 \, \text{cm} + 8 \, \text{cm} - 3 \, \text{cm} = 13 \, \text{cm}$, the segment $A D$ has a length of $13 \, \text{cm}$.

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0071.jpg?height=60&width=1294&top_left_y=921&top_left_x=378)

Solutions of the 2nd Round 1975 taken from [5]

### 2.17 XVI. Olympiad 1976

### 2.17.1 1st Round 1976, Class 5