Even simpler than the clear tape mentioned in another answer:
Write or draw directly on the (bottom of) the trophy with a permanent marker, remove it with rubbing alcohol or nail polish remover when you get it back. Previously degreasing the area you are planning to write on can improve the durability of the marker.
You can protect the marker ...
Most left-handed people I know turn the page at an angle (which differs between individuals) that allows their left hand to slide above or below the line they just wrote.
This certainly needs some getting used to, but doesn't seem too unusual, considering one of my right-handed friends writes with the paper lying at a 90° angle to him.
The most important ...
Lateral thinking: use a different pen and/or paper.
There are pens designed for left handers to deal with the smudging problem.
There are also notebooks and paper available that allow ink to dry quickly.
I don't want to promote any particular product: you can do some searching yourself.
The ink winds up on the left edge of your hand, as it moves across the paper. You can put a small piece of paper (like a Post-It) on that edge while you write. I've done this for years and it works well. The Post-It (for example) absorbs any not-quite-dry ink and protects the edge of the hand. Instead of a Post-It, a piece of paper can be taped on pretty ...
Use a hard pencil (i.e. one that doesn't easily smear) and just brush off the shavings with your hand or - as has been suggested - a soft brush. Hold the paper on one side and brush away from there to avoid crinkling it.
For softer pencils you can use kneaded erasers (putty erasers) which avoids the problem in the first place. They don't work as well for ...
Actually, when you keep the pen upside down for a long time, they dry or ink flow is irregular. The gravitational pull is a reason, you can even try this with 'markers'. Sometimes the ball pens also dry off if they are not used regularly.
If you want to create a mark that will be “invisible” to the casual and unknowing observer, use clear nail polish, preferably in an unobtrusive spot.
Either apply a “dot” for a 3D-effect or choose something with a different sheen, e.g. a matte top coat on a glossy trophy. Almost as good is a polish with a very light tint or even better, a bit of iridescence....
Print your name, address, and student ID neatly and discretely on a small piece of clear cellophane tape using a fine-point permanent black marker and apply it to the base/bottom of the trophy.
An alternative is to do the same thing on the sticky side of the tape and do the same thing. Printing on that side is more resilient to being ...
GOJO Hand cleaner or any hand cleaner in a jar that's based on petroleum will dissolve the sticker's glue. It's safe and easy to work with and the sticker falls off without leaving a trace once it has soaked in it for a while.
Use a Vinyl eraser and a metal eraser guard. Use a kneaded eraser to pick up the pieces. Vinyl erasers can erase almost anything and kneaded erasers can be shaped to fit any space and they do not leave any residue behind. Also, if you use an electric eraser, this may help with the accuracy of what you are erasing.