Will there be any problem if I keep doing this? I tend to change the first three strings constantly where the thicker ones every 2 months or so.

OUCH i change mine once every two weeks... But back in the day I used to do the same as you. It is fine as long as the string guage and brand are the same as the one you are replacing. At one point I had DR's Ernie Balls and GHS boomers all on at once, it wasn't exactly pretty.

Strings come in a set for a reason. For best results change all the strings when you think it's time, but changing one everytime it breaks wont give them poor performance, just mismatched.

By mismatched do you mean it won't sound nicely in tune?

There is nothing wrong with what you are doing a long as they are the same gauge, but two months thats to long to go for a string change especialy if you are using your guitar steady as I believe you are because your wearing out the G-B and E strings. By going so long you might have tuning problems as half the strings are on the way out or should be dead. I used to do that if I played 50 hours in a week and was a bit lazy .If you are making a change after 2 months you have no excuse but to change them all.

Even mixing gauges works.

Just beware of mixing the occasional gut cello string with nylon, roundwound and flatwound.....

I have not been too experimental but I broke strings on occasion and replaced with different gauge. Never tried the banjo string though.

I think most of us have replaced our strings then a day or a few days later broke one and replaced only that one, it no problem.

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This is the first time I've replaced the first three strings while keeping the old kinda rusty 3 lower ones.
I can hear it's a bit out of tune and it's not the intonation. Could it be that the different brightness levels between old and new strings be causing this?

If the others are rusty........why not change those as well?

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Yes imc, it is not holding tune because you have 3 new and 3 dead strings. That is why everyone is kind of puzzled that after such a long time between string changes you just did not change them all.