#computability

Is there an algorithm that, given the code of any program and an input, correctly decides whether the program eventually halts on that input? Prove your answer.

Is there an algorithm that takes any polynomial equation with integer coefficients, in any number of unknowns, and decides whether it has a solution in integers?

Among all Turing machines with 5 states and 2 symbols that halt when started on a blank tape, what is the largest number of steps $S(5)$ taken before halting? Determine $S(5)$ and prove that every 5-state machine that runs longer never halts.