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Suppose that I know a normal distribution of randome varible $X$ satisfies the property that $P(20<X<30) = 0.9$, is that true that the lower quartile of $X$ must be between $20$ and $25$?

I think the answer is No since it might be possible that with some certain standard deviation, we can make $P(20<X<25)$ very very small but its summation with $P(25<X<30)$ still gives me $0.9$.

But my professor said it is true. Can we formally prove this?

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You are correct. You can try a mean of 29 and $\sigma=.780304$. Then $P(20<X<30)=.90$ and the first quartile is $28.4737>25$.

Mathematica graphics

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