Decidability of checking whether all words accepted by a TM have an even number of 1s
First, it can be shown easily by Rice's theorem.

Alternatively, you can show a reduction from $\overline{HP}$ as follows:

$$f(\left&lt; M \right&gt;,x)=\left&lt;M_x\right&gt;$$

$M_x$ on input $w$:


run $M$ on $x$ for $|w|$ steps. 
if M stopped then accept.
if the number of 1's in $w$ is even then accept, otherwise reject.


I'll let you verify the correctness.