Super Hard Task

[Papabile]

Hi, I have this super hard task on my homework and I dont know how to do it
"Let n be a positive integer. Inside a convex polygon of perimeter 1 afinite number of line segments is drawn such that their total length is strictly greater than n. Showthat there exists a line that intersects at least 2n + 1 of the segments."

 

[Dr.Peterson]

Hi, I have this super hard task on my homework and I dont know how to do it
"Let n be a positive integer. Inside a convex polygon of perimeter 1 a finite number of line segments is drawn such that their total length is strictly greater than n. Show that there exists a line that intersects at least 2n + 1 of the segments."

The first thing to do is to think about what you do know -- what have you learned that this homework is intended to exercise?

You haven't told us anything about the context, so we can't help you with that step. This is one of the things explained in this summary of the guidelines for submitting questions. Please read it, if you haven't done so yet.

Is this for a geometry class, or have you learned the pigeonhole principle, or what?

Since I have no idea how to start, I'd begin by experimenting with the situation, trying to get an idea how it works. Suppose the polygon is a square, each side 1/4. Take n=1, and suppose we draw four 1/4 unit segments in it (or maybe 8 1/8 unit segments). Does it make sense that some line must pass through at least 2(1)+1 = 3 of them? Try arranging the segments to try to prevent this from being possible.

I say all this without having done any of it; it is just my first thought of a strategy for thinking about it. See what you can do with it, and write back -- being sure to include not only your thoughts, but also the context of the question.

 

[Papabile]

I know everything bcs my lesson is really specific. The teacher give us homework (6 problems )and at lesson we presentation our solution. It can be everything. First problem is about divergent of series with any bijection, second about simple graph, third about matrix etc. And I solved every problem expect this one.

 

[Dr.Peterson]

Hi, I have this super hard task on my homework and I dont know how to do it
"Let n be a positive integer. Inside a convex polygon of perimeter 1 a finite number of line segments is drawn such that their total length is strictly greater than n. Show that there exists a line that intersects at least 2n + 1 of the segments."

Does anyone have any thoughts on this? Papabile's attempts to post apparently aren't getting through, so he is messaging me to try to answer my questions, but it appears that this problem has no real context that will provide any hints. It is purportedly a college-level review of high school material.

I'm having trouble thinking of anything to do. Clearly each segment will be less than 1/2 unit long, so there will be more than 2n segments (usually many more); but I can't even think how I'd arrange them to minimize the number of them that any line could cross.