Class with a single method

I'm making a problem solving program which takes as an input an ID which I later use to locate the problem in my list of problems.

I have the following hierarchy

Base class for all problems

public abstract class Problem
{
    public abstract int ID { get; }

    protected Problem()
    {
        Controller.Problems.Add(this);
    }

    public abstract ProblemOutput Solve();
}

Derived class Problem01

public sealed class Problem01 : Problem
{
    public override int ID => 1;
    private ProblemOutput problemOutput;

    public override ProblemOutput Solve()
    {
        //solving the problem here
    }
}

And this is where I run the main code

    internal static List<Problem> Problems = new List<Problem>();

    static void Main(string[] args)
    {
        Console.Write("Enter problem number");
        int problemNumber;
        while (!int.TryParse(Console.ReadLine(), out problemNumber))
        {
            Console.Write("Invalid input");
        }
        if (Problems.All(p => p.ID != problemNumber))
        {
            Console.WriteLine("Sorry this problem is not solved yet");
        }
        else
        {
            ProblemOutput problemOutput = Problems.Single(p => p.ID == problemNumber).Solve();
            Console.WriteLine(problemOutput.OutputText);
            Console.WriteLine();
            Console.WriteLine($@"Time took to calculate in milliseconds : {problemOutput.TimeToSolve}");
        }
    }

But there's a problem with this since Problem01 is never being instantiated the base constructor wont be triggered thus not getting added into the list but in the same time I never need to instantiate the class itself I just need to use the solve method from the respective 'Problem' class. Is there any design pattern or some better solution that I can apply here ?