OPM 1997

The dodo was a strange animal. As it has already become extinct, only conjectures can be made about its way of life. One of the most unique conjectures is linked to the way the dodo moved. It seems that an adult animal only moved by jumping, which could be of two types: type I: $1$ meter to the East and $3$ to the North; type II: $2$ meters to the West and $4$ to the South. a) Prove that it was possible for the diode to reach a point located $19$ meters to the East and $95$ to the North of it and determines the number of jumps for each type he needed to carry out. b) Prove that it was impossible for the diode to reach a point located $18$ meters to the East and $95$ meters to the North of it.

A square region of side $12$ contains a water source that supplies an irrigation system constituted by several straight channels forming polygonal lines. Considers the source as a point and each channel as a line segment. Knowing that a point is irrigated if it is not more than $1$ distance from any channel and that the system was designed so that the entire region is irrigated, proves that the total length of irrigation channels exceeds $70$ .

$n$ parallel segments of lengths $a_1 \le a_2 \le a_3 \le ... \le a_n$ were painted to mark an airport atrium. However, the architect decided that the $n$ segments should have equal length. If the cost per meter of extending the lines is equal to the cost of reducing them, how long should the lines be in order to minimize costs?