The satellite design is based on the first satellite humans put in orbit, namely Sputnik 1. An empty glass fiber sphere of the same diameter, covered in aluminum foil used in kitchen is all that’s needed. The approximate altitude and the number of orbits can be easily be determined with a telescope.
The satellite surface area is 1 square meter. Kitchen grade aluminum foil weighting about 1 gram is readily available. Fiberglass that thin might be a problem. It might be necessary to use a fiberglass skeleton instead of a sphere, to cut down the weight.
The aerodynamic drag becomes important for a satellite this size. During launch the satellite will be just thin flexible sheets lens-shaped. On release a piece of string will turn the sheets into a sphere by closing in the north and south pole of the sphere.
27.10.2015 Update
The Sputnik 1 model is not the best choice. First of all because the 20 grams limit in the same volume means about 4000 times higher drag. Sputnik 1 only made 1440 orbits, while the value needed for the competition would be above 36000 orbits. Secondly because even fiberglass is heavy for this structure.Some adjustments are needed for smaller diameter and higher visibility.
It was decided that a mirror like surface is not the best choice for visibility. Galileo realized that the Moon is not a mirror, yet it lights the night sky! A white ball of 10 cm diameter will have the same visibility and experience a lower decay, enough for nine orbits, theoretically 11. It is also easy now to work with fiberglass, since the final surface density of the material can be about 636 grams per square meter in order to fit the total mass of 20 grams.
Alchemy team blog 
Have you thought about a small balloon either white on the inside or aluminized that gets pressurized from a tiny capsule? It wouldn’t take much to inflate the balloon in the near vacuum though the aerodynamic drag would be considerable. We also thought of a simple sheet of reflector material that could be tracked by radar.
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I was afraid to use a sheet of reflective material because I won’t control it’s attitude. Three orthogonal sheets can do the trick, and maybe even get a visible size. I think the considerable aerodynamic drag is scientifically useful for fine orbit modelling. The nine orbits will be given by the high altitude. An inflated balloon has the advantage of omnidirectional drag, making it easier to interpret the data from observed trajectory. The inflating capsule could provide satellite separation too, with a low weight. So I don’t see any cons for the balloon idea.
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