You are given a pen and a large piece of paper. Without using any additional tools (eg., ruler, compass, string etc) how can you draw a perfect circle?
Make a compass:
Grasp the pen in your fist so that your thumb is uppermost and the pen points down, like holding a dagger.
Using the same arm, place your elbow's funny-bone at the centre of the large sheet of paper.
Rest the tip of the pen on the paper.
Keeping your arm still, drag the paper round with your other hand so that it rotates under your funny-bone.
The pen will draw a line as the circumference of a circle. Its radius is the length of your forearm.
Oh, I'm obviously thinking too far outside the box, but this is a cute solution:
Draw a straight line across the page. When the page is rolled up, the straight line becomes a circle:
Objection #1: The circle is imperfect because it is hard to draw a straight line.
Response: This is true to some degree of all the solutions, but we can mitigate by using the bottom edge of the paper as a rule to keep the line straight.
Objection #2: The circle is imperfect because it is hard to roll the paper into a perfect circle.
Response: Keep rolling it until the circle becomes more perfect.
If the piece of paper is rectangular and not too rigid:
You can draw two points which will be one diameter and then use a corner of the sheet and Thales's Theorem to draw as many points on the circle as you like.
A bit more detail for the non-geometers: Thales's theorem says that if you connect a point on a circle with the two ends of a diameter you will get a right angle and, conversely, if a point forms a right angle when connected with the ends of a diameter then the point lies on that circle. Therefore after choosing two points which shall be our diameter we use the corner of the sheet as a right angle and gently bend the sheet so that the two sides enclosing the corner align with our two points. We can do this in many ways and each will give us another point on the circle.
If the paper is large enough, you can use yourself as a compass. For example, you could make a mark in the centre of where you want the circle, and anchor a finger on it. With the pen in the other hand, and at full stretch, rotate about the anchored finger, leaving a circular trail as you go. You mighy want to experiment with other body parts to see which gives the best results. I reckon my nose would make a good anchor...
This answer is very different from the others and I'm not sure if it's too obscure, but technically it fits the rules I think.
Draw 5 stick figures. One holding a microphone, two holding a guitar, one holding a bass guitar and one sitting on a drumset. Underneath those, write the names Maynard, Billy, James, Matt and Jeff. You have now drawn A Perfect Circle.
It's debatable how many people would recognize this band. However if it is recognized, it arguably fits the criterion of being "a perfect circle" better than any attempt to draw an actual circle since those can never be absolutely perfect.
Just as an "off the top" answer to this scenario that should be fairly close to perfect:
(I want to take a moment as perfection is often something we don't realize how varied peoples accepted notions for this can be. One person may believe 100% is perfect, another may need the 100% to be done while 100% of other factors are in place a well, etc. That said, this response has taken some basic liberties such as leaving out the variance in paper thickness, airflow of the ink drying and the ink's viscosity due to a variety of factors from chemical elements of the ink itself to the temperature in the pen/around the pen/at the point of the paper, etc. The list goes on and on but I believe the point has been made.)
Simply bend the pen in a manner to where you can mark on the paper with the writing end and pin (hold in place or create a pivot if you will, not the utensil itself) the other end somewhere on the paper which would allow the pin to spin a full circle without leaving the paper surface. Now rotate the pen 360 degrees (or spin the paper but this is getting long winded) and you should have a 'perfect' circle.
p.s. A Perfect Circle is a vastly under rated band. Just saying :P.
A bit late to the game, but here's an answer that's "way out there":
Tear off a small piece of the paper, poke a tiny hole in the small piece of paper with the tip of the pen, and hold it up to the sun. Then trace the image of the sun's edge that comes through the "pinhole" lens of the tiny hole that projects onto the larger remaining piece of paper. You'd need to make sure that the large piece paper is perfectly perpendicular to the sun's rays, but per https://science.nasa.gov/science-news/science-at-nasa/2008/02oct_oblatesun, the sun is just about the roundest item in the universe.
Pick your writing implement carefully. It needs to be a simple round pencil, not one of those yellow octagonal pencils with a rubber on the end, nor one of those weird triangular ones that mysteriously appear in your pencil case, nor even one of those fancy mechanical pencils that your grandfather gives you. No, it needs to be a nice, uncomplicated round pencil.
Press hard on the piece of paper to break off a little of the lead. This will leave a mark, but unless you're exceptionally lucky, it won't be a circle. That's fine because nobody said you can't leave non-circular marks on the paper.
Now, use the blunt end of the pencil to turn the broken-off piece into powder. Take your time over this, not because you need to for the result, but because there's great pleasure to be taken in grinding things up. It may be a bit messy, but it will all be worth it in a moment.
Make sure the blunt end of the pencil is completely coated in powder, and then press it against the paper. When you remove it, you'll have a perfect circle.
Fame and fortune will inevitably follow as reward for your efforts. Well done, you!
I know, I know: the OP said I had to use a pen... well, I reckon something similar would work with a pen, too. Perhaps squeeze out a little ink, and take the pen apart and use half of it to print little circles?
I'll bet my granfather will be impressed with my ingenious solution. He might even reward me...
Carefully rip off a 1 inch wide strip of the paper. Jam the tip of the pencil through near one end of the strip and through the sheet of paper near its center. Break off the pencil, resharpen, and then poke it through the strip at desired radius. Use the pinned strip to maintain constant radius as you pivot around.
Eleven answers already! A record perhaps?
Here is my off base answer. I tried. Looked like a circle
I took a ball point pen and a thick paper. Just put the pen perpandicular to the paper and pressed hard to puncture the paper. The pen point was circular so it created a nice circular hole!
With a pencil it worked even better because the pencil end is conical in shape!
Here's my solution:
Use a hair (you can get one on your head). Tie one end of the hair to around the point of the pen, and you got yourself a hassle free, easy to use compass!
To use this compass, use your finger to press the hair at a certain distance from the end tied to the pen (distance depends on how big you want your circle to be), pull the pen away from the part that's pressed down until the hair straightens out, and draw around.
if the paper has arbitrary large mass with infinite young's modulus and the friction of the ball point pen writing on the paper is arbitrarily small relative to the papers mass, you can spin the paper in the air at an arbitrary fast speed. applying an arbitrary small force to the pen tip applied to the paper yields a perfect circle that is superior to other methods involving pivots on a compass derived from human fingers or point methods that rely on visual feedback and human motor control.
the trick to solving this problem is that the mass of the paper, the modulus of the paper, the affects of gravity over time, the friction of the pen tip, air friction etc.. are not constrained. this makes any high precision method using physics completely possible and superior to methods that rely on imprecise instruments such as the human hand or the human eye. even if the hand was shaking when the pen tip was applied, the error would not manifest immediately if the spin of the paper was sufficiently much faster than the movement of the pen in non axial trajectory. there would be a moment in time where the circle would be ideal considering the forces used to setup the experiment were ideal. even the affects of axial force with respect to the pen would take time to integrate and act on the paper to change the papers trajectory. therefor all errors are minimized and accounted for. you can for example, apply the pen from the bottom of the paper and use the paper as a gyroscope to completely stabilize the experiment.