## Do \$P/O(1)\$ equal to \$P\$ if solver need to consider smaller input?

If, for each $$n$$, there is program of length $$O(1)$$ that run for $$Poly(n)$$ time, and solve any $$F(x)$$ with $$size(x) (with a larger $$n$$ it still solves well, but maybe slower), then is there one program that solve all $$F(x)$$ in $$Poly(size(x))$$ time?