Clean up and improve simple graphic object

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This customer received 56 graphic designs from 20 designers. They chose this graphic design from bestwork as the winning design.
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I have a graphic object, a simple target. The second is just the first one rotated 45 degrees. Both need to be cleaned up to appear as clean as possible on a smartphone screen and when printed on a black and white printer (no grayscale!) Given the very small sizes, it may be a challenge to make perfect curves, but I need the best that can be done. The originals are 80 pixels square. I need both of them in 40, 80 (original), 88, 102, 106 and 118 pixels. That 's a total of 12 different images. They will probably all need to be hand tuned.
Because there are so many excellent entries in order to help me choose I will add an additional optional version of the graphic. This white but with a thin (thinnest possible) black border around the white. That would include the circle and the lines, inside and out. Please keep in mind that the background is always transparent, so on a white background you would simply see an outline of the graphic.
Again, this is optional, but might make the difference in my selection!
Updates
To expand on the question that was just asked. The final result will be used primarily as a png file and like the sample needs to be transparent, so I can render it on different backgrounds.
Added Saturday, January 31, 2015
Thank you for all the responses, they are looking great. I would like to add one more requirement: Sometimes I will need this image to be white on transparent background, not just black. Hopefully this will just be changing the colors and won't add much and some designers have already added examples.
I like what I see so far, choosing a winner will be hard.
Thanks again,
Ken
Added Monday, February 02, 2015
Because there are so many excellent entries in order to help me choose I will add an additional optional version of the graphic.
This white but with a thin (thinnest possible) black border around the white. That would include the circle and the lines, inside and out.
Please keep in mind that the background is always transparent, so on a white background you would simply see an outline of the graphic.
Again, this is optional, but might make the difference in my selection!
Thank you,
Ken
Added Thursday, February 5, 2015
Thank you all for your contest submissions. I am reviewing them now and will pick a winner shortly.
All of the work is great and I'm sorry I don't need to pick more than one.
Yours,
Ken
Added Sunday, February 08, 2015
Industry/Entity Type
Printer