= Solution
Reliable transmission at rate $r$ means that there is a sequence of binary block codes of lengths $n$, message counts $M_n$ with $\liminf n^{-1}\log_2M_n\ge r$, and decoding error <probabilities> tending to zero. The <Shannon second coding theorem> identifies the supremum of reliable rates with the <channel capacity>.
For a <binary symmetric channel>, the <conditional entropy> of the output given any input bit is $h_2(p)=-p\log_2p-(1-p)\log_2(1-p)$. The output <entropy> is at most one bit and attains one for uniform input, so the maximum <mutual information> is
$$
\boxed{C=1-h_2(p)}.
$$
All rates strictly below $C$ are achievable, and rates above $C$ are not reliable; this states the supremum without claiming achievability of every boundary rate under every convention.
For very small $p$, the capacity is close to one, with $h_2(p)=p\log_2(1/p)+p/\log 2+O(p^2)$. At $p=1/2$, the output is independent of the input and capacity is zero. For $p>1/2$, flipping every received bit converts the channel to crossover <probability> $1-p<1/2$; its capacity is unchanged because $h_2(p)=h_2(1-p)$. At $p=1$ the deterministic bit flip is perfectly reversible.
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