  Recently the first author proved a congruence proposed in 2006 by Adamchuk:
$\sum_{k=1}^{\lfloor\frac{2p}{3}\rfloor}\binom{2k}{k}\equiv 0\pmod{p^2}$ for
any prime $p=1 \pmod{3}$. In this paper, we provide more examples (with proofs)
of congruences of the same kind
$$\sum_{k=1}^{\lfloor\frac{ap}{r}\rfloor}\binom{2k}{k}x^k \pmod{p^2}$$ where
$p$ is a prime such that $p\equiv 1 \pmod{r}$, $a/r$ is a fraction in $(1/2,1)$
and $x$ is a $p$-adic integer. The key ingredients are the $p$-adic Gamma
functions $\Gamma_p$ and a special class of computer-discovered hypergeometric
identities.
