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.
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