## Task 4 - 280514

a) Draw the points $A(1 ; 9), B(4 ; 6)$, and $C(6 ; 10)$ in a coordinate system!

Connect each pair of these three points with a line segment! How many connecting segments are there in total?

b) Draw two more points $D$ and $E$; choose them so that no connecting segment between any two of the five points $A, B, C, D, E$ contains any other of the five points! Connect each pair of the five points with a line segment!

How many connecting segments are there in total?

c) The number of connecting segments sought in b) can also be determined by a reasoning process without drawing the points and the segments.

Describe such a reasoning process!

d) Determine the number of all connecting segments between any two of ten points, for which the same condition as in b) applies, using the reasoning process described in c)!
d) By similar considerations as in c), one obtains that under the given conditions, because

$$
9+8+7+6+5+4+3+2+1=45
$$

(or because $9 \cdot 10=90$ and $90: 2=45$) there are exactly 45 lines that connect the ten points in the required manner.

Solutions of the 1st Round 1988 adopted from [5]

### 2.29.2 2nd Round 1988, Class 5