I tried to play around with the question and came up with some (possibly wrong) observations.
Firstly, I noticed that for there to be a ratio which is a power of 3, you must have $\frac{a}{b} = 3^n$. I counted all the powers of 3 lesser than 301.
These are $3,9,27,81,243$. Next, I realised that there are a 100 possible pairs with 3 as a common ratio, 33 possible pairs with 9 as a common ratio, 11 possible pairs with 27 as a common ratio, 3 possible pairs with 81 as a common ratio and 1 possible pair with 243 as a common ratio.
Eliminating the common pairs, I realised there must be $100 + 1 + 1 + 1 +1 + 1$ possible pairs in total. That is there are $105$ possible pairs.
Now I can use the pigeonhole principle to prove that at least one of the pairs must be filled. Is this proof wrong and if so where is it wrong?