Is the gift that I receive for solving religious competitions and distributing it to friends, and stipulating that they give me a small portion of the prize if they win, considered unlawful?

What you are doing involves clear deception, for your act of solving competitions and distributing answers to your friends, and urging them to submit the solution, leads to a number of participants winning prizes, even though the solver is one person.

As for your condition upon your friends to give you a percentage of the prize if they win, it is an unknown wage, which makes it an ambiguous and uncertain transaction (gharar). The Prophet (peace and blessings be upon him) forbade gharar. Based on this, what you are doing is impermissible.