  Let p be a prime and let a be a positive integer. In this paper we determine
$\sum_{k=0}^{p^a-1}\binom{2k}{k+d}/m^k$ and
$\sum_{k=1}^{p-1}\binom{2k}{k+d}/(km^{k-1})$ modulo $p$ for all d=0,...,p^a,
where m is any integer not divisible by p. For example, we show that if
$p\not=2,5$ then
$$\sum_{k=1}^{p-1}(-1)^k\frac{\binom{2k}k}k=-5\frac{F_{p-(\frac p5)}}p (mod
p),$$ where F_n is the n-th Fibonacci number and (-) is the Jacobi symbol. We
also prove that if p>3 then $$\sum_{k=1}^{p-1}\frac{\binom{2k}k}k={8/9}
p^2B_{p-3} (mod p^3),$$ where B_n denotes the n-th Bernoulli number.
