It is more fun.
When ideas are bounced around, math gets better.
The best sources of good problems and new tricks are your coauthors.
Coauthorship is a closer relationship than friendship. The ultimate situation is when the collaboration evolves into a long term relation, resulting a series of joint-work. That is to be cherished year after year.

In math, we use the Hardy-Littlewood rule. That is, authors are alphabetically ordered and everyone gets an equal share of credit.
The one who has worked the most has learned the most and is therefore in the best position to write more papers on the topic.

Dig the brain of your advisor who is usually willing to help. Try to befriend at least one faculty each quarter. Apply the same spirit to other graduate students. Prepare something mathematical to say.
Go to seminars and colloquiums. Talk to departmental visitors and join the lunch/dinner with them.
Check announcements of meetings/workshops of mathematics and CS societies as well as numerous research centers. Participate at these professional meetings and find out the problems that other researchers are currently working on or interested in.
Some problems are particularly good for sharing, for example, those with a game flavor or in need of developing concepts. Try to have one such problem for social purposes.
Pick a problem that you really wish that you alone can solve and then enjoy/endure the frustration in secret.

Not to do:
If you have any bad feeling about sharing the work or the credit, don't collaborate. In mathematics, it is quite okay to do your research independently. (Unlike other areas, you are not obliged to include the person who fund your research.) If the collaboration already has started, the Hardy-Littlewood rule says that it stays a joint work even if the contribution is not of the same proportion. You have a choice of not to collaborate the next time. (If you have many ideas, one paper doesn't matter. If you don't have many ideas, then it really doesn't matter.) You might miss the opportunity for collaboration which can enhance your research and enrich your life. Such opportunity is actually not so easy to cultivate but worths all the efforts involved.