A random variable X is normally distributed with $\mu = 60$ and $\sigma$ = 3. What is the value of 2 numbers a,b so that $P(X=a) = P(X=b)$.
The solution is $a = 60$ and $b = 65$.
However, I do not know how to come up with that answer. As far as I understand $P(X=a)$ and $P(X=b)$ have to be both 0 since you always have to give a range e.g. $P(a<X)$. Moreover if I insert the values 60 and 65 in the formula $Z = (X-\mu)/\sigma$ than I would end up with 0,1.667 and z-scores 0.5, 0.952 respectively.
Your reasoning is valid -- asking for $P(X=a)=P(X=b)$ makes essentially no sense because such a probability is always $0$ for a continuous distribution.
Indeed, this seems to be the only way to make $a=60, b=65$ a valid answer.
One might try to be charitable and "correct" the question into asking for two points where the probability density is the same -- but that wouldn't lead to $a=60, b=65$ being a solution; instead we would have $a=60+t, b=60-t$ for some $t$ (since the distribution is symmetric around $\mu=60$).
My tentative conclusion would be that (a) it's a trick question, (b) your understanding is correct, and (c) the solution you quote is just meant to be one possible answer.