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Is it possible to place 1000 line segments on the page so that the two ends of each line segment are on the inner points of other line segments? (By the inner point of the line segment, we mean a point other than its two ends)

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    $\begingroup$ Can the page be bent, shaped like a cone, taurus, etc.? $\endgroup$ – Mathaddict Jan 6 at 19:22
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I claim the answer is

no

and here's why:

Ignore the segments for now and just look at the leftmost endpoint(s). Any of these must be on a perfectly vertical line segment - if they weren't, there would need to be an endpoint even further left.

Of these, consider the topmost endpoint. It cannot be on a perfectly vertical line segment, because there is no other endpoint above it.

So that particular endpoint will not be on a different line segment; therefore it is not possible.

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  • $\begingroup$ I did not understand your reason! topmost endpoints could be on a horizental and vertical line. couldn't they? $\endgroup$ – MathR Jan 6 at 7:13
  • $\begingroup$ @Reza No, the topmost endpoint on the left cannot be on a horizontal or diagonal line because it's as far left as possible. And it can't be on a vertical line because it's the topmost of its 'column'. $\endgroup$ – Deusovi Jan 6 at 7:34
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Partially inspired by Deusovi's answer, here's another argument.

First,

Rotate the page such that out of the $2000$ endpoints of the $1000$ line segment, one is the furthest to the left, without ties. How do we know this is possible? Well, there's only $2000$ endpoints, so there's only $\binom{2000}{2}$ pairs of endpoints, and for every pair of endpoint, there's only two ways to rotate the page so that they're tied for how far left they are. Eliminate all those $2 \binom{2000}{2}$ ways to orient the page, and you still have infinitely many ways to orient the page that produce no ties.

Then,

The leftmost endpoint cannot be the inner point of any line segment. If $X$ is the inner point of line segment $AB$, then either $A$ is to the left of $X$, or $B$ is to the left of $X$, or $AB$ is vertical and all three are equally far left. None of these are possible when $X$ is the unique leftmost endpoint.

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Yes, as long as you're not too picky about the shape of your page.

If you can curl your page so that it forms a cylinder, then you can draw a line forming a ring around the cylinder and put line segments on just that ring so that they all overlap and hit each others' interiors. Then the arguments in the other answers don't make sense, as there is no longer a leftmost point as you can keep going left forever.

enter image description here

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