Rust particles in hot water

I have recently noticed tiny rust particles in a cup after running hot water from my sink. The water is clear. I have the original boiler which is 35 yrs. old which was put in when the house was built and still running ok. Its a combination boiler for hot water gas heated baseboard heat and hot water for sink, shower, etc. I have copper pipes. There has been no water pipe repair work being done outside the home by the town and we use town water.
I would appreciate everyones comments as to where the rust might be coming from.
Could it be that the old boiler is discharging the rust particles and needs replacement?
Thank you.

Sounds like you may be right...could be the heat exchanger in the boiler is going. See what the pros have to say. Sure there's no galvanized piping anywhere? Do you see this if you run hot from other sinks or say the tub? If you get it everywhere, it's in the supply. If it is only on one sink, it's probably not the boiler.

I don't have any boiler experience, but if you have rust appearing in your water, it has to be coming from iron or steel and it sounds like the only critter that has iron or steel is the boiler. Can't even guess if it needs immediate replacement or what else might be done. Hang in here, there are people that know about boilers that will probably chime in.