Greece National Olympiad Problem 1

Probability Level 5

On the plane are given $k+n$ distinct lines , where $k>1$ is integer and $n$ is integer as well.Any three of these lines do not pass through the same point . Among these lines exactly $k$ are parallel and all the other $n$ lines intersect each other. All $k+n$ lines define on the plane a partition of triangular , polygonic or not bounded regions. Two regions are called different, if they don't have common points or if they have common points only on their boundary. A region is called ''good'' if it is contained between two consecutive parallel lines .

If in a such given configuration the minimum number of ''good'' regions is $176$ and the maximum number of these regions is $221$ , find $k+n$ .This problem is part of this set .

The answer is 27.

1 solution

Parth Lohomi
Mar 22, 2015

Let's call the region between the $k$ parallel lines the "Q". Fixing $k$ and increasing $n$ , the minimum of good regions is attained if all intersections between the $n$ lines are outside Q. So $f_{min} = (n + 1)(k - 1)$ .

For the maximum, each of the $n$ lines must intersect all the others inside Q. If the $n$ -th line is added (not counting the $k$ parallel lines), it is cut by the already existing lines into at most $(k + n - 2)$ segments lying inside Q which is thus the maximal number of good regions added. So $f_{max} = 2(k - 1) + \sum_{i = 2}^n (k + i - 2) = nk + k + \dfrac {n^2 - 3n - 2}2 = f_{min} + \dfrac {n^2 - n}2$ . So we have $\dfrac {n^2 - n}2 = 221 - 176 = 45$ , thus $n = 10$ , hence $k = 17$ .

@Andrei Golovanov I love this question. Can you define what a "zone" is? In such questions, these terms can be heavily loaded with the author's intent, so it's best to make it clear. I think it's fine to just say "contained between two parallel lines.

The initial setup is also slightly confusing to read. I would suggest setting up the $n$ in general position first, and then adding in the $k \geq 2$ parallel lines (not parallel to any of these $n$ ) lines. This makes it easier to understand the first 3 sentences (though I agree that it lengthens the description).

Calvin Lin Staff - 4 years, 4 months ago

question is not clear. what is meant by zone.

Srikanth Tupurani - 1 year, 8 months ago

Greece National Olympiad Problem 1

The answer is 27.

1 solution

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